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Efficient Solution Techniques for Axisymmetric Problems

Permanent Link: http://ufdc.ufl.edu/UFE0041576/00001

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Title: Efficient Solution Techniques for Axisymmetric Problems
Physical Description: 1 online resource (122 p.)
Language: english
Creator: Oh, Minah
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: axisymmetric, fem, maxwell, multigrid, nedelec, sobolev, weighted
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Consider a three-dimensional (3D) problem defined on a domain symmetric by rotation around an axis with data independent of the angular component. By using cylindrical coordinates, we can then reduce this axisymmetric 3D problem into a two-dimensional (2D) one. The advantage of such dimension reduction is that the discretization of the 3D problem results in a linear system of the same size as the 2D one saving computational time significantly. Due to the Jacobian arising from change of variables, however, we must work in weighted Sobolev spaces, where the weight function is the radial component r , once this dimension reduction is done. In this dissertation, we analyze the time harmonic Maxwell equations under axial symmetry. In particular, we provide an edge finite element analysis and a multigrid analysis of the so-called ?meridian? problem, a problem arising from the axisymmetric Maxwell equations. New commuting projectors in weighted spaces are introduced, and a dual mixed problem in weighted spaces, which is interesting in its own right, is analyzed. These will provide the main ingredients for the analysis of the meridian problem.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Minah Oh.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Gopalakrishnan, Jayadeep.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041576:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041576/00001

Material Information

Title: Efficient Solution Techniques for Axisymmetric Problems
Physical Description: 1 online resource (122 p.)
Language: english
Creator: Oh, Minah
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: axisymmetric, fem, maxwell, multigrid, nedelec, sobolev, weighted
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Consider a three-dimensional (3D) problem defined on a domain symmetric by rotation around an axis with data independent of the angular component. By using cylindrical coordinates, we can then reduce this axisymmetric 3D problem into a two-dimensional (2D) one. The advantage of such dimension reduction is that the discretization of the 3D problem results in a linear system of the same size as the 2D one saving computational time significantly. Due to the Jacobian arising from change of variables, however, we must work in weighted Sobolev spaces, where the weight function is the radial component r , once this dimension reduction is done. In this dissertation, we analyze the time harmonic Maxwell equations under axial symmetry. In particular, we provide an edge finite element analysis and a multigrid analysis of the so-called ?meridian? problem, a problem arising from the axisymmetric Maxwell equations. New commuting projectors in weighted spaces are introduced, and a dual mixed problem in weighted spaces, which is interesting in its own right, is analyzed. These will provide the main ingredients for the analysis of the meridian problem.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Minah Oh.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Gopalakrishnan, Jayadeep.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041576:00001


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EFFICIENTSOLUTIONTECHNIQUESFORAXISYMMETRICPROBLEMS By MINAHOH ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010

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c r 2010MinahOh 2

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Dedicatedtomyparents,KwangSeongOhandSeongHyeOh;mysi ster,MinjeongOh; andmysignicantother,JoshuaDucey 3

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ACKNOWLEDGMENTS IrstwanttothankmyPh.D.dissertationadvisor,Dr.JayGo palakrishnanfor guidingmetobecomeatruemathematician.Yougavemeinvalu ablesupportsinceday oneofgraduateschool,andIalwaysfeltsoluckyforhavingy ouasmyadvisor.Ialso wanttothankDr.PaulRobinsonforteachingmedifferential geometryandforgivingme sincereadvicewheneverIhadanyproblems.Isendmygratitu detoDr.HenryZmuda forhelpingmendgreatapplicationsformydissertation,a ndtoDr.PatrickDeLeenheer andDr.WilliamHagerfortheirconstantadviceingraduates choolandinthejobmarket. Iamgratefultohavealloftheabovementionedprofessorsin mycommittee,andIvery muchappreciatetheircontributiontothisdissertation.L astlybutnottheleast,Iwant tothankDr.PaulHumkeforhelpingmecometograduateschool andforgivingme continuousencouragementsincethen. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................7 LISTOFFIGURES .....................................8 ABSTRACT .........................................9 CHAPTER 1INTRODUCTION ...................................10 1.1TheTimeHarmonicMaxwellEquations ...................10 1.2TheMaxwellEquationsunderAxialSymmetry ...............13 2PRELIMINARIESONFUNCTIONSPACES ....................15 2.1Distributions ...................................15 2.2StandardSobolevSpaces ...........................16 2.3WeightedSobolevSpaces ...........................19 3PRELIMINARIESONALGORITHMS .......................25 3.1FiniteElementMethods ............................25 3.2MultigridSolver .................................28 4COMMUTINGPROJECTORSINWEIGHTEDSPACES .............32 4.1AGlobalProjectorinWeightedSpaces ...................32 4.2CommutingSmoothedProjectorsinWeightedSpaces ...........39 4.2.1Denitions ................................40 4.2.2Construction ..............................46 5ANALYSISOFADUALMIXEDPROBLEMINWEIGHTEDSPACES ......57 5.1ProblemStatementandAnalysis .......................57 5.2NumericalResults ...............................65 6FINITEELEMENTANALYSISFORTHEMERIDIANPROBLEM ........67 6.1TheEdgeFiniteElementMethod .......................67 6.2ProofoftheQuasi-OptimalityResult .....................69 6.3NumericalResults ...............................74 7MULTIGRIDANALYSISFORTHEMERIDIANPROBLEM ............76 7.1MultigridAnalysisfortheRelatedPositiveDenitePro blem ........77 7.1.1TheMultigridAlgorithm ........................77 5

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7.1.2MultigridAnalysis ............................81 7.1.3NumericalResults ...........................90 7.2MultigridAnalysisfortheIndeniteProblem .................91 7.2.1TheMultigridAlgorithm ........................92 7.2.2MultigridAnalysis ............................99 7.2.3NumericalResults ...........................105 8ABIOMEDICALAPPLICATIONOFTHEMERIDIANPROBLEM ........108 APPENDIX APROOFOFTHEEXACTSEQUENCEPROPERTY ...............112 BPROOFOFPROPOSITION4.2.1 .........................114 REFERENCES .......................................119 BIOGRAPHICALSKETCH ................................122 6

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LISTOFTABLES Table page 5-1Mixedproblemconvergencerates .........................66 6-1FEMconvergencerates ...............................75 7-1V-cycleconvergencerates ..............................91 7-2Backslash-cycleconvergencerates: =1 .....................106 7-3Backslash-cycleconvergencerates: =10 ....................106 7

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LISTOFFIGURES Figure page 4-1Domain D a correspondingtopoint a ........................41 7-1Domains .......................................91 7-2Wavenumbervscoarsestmeshsize ........................107 8-1Coaxial-baseddoubleslotchokedprobe ......................109 8-2Distributionoftheelectromagneticpower .....................110 8

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy EFFICIENTSOLUTIONTECHNIQUESFORAXISYMMETRICPROBLEMS By MinahOh May2010 Chair:JayGopalakrishnanMajor:Mathematics Considerathree-dimensional( 3 D )problemdenedonadomainsymmetricby rotationaroundanaxiswithdataindependentoftheangular component.Byusing cylindricalcoordinates,wecanthenreducethisaxisymmet ric 3 D problemintoa two-dimensional( 2 D )one.Theadvantageofsuchdimensionreductionisthatthe discretizationofthe 3 D problemresultsinalinearsystemofthesamesizeasthe 2 D onesavingcomputationaltimesignicantly.DuetotheJaco bianarisingfromchange ofvariables,however,wemustworkinweightedSobolevspac es,wheretheweight functionistheradialcomponent r ,oncethisdimensionreductionisdone.Inthis dissertation,weanalyzethetimeharmonicMaxwellequatio nsunderaxialsymmetry. Inparticular,weprovideanedgeniteelementanalysisand amultigridanalysisof theso-called“meridian”problem,aproblemarisingfromth eaxisymmetricMaxwell equations.Newcommutingprojectorsinweightedspacesare introduced,andadual mixedprobleminweightedspaces,whichisinterestinginit sownright,isanalyzed. Thesewillprovidethemainingredientsfortheanalysisoft hemeridianproblem. 9

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CHAPTER1 INTRODUCTION Theproblemtobeanalyzedinthisdissertationisthetimeha rmonicMaxwell equationsunderaxialsymmetry.Itiswell-knownthattheti meharmonicMaxwell equationsdecouplesintotwo 2 D problemsunderaxialsymmetry:onecalledthe azimuthalproblemandtheothercalledthemeridianproblem [ 2 9 21 ].Thischapter consistsoftwosections.Therstsectiondescribeshowthe Maxwellequationsget reducedtoitstimeharmonicform.Thesecondchapterpresen tsthedecouplingofthe timeharmonicMaxwellequationsunderaxialsymmetry. 1.1TheTimeHarmonicMaxwellEquations Inthissection,wedescribeindetailhowtheMaxwellequati onsgetreducedtoits timeharmonicformbyassumingpropagationatasinglefrequ ency.Wecloselyfollow theargumentin[ 31 ,Chapter1].TheMaxwellequationscanbestatedasfollows: @ B @ t + curl E =0, div D = @ D @ t curl H = J div B =0, (1–1) where D istheelectricdisplacement, B isthemagneticinduction, E istheelectric eld, H isthemagneticeld, isthechargedensityfunction,and J isthecurrent densityfunction.TheneitherbyFouriertransformintimeo rtoanalyzeelectromagnetic 10

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propagationatasinglefrequency,wecanwrite E = R ( e i t ^ E ( x )), D = R ( e i t ^ D ( x )), H = R ( e i t ^ H ( x )), B = R ( e i t ^ B ( x )), J = R ( e i t ^J ( x )), = R ( e i t ^ ( x )), where R ( e i t ^ E ( x )) denotestherealpartof e i t ^ E ( x ) ,etc.,and denotestheangular frequency,i.e., =2 f ,where f isthefrequency.Bysubstitutingtheseinto( 1–1 ),we reachthetimeharmonicMaxwellequations: i ^ B + curl ^ E =0, div ^ D =^ i ^ D curl ^ H = ^J div ^ B =0. (1–2) Invacuumorfreespace,theeldsarerelatedbytheequation ^ D = 0 ^ E and ^ B = 0 ^ H where 0 8.854 10 12 Fm 1 iscalledtheelectricpermittivity,and 0 =4 10 7 Hm 1 iscalledthemagneticpermeability.Furthermore,thespee doflightinvacuumisgiven by c = p 0 0 1 ( 2.998 10 8 ms 1 ) Inthecasewhenvariousdifferentmaterialsoccupythedoma inoftheelectromagnetic eld,andifthematerialpropertiesdonotdependonthedire ctionoftheeldandthe materialislinear( inhomogeneous,isotropicmaterials) ,thenwehavetherelation ^ D = ^ E and ^ B = ^ H (1–3) 11

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where and arepositive,bounded,scalarfunctionsofpositionwhichd ependonthe material.Additionally,inaconductingmaterial,theelec tromagneticelditselfgivesrise tocurrents.ByassumingthattheOhmslawholds,wehavethat ^J = ^ E + ^J a (1–4) where 0 iscalledtheconductivity,and ^J a istheappliedcurrentdensity.Regions where > 0 arecalledconductors. =0, = 0 ,and = 0 invacuum. Thus,ininhomogeneous,isotropicmaterials,bysubstitut ing( 1–3 )and( 1–4 ) into( 1–2 )andsimplifying,weobtainthenalversionoftherst-ord erMaxwellsystem: i r H + curlE =0 i r E + curlH = 1 i F (1–5) where = p 0 0 isthewavenumber, E = p 0 ^ E H = p 0 ^ H ,and F = i p 0 ^J a r and r arecalledtherelativepermittivityandrelativepermeabi lityrespectively,anditis denedby r = 1 0 ( + i ) and r = 0 ( r =1= r invacuum). Remark 1.1.1 Thewavenumber isrelatedtothewavelengh = c f inthefollowing way: = p 0 0 =2 fc 1 = 2 cf 1 = 2 Finally,in( 1–5 ),bywriting H intermsof E usingtherstequation,andsubstituting intothesecondequation,weobtainthesecond-orderMaxwel lsystem: curl ( 1 r curlE ) 2 r E = F (1–6) 12

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1.2TheMaxwellEquationsunderAxialSymmetry Throughoutthisdissertation,weassumethatwearesolving thetimeharmonic Maxwellequationsonanaxisymmetric 3 D -domainwithaxisymmetricdata F .Vector valuedfunctionsarecalledaxisymmetricwheneachoftheir componentswithrespectto the e r e e z basisareindependentofthe -variable.Thenthecorrespondingsolution E inthiscasewillalsobeaxisymmetric[ 6 ]. Letuswrite E = E r e r + E e + E z e z .Thenincylindricalcoordinates,underaxial symmetry( @ =0 ),the curl operatorsreads: curlE = @ z E e r +( @ z E r @ r E z ) e + 1 r @ r ( rE ) e z Wewillusethefollowingnotationinrelationwiththeabove formulafor curlE : curl rz ( E r E z )= @ z E r @ r E z curl rz E =( @ z E 1 r @ r ( rE )). (1–7) Noticethat E r and E z onlyaffectsthe -componentof curlE while E onlyaffectsthe r and z componentsof curlE .Therefore,( 1–6 )decouplesintotwo 2 D problemsbywriting E =( E r ,0, E z )+(0, E ,0) .Oneiscalledthemeridianproblemandtheotheriscalledth e azimuthalproblem: MeridianProblem: curl rz ( 1 r curl rz E rz ) 2 r E rz = F rz (1–8) AzimuthalProblem: curl rz ( 1 r curl rz E ) 2 r E = F (1–9) where E rz denotes ( E r E z ) ,andsimilarnotationholdsfor F .Therefore,bysolving themeridianproblemweobtainthe rz -componentof E andbysolvingtheazimuthal problemwegetthe -componentof E .Inotherwords,bysolvingtwo2Dproblems,we ndthesolutionforthe3DMaxwellequations( 1–6 ). 13

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Inthisdissertation,wewillanalyzethemeridianproblem. Inparticular,themain resultofthisdissertationistheniteelementanalysisan dthemultigridanalysisofthe meridianproblem. 14

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CHAPTER2 PRELIMINARIESONFUNCTIONSPACES OurpurposehereistodenesomestandardandweightedSobol evspacesand torecalltheirproperties.Tosimplifythediscussion,wes hallworkfromnowonwith real-valuedfunctions,buttheresultsstatedherecanbeex tendedtocomplex-valued functionsaswell.Notethatweusethenotation R todenotetherealnumbereldand N todenotethesetofnon-negativeintegers. 2.1Distributions Let n beanopensubsetof R n ,anddene D (n) tobethelinearspaceofinnitely differentiablefunctionsthathavecompactsupportin n .Weoftencallthisspacethetest functionspace.Next,let D 0 (n) denotethedualspaceof D (n) inthesensethatalinear functional T : D (n) R iscontainedin D 0 (n) ,providedthatforeverycompactset K n thereexistconstants C and k suchthat j T ( ) j C X j j k sup K j @ j forall 2 D (n) ,whereweusedthestandardmulti-indexnotationforderiva tives: =( 1 2 n ) 2 N n j j = P ni =1 i ,and @ = @ j j @ x 1 1 @ x n n Thisspaceiscalledthespaceofdistributions,andif f 2 D 0 (n) islocallyintegrable, then < f > = Z n f dV forall 2 D (n), where < > denotesthedualitypairing.Therefore, L 1loc (n) D 0 (n) ,where L 1loc (n) denotesthespaceoflocallyintegrablefunctionsdenedon n If f 2 L 1loc ( R ) ,thederivativeof f exists,anditisalsolocallyintegrable,thenforall 2 D ( R ) < f 0 > = Z 1 1 f 0 ( x ) ( x ) dx = Z 1 1 f ( x ) 0 ( x ) dx = < f 0 > 15

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bytheintegrationbypartsformulaandbythefactthatall 2 D ( R ) havecompact supportin R .Weadoptthisruleforall D 0 (n) ingeneralanddenethederivativeof distributionsinthefollowingway. For f 2 D 0 (n) ,dene @ f 2 D 0 (n) by <@ f > =( 1) j j < f @ > forall 2 D (n). Notethattheabovedenitionmakessense,sincefunctionsi n D (n) areinnitely differentiable.Foramoredetaileddiscussionofdistribu tionssee[ 36 ]. Additionally,wewilllaterusethenotation D ( n) todenotethelinearspaceof restrictionsoffunctionsin D ( R n ) to n .Notethatfunctionsin D (n) vanishonthe boundaryof n duetoitscompactsupporton n ,butfunctionsin D ( n) maynotvanishon theboundary. 2.2StandardSobolevSpaces Nowwiththisnotionoftakingderivativesinthesenseofdis tribution,wecandene somebasicSobolevspacesinourinterest.Recallthat L 2 (n)= u : Z n j u j 2 dV < 1 isaHilbertspacewiththeinnerproductbeing ( u v )= Z n uvdV Wewilldenote kk forthe L 2 -norm.Since L 2 (n) L 1loc (n) D 0 (n) ,wecandene H 1 (n)= u 2 L 2 (n): grad u 2 L 2 (n) n 16

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wherewearetakingthegradientof u 2 L 2 (n) inthesenseofdistribution.The semi-norm,normandinnerproductassociatedtothisHilber tspaceare j u j H 1 (n) =( Z n j grad u j 2 dV ) 1 2 k u k H 1 (n) =( k u k 2 + j u j 2H 1 (n) ) 1 2 ( u v ) H 1 (n) = Z n uv + grad u grad vdV Remark 2.2.1 Foranyinteger m 0 ,wecandene H m (n)= u 2 L 2 (n): @ u 2 L 2 (n) forall j j m withtheinnerproduct ( u v ) H m (n) =( u v )+ X j j = m ( @ u @ v ). Foranypositiverealnumber s H s (n) isdenedbymeansofinterpolation[ 4 ]. Fortheremainderofthissection,weassumethat n R 3 ,andwedenetwo Sobolevspacesofvectorvaluedfunctions,i.e., H ( curl ,n)= u 2 L 2 (n) 3 : curlu 2 L 2 (n) 3 H (div,n)= u 2 L 2 (n) 3 :div u 2 L 2 (n) TheseareHilbertspaceswiththeinnerproduct ( u v ) H(curl,n) =( u v )+( curlu curlv ), ( u v ) H(div,n) =( u v )+(div u ,div v ). Thesearespacesofequivalenceclassesoffunctionsdened uptomeasurezero. Therefore,wemustdeneappropriatetracemapsinordertot alkabouttheboundary (setofmeasurezero)valueoffunctionsintheseSobolevspa ces.Detailsonthesetrace 17

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mapscanbefoundedin[ 24 ,Chapter1],butabriefdescriptionofthemwillbegiven here. Intheremainderofthischapter,forsimplicity,weassumet hat n R 3 issimply connected,bounded,andLipschitz-continuouswithconnec tedboundary.Let C 1 ( n) denotethespaceofinnitelydifferentiablefunctionsde nedon n ,andlet tr : C 1 ( n) C 1 ( @ n) betherestrictionmap,where @ n denotestheboundaryof n .Since C 1 ( n) is densein H 1 (n) ,andthismapiscontinuouswithrespecttothe H 1 -norm,thereexistsa uniquecontinuousextensionofthemap tr to H 1 (n) .Weshallstilldenotethisextension by tr .Infact, tr mapsinto H 1 2 ( @ n) inthiscase,i.e., tr : H 1 (n) H 1 2 ( @ n) iscontinuous. Similarly,wecandenethetraceon H ( curl ,n) and H (div,n) viacontinuoustrace maps.Inparticular, u 7! u n j @ n and u 7! u n j @ n arethecontinuoustracemaps H ( curl ,n) H 1 2 ( @ n) 3 and H (div,n) H 1 2 ( @ n) respectively,where n denotesthe unitoutwardnormaloftheboundary,and H 1 2 ( @ n) denotesthedualspaceof H 1 2 ( @ n) Theseresultsareprovedin[ 24 ,Chapter1.2]. Byusingthesetracemaps,wecandeneclosedsubspacesof H 1 (n) H ( curl ,n) and H (div,n) respectively: H 1 0 (n)= u 2 H 1 (n): u j @ n =0 H 0 ( curl ,n)= f u 2 H ( curl ,n): u n j @ n =0 g H 0 (div,n)= f u 2 H (div,n): u n j @ n =0 g (2–1) Equivalently,wemaydene H 1 0 (n) H 0 ( curl ,n) ,and H 0 (div,n) astheclosureof D (n) withrespecttothenormon H 1 (n) H ( curl ,n) ,and H (div,n) respectively.These subspacesareimportanttheoreticallyandpractically.Fo rexample,asweshall seeinlaterchapters,ifweconsidertheMaxwellequationsw ithperfectlyconducting boundaryconditions,theelectriceldliesinthespace H 0 ( curl ,n) .Thenextcontinuous embeddingresultisstatedandprovedin[ 24 ,Chapter1.3]. 18

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Theorem2.2.1. If n isconvex,thenthespaces H 0 ( curl ,n) \ H (div,n) and H ( curl ,n) \ H 0 (div,n) arebothcontinuouslyembeddedin H 1 (n) 3 .Thus, k u k H 1 (n) 3 C ( k u k + k curlu k + k div u k ), if u 2 H 0 ( curl ,n) \ H (div,n) or u 2 H ( curl ,n) \ H 0 (div,n) Thefollowingmoregeneralresultisprovedin[ 22 ]. Theorem2.2.2. If n isaboundedLipschitzdomain,thenthespaces H 0 ( curl ,n) \ H (div,n) and H ( curl ,n) \ H 0 (div,n) arebothcontinuouslyembeddedin H 1 2 (n) 3 .Thus, k u k H 1 2 (n) 3 C ( k u k + k curlu k + k div u k ), if u 2 H 0 ( curl ,n) \ H (div,n) or u 2 H ( curl ,n) \ H 0 (div,n) Above,andintherestofthedissertation,weuse C todenoteagenericconstant independentofthefunctionsinvolvedinnormestimates,wh ichmaytakedifferentvalues atdifferentoccurrences. Weendthissectionbystatingtwointegrationbypartsformu las[ 24 ,Chapter1.2]. Noticethatfortheclosedsubspaces( 2–1 ),theformulascouldbewritteninasimpler formduetozeroboundaryconditions.Theorem2.2.3. (Green'sFormula) ( v grad )+(div v )= < v n > @ n forall v 2 H (div,n), 2 H 1 (n), ( curlu ) ( u curl )= < u n > @ n forall u 2 H ( curl ,n), 2 H 1 (n) 3 where < > @ n denotesthedualitypairingbetweeneither H 1 2 ( @ n) and H 1 2 ( @ n) or H 1 2 ( @ n) 3 and H 1 2 ( @ n) 3 2.3WeightedSobolevSpaces Inthissection,weintroducesomeimportantweightedSobol evspacesthatwill bethefunctionspacesinourmaininterest.Ifwehavea 3 D problemdenedonan axisymmetricdomain,andifthegivendataisalsoaxisymmet ric,thenthis 3 D problem 19

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canbereducedtoa 2 D oneviacylindricalcoordinates ( r z ) .Thisisaremarkable feature,asitreducescomputationaltimesignicantly,bu tduetotheJacobianarising fromchangeofvariables,wearenowinweightedSobolevspac eswheretheweight functionistheradialcomponent r Inparticular,suppose n R 3 issymmetricwithrespecttothe z -axis,and D isthe restrictionof n tothe rz -planewhere r 0 (oftencalledthemeridianhalf-planewhich willbedenotedby R 2+ ).Therefore, n isobtainedbyrotating D aroundthe z -axis.Here andthroughoutthisdissertationexceptinChapter 3 n R 3 and D R 2+ willalways indicatedomainsrelatedinsuchkindofway. Let L 2 (n) denotethesubspaceof L 2 (n) thatconsistsoffunctionsthatareinvariant underrotation.Weadoptthisnotation X (n) ingeneralforanySobolevspace X (n) Vector-valuedfunctionsarecalledaxisymmetricifeachof itscomponentsareinvariant underrotation.Wewilluse f D todenotetherestrictionof f denedon n onto D ,i.e., f D ( r z )= f ( r ,0, z ) .Similarnotationfollowsforvector-valuedfunctionsasw ell.Then,for f 2 L 2 (n) Z D j f D ( r z ) j 2 rdrdz = 1 2 Z n j f ( r z ) j 2 rdrd dz = 1 2 Z n j f ( x y z ) j 2 dxdydz < 1 Therefore,itisclearthat f 2 L 2 (n) ifandonlyif f D 2 L 2r ( D ), (2–2) where L 2r ( D )= u : Z D j u ( r z ) j 2 rdrdz < 1 20

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ThisisaHilbertspacewiththeinnerproduct ( u v ) r = Z D uvrdrdz andtheinducednormwillbedenotedby jjjj r Remark 2.3.1 If D R 2+ intersectstheaxisofsymmetry( r =0 )then L 2 ( D ) isstrictly includedin L 2r ( D ) .Forexample,if D istheunitsquarein R 2+ and f ( r z )= 1 p r then f ( r z ) = 2 L 2 ( D ) but f ( r z ) 2 L 2r ( D ) .Intheniteelementanalysisandthemultigrid analysis,weoftenusespecicmembersinweightedSobolevs paces.Therefore, extendingtheresultsknownforthestandardSobolevspaces toweightedSobolev spacesrequiresattention,andthatisthefocusofthisdiss ertation. Notethatincylindricalcoordinates,underaxialsymmetry ( @ =0 ),wehave grad = @ r e r + @ z e z (2–3) div v = 1 r @ r ( rv r )+ @ z v z (2–4) curlu = @ z u e r +( @ z u r @ r u z ) e + 1 r @ r ( ru ) e z (2–5) Wedenetwo curl valuesin 2 D inthefollowingway. curl rz ( u r u z )= @ z u r @ r u z curl rz =( @ z 1 r @ r ( r )). (2–6) Noticethat curl rz takesavectorvaluedfunctionandreturnsascalarvaluedfu nction, while curl rz doesviceversa. curl rz and curl rz arethenrelatedthroughthe 3 D curl under axialsymmetryby( 2–5 ),i.e., curl rz ( u r u z ) and curl rz u returnthe -componentandthe rz -componentof curlu respectivelyunderaxialsymmetry.Therefore,if u =( u r ,0, u z ) 2 H ( curl ,n) thenwehavethat ( u r u z ) D 2 L 2r ( D ) 2 by( 2–2 ), curl rz ( u r u z ) D 2 L 2r ( D ) by( 2–5 ). 21

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WegiveanametosuchweightedSobolevspace: H r (curl, D )= w 2 L 2r ( D ) 2 :curl rz w 2 L 2r ( D ) TheinnerproductassociatedwiththisHilbertspaceis ( u v )=( u v ) r +(curl rz u ,curl rz v ) r andtheinducednormwillbedenotedby jjjj Similarly,byusing( 2–3 ),wehavethatif 2 H 1 (n) ,then D 2 H 1 r ( D ) ,where H 1 r ( D )= 2 L 2r ( D ): grad rz 2 L 2r ( D ) 2 and grad rz =( @ r @ z ) .Ingeneral, H k r ( D ) isdenedtobefunctionsin L 2r ( D ) whosedistributionalderivativesuptoorder k 1 isalsoin L 2r ( D ) .Furthermore, foranypositiverealnumber s H s r ( D ) isdenedastheHilbertinterpolationspace [ H [ s ]+1 r ( D ), H [ s ] r ( D )] [ s ]+1 s ofindex [ s ]+1 s betweenthespaces H [ s ]+1 r ( D ) and H [ s ] r ( D ) where [ s ] standsfortheintegerpartof s [ 6 ].Thenormandsemi-normof H s r ( D ) is denedintheusualway,andtheywillbedenotedby jjjj H s r ( D ) and jj H s r ( D ) respectively. Wedeneadditionalweightedfunctionspacesherefortheco nvenienceofthe reader.Let e H 1 r ( D )= L 21 = r ( D ) \ H 1 r ( D ) ,where L 21 = r ( D )= f u 2 L 2 ( D ): k u k 2L 21 = r ( D ) = Z D 1 r j u j 2 drdz < 1g Then k v k e H 1 r ( D ) =( k v k 2H 1 r ( D ) + k v k 2L 21 = r ( D ) ) 1 2 denesanormon e H 1 r ( D ) .Furthermore,thesemi-normandnorm j v j e H 2 r ( D ) = 1 r @ r ( rv ) 2H 1 r ( D ) + j @ z v j 2H 1 r ( D ) 1 2 k v k e H 2 r ( D ) =( j v j 2 e H 2 r ( D ) + k v k 2 e H 1 r ( D ) + k @ z v k 2L 21 = r ( D ) ) 1 2 22

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denetheHilbertspace e H 2 r ( D )= f v 2 e H 1 r ( D ): k v k e H 2 r ( D ) < 1g WesummarizetheisomorphismsbetweentheseweightedSobol evspacesand standardSobolevspacesunderaxialsymmetry.Thefollowin gresultscanbefound in[ 2 ,Section3.2]. Theorem2.3.1. Thefollowingisomorphismtheoremshold: 1..Thetracemapping f f D isanisometry(uptoafactor 1 p 2 )from L 2 (n) to L 2r ( D ) .Thesameholdsforthereciprocallifting, L 2r ( D ) L 2 (n) 2..For s 2 (0,2] ,thetraceoperatorisanisomorphismfrom H s (n) to H s + ( D ) ,where H s + ( D )= H s r ( D ) if s 6 =2, H 2 + ( D )= w 2 H 2 r ( D ): @ rr w 2 L 21 = r ( D ) 3..For s 2 (0,2] ,thetraceoperatorisanisomorphismfrom H s (n) 3 to H s ( D ) H s ( D ) H s + ( D ) ,where H s ( D )= H s r ( D ) if s 6 =1, H 1 ( D )= e H 1 r ( D ). 4..Therangeofthetraceoperatorfrom H ( curl ,n) is ( w r w w z ):( w r w z ) 2 L 2r ( D ) 2 ,curl rz ( w r w z ) 2 L 2r ( D ), rw 2 H 1 1 = r ( D ) Furthermore,letusobservehowtheboundaryconditionsoft he 3 D problem transferintoboundaryconditionsofthe 2 D problem.Let @ D = 0 [ 1 where 0 is thepartoftheboundarythatisontheaxisofsymmetry( r =0 ),and 1 denotesthe remainderoftheboundary.Functionsin e H 1 r ( D ) arewellknowntohavezerotraceon 0 [ 2 25 ].Itisalsoknownthatfunctionsin H 1 r ( D ) havetracesin L 2r ( 1 ) ,i.e.,for in H 1 r ( D ) ,thetrace j 1 makessenseasafunctionin L 2r ( 1 ) ,buttraceon 0 isnotdened 23

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ingeneral[ 28 ].Wecandenethetangentialtraceoperator r t : D ( D ) 2 7! e H 1 r ( D ) 0 by h r t ( v ), i = Z 1 r v t ds forall 2 e H 1 r ( D ), (2–7) where h i denotesdualitypairingin e H 1 r ( D ) ,and t istheunittangentvectoron @ D orientedcounterclockwise.In[ 21 ,Proposition2.2],itisshownthat r t extendstoa continuouslinearmapfrom H r (curl, D ) to e H 1 r ( D ) 0 .Moreover,theintegrationbyparts formula h r t ( v ), i =( v curl rz ) r (curl rz v ) r (2–8) holdsforall v in H r (curl, D ) and in e H 1 r ( D ) Naturally,theboundaryconditionsforthe3Dproblemgiver isetoboundary conditionson 1 .Weconsiderperfectlyconductingboundaryconditionson @ n in theMaxwellEquations,whichassertsthatthe3Delectrice ldhaszerotangential componenton @ n .Intheaxisymmetriccase,thisboundarycondition,loosel yspeaking, translatesintoanessentialboundaryconditionoftheform E rz t =0 on 1 .Therefore,if E 2 H 0 ( curl ,n) then ( E r E z ) D 2 H r (curl, D ) ,where H r (curl, D ) isaclosedsubspace of H r (curl, D ) denedas: H r (curl, D )= f v 2 H r (curl, D ): r t ( v )=0 g Similarly,dene H 1 r ( D )= 2 H 1 r ( D ): j 1 =0 andnoticethatif f 2 H 1 0 (n) then f D 2 H 1 r ( D ) .Theniteelementanalysisand themultigridalgorithmweshallgiveinlaterchaptersaref oranequationposedin H r (curl, D ) 24

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CHAPTER3 PRELIMINARIESONALGORITHMS FiniteElementMethodisamethodofndinganapproximation oftheexact solutiontoapartialdifferentialequation(PDE).Itisame thodthatchangesaPDE systemdenedonsomeinnitedimensionalspaceintoamatri xsystem,i.e.,anite dimensionalproblem.Multigridisanefcientwaytosolvet hematrixsystemobtained bytheniteelementmethod.Bothniteelementmethodandmu ltigridarewell-known notonlyforitsefciencybutalsoforitswell-established mathematicaltheorybehindit. Inthischapter,wegiveabriefintroductiontotheniteele mentmethodandthemultigrid V -cyclealgorithm. 3.1FiniteElementMethods Let n R 2 and f 2 L 2 (n) .SupposewewanttosolvethefollowingPoisson equation: Find u 2 H 1 (n) suchthat 4 u = f on n, u =0 on @ n. (3–1) Bymultiplyinganarbitrary v 2 H 1 (n) withhomogeneousDirichletboundaryconditions ( v 2 H 1 0 (n) )onbothsideoftheequation,andbyapplyingTheorem 2.2.3 inthe 2 D case,wegettheweakformulationof( 3–1 ): Find u 2 H 1 0 (n) suchthat Z n grad u grad vdV = Z n fvdV forall v 2 H 1 0 (n). (3–2) Notethat( 3–2 )hasauniquesolutionbytheLax-MilgramLemma. Theniteelementmethodwantstochangethisproblem( 3–2 )intoanite dimensionalproblem.Therefore,thenextstepistoconstru ctanitedimensional subspace M h H 1 0 (n) 25

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Werstdividethedomainintonitelymanytrianglessatisf yingsomeassumptions. Inparticular,weassumethattheniteelementtriangulati on T h of n satisestheusual geometricalconformityconditions[ 18 ]witharepresentativemeshsize h ,whichisthe following. Suppose n R n isapolyhedraldomainsubdividedinto n -simplices K collectedinto T h .Wesaythat T h isaniteelementmeshifitsatisesthefollowingconditio ns: 1.Theinteriorof K i and K j aredisjointwhenever i 6 = j forall K i K j 2 T h 2. S K 2 T h K = n 3.Anyfaceofany K 2 T h iseitherasubsetof @ n orafaceofanother K 0 2 T h Hereandthroughtoutthepaper,weassumethatthemesh T h isquasiuniform.A meshisquasiuniformifitsatisesthenexttwoconditions:1.shaperegularity:Thereexists > 0 suchthat h k k forall k 2 T h ,where h k is thediameterofthetriangleK,and k ismaximumradiusoftheinscribedcirclesin K. 2.Thereexists > 0 suchthat h h k forall K 2 T h ,where h =max K h k Next,weconstructanitedimensionalsubspaceof H 1 0 (n) inthefollowingway: M h = v 2 H 1 0 (n): v j K = a K x + b K y + c K forsome a K b K c K 2 R forall K 2 T h Theninsteadofsolving( 3–2 )wesolvethenitedimensionalproblemthatlooks justlike( 3–2 )butwiththeinnitedimensionalspace H 1 0 (n) replacedbythenite dimensionalsubspace M h : Find u h 2 M h suchthat Z n grad u h grad v h dV = Z n fv h dV forall v h 2 M h (3–3) Wewilldiscusshowthisproblemcanbechangedintoamatrixs ystem,butletus rstseehowwelltheniteelementapproximation u h of( 3–3 )approximatestheexact solution u of( 3–2 ).Thefollowingresultsareprovedin[ 18 ]. 26

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Theorem3.1.1. Let u 2 H 1 0 (n) and u h 2 M h betheuniquesolutionof( 3–2 )and( 3–3 ) respectively.Then 1..(Quasi-OptimalityResult) jj u u h jj H 1 (n) C inf v h 2 M h jj u v h jj H 1 (n) 2..(ErrorEstimate) jj u u h jj H 1 (n) Ch j u j H 2 (n) 3..(RegularityResult)If n isconvexthen jj u jj H 2 (n) C jj f jj Thequasi-optimalityresultfollowsdirectlyfromtheCea' sLemma,andthistells usthattheerrorbetween u and u h isboundedbyaconstantmultipleofthebest approximationof u in M h .Weobtainitem 2 byusingitem 1 andtheerrorestimate jj u u jj Ch j u j H 2 (n) ,where : H 1 0 (n) M h isthecanonicalprojectionoperator. Item 2 ismeaningfulonlywhen u 2 H 2 (n) ,whichitem 3 impliesundertheconvexity assumptionof n .Inthecasewhen n isnon-convex, u haslessregularity,butsimilar resultsasitem 2 continuetoholdwithaweakernormandalowerdegreeof h onthe righthandside.Item 2 impliesthat jj u u h jj H 1 (n) approacheszeroasthemeshsize h approacheszero,i.e.,thediscretesolutionconvergestot heexactsolutionaswe continuetorenethemesh. Since M h isanitedimensionalsubspace,weusebasisfunctions f i g Ni =1 to solve( 3–3 ).Let V h denotethesetofinteriornodesof T h .Foreach v i 2 V h ,let i betheuniquefunctionin M h thatsatises i ( v j )= ij ,where istheKroneckerdelta here.Thesefunctionsareoftencalledhatfunctions,andit iseasytoseethatthese hatfunctionsformabasisin M h .Thuswecanwritethediscretesolutionof( 3–3 )as u h = P Ni =1 u i i ,where N isthenumberofinteriornodes,andsince( 3–3 )holdsforall 27

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v h 2 M h ,itmustholdforall f i g Ni =1 inparticular.Therefore, Z n grad ( N X i =1 u i i ) grad j dV = Z n f j dV forall j =1 N N X i =1 ( Z n grad i grad j dV ) u i = Z n f j dV forall j =1 N Hence b A ~ u = ~ b where b A isthe N N matrixsuchthat b A ji = R n grad i grad j dV ,and ~ b isthelength N vectorwhose j -thcomponentis R n f j dV .Notonlyisthematrix b A invertible,itis alsosymmetricpositivedenite.Bysolvingthismatrixsys tem(bynding ~ u )wendthe coefcients f u i g Ni =1 ,andsoweobtaintheapproximation u h byusing u h = P Ni =1 u i i Asseenabove,thesizeofthematrixsystemobtainedbythen iteelementmethod correspondstothedimensionoftheniteelementsubspace, Thus,asthemeshgets nerthecorrespondingmatrixproblemthatneedstobesolve dgetslarger.Thenerthe meshis,however,theclosertheapproximationsolutionist otheexactone,soweare nowinterestedinsolvingthislargesparsematrixsystemef ciently. 3.2MultigridSolver Wehaveseenintheprevioussectionthat,byusingniteelem entmethods,we canchangeacontinuousPDEsystemintoamatrixsystem.Noww eareinterestedin solvingthismatrixsysteminanefcientway.Multigridtec hniquesgiveiterativemethods forsolvingmatrixsystemsobtainedbytheniteelementmet hod,byusingasequence ofmeshes.Itisverypowerful,sinceoftentheconvergencei suniformwithrespectto themeshsize.Inotherwords,thenumberofiterationstocon vergencestaysnearly constantnomatterhowlargethematrixbecomes.Wereferto[ 11 14 ]fordetailson multigridtheory. Suppose A isanoperatorthatsatises ( Au v ) q = a ( u v ) forsomebilinearform a ( ) andsomeinnerproduct ( ) q onsomefunctionspace M .Considerasequenceof 28

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nestedniteelementsubspacesof M ,i.e., M 1 M 2 M J Inparticular, f T i g Ji =1 isasequenceofmeshessuchthat T i +1 isarenementof T i forall 1 i J 1 ,and M i istheniteelementsubspacewithrespecttothemesh T i forall 1 i J .Dene A k : M k M k by ( A k u k v k ) q = a ( u k v k ) forall u k v k 2 M k Wewanttoconstructanefcientmultigriditerationtosolv eequationsoftheform A J x = f on M J Let R k : M k M k beasmoother,anddene Q k : M k +1 M k by ( Q k u k +1 v k ) q =( u k +1 v k ) q forall v k 2 M k Inotherwords, Q k isthe ( ) q -orthogonalprojectionfrom M k +1 onto M k .Thenthe standardV-cyclemultigridalgorithmisasfollows:Algorithm 3.2.1(V-cycle) Given u and f in M k ,denetheoutput MG k ( u f ) in M k bythe followingrecursiveprocedure: 1..Set MG 1 ( u f )= A 1 1 f 2..For k > 1 ,dene MG k ( u f ) recursively: (a) v (1) = u + R k ( f A k u ) .(Presmoothing) (b) v (2) = v (1) + MG k 1 (0, Q k 1 ( f A k v (1) )) .(Correctionusingcoarsemeshes) (c) v (3) = v (2) + R t k ( f A k v (2) ) .(Postsmoothing) (d)Set MG k ( u f )= v (3) ThefollowingTheoremgivesconditionsonthesmoother R k whichassuresthe uniformconvergenceofthemultigridV-cycle[ 10 11 14 ]. Theorem3.2.1. (UniformV-cycleConvergence) 29

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Let E k : M k M k betheerrorreductionoperatorof MG k ,andlet P k 1 bethe a ( ) orthogonalprojectionfrom M k to M k 1 .Supposethereexistsaconstant C independent ofksuchthat k ( I P k 1 ) K k u k 2a C ( k u k 2a k K k u k 2a ) forall u 2 M k (3–4) where K k = I R k A k .Then, 0 a ( E k u u ) a ( u u ) forall u 2 M k with = C 1+ C Intheremainderofthissection,weexplainwhytheconclusi onofTheorem 3.2.1 impliestheuniformconvergenceofthemultigridV-cycle.R ecallthattheerrorreduction operator,bydenition,satises x x n = E k ( x x n 1 ), where x istheexactsolutionand x n istheresultofthe n -thiterationofthemultigrid solver.Hence, k x x n k a k E k k na k x x 0 k a on M k .Theroem 3.2.1 impliesthat k E k k 2a =sup v 6 =0 a ( E k v E k v ) a ( v v ) max ( E k ) 2 2 where max ( E k ) denotesthemaximumeigenvalueof E k .Therefore, k E k k a sothat k x x n k a n k x x 0 k a 30

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Since isindependentofthemeshsize,thisshowsthatthemultigri dalgorithm convergesatauniformrateindependentofthemeshsize. 31

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CHAPTER4 COMMUTINGPROJECTORSINWEIGHTEDSPACES Inthischapter,wewillconstructprojectorsinweightedsp aceswithcommuting propertiesandapproximationproperties.Theseprojector swillplayanessentialrole inChapters 5 6 ,and 7 .Weconstructtwodifferenttypeofprojectorson H r (curl, D ) whichwillbepresentedintwoseparatesections.Hereandin theremainingofthis dissertation,wewillassumethattherotationaldomain n R 3 isasimplyconnected, boundedLipschitzdomainwithconnectedboundary.Additio nally,wewillassume that D issimplyconnectedand 1 isconnectedinordertousetheexactsequence property( 4–3 ). 4.1AGlobalProjectorinWeightedSpaces Thepurposeofthissectionistoexhibitaprojector Wh intotheN ed elecnite element[ 32 ]subspaceof H r (curl, D ) thathasacommutativitypropertyinvolvingthe L 2r ( D ) -orthogonalprojection Sh intoaspaceofpiecewiseconstantfunctions.Theresults inthissectionarecontainedin[ 20 ]. First,letusdenetheniteelementsubspacesontowhichth eprojectionsmap.Let N 1 = f ( a bz c + br ): a b c 2 R g P 1 = f c 0 + c 1 r + c 2 z : c i 2 R for i =0,1,2 g Theniteelementspacesweshalluseare V h = u 2 H 1 r ( D ): u j K 2 P 1 forall K 2 T h W h = f v 2 H r (curl, D ): v j K 2 N 1 forall K 2 T h g V h = f v 2 V h : v j 1 =0 g W h = f v 2 W h : r t ( v )=0 g S h = u 2 L 2r ( D ): u j K isconstantforall K 2 T h 32

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whereweassumethat D ismeshedbyaniteelementtriangulation T h satisfying theusualgeometricalconformityconditions(seeChapter 3 section 3.1 )witha representativemeshsize h .Projectorsintotheseniteelementspaceswithcommutati vity propertieshavebeenconstructedpreviously.Indeed,in[ 21 ],wendprojectors b Vh b Wh and b Sh onto V h W h ,and S h respectively.Inparticular,itisprovedin[ 21 ]thatthey satisfy curl rz b Wh v = b Sh curl rz v (4–1) k b Wh v v k r Ch j v j H 1 r ( D ) 2 (4–2) forall v in H 1 r ( D ) 2 (see[ 21 ,Lemma5.1]for( 4–1 )and[ 21 ,Lemma5.3]for( 4–2 )).The projection b Sh equalsthe L 2r ( K ) -orthogonalprojectionof forall K intersecting 0 whilefortheremainingelements K 0 ,itequalsthe(unweighted) L 2 ( K 0 ) -orthogonal projectionof Unfortunately,theseprojectorsareinadequateforourpur posesinChapter 5 .Let Sh denotethe L 2r ( D ) -orthogonalprojectioninto S h .Forouranalysislater,weneeda projector Wh thatsatisesthecommutativitypropertyin( 4–1 )with Sh .Theprojector b Sh of[ 21 ]isnotequalto Sh .Therefore,theremainderofthissectionisdevotedtothe constructionoftheprojector Wh withthepropertiesweneed,aslistedinthefollowing theorem.Theorem4.1.1. Let Sh : L 2r ( D ) S h bethe L 2r ( D ) -orthogonalprojection.Thereisa projector Wh : H r (curl, D ) W h suchthat 1.. Wh iswelldenedandcontinuouson H r (curl, D ) 2..thecommutativityproperty Sh curl rz u =curl rz Wh u holdsforall u in H r (curl, D ) 33

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3..theapproximationproperty rr u Wh u rr C inf u h 2 W h k u u h k holdsforall u in H r (curl, D ) Weremarkthattheprojector Wh iscontinuouson H r (curl, D ) ,whereastypical projectorsintotheN ed elecspace,suchasN ed elec'soriginalprojector[ 32 ],aswellas theprojector b Wh ,requiremoreregularityduetoedge-baseddegreesoffreed om.This isachievedbyaglobaldenitionof Wh withoutlocaldegreesoffreedom,giveninthe proofofthistheorembelow. Fortheproofandsubsequentanalysis,wewillneedan“exact sequenceproperty” andtheso-called“discreteHelmholtzdecomposition,”but adaptedtoourweighted innerproductsetting.Becausewehaveassumedthat 1 isconnectedand D issimply connected,itfollowsthatthesequence 0 V h grad rz W h curl rz S h 0 (4–3) isexact,asprovedinAppendix A .Thismeansthatthemap curl rz : W h 7! S h is surjectiveanditsnullspacecoincideswiththe grad rz ( V h ) .Suchresultsarestandard inthecaseofnoboundaryconditionsorwhenboundarycondit ionsholdontheentire boundary.Inourapplication,aboundaryconditionispresc ribedonlyonpartof @ D namely 1 .Sincewehavenotbeenabletolocateareferencefortheproo fofexactness inthiscase,weincludeashortproofinAppendix A Next,letusadaptthewell-knowndiscreteHelmholtzdecomp ositiontoourweighted norms.Givena v h in W h ,thereisaunique h in V h satisfying ( grad rz h grad rz h ) r =( v h grad rz h ) r forall h in V h Theuniqueexistenceof h isguaranteedbytheLax-Milgramlemma,whichmaybe invokedforthisvariationalproblemthanksto[ 25 ,Proposition2.1].Itistrivialtoverify 34

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thestabilityestimate k grad rz h k r k v h k r (4–4) Let r h = v h grad rz h .ThentheweighteddiscreteHelmholtzdecompositionis v h = grad rz h + r h Notethatthecomponentsofthedecompositionareorthogona lwithrespecttothe weightedinnerproductsofboth L 2r ( D ) 2 and H r (curl, D ) Tocharacterize r h further,let curl 0rz : S h 7! W h bedenedby (curl 0rz s h w h ) r =( s h ,curl rz w h ) r forall s h in S h and w h in W h i.e., curl 0rz istheadjointof curl rz : W h S h withrespecttotheweightedinner product ( ) r .Bytheexactnessof( 4–3 ), grad rz ( V h )=ker(curl rz ) where ker(curl rz ) denotesthenullspaceof curl rz in W h .Hencetheorthogonalityof r h with grad rz ( V h ) impliesthat r h isintherangeoftheadjoint curl 0rz ,i.e.,thereisan element a h in S h suchthat r h =curl 0rz a h andmoreover, a h isuniqueduetotheinjectivityof curl 0rz (whichfollowsfromthe surjectivityof curl rz intheexactsequence( 4–3 )).Inotherwords,analternatewayof statingthedecompositionisthatforall v h in W h ,thereisaunique a h in S h andaunique h in V h suchthat v h = grad rz h +curl 0rz a h (4–5) WeshallnowusethisdecompositiontoproveTheorem 4.1.1 35

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ProofofTheorem 4.1.1 Dene Wh : H r (curl, D ) W h by ( Wh v grad rz h ) r =( v grad rz h ) r forall h in V h (4–6a) (curl rz Wh v s h ) r =(curl rz v s h ) r forall s h in S h (4–6b) Wewillnowverifythattheassertedstatementsholdforthis Wh 1 .Firstofall,observethat( 4–6 )isasquaresystemofequations.Indeed,dueto theexactnessof( 4–3 ),thenumberofequationsin( 4–6 )equal dim( V h )+dim( S h )=dim( grad rz ( V h ))+dim(curl rz ( W h )), =dim(ker(curl rz ))+dim(curl rz ( W h )), =dim( W h ), bytherank-nullitytheorem.Thus,weonlyneedtoshowthatt hekernelofthelinear system( 4–6 )istrivial.If v =0 ,thentherighthandsideof( 4–6 )iszero,so ( Wh v grad rz h +curl 0rz s h ) r =0 forall h in V h and s h in S h Thisimpliesthat Wh v =0 ,bytheweighteddiscreteHelmholtzdecompositionof W h Therefore, Wh iswelldened. Beforeweproceedtoprovethecontinuityof Wh on H r (curl, D ) ,letusnotethat k s h k r C k curl 0rz s h k r forall s h 2 S h (4–7) Thisfollowsfrom[ 21 ,Theorem6.1(2)],whichassertsthat k v h k r C k curl rz v h k r forall v h 2 R ?h (4–8) where R ?h denotestheorthogonalcomplementof grad rz ( V h ) in W h intheweighted L 2r ( D ) -norm.Indeed,( 4–8 )impliesthat k curl 0rz a h k r =sup v h 2 R ?h (curl 0rz a h v h ) r k v h k r sup v h 2 R ?h ( a h ,curl rz v h ) r C k curl rz v h k r = 1 C k a h k r 36

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whichproves( 4–7 ). Now,toprovethecontinuityof Wh ,letusrstusetheweighteddiscreteHelmholtz decomposition( 4–5 )andwrite Wh v = grad rz h +curl 0rz a h with h in V h and a h in S h Setting h = h in( 4–6a )and s h = a h in( 4–6b )andapplyingCauchy-Schwarzinequality, k grad rz h k r k v k r and k curl 0rz a h k 2r =(curl rz v a h ) r k curl rz v k r k a h k r k curl rz v k r C k curl 0rz a h k r by( 4–7 ).Hence,thestatedcontinuityof Wh followsbythestabilityoftheweighted discreteHelmholtzdecomposition. 2 .Commutativityisclearfrom( 4–6b )andthedenitionof Sh 3 .Toprovetheerrorestimate,consideranarbitrary u in H r (curl, D ) and u h in W h UsetheweighteddiscreteHelmholtzdecompositiontosplit Wh u u h = grad rz h +curl 0rz b h with h in V h and b h in S h .Then,by( 4–6 ), ( Wh u u Wh u u h ) r =( Wh u u grad rz h +curl 0rz b h ) r =( Wh u u ,curl 0rz b h ) r =(curl rz Wh u b h ) r ( u h ,curl 0rz b h ) r ( u u h ,curl 0rz b h ) r =(curl rz u b h ) r (curl rz u h b h ) r ( u u h ,curl 0rz b h ) r andhence rr Wh u u h rr 2r =( u u h Wh u u h ) r +(curl rz ( u u h ), b h ) r ( u u h ,curl 0rz b h ) r 37

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Now,usingtheCauchy-Schwarzinequality, rr Wh u u h rr 2r k u u h k k Wh u u h k r + k b h k r + k curl 0rz b h k r C k u u h k k Wh u u h k r wherewehaveapplied( 4–7 )to b h ,andusedthestabilityofthediscreteHelmholtz decomposition.Thetriangleinequalitynowyieldstheesti mate rr u Wh u rr r C k u u h k Finally,since k curl rz ( u Wh u ) k r = k curl rz u Sh curl rz u k r k curl rz u curl rz u h k r wehave k u Wh u k C k u u h k Since u h in W h isarbitrary,thisprovesthestatedapproximationpropert y. Thefollowingcorollarygivesmorespecicestimatesinter msofthemeshsize h undercertainconditions.Corollary4.1.1. Let u bein H s r ( D ) 2 forany 0 s 1 .Theapproximationproperty rr u Wh u rr Ch s k u k H s r ( D ) 2 + k curl rz u k H s r ( D ) holdsprovidedthat curl rz u isin H s r ( D ) ,and rr u Wh u rr r Ch s k u k H s r ( D ) 2 (4–9) holdsprovidedthat curl rz u isin S h .If s =1 thenormsontherighthandsideofboth inequalitiesabovecanbereplacedbysemi-norms.Proof. Inthecasethat curl rz u isin H s r ( D ) ,therstestimatefollowsdirectlyfrom Theorem 4.1.1 item 3 andCorollary 4.2.1 ,bytaking u h = ch u (seesection 4.2 ). 38

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Thisprojectorwillbeconstructedinthenextsection,butw eusethisresulthereforthe completionofthisproof. Nowsuppose curl rz u isin S h .ThenthefollowingestimateofTheorem 4.1.1 item 3 k u Wh u k C k u ch u k reducestosimply k u Wh u k r C k u ch u k r becauseofthecommutativityproperties.Indeed, curl rz ch u = oh curl rz u =curl rz u and similarly curl rz Wh u =curl rz u .UsingTheorem 4.2.1 ,theestimate( 4–9 )thenfollows. 4.2CommutingSmoothedProjectorsinWeightedSpaces Inthissection,weconstructtheprojectorusedintheproof ofCorollary 4.1.1 inthe previoussection.Inordertoprovetheedgeniteelementap proximationforthereduced Maxwellsystem,whentheoriginalrotationaldomainisabou ndedLipschitzdomain, weneedcommutingprojectorsinweightedspaces,thatrequi relower-orderregularity thanthosethatarealreadyknown[ 20 21 ].AweightedCl ementoperatorhasbeen constructed[ 3 ]foritsapplicationtotheaxisymmetricStokesproblem,bu tthisoperator isinsufcientfortheanalysisoftheaxisymmetricMaxwell equations. Inthissection,weconstructcommutingprojectorsinweigh tedspacesthatare denedontheweighted L 2 -space.Wemodifythecommutingquasi-interpolators bySch ¨ oberl[ 35 ]thatwasconstructedinstandardSobolevspaces,sothatth ey areappropriateforweightedSobolevspaces.Additionally ,inordertochangethese operatorsintoprojectors,weadaptthemethodusedin[ 17 ]whichintroducestheinverse operatoroftherestrictionofthequasi-interpolatortoit sprojectedspace. Notethatthroughoutthissection,welet curl rz ( v r v z )= @ r v z @ z v r whichisa multipleof 1 toouroriginaldenitionof curl rz (see( 2–6 )).Thisistoavoidtheuseof extranegative( )symbolsintheproofsinthissectionwheneverweapplythei ntegration bypartsformulaTheorem 2.2.3 inthe 2 D -case[ 24 ,Theorem2.11].Ofcourse,the 39

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resultsprovedherestillholdtruewhenweconsider curl rz accordingtoouroriginal denition.4.2.1Denitions Forsmoothfunctions,classicalnodalinterpolationopera torscanbeapplied,and theseoperators I g h I c h ,and I o h makethefollowingdiagramcommute: H 1 r ( D ) grad rz H r (curl, D ) curl rz L 2r ( D ) ??y I g h ??y I c h ??y I o h V h grad rz W h curl rz S h I g h and I c h cannotbeapplied,however,toallfunctionsin H 1 r ( D ) and H r (curl, D ) respectively.Theycanonlybeappliedtofunctionsinthese spaceswithextraregularity duetothelocaldegreesoffreedomusedintheirconstructio ns: ( I g h )( x )= X v 2 V ( v ) v ( x ), ( I c h z )( x )= X e 2 E ( Z e z t ds ) e ( x ), ( I o h s )( x )= X K 2 T h ( 1 j K j Z K sd x ) K where T h isatriangulationof D V denotesthesetofverticesin T h noton 1 E denotes thesetofedgesin T h noton 1 ,and K 2 T h denotesalltrianglesin T h Ourgoalistoconstructcommutingprojectorsthatcanbeapp liedtoallfunctions in L 2r ( D ) or L 2r ( D ) 2 .Thereforewewilldenemeshdependentsmoothersforfunct ions in L 2r ( R 2+ ) sothatwecanapplytheclassicalnodalinterpolationopera torsafterwe applythesesmoothersto L 2r -functions.Beforedeningthesesmoothers,werststatea propositionthatwillbeusedintheirdenitions.Theproof ofProposition 4.2.1 isgivenin Appendix B Proposition4.2.1. Let a =( a r a z ) beapointin R 2+ andlet D a bethedomainassociated to a asinoneofthecasesinFigure 4-1 Inallthreecases,forany k 0 ,thereexistsafunction a ( r z ) 2 P k suchthat 40

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ACase1 BCase2 CCase3 Figure4-1.Domain D a correspondingtopoint a 1.. ( a p ) r D a = p ( a ) forall p 2 P k ,where P k isthespacesofpolynomialsoforderup to k on D a 2.. k a k 2L 2r ( D a ) C 2 r a ,where r a = incase1. min y 2 D a r ( y ) incase2andcase3. InlinewithProposition 4.2.1 ,foragiventriangulation T h of D wedenean associateddomain D h a foreachmeshvertex a 2 T h .Wewrite h =max K 2 T h h K where h K isthediameterof K 1.If a ison 0 then D h a ischosenasinCase1inFigure 4-1 with = h 2.If a isintheinteriorof D then D h a ischosenasinCase2inFigure 4-1 with = h 3.If a ison 1 then D h a ischosenasinCase3inFigure 4-1 with = h ,i.e., D h a = y 2 R 2+ : j y ~ a j < h where ~ a =(~ a r ,~ a z ) isobtainedby ~ a r ~ a z = a r a z + cos sin sin cos 2 h 0 forsomexedangle 0 Next,wechoose 0 << 1 andanangle inthefollowingway. 41

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1.For a noton 1 :Thereexists 0 < 0 < 1 suchthat D h a D a k forall a ,where D a k denotesthe“vertexpatch”domainformedbytheunionofallt rianglesin T h connectedtothemeshvertex a ,whenever 0 < 0 .Choose 2 (0, 0 ] 2.For a on 1 :Foreach a ,choose 0 sothat D h a R 2+ n D Here,wesummarizesomenotationsthatwillbeusedintherem ainderofthepaper. Wewrite K =[ a 1 a 2 a 3 ] toindicatethat K 2 T h isatrianglewithvertices a 1 a 2 ,and a 3 .Additionally, e 1 =[ a 2 a 3 ] e 2 =[ a 3 a 1 ] ,and e 3 =[ a 1 a 2 ] denotethethreeedgesof K withaxedorientation(counter-clockwise),where [ a 2 a 3 ] denotestheedgefrom a 2 to a 3 ,etc. Dene i ( y i )= r ( y i ) a i ( y i ) for 1 i 3 ,where a i isthefunctionintroducedin Proposition 4.2.1 .Wewrite 123 = 1 2 3 and 12 = 1 2 ,etc.Dene ~ x y bythesame barycentriccoordinates i ( x ) ,for 1 i 3 ,as x 2 K withrespecttothetriangle [ y 1 y 2 y 3 ] ,i.e., ~ x y ( x y 1 y 2 y 3 )= 3 X i =1 i ( x ) y i (4–10) Wenowdenethefollowingmeshdependentsmootherswhichar esimilartothose bySch ¨ oberl[ 35 ].Let u w 2 L 2r ( R 2+ ) and v 2 L 2r ( R 2+ ) 2 .Then S g u ( x )= Z D h a 1 Z D h a 2 Z D h a 3 123 u (~ x y ) d y 3 d y 2 d y 1 S c v ( x )= Z D h a 1 Z D h a 2 Z D h a 3 123 [ 3 X i =1 y i v (~ x y ) grad rz i ( x )] d y 3 d y 2 d y 1 S o w ( x )= Z D h a 1 Z D h a 2 Z D h a 3 123 [( 3 X m =1 ( @ r m ( x ))) y m ) ( 3 X n =1 ( @ z n ( x )) y n )] w (~ x y ) d y 3 d y 2 d y 1 (4–11) wherefortwodimensionalvectors c =( c r c z ) and d =( d r d z ) ,wewrite c d todenote c r d z c z d r Insection 4.2.2 ,wewilloftenneedtocomputetheunisolventnode,edge,and elementdegreesoffreedomfor S g u S c v ,and S o w respectively,soletusdothis computationhere. 42

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Let K 2 T h suchthat K =[ a 1 a 3 a 3 ] .Then S g u ( a 1 )= Z D h a 1 Z D h a 2 Z D h a 3 123 u ( y 1 ) d y 3 d y 2 d y 1 since ~ x y = y 1 if x = a 1 = Z D h a 1 1 u ( y 1 ) d y 1 (4–12) Similarly, S g u ( a i )= Z D h a i i u ( y i ) d y i forall 1 i 3. Next,for q ( s )=(1 s ) a 2 + s a 3 ,0 s 1 ,and e 1 =[ a 2 a 3 ] ,wehavethat Z e 1 S c v t ds = Z 1 0 S c v ( q ( s )) q 0 ( s ) ds = Z 1 0 Z D h a 1 Z D h a 2 Z D h a 3 123 [( X 1 i 3 y i v ((1 s ) y 2 + s y 3 ) grad rz i ) ( a 3 a 2 )] d y 3 d y 2 d y 1 ds = Z 1 0 Z D h a 1 Z D h a 2 Z D h a 3 123 ( y 3 y 2 ) v ((1 s ) y 2 + s y 3 ) d y 3 d y 2 d y 1 ds = Z D h a 1 Z D h a 2 Z D h a 3 123 Z [y 2 ,y 3 ] v t dsd y 3 d y 2 d y 1 = Z D h a 2 Z D h a 3 23 Z [y 2 ,y 3 ] v t dsd y 3 d y 2 (4–13) Thethirdequalityholds,since grad rz 1 ( a 3 a 2 )=0, grad rz 2 ( a 3 a 2 )= 1, grad rz 3 ( a 3 a 2 )=1. Similarly, Z e 2 S c v t ds = Z D h a 1 Z D h a 3 13 Z [y 3 ,y 1 ] v t dsd y 3 d y 1 and Z e 3 S c v t ds = Z D h a 1 Z D h a 2 12 Z [y 1 ,y 2 ] v t dsd y 2 d y 1 43

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Finally,itisstraightforwardtoshowthat 1 ( P 3m =1 ( @ r m (x))y m ) ( P 3n =1 ( @ z n (x))y n ) isthe Jacobianarisingfromchangeofvariablesfrom x to ~ x y .Therefore, 1 j K j Z K S o wd x = 1 j K j Z D h a 1 Z D h a 2 Z D h a 3 123 Z [y 1 ,y 2 ,y 3 ] w ( z ) d z d y 3 d y 2 d y 1 (4–14) Afterapplyingthesesmootherstofunctionsinappropriate spaces,wecanthen applytheclassicalnodalinterpolationoperatorswhichre quireshigher-orderregularity. Thus,wemaydenethefollowingSch ¨ oberlquasi-interpolatorsinweightedspaces. Fromhereon,letusassumethat u w 2 L 2r ( D ) and v 2 L 2r ( D ) 2 .Theextensionof thesefunctionsto L 2r ( R 2+ ) and L 2r ( R 2+ ) 2 willbedenotedas ~ u ~ w ,and ~ v respectively. Dene R g h R c h ,and R o h on H 1 r ( D ) H r (curl, D ) ,and L 2r ( D ) respectivelyby R g h u = I g h S g ~ u R c h v = I c h S c ~ v R o h w = I o h S o ~ w where ~ denotestheextensionbyzero,i.e., ~ u 2 L 2r ( R 2+ ) suchthat ~ u j D = u andzero elsewhere. Notethattheseoperatorsarenotprojectorsastheydonotpr eservefunctions thatarealreadyintheirprojectedspaces.Insection 4.2.2 ,wewillmodifythese quasi-interpolatorsintoprojectors.Beforeweendthisse ction,weverifythatthese quasi-interpolatorscommute.Lemma4.2.1. R g h R c h ,and R o h satisfythefollowingcommutingdiagramproperties: 1.. R c h ( grad rz u )= grad rz ( R g h u ) forall u 2 H 1 r ( D ) 2.. curl rz ( R c h v )= R o h (curl rz v ) forall v 2 H r (curl, D ) Proof. 1..Sinceboththelefthandsideandtherighthandsidearefu nctionsin W h itsufcestoshowthattheunisolventedgefunctionalsagre eon K =[ a 1 a 2 a 3 ] i.e., Z e i ( grad rz R g h u ) t ds = Z e i R c h ( grad rz u ) t ds 44

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forall 1 i 3 .Itsufcestocheckfor e 1 =[ a 2 a 3 ] ,astheresultwillholdfor e 2 and e 3 inanidenticalway.Thisistrue,since Z e 1 ( grad rz R g h u ) t ds = R g h u ( a 3 ) R g h u ( a 2 ), = S g ~ u ( a 3 ) S g ~ u ( a 2 ), = Z D h a 3 3 ~ u ( y 3 ) d y 3 Z D h a 2 2 ~ u ( y 2 ) d y 2 by( 4–12 ), andsince Z e 1 ( R c h grad rz u ) t ds = Z e 1 ( S c grad rz ~ u ) t ds = Z D h a 2 Z D h a 3 23 Z [y 2 ,y 3 ] ( grad rz ~ u ) t dsd y 3 d y 2 by( 4–13 ), = Z D h a 2 Z D h a 3 23 [~ u ( y 3 ) ~ u ( y 2 )] d y 3 d y 2 = Z D h a 3 3 ~ u ( y 3 ) d y 3 Z D h a 2 2 ~ u ( y 2 ) d y 2 forallsmoothfunctions u 2 H 1 r ( D ) .Theresultfollowsfromthedensityofsmooth functionsin H 1 r ( D ) 2..Now,sincebothquantitiesarein S h ,itisenoughtocheckthat 1 j K j Z K curl rz ( R c h v ) d x = 1 j K j Z K R o h (curl rz v ) d x Byusingintegrationbyparts, 1 j K j Z K curl rz ( R c h v ) d x = 1 j K j 3 X i =1 Z e i R c h v t ds = 1 j K j 3 X i =1 Z e i S c ~ v t ds = 1 j K j Z D h a 1 Z D h a 2 Z D h a 3 123 Z [y 1 ,y 2 ]+[y 2 ,y 3 ]+[y 3 ,y 1 ] ~ v t dsd y 3 d y 2 d y 1 by( 4–13 ). 45

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Additionally, 1 j K j Z K R o h (curl rz v ) d x = 1 j K j Z K S o (curl rz ~ v ) d x = 1 j K j Z D h a 1 Z D h a 2 Z D h a 3 123 Z [y 1 ,y 2 ,y 3 ] (curl rz ~ v )( z ) d z d y 3 d y 2 d y 1 by( 4–14 ), = 1 j K j Z D h a 1 Z D h a 2 Z D h a 3 123 Z [y 1 ,y 2 ]+[y 2 ,y 3 ]+[y 3 ,y 1 ] ~ v t dsd y 3 d y 2 d y 1 forallsmoothfunctions v 2 H r (curl, D ) .Theproofisthencompletebythedensity ofsmoothfunctionsin H r (curl, D ) 4.2.2Construction Inthissection,wewillshowthatthequasi-interpolators R g h R c h ,and R o h are uniformlyboundedinthe L 2r -norm,andthattheseoperatorsareinvertiblewhen restrictedtotheirprojectedspaces.Byusingtheseresult s,wewillthenmodifythese operatorsintoprojectorsandverifytheirerrorestimates TheproofsofLemma 4.2.2 4.2.3 ,and 4.2.4 followthelinesof[ 35 ].Wemust, however,paycloseattentiontotheweightfunction r ,sincewearemodifyingthese proofstoweightedspaces.Lemma4.2.2. Thereexistsaconstant C independentof h and suchthat 1.. k R g h u k 2L 2r ( D ) C 3 k u k 2L 2r ( D ) ,forall u 2 L 2r ( D ) 2.. k R g h u h u h k L 2r ( D ) C k u h k L 2r ( D ) ,forall u h 2 V h Proof. Let K =[ a 1 a 2 a 3 ] beaxedtrianglein T h ,andlet D K denotetheunionof D h a 1 D h a 2 ,and D h a 3 .Then,duetotheshaperegularitypropertyof T h andthefactthat ~ u and ~ u h areextensionsof u and u h byzerorespectively,tocompletetheproofitsufcestopro ve localestimates: k R g h u k 2L 2r ( K ) C 3 k ~ u k 2L 2r ( D K ) k R g h u h u h k L 2r ( K ) C k ~ u h k L 2r ( D K ) 46

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Wehavethat R g h u j K = 3 X i =1 S g ~ u ( a i ) i (4–15) andthatforeach 1 i 3 j S g ~ u ( a i ) j = j Z D h a i i ~ u ( y i ) d y i j by( 4–12 ), = j ( a i ,~ u ) r D h a i j bydenitionof i k a i k L 2r ( D h a i ) k ~ u k L 2r ( D h a i ) C p ( h ) 2 r a i k ~ u k L 2r ( D h a i ) byProposition 4.2.1 item 2 C p ( h ) 3 k ~ u k L 2r ( D h a i ) (4–16) Thelastinequalityholds,since r a i h foreach 1 i 3 ,bydenitionof r a i andbythe criterionofchoosing .Therefore,itfollowsthat k R g h u k 2L 2r ( K ) = Z K j 3 X i =1 S g ~ u ( a i ) i ( x ) j 2 r ( x ) d x by( 4–15 ), C 3 X i =1 j S g ~ u ( a i ) j 2 Z K j i ( x ) j 2 r ( x ) d x C 3 X i =1 C ( h ) 3 k ~ u k 2L 2r ( D h a i ) Ch 2 max x 2 K r ( x ) by( 4–16 ), C 3 k ~ u k 2L 2r ( D K ) Thelastinequalityholds,since max x 2 K r ( x ) Ch .Fromnowon,wewillwrite r K to denote max x 2 K r ( x ) forsimplicityofthenotation.Therefore,thiscompletest heproofof item 1 47

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Foritem 2 ,wenotethatforall 1 i 3 (( S g ~ u h )( a i ) u h ( a i )) j K = Z D h a i i (~ u h ( y i ) u h ( a i )) d y i max y i 2 D h a i j grad rz ~ u h ( y i ) j max y i 2 D h a i j a i y i j Ch k grad rz ~ u h k L 1 ( D h a i ) (4–17) Hence,k R g h u h u h k 2L 2r ( K ) = Z K j 3 X i =1 (( S g ~ u h )( a i ) u h ( a i )) i ( x ) j 2 r ( x ) d x C 3 X i =1 j ( S g ~ u h )( a i ) u h ( a i ) j 2 Z K j i ( x ) j 2 r ( x ) d x C 3 X i =1 ( h ) 2 k grad rz ~ u h k 2L 1 ( D h a i ) h 2 r K by( 4–17 ), C ( h ) 2 k grad rz ~ u h k 2L 2r ( D K ) C 2 k ~ u h k 2L 2r ( D K ) bytheinverseinequality. Thiscompletestheproof. Lemma4.2.3. Thereexistsaconstant C independentof h and suchthat 1.. k R c h v k 2L 2r ( D ) C 3 k v k 2L 2r ( D ) forall v 2 L 2r ( D ) 2 2.. k R c h v h v h k L 2r ( D ) C k v h k L 2r ( D ) forall v h 2 W h Proof. Fix K 2 T h suchthat K =[ a 1 a 2 a 3 ] ,andlet C K denotetheconvexhullof D h a 1 D h a 2 ,and D h a 3 .AsintheproofofLemma 4.2.2 ,itsufcestoprovethelocalestimates: k R c h v k 2L 2r ( K ) C 3 k ~ v k 2L 2r ( C K ) and k R c h v h v h k L 2r ( K ) C k ~ v h k L 2r ( C K ) 48

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Wehavethat R c h v j K = 3 X i =1 ( Z e i S c ~ v t ds ) e i andthat k e i k 2L 2r ( K ) Cr K forall 1 i 3 ,since e i = ( j grad rz k k grad rz j ) ,where a j and a k arethetwoverticesoftheedge e i .Therefore, k R c h v k 2L 2r ( K ) = Z K j 3 X i =1 ( Z e i S c ~ v t ds ) e i ( x ) j 2 r ( x ) d x C 3 X i =1 j Z e i S c ~ v t ds j 2 Z K j e i ( x ) j 2 r ( x ) d x Cr K 3 X i =1 j Z e i S c ~ v t ds j 2 (4–18) Wewillrstboundthesummandinvolving e 1 =[ a 2 a 3 ] sincetheotherswillfollowby thesameway. j Z e 1 S c ~ v t ds j = j Z D h a 2 Z D h a 3 23 Z 1 0 ~ v ((1 s ) y 2 + s y 3 ) ( y 3 y 2 ) dsd y 3 d y 2 j by( 4–13 ), Ch j Z D h a 2 Z D h a 3 23 Z 1 0 ~ v ((1 s ) y 2 + s y 3 ) dsd y 3 d y 2 j (4–19) Next,weanalyzetheaboveintegralwithrespectto s intwoseparatepieces,i.e., where 0 s 1 2 and 1 2 < s 1 .Firstofall, j Z D h a 3 3 Z 1 2 0 Z D h a 2 2 ~ v ((1 s ) y 2 + s y 3 ) d y 2 dsd y 3 j = j Z D h a 3 3 Z 1 2 0 ( a 2 ,~ v ((1 s ) y 2 + s y 3 )) r D h a 2 dsd y 3 j j Z D h a 3 3 Z 1 2 0 k a 2 k L 2r ( D h a 2 ) k ~ v ((1 s ) y 2 + s y 3 )) k L 2r ( D h a 2 ) dsd y 3 j C p ( h ) 3 j Z D h a 3 3 Z 1 2 0 ( Z D h a 2 j ~ v ((1 s ) y 2 + s y 3 )) j 2 r ( y 2 ) d y 2 ) 1 2 dsd y 3 j byProposition 4.2.1 C p ( h ) 3 j Z D h a 3 3 Z 1 2 0 ( Z D h a 2 j ~ v ((1 s ) y 2 + s y 3 )) j 2 r ((1 s ) y 2 + s y 3 )) d y 2 ) 1 2 dsd y 3 j (4–20) 49

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Thelastinequalityholds,sincewhen 0 s 1 2 r ( y 2 ) 2(1 s ) r ( y 2 ) 2((1 s ) r ( y 2 )+ sr ( y 3 )). Bychangeofvariablesfrom y 2 to z =(1 s ) y 2 + s y 3 ,wehave Z D h a 2 j ~ v ((1 s ) y 2 + s y 3 )) j 2 r ((1 s ) y 2 + s y 3 )) d y 2 = Z g D h a 2 j ~ v ( z ) j 2 r ( z ) 1 (1 s ) 2 d z where gD h a 2 C K .Therefore,bycontinuingfrom( 4–20 ), j Z D h a 3 3 Z 1 2 0 Z D h a 2 2 ~ v ((1 s ) y 2 + s y 3 ) d y 2 dsd y 3 j C p ( h ) 3 j Z D h a 3 3 Z 1 2 0 ( Z C K j ~ v ( z ) j 2 r ( z ) 1 (1 s ) 2 d z ) 1 2 dsd y 3 j C p ( h ) 3 j Z D h a 3 3 ( Z C K j ~ v ( z ) j 2 r ( z ) d z ) 1 2 d y 3 j since 1 (1 s ) 2 4 = C p ( h ) 3 j Z D h a 3 3 k ~ v k L 2r ( C K ) d y 3 j C p ( h ) 3 k ~ v k L 2r ( C K ) j Z D h a 3 3 d y 3 j = C p ( h ) 3 k ~ v k L 2r ( C K ) Similarly,wegetsuchaboundfor j R D h a 3 3 R 1 1 2 R D h a 2 2 ~ v ((1 s ) y 2 + s y 3 ) d y 2 dsd y 3 j aswell, sowehavethat j Z D h a 2 Z D h a 3 23 Z 1 0 ~ v ((1 s ) y 2 + s y 3 ) dsd y 3 d y 2 j C p ( h ) 3 k ~ v k L 2r ( C K ) (4–21) Therefore,by( 4–19 ), j Z e 1 S c ~ v t ds j 2 C h 3 k ~ v k 2L 2r ( C K ) 50

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Clearly,bysimilararguments,suchestimateholdsfor j R e 2 S c ~ v t ds j 2 and j R e 3 S c v t ds j 2 aswell.Therefore,by( 4–18 ), k R c h v k 2L 2r ( K ) Cr K h 3 k ~ v k 2L 2r ( C K ) C 3 k ~ v k 2L 2r ( C K ) Toproveitem 2 ,weagainuse( 4–13 ).Wendaboundfor j R e 1 ( S c ~ v h v h ) t ds j asin theproofofitem 1 j Z e 1 ( S c ~ v h v h ) t ds j = Z D h a 2 Z D h a 3 23 ( Z [y 2 ,y 3 ] ~ v h t ds Z [a 2 ,a 3 ] v h t ds ) d y 3 d y 2 (4–22) Denote L fortheareaenclosedbythelinesegments [ a 2 a 3 ],[ a 3 y 3 ],[ y 3 y 2 ] and [ y 2 a 2 ] Then,byintegrationbyparts,wehavethat Z L curl rz ~ v h d x = Z [a 2 ,a 3 ]+[a 3 ,y 3 ]+[y 3 ,y 2 ]+[y 2 ,a 2 ] ~ v h t ds Therefore,from( 4–22 ),wehavethat j Z e 1 ( S c ~ v h v h ) t ds j Z D h a 2 Z D h a 3 23 ( j Z L curl rz ~ v h d x j + j Z [a 2 ,y 2 ] ~ v h t ds j + j Z [a 3 ,y 3 ] ~ v h t ds j ) d y 3 d y 2 Z D h a 2 Z D h a 3 23 ( k curl rz ~ v h k L 1 ( L ) Ch ( h )+ k ~ v h k L 1 ([a 2 ,y 2 ] [ [a 3 ,y 3 ]) h ) d y 3 d y 2 Z D h a 2 Z D h a 3 23 Ch ( k ~ v h k L 1 ( L ) + k ~ v h k L 1 ([a 2 ,y 2 ] [ [a 3 ,y 3 ]) ) d y 3 d y 2 bytheinverseinequality, Ch k ~ v h k L 1 ( C K ) (4–23) Thelastinequalityholds,since L C K and [ a 2 y 2 ] [ [ a 3 y 3 ] C K .Obviously,suchresult holdsfor j R e 2 ( S c ~ v h v h ) t ds j and j R e 3 ( S c ~ v h v h ) t ds j aswellbysimilararguments. Thus, 3 X i =1 j Z e i ( S c ~ v h v h ) t ds j 2 Ch 2 2 k ~ v h k 2L 1 ( C K ) 51

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Hence,bysimilarargumentsasin( 4–18 ),wereach, k R c h v h v h k 2L 2r ( K ) Cr K h 2 2 k ~ v h k 2L 1 ( C K ) C 2 k ~ v h k 2L 2r ( C K ) Lemma4.2.4. Thereexistsaconstant C independentof h and suchthat 1.. k R o h w k 2L 2r ( D ) C k w k 2L 2r ( D ) forall w 2 L 2r ( D ) 2.. k R o h w h w h k L 2r ( D ) C k w h k L 2r ( D ) forall w h 2 S h Proof. Fix K =[ a 1 a 2 a 3 ] 2 T h ,andlet C K denotetheconvexhullof D h a 1 D h a 2 ,and D h a 3 AsintheproofsofLemmas 4.2.2 and 4.2.3 ,itisenoughtoshowthat k R o h w k 2L 2r ( K ) C k ~ w k 2L 2r ( C K ) and k R o h w h w h k L 2r ( K ) C k ~ w h k L 2r ( C K ) Wewillwrite J =( P 3m =1 ( @ r m ) y m ) ( P 3n =1 ( @ z n ) y n ) inthedenitionof S o .Then, j R o h w j K j 1 j K j Z K j Z D h a 1 Z D h a 2 Z D h a 3 123 w (~ x y ) Jd y 3 d y 2 d y 1 j d x 1 j K j 3 X i =1 Z T i j Z D h a 1 Z D h a 2 Z D h a 3 123 w (~ x y ) Jd y 3 d y 2 d y 1 j d x (4–24) 52

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where T i = x 2 K : i ( x ) > 1 3 ,for 1 i 3 .Werstboundthesummandinvolving T 1 Theothersummandsarealsoboundedinanidenticalway. Z T 1 j Z D h a 1 Z D h a 2 Z D h a 3 123 ~ w (~ x y ) Jd y 3 d y 2 d y 1 j d x Z T 1 Z D h a 2 Z D h a 3 23 j ( a 1 ,~ w (~ x y ) J ) r D h a 1 j d y 3 d y 2 d x Z T 1 Z D h a 2 Z D h a 3 23 k a 1 k L 2r ( D h a 1 ) k ~ w (~ x y ) J k L 2r ( D h a 1 ) d y 3 d y 2 d x Z T 1 Z D h a 2 Z D h a 3 23 C p ( h ) 3 ( Z D h a 1 r ( y 1 ) j ~ w (~ x y ) j 2 j J j 2 d y 1 ) 1 2 d y 3 d y 2 d x byProposition 4.2.1 C p ( h ) 3 Z D h a 2 Z D h a 3 23 j T 1 j 1 2 ( Z T 1 Z D h a 1 r (~ x y ) j ~ w (~ x y ) j 2 j J j 2 d y 1 d x ) 1 2 d y 3 d y 2 (4–25) r ( y 1 ) Cr (~ x y ) inthelastinequality,since x 2 T 1 andbydenitionof T 1 : r ( y 1 )= 1 1 ( x ) 1 ( x ) r ( y 1 ) 3 r ( 1 ( x ) y 1 ) 3( r ( 1 ( x ) y 1 )+ r ( 1 ( x ) y 2 )+ r ( 1 ( x ) y 3 ))=3 r (~ x y ). Now, j T 1 j 1 2 Ch ,and J C ,since J = j ~ K j j K j ,where ~ K =[ y 1 y 2 y 3 ] .Therefore,by( 4–25 ), Z T 1 j Z D h a 1 Z D h a 2 Z D h a 3 123 ~ w (~ x y ) Jd y 3 d y 2 d y 1 j d x Ch p ( h ) 3 Z D h a 2 Z D h a 3 23 ( Z T 1 Z D h a 1 r (~ x y ) j ~ w (~ x y ) j 2 j J j d y 1 d x ) 1 2 d y 3 d y 2 = Ch p ( h ) 3 Z D h a 2 Z D h a 3 23 ( Z D h a 1 Z T 1 r (~ x y ) j ~ w (~ x y ) j 2 j J j d x d y 1 ) 1 2 d y 3 d y 2 = Ch p ( h ) 3 Z D h a 2 Z D h a 3 23 ( Z D h a 1 Z ~ T 1 r ( z ) j ~ w ( z ) j 2 d z d y 1 ) 1 2 d y 3 d y 2 bychangeofvariables, Ch p ( h ) 3 j D h a 1 j 1 2 k ~ w k L 2r ( ~ T 1 ) Ch p h k ~ w k L 2r ( C K ) (4–26) 53

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Therefore,by( 4–24 ),wehavethat j R o h w j K j 2 1 j K j 2 Ch 2 h k ~ w k 2L 2r ( C K ) C h 3 k ~ w k 2L 2r ( C K ) Hence, k R o h w k 2L 2r ( K ) Ch 2 r K j R o h w j 2K j Ch 3 h 3 k ~ w k 2L 2r ( C K ) C k ~ w k 2L 2r ( C K ) Thisprovesitem 1 oftheLemma. Nowweproveitem 2 .Itisclearthat j R o h w h w h j K j = j 1 j K j Z K S o ~ w h d x 1 j K j Z K w h d x j = j 1 j K j Z D h a 1 Z D h a 2 Z D h a 3 ( Z ~ K ~ w h ( z ) d z Z K w h ( x ) d x ) d y 3 d y 2 d y 1 j by( 4–14 ), 1 j K j Z D h a 1 Z D h a 2 Z D h a 3 Z ( ~ K n K ) [ ( K n ~ K ) j ~ w h ( x ) j d x d y 3 d y 2 d y 1 (4–27) where ~ K =[ y 1 y 2 y 3 ] .Noticethatboth ~ K n K and K n ~ K willalwaysbeinsidetheunion of conv ( D h a 1 D h a 2 ) ,and conv ( D h a 2 D h a 3 ) ,and conv ( D h a 3 D h a 1 ) ,where conv ( A B ) denotes theconvexhullof A and B .Therefore, area (( ~ K n K ) [ ( K n ~ K )) Ch ( h ) ,andso continuing( 4–27 ), j R o h w h w h j K j 1 j K j Z D h a 1 Z D h a 2 Z D h a 3 Ch ( h ) k ~ w h k L 1 ( C K ) d y 3 d y 2 d y 1 C k ~ w h k L 1 ( C K ) (4–28) 54

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Hence, k R o h w h w h k 2L 2r ( K ) Ch 2 r K j R o h w h w h j K j 2 Ch 2 r K 2 k ~ w h k 2L 1 ( C K ) by( 4–28 ), C 2 k ~ w h k 2L 2r ( C K ) DuetothesecondresultofLemmas 4.2.2 4.2.3 ,and 4.2.4 ,thereisa 0 < 1 0 suchthattheoperators R g h j V h R c h j W h ,and R o h j S h areinvertibleforall 0 < 1 .We denotetheseinverseoperatorsby J g h : V h V h J c h : W h W h ,and J o h : S h S h respectively.Fortherestofthepaper,wex 2 (0, 1 ] .Thenbyconstruction,these inverseoperatorspreservethecommutingdiagramproperti es,andtheyarealso uniformlyboundedinthe L 2r -norm.Hence,wemodifythequasi-interpolatorsasin[ 17 ]to constructprojectors. Dene gh : L 2r ( D ) V h ch : L 2r ( D ) 2 W h ,and oh : L 2r ( D ) S h by 1. gh = J g h R g h 2. ch = J c h R c h 3. oh = J o h R o h Theorem4.2.1. Projectors gh ch ,and oh satisfythefollowingerrorestimatesforall 0 s 1 1.. k u gh u k r Ch s k u k H s r ( D ) forall u 2 H s r ( D ) 2.. k v ch v k r Ch s k v k H s r ( D ) 2 forall v 2 H s r ( D ) 2 3.. k w oh w k r Ch s k w k H s r ( D ) forall w 2 H s r ( D ) If s =1 thenormsontherighthandsideofallthreeinequalitiesabo vecanbereplaced bysemi-norms. 55

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Proof. Wewillonlyprovetheerrorestimatefor ch astheprooffollowssimilarlyforthe otherprojectorsaswell.Let h : L 2r ( D ) 2 W h bethe L 2r -orthogonalprojection.Since k v ch v k r (1+ C ) k v k r byLemma 4.2.3 item 1 and k v ch v k r = k ( v h v ) ch ( v h v ) k r (1+ C ) k ( v h v ) k r (1+ C ) Ch j v j H 1 r ( D ) 2 itfollowsthat k v ch v k r Ch s k v k H s r ( D ) 2 forall 0 s 1 ,byinterpolationtheory[ 4 ]. Thefollowingcorollaryisimmediatebythecommutingdiagr amproperty. Corollary4.2.1. k v ch v k Ch s ( k v k H s r ( D ) 2 + k curl rz v k H s r ( D ) ) if v 2 H s r ( D ) 2 and curl rz v 2 H s r ( D ) forall 0 s 1 56

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CHAPTER5 ANALYSISOFADUALMIXEDPROBLEMINWEIGHTEDSPACES Inthischapter,weanalyzeamixedprobleminweightedspace s,thatwillprovide mainingredientsfortheanalysisofthemeridianproblemin Chapter 6 and 7 .This problemisinterestinginitsownright,sinceitisrelatedt otheazimuthalproblemaswe shallsee.Wewillprovethattheproblemiswellposedandpro videerrorestimatesfor thediscretesolution.Muchoftheresultsinthischapterar econtainedin[ 20 ]. 5.1ProblemStatementandAnalysis Theproblemcanbestatedasfollows:Find z in H r (curl, D ) and p in L 2r ( D ) satisfying ( z w ) r ( p ,curl rz w ) r =0, forall w in H r (curl, D ), ( s ,curl rz z ) r =( s f ) r forall s in L 2r ( D ). (5–1) Observethatthisisavariationalformulationofthebounda ryvalueproblem z = curl rz p on D curl rz z = f on D z t =0 on 1 whichcanalsobewrittenasthesecond-orderboundaryvalue problem curl rz curl rz p = f on D and curl rz p t =0 on 1 (5–2) Thedifferentialoperatorappearinghereisthesameasthes econdorderoperator appearingintheazimuthalproblem( 1–9 )deningthe -componentoftheelectric eldinthetimeharmonicMaxwellequationsunderaxialsymm etry.Interestingly,this “azimuthal”operatorplaysanimportantroleintheniteel ementanalysisandthe multigridanalysisofthe“meridian”operator.Problem( 5–1 )isindependentlyinteresting becauseoftheabovementionedconnectiontotheazimuthalM axwellsystem.Indeed,a 57

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primalvariationalformulationof( 5–2 ),butwithdifferentboundaryconditions,isanalyzed in[ 25 ].Inthissection,wewillanalyzethedualvariationalform ulation( 5–1 ). Webeginwiththefollowinglemma,whichwillhelpinproving thatthemixed problem( 5–1 )iswellposed(cf.Lemma A.0.6 inAppendix A ). Lemma5.1.1. Themap curl rz : H r (curl, D ) 7! L 2r ( D ) issurjective. Proof. Let s bein L 2r ( D ) .Itisshownin[ 5 25 ]thatthereexistsaunique u in V satisfying ( curl rz u curl rz v ) r =( s v ) r forall v 2 V (5–3) where V := f v 2 e H 1 r ( D ): v =0 on @ D g .Thisimpliesthat s =curl rz curl rz u in L 2r ( D ) by thedensityof D ( D ) in L 2r ( D ) .Hence,setting w = curl rz u ,wendthat s =curl rz w (5–4) Notethat w isin L 2r ( D ) 2 ,since k curl rz u k r C k u k e H 1 r ( D ) by[ 25 ,Proposition3.1].In fact,thesetwonormsareequivalentas u isin V .Moreover,by( 5–4 ) curl rz w isin L 2r ( D ) ,so w isin H r (curl, D ) .However,wewanttoexpress s asthecurlofafunctionin H r (curl, D ) Tothisend,werstdene W 0 by W 0 := w 2 H r (curl, D ):( w grad rz q ) r =0 forall q 2 H 1 r ( D ) andlet P 0 denotetheorthogonalprojectionfrom H r (curl, D ) to W 0 inthe ( ) r -inner product.Clearly,ifweshowthat s =curl rz ( P 0 w ), (5–5) theproofofthelemmawillbecomplete. Consideringany in D ( D ) ,itiseasytocheckthat curl rz isin W 0 (5–6) 58

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Furthermore,since isin D ( D ) (curl rz ( P 0 w ), ) r =( P 0 w curl rz ) r by( 2–8 ) =( w curl rz ) r by( 5–6 ) =( s ) r by( 2–8 )and( 5–4 ) Since D ( D ) isdensein L 2r ( D ) (seee.g.[ 28 ]),wehaveproved( 5–5 ). Theorem5.1.1. Thereexistsaunique z in H r (curl, D ) andaunique p in L 2r ( D ) satisfying ( 5–1 ) .Moreover,thereisaconstant C stability > 0 independentof f suchthat k z k + k p k r C stability k f k r Proof. BytheBabu ska-Brezzitheoryofmixedmethods[ 16 ],thetheoremwillfollow onceweverifytheinf-supcondition C 1 k s k r sup v 2 H r (curl, D ) (curl rz v s ) r k v k forall s in L 2r ( D ), (5–7) andcoercivityonthekernel, k v k C 2 k v k r forall v in G (5–8) where G isthekerneldenedby G = w 2 H r (curl, D ):(curl rz w s ) r =0 forall s in L 2r ( D ) Above, C 1 and C 2 aretwoconstantsindependentofthefunctionsinvolved. Theinf-supcondition( 5–7 )isequivalenttoassertingthattheadjointoftheoperator curl rz : H r (curl, D ) 7! L 2r ( D ) isboundedfrombelow,whichisequivalenttothesurjectivi tyoftheabovecurlmap (bystandardargumentsusingtheClosedRangeTheorem,seee .g.,[ 16 x II.1]). ThissurjectivityispreciselytheassertionofLemma 5.1.1 .Hence,itonlyremainsto 59

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verify( 5–8 ),whichisobvious,since v isin G ifandonlyif curl rz v =0 ,sothat k v k = k v k r forall v in G Next,weprovearegularityresult.Recallthat n istherevolutionof D ,andthat wedenoteby L 2 (n) and H k (n) thesubspacesofaxisymmetricfunctionsof L 2 (n) and H k (n) ,respectively,for k 1 .Therestrictionmap g ( r z ) 7! g D ( r z ) givenby g D ( r z )= g ( r ,0, z ), forall ( r z ) in D isanisometry(uptoafactorof p 2 )from L 2 (n) onto L 2r ( D ) (SeeTheorem 2.3.1 ).The reverseoperationwillbedenotedbysuperscriptingfuncti onswith n ,i.e.,given ( r z ) on D ,thefunction n isdenedby n ( r z )= ( r z ) .Thus ( g D ) n = g ,for g in L 2 (n) Withtheuseofsuchnotations,wewillnowprovethefollowin gestimates,whichwill beusefulinourmultigridanalysis.Theorem5.1.2. Thesolution ( z p ) of ( 5–1 ) satises k z k H s r ( D ) 2 C regularity k f k r ( k p k 2H s r ( D ) + k curl rz p k 2H s r ( D ) 2 ) 1 2 C regularity k f k r foranydata f in L 2r ( D ) ,where s = 1 2 if n isaboundedLipschitzdomain,and s =1 if n isconvex.Infact,if n isconvex k p k e H 2 r ( D ) C regularity k f k r Proof. Let ( z p ) solve( 5–1 ).Dene p = p n e .Thenrecallingtheexpression div q = 1 r @ r ( rq r )+ 1 r @ q + @ z q z (5–9) fordivergenceincylindricalcoordinates,wendthat div p =0 on n, p n =0 on @ n. 60

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Thelastequalityfollowsbecausetheunitoutwardnormal n on @ n isorthogonalto e Furthermore,fromtherstequationof( 5–1 ),weknowthattheequality z curl rz p =0 holdsinthedistributionalsense.Since z isin L 2r ( D ) 2 ,theequality z = curl rz p (5–10) infactholdsin L 2r ( D ) 2 Bywritingoutthethree-dimensionalcurlincylindricalco ordinates, weobservethat curlp =( curl rz p ) n = z n (5–11) whereforaxisymmetricvectorelds v = v r e r + v z e z ,therevolutionisdenedby v n = v n r e r + v n z e z .Thus curlp isin L 2 (n) 3 .Combiningtheseobservations,wend that p isin H ( curl ,n) \ H 0 (div,n) ,aspacewhichiswell-knowntobecontinuously embeddedin H 1 2 (n) 3 forboundedLipschitzdomains(Theorem 2.2.2 ).Thus,wehave k p k H 1 2 (n) 3 C k p k H(curl,n) + k p k H(div,n) C k p k r + k curl rz p k r C k p k r + k z k r C k f k r (5–12) whereinthelaststepwehaveusedTheorem 5.1.1 Thesecondequalityofthevariationalproblem( 5–1 )showsthat curl rz z = f holdsin L 2r ( D ) .Translatingthisfor z n ,wehave curlz n = f n e on n, div z n =0 on n, z n n =0 on @ n. Thelastequalityholdsbecause r t ( z )=0 ,andthesecond, div z n =0 ,follows from( 5–11 ).Nowusingthecontinuousembeddingof H 0 ( curl ,n) \ H (div,n) into 61

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H 1 2 (n) 3 [ 22 ],weobtain k z n k H 1 2 (n) 3 = k curlp k H 1 2 (n) 3 C k f k r (5–13) ByusingTheorem 2.3.1 item 3 whichstatestheisomorphismbetween H 1 2 (n) 3 and H 1 2 r ( D ) H 1 2 r ( D ) H 1 2 r ( D ) ,therstinequalityofthetheoremfollowsimmediatelyfro mthe aboveestimate,since k z k H 1 2 r ( D ) 2 C k z n k H 1 2 (n) 3 Againbyusingthisisomorphism(Theorem 2.3.1 item 3 ),( 5–12 )and( 5–13 )weget jj curl rz p jj 2H 1 2 r ( D ) 2 + jj p jj 2H 1 2 r ( D ) C ( jj curlp jj 2H 1 2 (n) 3 + jj p jj 2H 1 2 (n) 3 ), C jj f jj 2r whichprovesthesecondestimateofthetheorem.If n isconvexthenwereachthe resultbyusingthecontinuousembeddingresultof H 0 ( curl ,n) \ H (div,n) and H ( curl ,n) \ H 0 (div,n) into H 1 (n) 3 (Theorem 2.2.1 ),andtheisomorphismbetween H 1 (n) 3 and e H 1 r ( D ) e H 1 r ( D ) H 1 r ( D ) (Theorem 2.3.1 item 3 ). Letusnowconsiderthemixedniteelementapproximationof ( 5–1 ).Thediscrete problemistond z h in W h and p h in S h satisfying ( z h w h ) r ( p h ,curl rz w h ) r =0, forall w h in W h ( s h ,curl rz z h ) r =( s h f ) r forall s h in S h (5–14) Atthispoint,wecanproceedtoanalyzethediscretemixedme thodbyverifying theconditionsoftheBabu ska-Brezzitheory,whichwouldyield apriori errorestimates. However,forourmultigridanalysis,wewillneederroresti matesinaslightlymore specializedform,sowewillprovideadirecterroranalysis .Wewillalsoproveahigher orderestimateobtainedviaduality.Theseresultsarecoll ectedinthenexttheorem. Theorem5.1.3. Suppose z in H r (curl, D ) and p in L 2r ( D ) solve ( 5–1 ) 1..Thereisaunique z h in W h andaunique p h in S h satisfying ( 5–14 ) 62

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2..Thefollowingerrorestimateholds: k z z h k r = rr z Wh z rr r 3..If f isin S h ,then rr Sh p p h rr r Ch 2 s k f k r where s = 1 2 if n isaboundedLipschitzdomain,and s =1 if n isconvex. Proof. Proofof 1 :Suppose f =0 in( 5–14 ).Thenbysetting w h = z h ,weget z h =0 Then ( p h ,curl rz w h ) r =0 forall w h in W h whichimpliesthat p h =0 bytheexactnessof( 4–3 ),andthiscompletestheproof. Proofof 2 :Since,by( 5–14 ), ( z h grad rz h ) r =0=( z grad rz h ) r forall h 2 V h and (curl rz z h s h ) r =( f s h ) r =(curl rz z s h ) r forall s h 2 S h bytheuniquesolvabilityof( 4–6 ), z h = Wh z .Therefore,item 2 holds. Proofof 3 :Weproceedbyadualityargument,suitablymodied.Let z in H r (curl, D ) and p in L 2r ( D ) solve( 5–1 )with f setto Sh p p h .Lettheirdiscrete counterpartsbe z h in W h and p h in S h ,whichsolve( 5–14 )with f setto Sh p p h Thenby( 5–14 )and( 5–1 ), rr Sh p p h rr 2r =( Sh p p h ,curl rz z h ) r =( p p h ,curl rz z h ) r =( z z h z h ) r =( z z h z h z ) r +( z z h z ) r (5–15) Now,since f isgiventobein S h curl rz z =curl rz z h = f 63

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Thistogetherwiththedenitionof f z p g implythatthelasttermin( 5–15 )vanishes: ( z z z h ) r =( p ,curl rz ( z z h ))=0. Usingthisin( 5–15 )andcontinuing, rr Sh p p h rr 2r =( z z h z h z ) r k z z h k r k z h z k r = rr z Wh z rr r rr z Wh z rr r byitem 2 Ch 2 s k z k H s r ( D ) 2 k z k H s r ( D ) 2 byCorollary 4.1.1 forall 0 s 1 .Wechoose s = 1 2 forboundedLipschitz n ,and s =1 forconvex n in ordertoapplytheregularityresultofTheorem 5.1.2 k z k H s r ( D ) 2 C k Sh p p h k r Thus, rr Sh p p h rr 2r Ch 2 s k z k H s r ( D ) 2 k Sh p p h k r CancelingthecommonfactorandapplyingTheorem 5.1.2 again,weobtaintherequired estimate. Remark 5.1.1 If n isconvexthenitem 3 ofTheorem 5.1.3 canbethoughtofas asuperconvergenceresult,asitshowsthatweobtainquadra ticconvergencefor Sh p p h evenwhenusingpiecewiseconstantapproximationspaces.I nthisrespect, thisresultissimilartocertainknownsuperconvergenceer rorestimatesderivedvia dualityargumentsfortheRaviart-Thomasmixedmethod[ 19 23 34 ],albeitwithouta degenerateweightfunction.Remark 5.1.2 Thereisananalogueofthemixedproblem( 5–1 )inthecaseofthefully three-dimensional curlcurl operator,sometimescalledthedualmixedformulation(see e.g.[ 8 ]whereitusedforeigenvalueanalysis).However,thismeth odisnotpractically 64

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popularinthe3Dcaseasitsimplementationrequiresabasis forexactlydivergence-free niteelementspaces,whichisnoteasytoconstruct.Thisdi fcultyisabsentinthe axisymmetriccase. 5.2NumericalResults Inthissection,wewillreporttheobservedconvergencerat efortheapproximate solutionofthedualmixedproblem.Thiswillserveasatesto fthesharpnessofour theoreticalerrorestimates. Forcomputerimplementationofthemixedmethod,weneedtoa ssemblethematrix representationsoftheoperators A h : W h 7! W 0 h and B h : W h 7! S 0 h denedby A h u h ( w h )=( u h w h ) r forall u h w h 2 W h B h u h ( s h )= (curl rz u h s h ) r forall u h 2 W h s h 2 S h Let A and B denotethematrixrepresentationsof A h and B h ,respectively,intermsof thestandardlocalbasesfor W h and S h (consistingoftheWhitneyfunctions e ,andthe indicatorfunctionsoftriangles).Then( 5–14 )canberewrittenasthelinearsystem A z + B t p =0, B z = f where z and p denotethevectorsofcoefcientsinthebasisexpansionsof z h and p h respectively.Thevector f iscomputedfromtherighthandsideof( 5–14 )asusual.In practice,wecompute p and z bysolving C p = f A z = g where C = BA 1 B t and g = B t p .Boththesesystemscanbesolvedviatheconjugate gradientmethodas C and A aresymmetricandpositivedenite.Notethatwhensolving 65

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Table5-1.Mixedproblemconvergencerates level k z z h k r order k p p h k r order rr Sh p p h rr r order 1 0.305150 0.123463 0.044430 2 0.152784 0.998 0.059647 1.05 0.011144 1.995 3 0.076486 0.998 0.029553 1.013 0.002791 1.997 4 0.038263 0.999 0.014743 1.003 0.000698 1.999 5 0.019135 1 0.007367 1.001 0.000175 2 6 0.009568 1 0.003683 1 0.000044 2 7 0.004784 1 0.001842 1 0.000011 1.999 8 0.002392 1 0.000921 1 0.000003 2 therstequation,foreachapplicationof C ,weuseanotherinnerconjugategradient iterationtoobtaintheresultofmultiplicationby A 1 InTable 5-1 ,wereportthe L 2r ( D ) -normoftheobservederrorsinthemixedmethod approximationsof z p ,and Sh p .Inthiscase, f = 3 p = r 2 and z =(0,3 r ) .The domain D wasthechosentobetheunitsquare.Notethatinthiscase, n isconvex. Thecoarsestmeshisobtainedbydividingtheunitsquareint otwouniformtrianglesby connectingthepoints(0,0)and(1,1).Thisismeshlevel0.H igherlevelsareobtained bysuccessiverenements.Eachrenementisperformedbyco nnectingthemidpoints ofeachedge,sothemeshsizereducesby 1 = 2 ,andthenestmesh(level 8 )isroughly ofsize 1 = 256 .Theorderofconvergenceiscomputedas log 2 ( e j 1 = e j ) ,where e j isthe computed L 2r ( D ) -normoftheerroratmeshlevel j Fromthetable,weobservethattheapproximationsfor z and p convergeat rstorder.Thisconvergencefor z isinaccordancewithTheorem 5.1.3 item 2 .The convergencefor p isalsoinaccordancewiththetheorem,becausebytrianglei nequality k p p h k r k p Sh p k r + k Sh p p h k r andalthoughTheorem 5.1.3 item 3 assertsthatthelasttermis O ( h 2 ) ,thersttermon therighthandside,being O ( h ) ,dominates.Thatthelasttermindeedsuperconvergesat doubletheorderisveriedinthelastrowofthetable. 66

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CHAPTER6 FINITEELEMENTANALYSISFORTHEMERIDIANPROBLEM Inthischapter,weusetheedgeniteelementmethodtondan approximate solutiontothemeridianproblem( 1–8 ).Wewillshowthatthetheedgeniteelement methodprovidesagoodapproximationtothisproblemunderc ertainconditions. 6.1TheEdgeFiniteElementMethod Theweakformulationof( 1–8 )inthesimplecaseofunitmaterialpropertiesreads: Find u =( E r E z ) 2 H r (curl, D ) suchthat (curl rz u ,curl rz v ) r 2 ( u v ) r =( F v ) r (6–1) forall v 2 H r (curl, D ) ,wherehereandintheremainderofthisdissertationwedeno te F for ( F r F z ) .Notethatthereisacountablesetofrealvaluesfor forwhich( 6–1 )does nothaveauniquesolution[ 29 ].Fortheremainderofthisdissertation,weassumethat isnotoneofsuchvaluessothat( 6–1 )isuniquelysolvable. Theniteelementmethodreducesaninnitedimensionalpro blemintoanite dimensionalone(SeeChapter 3 section 3.1 ).Therststepistoconstructanite dimensionalsubspaceoftheinnitedimensionalspace H r (curl, D ) .Wewillusethe lowestorderN ed elecspace W h H r (curl, D ) fortheniteelementsubspace.Recall that W h = f v h 2 H r (curl, D ): v h j K =( b az c + ar ) forsome a b c 2 R forall K 2 T h g Sincefunctionsin W h havetheform ( b az c + ar ) forsome a b c 2 R whenrestricted toeachtriangleinthetriangulationofthedomain,andther earenitelymanytrianglesin themesh, W h isclearlynitedimensional. Nextwesolvetheproblem( 6–1 )on W h insteadof H r (curl, D ) : Find u h 2 W h suchthat (curl rz u h ,curl rz v h ) r 2 ( u h v h ) r =( F v h ) r (6–2) 67

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forall v h 2 W h Nowwearereadytochangethisnitedimensionproblem( 6–2 )intoamatrix systembyusingbasisfunctions.Let f i g Ni =1 denotethebasisof W h .Thenthesolution u h = P Ni =1 c i i forsome c i 2 R ,1 i N .Additionally,since( 6–2 )holdsforall v h 2 W h ,itholdsforall j Nj =1 inparticular.Therefore, (curl rz N X i =1 c i i ,curl rz j ) r 2 ( N X i =1 c i i j ) r =( F j ) r forall 1 j N N X i =1 ((curl rz i ,curl rz j ) r 2 ( i j ) r ) c i =( F j ) r forall 1 j N Hence, f c i g Ni =1 canbeobtainedbysolvingthematrixsystem A ~ c = ~ b where A isthe N N matrixwhose ji -thentryis A ji =(curl rz i ,curl rz j ) r 2 ( i j ) r and ~ b isthevectoroflength N whose j -thentryis ( F j ) r .Oncewendthisvector ~ c wecanusetheformula u h = P Ni =1 c i i toobtaintheapproximatesolution u h Remark 6.1.1 Sincetherearethreeunknownsforeachtriangleandallfunc tionsin W h H r (curl, D ) musthavecontinuoustangentialcomponentsalongeachedge ,it followsthatthedimension N of W h isthenumberofedgesinthecorrespondingmesh thatisnoton 1 .Thisiswhywecallsuchmethodthe“edge”niteelementmeth od. Foreachedgenoton 1 weconstructabasisfunctioninthefollowingway.Let i be associatedwiththe i -thedgenoton 1 denotedby e i .Then i j K = C ( m grad rz n n grad rz m ) foreachtriangle K ,where m and n denotesthebarycentriccoordinatesof thetwoverticesof e i withrespecttothetriangle K .Wechoose C sothatthetangential componentof i willbeoneonthe i -thedge.Thus, i hassupportontwotrianglesif e i isnoton 0 ,anditisdenedpiecewise,butthetangentialcomponentso n e i willagree. Thisisthewell-knownWhitneybasisfunction. 68

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Wenowstatetheresultthatillustrateshowaccuratelythed iscretesolution u h of( 6–2 )approximatestheexactsolution u of( 6–1 ). Theorem6.1.1 (Quasi-Optimality) Suppose n ,therevolutionof D ,isabounded Lipschitzdomain.If( 6–1 )hasauniquesolution u 2 H r (curl, D ) thenthereisaconstant h 0 and C suchthat,forall 0 < h < h 0 ,( 6–2 )alsohasauniquesolution u h ,and k u u h k C inf w h 2 W h k u w h k Theseconstantsareindependentof u and u h Thisresultwillbeprovedinthenextsection.Byusingthist heoremandCorollary 4.2.1 wehavethefollowingresult.Corollary6.1.1. UnderthesameconditionsofTheorem 6.1.1 ,wehave k u u h k Ch s ( k u k H s r ( D ) 2 + k curl rz u k H s r ( D ) ), forall 0 s 1 Wewilllaterseethat k u k H 1 2 r ( D ) 2 + k curl rz u k H 1 2 r ( D ) C k F k r when n isabounded Lipschitzdomain.Thisresultshowsthatwecanmaketheappr oximation u h ascloseas wewanttotheexactsolution u byreningthemeshandmakingthemeshsizesmall. 6.2ProofoftheQuasi-OptimalityResult BeforeweproveTheorem 6.1.1 ,weprovetwoimportantLemmasthatwillnot onlybeusedfortheniteelementapproximationbutalsofor themultigridanalysisin Chapter 7 .Recallthat n isthethreedimensionalrotationaldomainof D Lemma6.2.1. Let w h 2 W h beadiscretedivergencefreefunction,i.e., ( w h grad rz h ) r =0 forall h 2 V h 69

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andsuppose Sw h 2 H r (curl, D ) and p 2 L 2r ( D ) isthesolutionofthefollowingdual mixedproblem. ( Sw h x ) r ( p ,curl rz x ) r =0, forall x in H r (curl, D ), ( s ,curl rz Sw h ) r =( s ,curl rz w h ) r forall s in L 2r ( D ). (6–3) Then k Sw h w h k r Ch s k curl rz w h k r where s = 1 2 if n isaboundedLipschitzdomain,and s =1 if n isconvex. Proof. Since w h isdiscretedivergencefree,bythediscreteHelmholtzdeco mpositionin weightedspaces, w h =curl 0rz p h ,forsome p h 2 S h .Thus, ( w h x h ) r ( p h ,curl rz x h ) r =0 forall x h 2 W h Therefore, k Sw h w h k r = rr Sw h Wh Sw h rr r byTheorem 5.1.3 item 2 Ch s k Sw h k H s r ( D ) 2 byCorollary 4.1.1 Ch s k curl rz w h k r byTheorem 5.1.2 InusingTheorem 5.1.2 inthelastinequality,wehavethat s = 1 2 when n isabounded Lipschitzdomain,and s =1 when n isconvex.Thiscompletestheproof. Letusdenoteby A : H r (curl, D ) H r (curl, D ) thebilinearform: A ( u v )=(curl rz u ,curl rz v ) r 2 ( u v ) r Thenwehavethefollowingregularityresultforthemeridia nproblem( 6–1 ).Recallthat given v ( r z ) on D thatisinvariantunderrotation,thefunction v n on n isdenedby v n ( r z )= v ( r z ) .Foraxisymmetricvectorelds v ( r z )=( v r ( r z ), v z ( r z )) onD,we dene v n on n by v n =( v n r ,0, v n z ) 70

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Lemma6.2.2. Given F 2 L 2r ( D ) 2 suchthat ( F grad rz ) r =0 forall 2 H 1 r ( D ), (6–4) suppose u 2 H r (curl, D ) isthesolutionof A ( u v )=( F v ) r forall v 2 H r (curl, D ). (6–5) Then k u k H s r ( D ) 2 + k curl rz u k H s r ( D ) C k F k r where s = 1 2 if n isaboundedLipschitzdomain,and s =1 if n isconvex.Infact,if n is convex,then curl rz u 2 e H 1 r ( D ) Proof. Itwasshownin[ 21 ]thatcondition( 6–4 )impliesthat ( F n grad ) r =0 forall 2 H 1 0 (n). Therefore,bytakingderivativeinthesenseofdistributio ns, (div F n )= ( F n grad )=0 forall 2 D (n), andso div F n =0 in L 2 (n) Since curl rz curl rz u 2 u = F directcalculationshowsthat curlcurlu n 2 u n = F n Notethat u n 2 H 0 (curl, D ) and F n 2 L 2 (n) 2 .Therefore,by[ 22 ], jj u n jj H 1 2 (n) 3 + jj curlu n jj H 1 2 (n) 3 jj F n jj (6–6) 71

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Then,sincethe -componentof curlu n is curl rz u ,itfollowsfromtheisomorphism between H 1 2 (n) and H 1 2 r ( D ) H 1 2 r ( D ) H 1 2 r ( D ) (Theorem 2.3.1 item 3 )andthe isomorphismbetween L 2 (n) and L 2r ( D ) (Theorem 2.3.1 item 1 )that k u k H 1 2 r ( D ) 2 + k curl rz u k H 1 2 r ( D ) C k F k r If n isconvexthenweusethewell-knownresult[ 30 ] jj u n jj H 1 (n) 3 + jj curlu n jj H 1 (n) 3 jj F n jj insteadof( 6–6 ).Thiscompletestheproof. NotethatinChapter 7 ,weassumethattherotationof D ,namely n ,isconvex. Therefore,inChapter 7 wewilluseLemmas 6.2.1 and 6.2.2 undertheconvexity assumption. Wearenowreadytoprovethemainresultofthischapter. ProofofTheorem 6.1.1 Iftheresultholds,thenthewell-posednessofproblem( 6–2 ) follows,soweonlyneedtoprovetheerrorestimategiventha tthemeshsizeis sufcientlysmall.Let e = u u h ,andlet w h 2 W h bearbitrary.Then A ( e w h )=0 .Thus, k e k 2 =( e u w h )+( e w h u h ), k e k k u w h k + A ( e w h u h )+(1+ 2 )( e w h u h ) r = k e k k u w h k +(1+ 2 )( e w h u h ) r (6–7) Weapproximate ( e w h u h ) r Let e = grad rz + bethecontinuousHelmholtzdecompositionof e where 2 H 1r ( D ) and 2 H r (curl, D ) .Let w h u h = grad rz h +curl 0rz s h bethediscrete 72

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Helmholtzdecompositionof w h u h ,andlet S (curl 0rz s h ) beasin( 6–3 ).Then ( grad rz w h u h ) r =( grad rz ,curl 0rz s h ) r =( grad rz ,curl 0rz s h S (curl 0rz s h )) r k e k r k curl 0rz s h S (curl 0rz s h ) k r Ch 1 2 k e k r k curl rz curl 0rz s h k r byLemma 6.2.1 = Ch 1 2 k e k r k curl rz ( w h u h ) k r Therstequalityholds,since ( grad rz grad rz h ) r =( e grad rz h ) r = 1 2 A ( e grad rz h )=0. Therefore, ( grad rz w h u h ) r Ch 1 2 k e k r k curl rz ( w h u h ) k r (6–8) Next,let z 2 H r (curl, D ) bethesolutionof A ( z x )=( x ) r forall x 2 H r (curl, D ). Then, k k 2r =( e ) r = A ( z e ), = A ( z ch z e ), C k z ch z k k e k Ch 1 2 ( k z k H 1 2 r ( D ) 2 + k curl rz z k H 1 2 r ( D ) ) k e k byCorollary 4.2.1 Ch 1 2 k k r k e k byLemma 6.2.2 Therefore, k k r Ch 1 2 k e k (6–9) 73

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Thus,itfollowsfrom( 6–7 )that k e k 2 k e k k u w h k +(1+ 2 )( e w h u h ) r = k e k k u w h k +(1+ 2 )(( grad rz w h u h ) r +( w h u h ) r ), k e k k u w h k + Ch 1 2 k e k k w h u h k by( 6–8 )and( 6–9 ), k e k k u w h k + Ch 1 2 k e k k w h u k + Ch 1 2 k e k k u u h k (1+ Ch 1 2 ) k e k k u w h k + Ch 1 2 k e k 2 Therefore, jj u u h jj 1+ Ch 1 2 1 Ch 1 2 jj u w h jj if 1 Ch 1 2 > 0 .Hence,thereexistssome h 0 > 0 suchthat,forall 0 < h < h 0 k u u h k C k u w h k forall w h 2 W h Inconclusion,forall 0 < h < h 0 k u u h k C inf w h 2 W h k u w h k 6.3NumericalResults Inthissection,wereportnumericalresultsthatprovideem piricalsupporttothe convergenceofedgeniteelementapporximationwhenappli edtotheindenitebilinear form A ( ) .Thedomain D istheunitsquareandmeshlevel1consistoftwouniform righttriangleswiththecommonedgeconnecting (0,0) and (1,1) .Thenextlevelmeshis obtainedbyconnectingthemidpointsofeachedge.Notethat ,inthiscase,therotational domainisconvex. Table 6-1 reportsthe L 2r ( D ) -normoftheerrorbetweentheexactsolution u and discretesoltuion u h of( 6–1 )and( 6–2 )respectivelywith f =( 2 r sin( z ),( 49 r 2 + 74

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Table6-1.FEMconvergencerates level k u u h k r order 2 0.17539 3 0.0987347 0.83 4 0.0456298 1.11 5 0.0213924 1.10 6 0.0104596 1.03 7 0.0051979 1.01 8 0.0025949 1.00 9 0.0012969 1.00 45)cos( z )) andwavenumber =7 .Theorderofconvergenceiscomputedas log 2 ( e j 1 = e j ) ,where e j isthecomputed L 2r ( D ) -normoftheerroratmeshlevel j Fromthetable,weobservethattheapproximationsfor u convergeatrstorder. ThisconvergenceisinaccordancewithTheorem 6.1.1 when n isconvex. 75

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CHAPTER7 MULTIGRIDANALYSISFORTHEMERIDIANPROBLEM WehaveseeninChapter 6 thattheerrorbetweentheniteelementapproximation of( 6–2 )andtheexactsolutionof( 6–1 )approacheszeroasthemeshsizeapproaches zero(Corollary 6.1.1 )undercertainassumptions.Thismeansthatwehavetosolve a largematrixsysteminordertogetagoodapproximationtoth emeridianproblem,since thesizeofthematrixsystemobtainedbytheedgeniteeleme ntmethodcorresponds tothenumberofedgesinthemesh.Multigridisanefcientit erativemethodthatis usedtosolveamatrixsystemobtainedbytheniteelementme thod(SeeChapter 3 section 3.2 ).Inthischapter,wewillshowthatthemultigrid“backslas h”cycleconverges atauniformrateindependentofthemeshsizewhenappliedto themeridianproblem undertheassumptionthattherotationofthedomain D ,namely n ,isconvex.Noticethat themeridianproblem( 6–1 )isanindeniteproblem.Techniquesfrom[ 12 26 ]provide awaytoanalyzetheerrorreductionoperatoroftheindenit eproblembyusingthe errorreductionoperatoroftherelatedpositivedenitepr oblem.Inparticular,itiswell known[ 11 ]thatmultigrid,asaniterativemethod,isconnectedthrou ghalinearerror reductionoperator E ,i.e., x x n +1 = E ( x x n ), where x istheexactsolutionand x n denotesthe n -thiterationresult.Inotherwords, jj x x n jj jj E jj n jj x x 0 jj where x 0 istheinitialvalueoftheiteration.Therefore,ifweshowt hat jj E jj < where 0 << 1 isaconstantindependentofthemeshsize,thentheuniform convergenceresultofthemultigridwillfollow. 76

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Now,let e E denotetheerrorreductionoperatorofthemultigridalgori thmwhen appliedtotherelatedpositivedeniteproblem ( u v )=( F v ) r .Inotherwords, therelatedpositivedeniteproblemisobtainedbyreplaci ng 2 inthemeridian problem( 6–1 )byone.Therststepistoshowthat jj e E jj < e where 0 < e < 1 isindependentofthemeshsize.Thenwewillshowthat rrr E e E rrr < Ch 1 ,where h 1 isthecoarsestmeshsizeinthesequenceofmeshesusedinthe multigrid algorithm,toconcludethat k E k rrr E e E rrr + rrr e E rrr < Ch 1 + e Hence, k E k < ,where = Ch 1 + e ,andweconcludethatthemultigridalgorithm convergesatauniformrate( 0 << 1 )whenappliedtothemeridianproblemgiventhat thecoarsestmeshsizeissufcientlysmall. Throughoutthischapter,wewillassumeadditionallythat n isconvex. 7.1MultigridAnalysisfortheRelatedPositiveDenitePro blem Inthissectionweprovideamultigridanalysisforthefollo wingpositivedenite problemrelatedtothemeridianproblem: Find u h 2 W h suchthat (curl rz u h ,curl rz v h ) r +( u h v h ) r =( F v h ) r forall v h 2 W h .Theresultsinthissectionarecontainedin[ 20 ]. 7.1.1TheMultigridAlgorithm Here,wepresentthemultigridV-cyclealgorithmandstatea uniformconvergence resultforthealgorithm.Wenowassumethetypicalgeometri calmultigridsetting, wherethediscretesolutionspaceisbasedonthenestmeshi nasequenceofnested 77

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renementsofacoarsemesh.Let T 1 bethecoarsestmeshsubdividing D .Typically T 1 issmallenoughsothatthecostofsolvingourniteelementp roblemonitisnegligible. For k =2,3,..., J ,themesh T k isobtainedfrom T k 1 byconnectingthemidpointsofall edges.Wewanttoefcientlysolveaniteelementproblemon themesh T J bymultigrid. ThemultilevelniteelementspacesareN ed elecspacesoneachofthemeshes, i.e.,let W k = f v 2 H r (curl, D ): v j K 2 N 1 forall K 2 T k g Dene k : W k W k by ( k u k v k ) r =( u k v k ) forall u k v k 2 W k Themultigridalgorithmwepresentisforsolvingalinearsy stemonthenestlevel,of thetype J u = f Todescribethealgorithm,werstneedtodenecertainsmoo thingoperators e R k : W k 7! W k .Thesecouldbeadditiveormultiplicativesubspacecorrec tionoperators basedonanyofthesubspacedecompositionsof[ 1 ]and[ 27 ].Todescribethem,rstlet D v k denotethe“vertexpatch”domainformedbytheunionofalltr ianglesin T k connected tothemeshvertex v .Dene W v k = f v 2 W k :supp( v ) D v k g .Foreverymesh edge e ,let e denotetheWhitneyedgebasisfunction,andlet W e k denote span( e ) .The decompositionof[ 1 ],adaptedtooursetting,is W k = X v 2 V k W v k + X e 2 E k W e k (7–1) where V k isthesetofnodesinthemesh T k thatarenoton 1 ,and E k isthesetof edgesofthemesh T k thatarenoton 1 buthavebothnodeson 1 .Thelastset E k may appear“pathological,”butitisneededinmixedboundaryco nditioncaseslikeours,asits edgesarenotcoveredbyanyofthevertexpatchesinthersts umof( 7–1 ). 78

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Tobeclear,letusexhibitthedecompositionfora u k in W k .Let E k denotethesetof edgesinthemeshthatarenoton 1 .Thenthebasisexpansionof u k is u k = X e 2 E k c e e Now,foreach v 2 V k ,dene E 1 k v and E 2 k v as E 1 k v = e 2 E k : oneendpointof e is v andtheotherison 1 E 2 k v = e 2 E k : oneendpointof e is v andtheotherisnoton 1 Whensummingoverthevertexpatches,theedgesof E 2 k v arecountedtwice.Hence, setting u vk := X e 2 E 1 k v c e e + X e 2 E 2 k v 1 2 c e e wehave u k = X v 2 V k u vk + X e 2 E k c e e Thisshowsthat W k canindeedbedecomposedasin( 7–1 ). Theothersubspacedecomposition,dueto[ 27 ],readsasfollowsinourapplication: W k = X e 2 E k W e k + X v 2 V k grad rz V v k (7–2) where V v k isthe(one-dimensional)spaceofcontinuousscalarfuncti onssupportedon D v k whicharelinearoneachtriangleof D v k andvanishon @ D v k Wecanuseeither( 7–1 )or( 7–2 )toconstructadditiveormultiplicativesmoothers. Thedetailsarestandard,sowepresentonlythealgorithmfo rtheblockGauss-Seidel typemultiplicativesmoothingiteration u i +1 = f GS k ( u i f ) wheretheprocedure f GS k ( ) isgivenbelow. 79

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Let W k i i =1,2,..., N k beanenumerationofthesubspacesineitherofthe decompositions( 7–1 )or( 7–2 ).Dene k i : W k i 7! W k i by ( k i v i w i ) r =( v i w i ), forall v i w i in W k i Letthe L 2r ( D ) 2 -orthogonalprojectiononto W k i bedenotedby Q k i Algorithm 7.1.1(Multiplicativesmoothing) Given u i in W k ,calculate u i +1 = f GS k ( u i f ) in W k asfollows: 1..Set u (0)i = u i 2..For j =1,2,..., N k ,compute u ( j ) i = u ( j 1) i + 1 k j Q k j ( f k u ( j 1) i ). 3..Settheresult u i +1 tobe u ( N k ) i Standardargumentsshowthatthisiterationcanberewritte nas u i +1 = u i + e R k ( f k u i ), with e R k =( I ( I e P k N k )( I e P k N k 1 ) ( I e P k ,1 )) 1 k where e P k j istheorthogonalprojectioninto W k j inthe ( ) -innerproduct. Withsuchasmoother e R k ,oranadditiveJacobitypesmootherbasedonthesame decompositions(whosedetailsweomit),wecannowdescribe themultigridalgorithm. Letthe L 2r ( D ) 2 -orthogonalprojectiononto W k bedenotedby Q k Algorithm 7.1.2(V-cycle) Given u and f in W k ,denetheoutput MG k ( u f ) in W k bythe followingrecursiveprocedure: 1..Set MG 1 ( u f )= 1 1 f 2..For k > 1 ,dene MG k ( u f ) recursively: (a) v (1) = u + e R k ( f k u ) (b) v (2) = v (1) + MG k 1 ( 0 Q k 1 ( f k v (1) )) 80

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(c) v (3) = v (2) + e R t k ( f k v (2) ) (d)Set MG k ( u f )= v (3) Itiswellknown[ 11 ]thattheV-cycleiterates x i +1 = MG k ( x i f ) ,approximatingthe exactsolution x = 1 k f ,areconnectedthroughalinearerrorreductionoperator e E k ,i.e., x x i +1 = e E k ( x x i ). ThefollowingisourmainresultontheconvergenceoftheV-c yclealgorithm.Itsproofis giveninthenextsection.Theorem7.1.1. Thereexistsapositivenumber e < 1 suchthat 0 ( e E k u u ) e ( u u ), forall u in W k andall k 1. Thenumber e isindependentofthemeshsizeandrenementlevel. 7.1.2MultigridAnalysis Inthissubsection,weproveTheorem 7.1.1 byverifyingtwoconditionsinastandard abstractframeworkformultigridanalysis[ 1 10 11 14 ].Westatetheconditionsandits implicationasthenextlemmaandomititswellknownproof.T heanalysisofthissection isheavilybasedonthetechniquesintroducedin[ 1 ].Let e P k denotetheorthogonal projectioninto W k inthe ( ) -innerproduct. Lemma7.1.1. TheassertionofTheorem 7.1.1 followsfromthetwoconditionsbelow: 1.. Existenceofastabledecomposition: Thereexistsaconstant C 1 > 0 independent ofthemeshsizesand k ,suchthatforall v in ( I e P k 1 ) W k ,thereisadecomposition v = N k X j =1 v j with v j in W k j satisfying N k X j =1 ( v j v j ) C 1 ( v v ). 81

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2.. Limitedinteraction: Thereexistsaconstant C 2 > 0 ,independentof k ,suchthat N k X j =1 N k X l =1 j ( v j w l ) j C 2 N k X j =1 ( v j v j ) 1 2 N k X l =1 ( w l w l ) 1 2 forall v j in W k j w l in W k l ,and k 1 Theremainderofthissectionisdevotedtothevericationo fthetwoconditions ofLemma 7.1.1 .Notethattherstconditiononlyinvolvesfunctionsontwo levels, k and k 1 .Thesecondinvolvesaninequalityoffunctionsinjustonel evel k .For thisreason,wecansimplifyournotationandusesubscripts H and h for k 1 and k ,respectively.Themesh T h isarenementof T H ,and H =2 h .Previouslydened notationswiththesenewsubscriptshavetheobviousdenit ions,e.g., SH denotesthe weighted L 2r ( D ) -orthogonalprojectioninto S H ,thespaceofpiecewiseconstantfunctions withrespecttothemesh T H ,etc.Beforeverifyingtheconditions,weneedanumberof preliminaryresults.Lemma7.1.2. Forall p h in S h rr p h SH p h rr r CH k curl 0rz p h k r (7–3) Proof. Given p h in S h ,dene z in H r (curl, D ) and p in L 2r ( D ) asthesolutionof( 5–1 ) with f =curl rz curl 0rz p h ,i.e., ( z w ) r ( p ,curl rz w ) r =0 forall w in H r (curl, D ), ( s ,curl rz z ) r =( s ,curl rz curl 0rz p h ) r forall s in L 2r ( D ). Then,with z h =curl 0rz p h ,thepair f z h p h g obviouslysatises ( z h w h ) r ( p h ,curl rz w h ) r =0 forall w h in W h ( s h ,curl rz z h ) r =( s h ,curl rz curl 0rz p h ) r forall s h in S h Moreover, curl rz z =curl rz z h in L 2r ( D ). (7–4) 82

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Bythetriangleinequality, rr p h SH p h rr r k p h p k r + rr p SH p rr r + rr SH p SH p h rr r (7–5) Wenowestimateeachofthetermsontherighthandsideabove. Beginningwiththemiddleterm,andusingastandardweighte dnormapproximation estimate(seee.g.[ 3 13 ]),wehave rr p SH p rr 2r CH 2 j p j 2H 1 r ( D ) CH 2 k p k 2H 1 r ( D ) + k r 1 p k 2r Therighthandsideisboundedbecause p isin e H 2 r ( D ) ,byTheorem 5.1.2 .Moreover,as in[ 25 ,Proposition3.1],itcanbeboundedfurtherby CH 2 ( k r 1 @ r ( rp ) k 2r + k @ z p k 2r ) ,which isthesameas CH 2 k curl rz p k 2r .Thus, rr p SH p rr r CH k curl rz p k r (7–6) Furthermore,since k curl rz p k 2r =( curl rz p z ) r cf.( 5–10 ) =( p ,curl rz z ) r by( 2–8 ), =( p ,curl rz z h ) r by( 7–4 ) =( curl rz p z h ) r by( 2–8 ) =( curl rz p ,curl 0rz p h ) r bytheCauchy-Schwarzinequality,weobtain k curl rz p k r k curl 0rz p h k r .Thus( 7–6 )yields rr p SH p rr r CH k curl 0rz p h k r (7–7) Next,considerthersttermontherighthandsideof( 7–5 ).Usingtriangle inequality, k p p h k r rr p Sh p rr r + rr Sh p p h rr r 83

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Theterm rr p Sh p rr r canbeboundedbythesametypeofargumentthatledto( 7–7 ). Theorem 5.1.3 item 3 providesaboundfortheotherterm.Thentheinverseestimat e[ 3 Lemma4]yields k p p h k r Ch k curl 0rz p h k r + Ch 2 k curl rz curl 0rz p h k r Ch k curl 0rz p h k r (7–8) Theonlyremainingtermontherighthandsideof( 7–5 )isboundedbyusing( 7–8 ) andthefactthatorthogonalprojectorshaveunitnorm: rr SH p SH p h rr r k p p h k r Ch k curl 0rz p h k r (7–9) Collectingtheestimatesof( 7–7 ),( 7–8 )and( 7–9 )in( 7–5 ),wehave rr p h SH p h rr r C ( H + h ) k curl 0rz p h k r Since h CH ,thiscompletestheproof. Thenextlemmaiscrucialinprovingtheuniformconvergence ofthemultigrid V-cycleandismodeledafterthelemmasin[ 1 ].Weshallmakesignicantuseofthe weighteddiscreteHelmholtzdecompositiondiscussedinSe ction 4.1 .Recallthatasper ourpreviousremarksonthenotation, e P H denotesthe -orthogonalprojectionintothe coarserofthetwospaces.Lemma7.1.3. Let w h bein W h .IftheweighteddiscreteHelmholtzdecompositionof w h e P H w h is w h e P H w h = grad rz h +curl 0rz a h with h in V h and a h in S h ,then k h k r CH rrr ( I e P H ) w h rrr r k curl 0rz a h k r CH rrr ( I e P H ) w h rrr 84

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Proof. Werstprovethesecondinequality.Dene z h in W h ,giventheabove a h ,by ( z h q h )=(curl 0rz a h q h ) r forall q h in W h (7–10) Observethat z h isorthogonalto grad rz V h .Bysetting q h =curl 0rz a h above, k curl 0rz a h k 2r =( z h ,curl 0rz a h ), =( z h ,( I e P H ) w h grad rz h ), (7–11) =( z h ,( I e P H ) w h ), =( z h z H ,( I e P H ) w h ), (7–12) k z h z H k rrr ( I e P H ) w h rrr (7–13) forany z H in W H .Next,wechooseasuitable z H andestimate k z h z H k Tothisend,rstdene z in H r (curl, D ) ,giventheabove z h ,by( 5–1 )with f = curl rz z h ,i.e., ( z w ) r ( p ,curl rz w ) r =0 forall w in H r (curl, D ), ( s ,curl rz z ) r =( s ,curl rz z h ) r forall s in L 2r ( D ). Thendene z H in W H bytheanalogueof( 5–14 )onthecoarserofthemeshes,i.e., ( z H w H ) r ( p H ,curl rz w H ) r =0 forall w H in W H ( s H ,curl rz z H ) r =( s H ,curl rz z h ) r forall s H in S H Since z h isorthogonalto grad rz V h ,itisclearthat z h isintherangeof curl 0rz bythe weighteddiscreteHelmholtzdecompositionof W h .Thus,thereisaunique p h in S h such that curl 0rz p h = z h .Inotherwords, ( z h w h ) r ( p h ,curl rz w h ) r =0 forall w h in W h 85

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whichistherstequationoftheformulation( 5–14 ).Thesecondequationof( 5–14 )is alsosatisedby z h triviallysince f =curl rz z h .Therefore,byTheorem 5.1.3 item 2 k z z h k r = rr z Wh z rr r k z z H k r = rr z WH z rr r whichimplies k z h z H k r rr z Wh z rr r + rr z WH z rr r bythetriangleinequality, CH j z j H 1 r ( D ) 2 byCorollary 4.1.1 CH k curl rz z h k r byTheorem 5.1.2 .(7–14) Wealsoneedtoestimate k curl rz ( z h z H ) k r .Bythedenitionof z H andLemma 7.1.2 k curl rz ( z h z H ) k r = rr curl rz z h SH curl rz z h rr r CH k curl 0rz curl rz z h k r Combiningthiswith( 7–14 ),weget k z h z H k 2 CH 2 k curl rz z h k 2r + k curl 0rz curl rz z h k 2r (7–15) Thisestimateisnotyetinaformwecanusein( 7–13 ).Tosimplifyitsrighthandside, dene h : W h W h by ( h v h w h ) r =( v h w h ) forall w h in W h 86

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weobservethat h z h =curl 0rz a h by( 7–10 ),and k h z h k 2r =( h z h h z h ) r =( z h h z h ), =( z h h z h ) r +(curl rz z h ,curl rz ( h z h )) r =( z h z h )+(curl 0rz curl rz z h z h ), = k z h k 2r +2 k curl rz z h k 2r + k curl 0rz curl rz z h k 2r Hence,returningto( 7–15 )andoverestimatingitsrighthandside, k z h z H k 2 CH 2 k h z h k 2r = CH 2 k curl 0rz a h k 2r Usingthisestimatein( 7–13 ),wehave k curl 0rz a h k 2r CH k curl 0rz a h k r k ( I e P H ) w h k fromwhichthesecondinequalityofthelemmafollows. Itnowonlyremainstoprovetherstestimateofthelemma.Le t in H 1 r ( D ) bethe uniquesolution(see[ 25 ])of ( grad rz grad rz ) r =( h ) r forall in H 1 r ( D ). (7–16) Then[ 25 ,Theorem2.1]givestheregularityestimate j j H 2 r ( D ) C k h k r (7–17) Observethatforany H in V H ( grad rz h grad rz H ) r =( grad rz h grad rz H ), =( w h e P H w h grad rz H )=0. (7–18) 87

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Wewillusethiswith H = b VH ,where b VH isthepreviouslymentionedprojectionof[ 21 ]. Proceedingbyastandarddualityargument[ 33 ], k h k 2r =( h h ) r =( grad rz grad rz h ) r by( 7–16 ), =( grad rz grad rz b VH grad rz h ) r by( 7–18 ), CH j j H 2 r ( D ) k grad rz h k r by[ 21 ,Lemma5.3], CH k h k r k grad rz h k r by( 7–17 ). Cancelingthecommonfactor,andusingthestabilityestima te( 4–4 ), k h k r CH k grad rz h k r CH rrr w h e P H w h rrr r whichnishestheproofofthelemma. Wecannowprovetheconvergenceofmultigridasaniterative method. ProofofTheorem 7.1.1 ByLemma 7.1.1 ,weonlyneedtoverifythetwoconditions there.Forverifyingthesecondconditiononthelimitedint eractionofsmoothing subspaces,wecanusestandardtechniques[ 1 14 ].Henceweomitit. Letusnowverifytherstconditionontheexistenceofastab ledecompositionfor thecaseofthesmoothingsubspacesof[ 1 ],namely( 7–1 ).Givenany w k in ( I e P k 1 ) W k let w k = grad rz k + r k beitsweighteddiscreteHelmholtzdecomposition,with k in V k = f v 2 H 1 r ( D ): v j K 2 P 1 forall K 2 T k g and r k in W k .ByLemma 7.1.3 k k k r Ch k 1 k w k k r (7–19) k r k k r Ch k 1 k w k k (7–20) 88

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Let V v k = f v 2 V k :supp( v ) D v k g .Then,byusingthedecomposition V k = X v 2 V k V v k (7–21) wesplit k = X v 2 V k vk with vk in V v k whilewesplit r k bythedecompositionof( 7–1 )as r k = X v 2 V k r vk + X e 2 E k r ek with r vk in W v k r ek in W e k Setting w vk = grad rz vk + r vk ,wewanttoshowthat X v 2 V k ( w vk w vk )+ X e 2 E k ( r ek r ek ) C ( w k w k ). (7–22) Expandingthetermsandusingtheorthogonalityofthediscr eteHelmholtzdecomposition andtheweightedinverseestimate[ 3 ,Lemma4],weobtain X v 2 V k k w vk k 2 + X e 2 E k k r ek k 2 = X v 2 V k k grad rz vk + r vk k 2 + X e 2 E k k r ek k 2 = X v 2 V k k grad rz vk k 2r + k r vk k 2r + k curl rz r vk k 2r + X e 2 E k k r ek k 2 C X v 2 V k h 2 k k vk k 2r +(1+ h 2 k ) k r vk k 2r + C X e 2 E k h 2 k k r ek k 2r Ch 2 k k k k 2r + k r k k 2r By( 7–19 )and( 7–20 ), X v 2 V k k w vk k 2 + X e 2 E k k r ek k 2 Ch 2 k h 2 k 1 k w k k 2 whichproves( 7–22 ).Thustheconditionontheexistenceofthestabledecompos itionis veriedforthesmoothingsubspacesof( 7–1 ). 89

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Asimilarandsimplerargumentveriestheexistenceofasta bledecompositionfor thesubspacesof( 7–2 )aswell.Weomitthedetails. 7.1.3NumericalResults Inthissubsection,wewillreporttheiterationcountsfort hemultigridV-cycleapplied to h x = f forafewchoicesofthedomainand f .NotethattheV-cycleoperatorcan alsobeusedasapreconditionerfor h andnumericalexperimentsusingitsohave alreadybeenreportedin[ 21 ]inthecontextofsolvingadiv-curlsystem. WewillverifytheuniformconvergenceofthemultigridV-cy clealgorithmfor h x = f WeapplytheV-cyclealgorithmtothethreedifferentdomain sshowninFigure 7-1 DomainIisconvexanditsrevolutionisalsoconvex,whileDo mainIIisconvex,butits revolutionisnonconvex,andDomainIIIanditsrevolutiona rebothnonconvex.The initialmesh(level1)fordomainIandIIconsistsoftwocong ruentrighttriangles,andfor domainIII,itconsistsoffourcongruentrighttriangles.I nallcases,weobtainthenext levelmeshbyconnectingtheedgemidpointsofalltriangles Table 7-1 reportstheconvergenceratewhen f =0 .WeapplytheGauss-Seidel smootherwiththesubspacedecomposition( 7–2 ).Forsuccesiveniteelementspaces W H W h ,theprolongationmatrixthatweusedforimplementationis thematrixwhose ( i j ) -thentryis R e i j t i ds ,where e i denotesthe i -thedgeofmesh T h thatisnoton 1 t i denotestheunittangentvectorofthisedge,and j isthe j -thbasisfunctionof W H .Therestrictionmatrixisthetransposeoftheprolongatio nmatrix.Foreachne levelmesh,theinitialvalue x 0 waschosenrandomlyinC++.Thestoppingcriterion is k x n k = k x 0 k < 10 7 ,where x n istheresultofthe n -thiteration(whichmeasures thereductionintheerrorsincetheexactsolutioniszero). Theconvergencerateis computedbytakingtheaverageof k x n k = k x n 1 k Asweseefromthetable,theconvergencerateisnearlyconst antandseems boundedindependentlyofthemeshsize.Additionally,alth oughweassumedthatthe revolutionofthetwo-dimensionaldomainisconvexthrough outthepaperinorderto 90

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0 1 2 2 1 r z ADomainI 0 1 2 2 1 r z BDomainII 0 1 2 1 2 r z CDomainIII Figure7-1.DomainsTable7-1.V-cycleconvergencerates Convergencerate Level DomainI DomainII DomainIII 2 0.27 0.29 0.32 3 0.37 0.35 0.41 4 0.43 0.41 0.40 5 0.42 0.40 0.40 6 0.41 0.41 0.41 7 0.41 0.41 0.41 8 0.41 0.41 0.41 9 0.41 0.41 0.41 provetheuniformconvergenceresult,itappearsthatevenw hentherevolutionofthe domainisnonconvex,theconvergencerateisindependentof themeshsize. 7.2MultigridAnalysisfortheIndeniteProblem Inthissection,wewillstudythemultigridbackslashcycle .Wewillalsostateour maintheoremthatimpliestheuniformconvergenceofthemul tigridbackslashcycle independentofthemeshsizewhenappliedtothediscretemer idianproblem( 6–2 ) providedthatthecoarsestmeshissufcientlyne. ItwasshowninChapter 6 thatiftheindeniteproblem( 6–1 )isuniquelysolvable, thenthediscreteproblem( 6–2 )isalsowell-posed,andthatthediscretesolution of( 6–2 )approximatestheexactsolutionof( 6–1 )giventhatthemeshsizeissufciently small.Therefore,throughoutthissection,weassumethat( 6–1 )hasauniquesolution andthatallmeshesconsideredherehavesmallenoughmeshsi zesothatthisresult (Theorem 6.1.1 )holds. 91

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7.2.1TheMultigridAlgorithm Asinsection 7.1 ,weassumethatwehaveasequenceofnestedtriangulationso f D ,whichwewilldenoteby T k forthe k th meshwhere k =1, J T 1 isthecoarsest meshwhile T J isthenestmesh.Let h k betherepresentativemeshsizeof T k .We willlaterseethatfortheoreticalandpracticalpurposes, theinitialmeshsize h 1 mustbe sufcientlysmall.Let W k betheniteelementsubspace W h withrespectto T k (See section 7.1.1 ).Dene P k : H r (curl, D ) W k tobetheorthogonalprojectionwith respectto A ( ) ,where A ( u v )=(curl rz u ,curl rz v ) r 2 ( u v ) r asinChapter 6 .Recallthat Q k : H r (curl, D ) W k istheorthogonalprojectionwith respecttothe L 2r -innerproduct,i.e., A ( P k u v k )= A ( u v k ) forall v k 2 W k ( Q k u v k ) r =( u v k ) r forall v k 2 W k Notethat P k iswell-denedsinceweareassumingtheuniquesolvability of( 6–1 ).We alsodene A k : H r (curl, D ) W k by ( A k u v k ) r = A ( u v k ) forall v 2 W k Sincethemeshesarenested, W 1 W J and W J istheniteelementsubspaceinwhichwewillsolvethegiven problem,butwe gothrougheach W k asthefollowingalgorithmillustrates. Algorithm 7.2.1(backslashcycle) Given u and f in W k ,denetheoutput MG k ( u f ) in W k bythefollowingrecursiveprocedure: 1..Set MG 1 ( u f )= A 1 1 f 92

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2..For k > 1 ,dene MG k ( u f ) recursively: (a) v (1) = u + R k ( f A k u ) (b) v (2) = v (1) + MG k 1 ( 0 Q k 1 ( f A k v (1) )) (c)Set MG k ( u f )= v (2) Here R k : W k W k isalinearsmoothingoperator.Notethatinthismultigrid algorithm(oftencalledthe“backslashcycle”)wesmoothon lyonceasweproceed tocoarsergrids.Oursmoothingoperatorswillalwaysbebas edonageneralized blockJacobiorblockGauss-Seideliterationasinsection 7.1 .Fortheseadditiveand multiplicativesubspacecorrectionsmoothers,weneedasu bspacedecomposition of W k .Asinsection 7.1.1 ,wewillusethedecomposition( 7–1 )or( 7–2 )toconstruct additiveormultiplicativesmoothers.Let W k = P N k i =1 W k i beeitherdecomposition( 7–1 ) or( 7–2 ),anddene A k i : W k W k i as ( A k i u v i ) r = A ( u v i ), anddene P k i : W k W k i as A ( P k i u v i )= A ( u v i ), forall v i 2 W k i .Itcanbeshown,bythesamewayasin[ 26 ,Proposition3.1]with Lemma 7.2.1 ,that A k i isinvertibleandthat P k i iswelldenedgiventhatthemeshsizeis sufcientlysmall. Nowwecanstatethefollowingalgorithmthatdenesthebloc kGauss-Seidel smootherfortheindeniteproblem,whichisthesameasAlgo rithm 7.1.1 with replacedby A ,i.e., Algorithm 7.2.2(IndeniteGauss-Seidel) Given u i in W k ,calculate u i +1 = GS k ( u i f ) in W k asfollows: 1..Set u (0)i = u i 93

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2..For j =1,2,..., N k ,compute u ( j ) i = u ( j 1) i + A 1 k j Q k j ( f A k u ( j 1) i ). 3..Settheresult u i +1 tobe u ( N k ) i Thisiterationcanberewrittenas u i +1 = u i + R k ( f A k u i ), with R k =( I ( I P k N k )( I P k N k 1 ) ( I P k ,1 )) A 1 k (7–23) andthisisthesmoothingoperatorthatwewilluseinthemult igridbackslashcycle. Thebackslashcycleiterates x i +1 = MG k ( x i f ) ,approximatingtheexactsolution x = A 1 k f ,areconnectedthroughalinearerrorreductionoperator E k [ 11 ],i.e., x x n +1 = E k ( x x n ). Thefollowingisourmainresultontheconvergenceofthebac kslashcyclealgorithm.Its proofisgiveninthenextsection.Theorem7.2.1. Thereexists h 0 > 0 suchthatif 0 < h 1 < h 0 then k E k k < forsome positivenumber < 1 forall k 1 .Thenumber isindependentofthemeshsizeand therenementlevel. Beforeweendthissubsection,letusprovealemmathatwillb eusedinprovingthe previoustheorem.Lemma7.2.1. If u 2 W k isadiscretedivergencefreefunctionsin W v k forany v 2 V k in decomposition( 7–1 )orifitisin W e k forany e 2 E k indecomposition( 7–2 ),then k u k r Ch k k curl rz u k r Proof. Werstshowthattheresultholdsforalledgeelementbasisf unctionsin W k Thentheresultwillholdforall u 2 W e k forany e 2 E k .Foredgebasisfunctionsthat 94

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correspondtoanedgeonatrianglethathaveemptyintersect ionwith 0 ,wecanextend suchresultonunweightednormstrivially,soweonlyhaveto provethelemmaforedge basisfunctionsfortrianglesthatdointersect 0 Weconsidertwodifferenttypeoftrianglesthatintersectt heaxisofsymmetry. Trianglesthatintersect 0 atexactlyonevertexarecatagorizedastypeonetriangle. Trianglesthatintersect 0 attwoverticesarecatagorizedastypetwo.Nowletus considertworeferencetriangles. b K 1 denotesthetrianglewithvertices (1, 1) (0,0) and (1,1) ,and b K 2 denotesthetrianglewithvertices (1,0) (0,1) ,and (0, 1) .Firstofall, let K beatypeonetriangle,andlet F beanafnehomeomorphismthatmaps b K 1 onto K suchthat F maps (0,0) totheonevertexof K ontheaxis.Map u ( r z ) covariantlyto denethefunction bu ( b r b z )= J ( F ) t u ( r z ) on b K 1 ,where J ( F ) istheFrechetderivativeof F .Observethat r = r 1 1 + r 2 2 on K ,where r i and i denotesthe r -coordinateandthe barycentriccoordinaterespectivelyofthei-thvertexof K notontheaxis.Also r 1 = h and r 2 = h areboundedaboveandbelowbyxedconstants(independento fthemeshsize,but dependingonelementanglesandtheangle 1 makeswith 0 ).Itisstaightforwardto showthat k u k 2r K Ch k b u k 2r b K 1 k curl rz bu k 2r b K 1 Ch k curl rz u k 2r K (7–24) Sinceeachedgebasisfunction b i for 1 i 3 on b K 1 satises rrr b i rrr r b K 1 rrr curl rz b i rrr r b K 1 byusing( 7–24 )wehavethat k k r K Ch k curl rz k r K (7–25) foralledgebasisfunctions onanytypeonetriangle K Similarly,foralltypetwotrianglesasimilarproofgoesth rough,exceptthatwenow useamappingfromtheotherreferenceelement b K 2 .Let K beatypetwotrianglenow, 95

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andlet v 3 denoteitsvertexnotontheaxis.Suppose 3 and h 3 denotethebarycentric coordinateandther-coordinateof v 3 respectively.Then r = h 3 3 ,andinequalities like( 7–24 )continuetohold.Sinceeachedgebasisfunction b i for 1 i 3 on b K 2 satises rrr b i rrr r b K 2 rrr curl rz b i rrr r b K 2 weconcludethat k k r K Ch k curl rz k r K (7–26) foralledgebasisfunctions onanytypetwotriangle K .Hence,by( 7–25 )and( 7–26 ),it followsthatforalledgebasisfunctions correspondingtoanedgeofatrianglethathas nonemptyintersectionwith 0 ,wehave k k r Ch k curl rz k r whichcompletestheprooffordecomposition( 7–2 ). Next,weprovethelemmafordiscretedivergencefreefuncti onsin W v k for decomposition( 7–1 ).Fixavertexpatch D v k ,andlet v 0 denotethevertexonwhich alltrianglesinthisvertexpatchmeet.Let w 2 W v k bediscretedivergencefree,andlet 0 bethenodalbasisfunctioncorrespondingto v 0 .Thendene z = grad rz 0 + w ,for some 2 R Foreachtriangleconnectedto v 0 ,weuselocalcoordinatesasintheproofof[ 26 Lemma3.1],toprovethattheresultholdsfor w .Therearenitelymanytrianglesinthis vertexpatchbytheshaperegularitypropertyof T k .Denotethe l -thtriangleby T l ,where 1 l N .Then,considerthelocalcoordiates ( r ( l ) z ( l ) ) on T l suchthatthe r ( l ) -axis containstheedgeof T l thatdoesnotintersect v 0 .Wewillcallthisedge e l .Also,let the z ( l ) -axisbetheperpendicularaxistosuch r ( l ) -axisthatmeets v 0 .Then,weusethe notation z ( l ) ,for 1 l N ,todenote z j T l inlocalcoordinates ( r ( l ) z ( l ) ) on T l 96

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Since z isin W v k ,ithassupportonlyintheinteriorofthevertexpatch,soth e tangentialcomponentof z on e l iszero,sowehavethat z ( l ) =( b ( l ) z ( l ) c ( l ) + b ( l ) r ( l ) ), (7–27) forsome c ( l ) b ( l ) 2 R .Wealsohavethat curl rz z ( l ) = 2 b ( l ) (7–28) forall 1 l N Now,letusconsiderthersttriangle T 1 inthevertexpatch W v k .Let v 1 denoteone oftheverticesof T 1 thatisnot v 0 .Withoutlossofgenerality,supposethattheedge connecting v 0 and v 1 isanedgeofboth T 1 and T 2 ,andlet t ( i ) betheunittangentvector oftheedgethatconnects v 0 and v 1 inlocalcoordinates ( r ( i ) z ( i ) ) for i =1,2 .Then, directcalculationshowsthat z (1) t (1) = c (1) v (1) 0, z 1 2 curl rz z (1) v (1)0 ^ v (1)1 j v (1)1 v (1)0 j (7–29) where v (1)i =( v (1) i r v (1) i z ) for i =1,2 ,and v (1)0 ^ v (1)1 = v (1) 0, z ( v (1) 1, z v (1) 1, r ) .Then,wechoose 2 R sothat c (1) =0 .Therefore,withsuchxed 2 R ,wehavethat z (1) =( b (1) z (1) b (1) r (1) ), sobyusing( 7–28 ),wehave j z (1) j Ch j curl rz z (1) j Additionally,since c (1) =0 ,by( 7–29 ),wehavethat z (1) t (1) = 1 2 curl rz z (1) v (1)0 ^ v (1)1 j v (1)1 v (1)0 j (7–30) Next,asin( 7–29 )butthistimeusinglocalcooridantes ( r (2) z (2) ) ,wederivethat z (2) t (2) = c (2) v (2) 0, z 1 2 curl rz z (2) v (2)0 ^ v (2)1 j v (2)1 v (2)0 j (7–31) 97

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Althoughweareinlocalcoordinatesystems,thetangential componentsstillagree,so by( 7–30 )and( 7–31 ),wehave c (2) = 1 2 curl rz z (1) v (1)0 ^ v (1)1 1 2 curl rz z (2) v (2)0 ^ v (2)1 v (2) 0, z Thus, j c (2) j Ch ( j curl rz z (1) j + j curl rz z (2) j ), anditfollowsfrom( 7–27 )and( 7–28 )that j z (2) j Ch ( j curl rz z (1) j + j curl rz z (2) j ). Ingeneral,byinduction,wehavethat j z ( l ) j Ch ( j curl rz z (1) j + j curl rz z (2) j + + j curl rz z ( l ) j ), (7–32) forall 1 l N Since j z ( r z ) j T l j C j z ( l ) ( r ( l ) z ( l ) ) j and j curl rz z ( l ) ( r ( l ) z ( l ) ) j C j curl rz z ( r z ) j T l j where C dependsontheshaperegularityconstant,itfollowsfrom( 7–32 )that j z j T l j Ch ( j curl rz z j T 1 j + j curl rz z j T 2 j + + j curl rz z j T l j ), forall 1 l N .Therefore,weconcludethat k z k r Ch k curl rz z k r Since z = grad rz 0 + w isanorthogonaldecomposition,thisimpliesthat k w k r Ch k curl rz w k r 98

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As w 2 W v k wereanarbitrarydiscretedivergencefreefunction,thisc ompletesthe proof. 7.2.2MultigridAnalysis Inthissubsection,wewillproveourmainresult,theunifor mconvergenceofthe multigridbackslashcycle.Ouranalysiswillbebasedonper turbationfromtheuniform multigridconvergenceestimatesforarelatedsymmetricpo sitivedeniteproblemas donein[ 12 ]and[ 26 ]. Insection 7.1 ,weanalyzedthemultigridconvergencewhenappliedtotheb ilinear form .AlthoughweanalyzedthemultigridV-cyclethere,suchres ultscanbeextended tothemultigridbackslashcycleaswell[ 15 ,Lemma2.2].Hereandintheremaining ofthischapter,let e E k denotetheerrorreductionoperatorofAlgorithm 7.2.1 with A k replacedby k and R k replacedby e R k describedinsection 7.1.1 .Weusethesame decomposition( 7–1 )or( 7–2 ). ThenthefollowingresultfollowsfromTheorem 7.1.1 .SeeChapter 3 section 3.2 for details.Theorem7.2.2. Thereexistsapositivenumber e < 1 suchthat rrr e E k rrr < e forall k 1 .Thenumber e isindependentofthemeshsizeandrenementlevel. Now,letusstudytheerrorreductionoperator E k and e E k further.Notethat,forerror reductionoperators,thesubscript k indicatesthat J = k .Dene T k : W J W k as T 1 = P 1 and T k = R k A k P k for k > 1 .Thenbyusingconsistencyandlinearityof MG k ( ) ,itisstraightforwardtoshowthat E k =( I T 1 )( I T 2 ) ( I T k ), (7–33) 99

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where I istheidentityoperator.Similarly,if e T 1 = e P 1 and e T k = e R k k e P k for k > 1 ,thenwe get e E k =( I e T 1 )( I e T 2 ) ( I e T k ). (7–34) Wewillanalyzethemultigridalgorithmbyusinganotherope rator Z k = T k e T k Supposewehave k Z 1 k Ch 1 k Z k k Ch k for k =2, J (7–35) Then,byanargumentof[ 12 ],itfollowsthat rrr e E k E k rrr issmall.Inparticular,since I T k = I e T k Z k and rrr I e T k rrr 1 k I T k k 1+ Ch k andsoby( 7–33 ), k E k k k Y i =1 (1+ Ch i ), whichcanbeboundedbyaconvergentinniteproduct.Thus, k E k k C .Additionally, since E k e E k =( E k 1 e E k 1 )( I e T k ) E k 1 Z k by( 7–33 )and( 7–34 ),itfollowsthat rrr E k e E k rrr Ch 1 (7–36) BycombiningTheorem 7.2.2 and( 7–36 ),itiseasytoshowthat k E k k isboundedby somepositiveconstantlessthan 1 giventhat h 1 issmallenough.Sowewillshowthat conditions( 7–35 )doinfacthold. Lemma7.2.2. Forall x J 2 W J and v 1 2 W 1 ( x J P 1 x J v 1 ) r Ch 1 k x J P 1 x J k k v 1 k 100

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Proof. Forgiven x J 2 W J and v 1 2 W 1 ,let x J P 1 x J = grad rz J +curl Jrz r J bethediscrete Helmholzdecompositionin W J ,andlet v 1 = grad rz 1 +curl 1rz s 1 ,bethediscreteHelmholz decompositionin W 1 Bydenitionof P 1 A ( x J P 1 x J v 1 )=0 so ( x J P 1 x J grad rz 1 )=0. Therefore, ( x J P 1 x J v 1 ) r =( x J P 1 x J ,curl 1rz s 1 ) r (7–37) Let Sw 1 2 H r (curl, D ) bethesolutionof( 6–3 ),where w 1 =curl 1rz s 1 .Then, ( grad rz J ,curl 1rz s 1 ) r =( grad rz J w 1 S w 1 ) r k x J P 1 x J k r Ch 1 k curl rz w 1 k r byLemma 6.2.1 Ch 1 k x J P 1 x J k r k v 1 k (7–38) Next,let z 2 H r (curl, D ) bethesolutionof A ( z x )=( w J x ) forall x 2 H r (curl, D ), where w J =curl Jrz r J .Since rr curl Jrz r J rr 2r =(curl Jrz r J x J P 1 x J ) r = A ( z x J P 1 x J ), = A ( z Wh 1 z x J P 1 x J ), C rr z Wh 1 z rr k x J P 1 x J k Ch 1 ( j z j H 1 r ( D ) 2 + j curl rz z j H 1 r ( D ) ) k x J P 1 x J k byCorollary 4.1.1 Ch 1 k w J k r k x J P 1 x J k byLemma 6.2.2 101

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Itfollowsthat (curl Jrz r J ,curl 1rz s 1 ) r rr curl Jrz r J rr r k v 1 k r Ch 1 k x J P 1 x J k k v 1 k r (7–39) Hence,by( 7–37 ), ( x J P 1 x J v 1 ) r =( grad rz J ,curl 1rz s 1 ) r +(curl Jrz r J ,curl 1rz s 1 ) r Ch 1 k x J P 1 x J k k v 1 k by( 7–38 )and( 7–39 ), whichisthedesiredresult. Nowweusethislemmatoprovetherstconditionof( 7–35 ). Lemma7.2.3. Thereexistsaconstant H 1 > 0 suchthat,forall 0 < h 1 < H 1 k Z 1 k Ch 1 Proof. Theprooffollowsasin[ 26 ,Lemma4.3].Let u J v J 2 W J .Then ( Z 1 u J v J )=( P 1 u J u J e P 1 v J ), = A ( P 1 u J u J e P 1 v J )+(1+ 2 )( P 1 u J u J e P 1 v J ) r =(1+ 2 )( P 1 u J u J e P 1 v J ) r Ch 1 k P 1 u J u J k rrr e P 1 v J rrr byLemma 7.2.2 Ch 1 k u J k k v J k Inthelaststep,weareagainusingLemma 7.2.2 .Thatis, k P 1 u J u J k 2 =( P 1 u J u J P 1 u J ) ( P 1 u J u J u J ), =(1+ 2 )( P 1 u J u J P 1 u J ) r ( P 1 u J u J u J ), Ch 1 k P 1 u J u J k k P 1 u J k + k P 1 u J u J k k u J k byLemma 7.2.2 Ch 1 k P 1 u J u J k 2 +( Ch 1 +1) k P 1 u J u J k k u J k 102

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Therefore,bycancelingthecommonfactor,weseethatthere existsaconstant H 1 > 0 suchthat k P 1 u J u J k C k u J k forall 0 < h 1 < H 1 .Thiscompletestheproof. Thenexttwolemmasverifythesecondconditionof( 7–35 ).Theseproofsproceed exactlyastheproofsof[ 26 ,Lemma4.4,4.5]byusingLemma 7.2.1 .Weincludethe outlineoftheproofhereforthesakeofcompleteness.Lemma7.2.4. Thereexistsaconstant C suchthat ( u k P k i u k v k i ) Ch k k u k P k i u k k r k curl rz v k i k r forall u k 2 W k and v k i 2 W k i ,where W k = P N k i =1 W k i k =2, J ,isdecomposition( 7–1 ) or( 7–2 ). Proof. Fordecomposition( 7–1 ), v k i 2 W v k forsome v 2 V k or v k i 2 W e k forsome e 2 E k If v k i 2 W v k thenlet v k i = grad rz w k i + x k i bethediscreteHelmholzdecompostionof v k i Then,since ( u k P k i u k grad rz w k i ) r = 1 2 A ( u k P k i u k grad rz w k i )=0 ( u k P k i u k v k i ) r =( u k P k i u k x k i ) r Ch k k u k P k i u k k r k curl rz x k i k r byLemma 7.2.1 = Ch k k u k P k i u k k r k curl rz v k i k r If v k i 2 W e k thentheresultfollowsdirectlyfromCauchy-Shwartzinequ alityand Lemma 7.2.1 .Fordecomposition( 7–2 ) v k i 2 grad rz V v k forsome v 2 V k or v k i 2 W e k forsome e 2 E k .Intheformercase,thereisnothingtoproveasbothsidesof the inequalitywewishtoprovearebothzero.Inthelattercase, theresultfollowsagainby Cauchy-ShwartzinequalityandLemma 7.2.1 103

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Lemma7.2.5. Thereexistsaconstant H 2 > 0 suchthat,forall 0 < h 1 < H 2 k Z k k Ch k for k =2, J Proof. Dene E i =( I P k i )( I P k i 1 ) ( I P k ,1 ), e E i =( I e P k i )( I e P k i 1 ) ( I e P k ,1 ), andlet E 0 = e E 0 = I .Then,by( 7–23 ), Z k = T k e T k = e E N k E N k Asintheproofof[ 26 ,Lemma4.5]itcanbeshownthat rrr ( e E N k E N k ) u rrr 2 Ch 2 k N k X i =1 k E i 1 u k 2 Ch 2 k k u k 2 Notethat,byusingLemma 7.2.4 ,thereexistsaconstant H 2 > 0 suchthattheabove secondinequalityholds,forall 0 < h 1 < H 2 .Seetheproofof[ 26 ,Theorem4.5]for details.Therefore,forall k > 1 k Z k k Ch k Byputtingallofthesepiecestogether,wecanproveourmain result. ProofofTheorem 7.2.1 ByLemma 7.2.3 andLemma 7.2.5 ,( 7–36 )holdsforsufciently small h 1 ,soforall k 1 k E k k rrr E k e E k rrr + rrr e E k rrr Ch 1 + e by( 7–36 )andTheorem 7.2.2 104

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Hence,thereexists h 0 > 0 suchthat h 0 < H 1 h 0 < H 2 ,andforall 0 < h 1 < h 0 ,wehave that k E k k < where = Ch 1 + e < 1 ,andthiscompletestheproof. 7.2.3NumericalResults Inthissection,wereportnumericalresultsthatprovideem piricalsupporttothe uniformconvergenceofthemultigridbackslashcyclewhena ppliedtotheindenite bilinearform A ( ) providedthatthecoarsestmeshissufcientlyne.Additio nally,for differentwavenumbers,wecomparethecoarsestmeshsizeth atguaranteesmultigrid convergence.Inallexperiments,ourdomainistheunitsqua reandmeshlevel1consist oftwouniformrighttriangleswiththecommonedgeconnecti ng (0,0) and (1,1) .The nextlevelmeshisobtainedbyconnectingthemidpointsofea chedge. Wecomputetheconvergencerateforthemultigridbackslash cyclewhen f =0 WeapplytheIndeniteGauss-Seidelsmootherwiththesubsp acedecomposition( 7–2 ). Inordertochoosetheinitialvalueindependentofthemeshl evel,werstchoose x 0 intheniteelementspaceofmeshlevel1tobethefunctionth atcorrespondstothe vectorwithoneoneachentrywithrespecttothestandardbas isfunctions.Thenwe usethis x 0 astheinitialvalueforalllevelmeshes,whichispossible, sincethenite elementspaceofmeshlevel1iscontainedintheniteelemen tspaceofallhigherlevel meshes.Thestoppingcriterionis k x n k = k x 0 k < 10 6 ,where x n istheresultofthe n -th iteration(whichmeasuresthereductionintheerrorsincet heexactsolutioniszero).The convergencerateiscomputedbytakingtheaverageof k x n k = k x n 1 k Table 7-2 reportstheconvergenceratewhenthewavenumber =1 .We consideredsixdifferentcoarsestmeshsizes h 1 ,andweusedthesparsematrixstructure inMATLABtosolvethelinearsystemonthecoarsestmesh.For axedcoarsestlevel mesh,theconvergencerateisnearlyconstantandclearlyin dependentofthemeshsize. 105

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Table7-2.Backslash-cycleconvergencerates: =1 h 1 Level 1 1/2 1/4 1/8 1/16 1/32 2 0.2 3 0.3 0.3 4 0.3 0.3 0.3 5 0.3 0.3 0.3 0.3 6 0.3 0.3 0.3 0.3 0.3 7 0.4 0.3 0.3 0.3 0.3 0.3 8 0.4 0.4 0.3 0.3 0.3 0.3 9 0.4 0.4 0.4 0.4 0.3 0.3 Table7-3.Backslash-cycleconvergencerates: =10 h 1 Level 1 1/2 1/4 1/8 1/16 1/32 2 1.8 3 1478.4 1165.8 4 736.7 743.8 12 5 787.6 1176.3 9.7 0.5 6 926.3 1300.5 9.2 0.3 0.3 7 948 1323.5 9.1 0.4 0.3 0.2 8 951.2 1327.8 9.1 0.4 0.3 0.3 9 951.9 1328.8 9.1 0.4 0.3 0.3 Itappearsthatwhen =1 ,wehaveuniformconvergenceforeachcoarestmeshsize considered.Thisisnolongerthecaseforhigherwavenumber s. Table 7-3 reportstheconvergenceratewhenthewavenumber =10 .Inthecase whenthemultigriddiverges,theconvergencerateiscomput edbytakingtheaverage of k x n k = k x n 1 k oftherstteniterations.Thisexampleillustratesthat,a sprovedin Theorem 7.2.1 ,thecoarsestmeshsizemustbesmallerthansomeconstantin orderto achieveaconvergencerateindependentofthemeshsizefort hemultigridbackslash cycle. Thenextexperimentgivesusanideaofhowthewavenumberand thecoarsest meshsizethatguaranteesmultigridconvergencearerelate d.Foraxedmeshlevel(level 9 ),wecomputedtheconvergencerateforintegerwavenumbers rangingfromoneto 106

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20 ,andforcoarsestmeshsizesrangingfrom (1 = 2) i where 0 i 5 .Allothersettings arethesameastheothermultigridexperimentsabove. InFigure 7.2.3 ,thecoarsestmeshsizewheremultigridstartstoconvergef oreach integerwavenumberwereplottedandthosedotswereconnect edbystraightlines. Roughlyspeaking,forlargerwavenumbers,thecoarsestmes hsizemustbesmaller inordertohavemultigridconvergence.Observefromthegra phthat,however,thisis notalwaysthecase.Itappearsthatsomewavenumbersbehave particularlynicerwith multigridwhilesomebehaveworse.Infact,whilethenumber ofmeshpointsinone waverangesfrom 5 to 10 formostofthewavenumbersconsidered,for =6 therewere approximately 2 meshpointsinonewave,andfor =18 therewereapproximately 22 meshpointsinonewavewhenthemultigridstartedtoconverg e. Theadvantageofthemultigridbackslashcycleunderaxials ymmetryisthatthe correspondingmatrixsystemthatneedstobesolvedontheco arsestlevelmeshfor eachiterationisstillrelativelysmallasweareintwodime nsionduetoaxialsymmetry. Figure7-2.Wavenumbervscoarsestmeshsize 107

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CHAPTER8 ABIOMEDICALAPPLICATIONOFTHEMERIDIANPROBLEM Inthischapter,wewillseeabiomedicalapplicationofthee dgeniteelement methodandtheaxisymmetricMaxwellequations.Theexperim entdoneinthischapter closelyfollowstheinformationgivenin[ 7 ]. HepaticMicrowaveAblation(MWA)isanalternativetreatme nttolivercancer.It isanexperimentalprocedureinwhichaprobeisinsertedthr oughtheskinorduring surgerytoinducecellnecrosisthroughtheheatingofdeepseatedtumors.Anefcient probeforsuchpurposeshouldbedesignedsothattheelectro magneticpowerpatternis highlylocalizednearthetipoftheprobe. Tocomputetheelectromagneticdistribution,wendanappr oximatesolutiontothe Maxwellequations.Assumingpropagationatasinglefreque ncy,wecanusethenite elementmethodtoapproximatethetimeharmonicMaxwellequ ations.Althoughaliver isnotaxisymmetric,weareonlyinterestedinthepartofthe liverthatcontainsthetumor. Inotherwords,sincealiverislocallyaxisymmetric,wecan takethecomputational domaintobeasmallcylindercontainingthetumorminusthep robe.Additionally,it isknownthattheelectriceldinsuchasetuphaszero -component.Sowecan approximatethemeridianproblem( 1–8 )tondtheelectricelddistributionoverthepart oftheliverinourinterest.Thiscomputationisbasedonthe dataandinformationgiven in[ 7 ]. Letusrecallthemeridianproblem( 1–8 ): curl rz ( 1 r curl rz E rz ) 2 r E rz = F rz SeeChapter 1 toseethedenitionoftheparametersinvolved.Wendan approximatesolutiontothemeridianproblembyusingthelo westorderN ed elec elements(seeChapter 6 ): 108

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Figure8-1.Coaxial-baseddoubleslotchokedprobe Find u h 2 W h suchthat ( 1 r curl rz u h ,curl rz v h ) r 2 ( r u h v h ) r =( ~ F rz v h ) r forall v h 2 W h Figure 8 isthecrosssectionoftheaxisymmetricprobedesignedforH epatic MicrowaveAblation(MWA)[ 7 ]. F rz = 0 ,andweputperfectlyconductingboundaryconditions( u h t =0 )onthepart oftheboundarythatisnotontheaxisofsymmetryexceptwher etherearethedouble slots.Asin[ 7 ],theelectriceldinsidetheprobeiscalculatedbythefor mula E ( r z )= C r e i z e r where C = 264.42 P in ln( r o = r i ) r o and r i aretheouterandinnerradiiofthecoaxialcable,and P in istheinputpower.Theelectriceldisintheradialdirecti ononlyinsidetheprobeand inbothradialandaxialinsidetheliver.Wecomputetheboun daryvaluesonthedouble slotsbyusingthisformula,and ~ F rz denotesthesourcecomingfromtheseslots. WeusethecommonlyusedMWAfrequencyof2.45GHz,sothewave number =51.31268 r and r dependonthepropertiesoftheliver,whichisgivenin[ 7 ]as: 109

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Figure8-2.Distributionoftheelectromagneticpower r = 0 =1 and r = 1 0 ( + i )=44.4+13.95 i .Inparticular, 0 =44.4 andthe conductivityatthegivenfrequencyis =1.9( S = m ) .Figure 8 showsthelogarithmic distributionof j E j 2 afterscaling. Notethatthedomainisnotarectangle,andthemissingparto ftherectanglenear theaxisofsymmetrycorrespondstotheprobe.Thetipofthep robeislocatedatthe point (0,7 10 3 ) ,andwecanseethelocalizationoftheelectromagneticpowe rright infrontofthetipoftheprobe.Inshort,theMaxwellequatio nsunderaxialsymmetry 110

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andtheedgeniteelementmethodcanbeusedfordesigningef cientprobesforthe HepaticMicrowaveAblation(MWA). 111

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APPENDIXA PROOFOFTHEEXACTSEQUENCEPROPERTY Weprovetheexactnessofthesequence( 4–3 )undertheassumptionthat 1 is connectedand D issimplyconnected. Theinjectivityof grad rz : V h 7! W h istrivial,soletusproceedtoprovethenext itemintheexactsequenceproperty,namely grad rz ( V h ) equalsthenullspaceof curl rz : W h 7! S h ,whichisdenotedby ker(curl rz ) .Itisobviousthat grad rz ( V h ) ker(curl rz ). Since D issimplyconnected,any w h 2 W h satisfying curl rz w h =0 coincideswitha gradient,say grad rz h ,andbycomparingthepolynomialdegrees, h isin V h .Moreover, h canbechosentobein V h ,because t grad rz h =0 on 1 andbecause 1 is connected. Thus,itonlyremainstoprovethat curl rz : W h 7! S h issurjective.Forthisweonly needtheconnectednessof D asweseebelow. LemmaA.0.6. Themap curl rz : W h S h issurjective. Proof. Thecollectionofindicatorfunctions K ,forallmeshelements K ,spans S h Therefore,toprovethelemma,itsufcestoshowthattherei sa u Kh in W h suchthat curl rz u Kh = K (A–1) forallmeshelements K .Todothisweneedacommutingprojector.Weusethe projectors b Wh and b Sh of[ 21 ]whichwealreadyintroducedearlier(althoughthisproof worksequallywell, mutatismutandis ,withN ed elec'soriginalprojector[ 32 ]). Tobegin,considerallmeshelements K near 0 ,specicallythosein T 0 h = f K 2 T h : K \ 0 isnon-empty g Let K p ( u )=( R K r p drdz ) 1 R K r p curl rz u drdz Then,forany K in T 0 h ,choosea function u K in D ( K ) 2 withnonzero K 1 ( u K ) .Byrescalingthisfunctionifnecessary,we canassumewithoutlossofgeneralitythat K 1 ( u K )=1 .Thenconsidertheinterpolant 112

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u Kh b Wh u K .Recallingthedenitionof b Wh in[ 21 ],wendthatallthedegreesof freedomdening b Wh appliedto u K vanish,excepttheinteriordegreeoffreedomon K namely R K r curl rz u K .Therefore,bythecommutativityproperty( 4–1 ), curl rz ( u Kh )= b Sh curl rz u K = K 1 ( u K ) K = K Thus,wehaveproved( A–1 )forall K in T 0 h Toconsidertheremainingelements,let K 0 beanelementsharingameshedgewith anelement K in T 0 h .Let u 0 denoteaninnitelydifferentiablevectorfunctionsuppor tedin K [ K 0 suchthat K 0 0 ( u 0 ) 6 =0. Then, curl rz ( b Wh u 0 )= b Sh (curl rz u 0 )= K 0 0 ( u 0 ) K 0 + K 1 ( u 0 ) K Thus,with u Kh assetpreviously,andwith u K 0 h =( b Wh u 0 K 1 ( u ) u Kh ) = K 0 0 ( u 0 ), curl rz ( u K 0 h )= curl rz ( b Wh u 0 ) K 1 ( u 0 ) K K 0 0 ( u 0 ) = K 0 sothat( A–1 )isprovedforallelementsin T 1 h = f K 1 2 T h n T 0 h : K 1 sharesanedgewith some K 2 T 0 h g aswell. Theproofiscompletedbygeneralizingto T j h = f K 0 2 T h n[ j 1 ` =0 T ` h : K 0 sharesan edgewithsome K 2 T j 1 h g ,for j 1 ,andformalizinganinductionargument. 113

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APPENDIXB PROOFOFPROPOSITION4.2.1 Incase1, a isapointonthe z -axis,andthecorrespondingsemi-disk D a is representedby r 2 +( z a z ) 2 < 2 ,where r 0 .Considerthemapping ^ r = r and ^ z = z a z or r =^ r and z = ^ z + a z Then,itisstraightforwardtoshowthatthismapsends D a inthe rz -planetothe semi-diskrepresentedby ^ r 2 +^ z 2 < 1 ,where ^ r 0 inthe ^ r ^ z -plane.Wecallthis semi-unit-disk ^ D 1 .NotethattheJacobianarisingfromchangeofvariablesfro m ( r z ) to (^ r ,^ z ) is 2 Letusdene ^ 1 2 ^ P k by Z ^ D 1 ^ r ^ 1 ^ pd ^ rd ^ z =^ p ( 0 ) forall ^ p 2 ^ P k (B–1) where ^ P k denotesthespaceofpolynomialsoforderupto k on ^ D 1 ,anddene a ( r z )= 1 3 ^ 1 (^ r ,^ z ). (B–2) Then,bychangeofvariables, Z D a r a ( r z ) p ( r z ) drdz = Z ^ D 1 (^ r ) 1 3 ^ 1 (^ r ,^ z )^ p (^ r ,^ z ) 2 d ^ rd ^ z bydenition( B–2 ), = Z ^ D 1 ^ r ^ 1 ^ pd ^ rd ^ z =^ p ( 0 ) bydenition( B–1 ), = p ( a ), forall p 2 P k ,whichprovesItem 1 114

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Item 2 followsdirectlyfromItem 1 : k a k 2r D a = Z D a j a ( r z ) j 2 rdrdz = a ( a ) byItem 1 andsince a 2 P k =^ 1 ( 0 ) 1 3 bydenition( B–2 ), C 3 Thelastinequalityfollows,since ^ 1 ( 0 ) onlydependsonthereferencedomain ^ D 1 .This completestheproofincase1. Next,weconsidercase2inwhich a isapointin R 2+ thatisnotonthe z -axis,and thecorrespondingdomain D a isthediskrepresentedby ( r a r ) 2 +( z a z ) 2 < 2 .Now, considerthemapping ^ r = r a r and ^ z = z a z or r = a r + ^ r and z = a z + ^ z Thismaptransfers D a inthe rz -planetothediskrepresentedby ^ r 2 +^ z 2 < 1 inthe ^ r ^ z -plane.Wecallthisunitdisk ^ D 2 .Underthismapping,theJacobianarisingfrom changeofvariablesfrom ( r z ) to (^ r ,^ z ) is 2 Dene ^ 2 (^ r ,^ z ) 2 ^ P k and ^ 2, (^ r ,^ z ) 2 ^ P k by Z ^ D 2 ^ 2 ^ pd ^ rd ^ z =^ p ( 0 ) forall ^ p 2 ^ P k (B–3a) Z ^ D 2 ^ 2, ^ pd ^ rd ^ z =^ p ( 0 ) forall ^ p 2 ^ P k (B–3b) foranypositiveandboundedfunction (^ r ,^ z ) on ^ D 2 ,where ^ P k isnowthespaceof polynomialsoforderupto k on ^ D 2 .Dene a 2 P k by a ( r z )= 1 2 ^ 2, (^ r ,^ z ) (B–4) 115

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with = a r + ^ r inthiscase.Notethatthefunction r ( r z ) isthenmappedto (^ r ,^ z ) Then, Z D a r a ( r z ) p ( r z ) drdz = Z ^ D 2 ( a r + ^ r ) 1 2 ^ 2, (^ r ,^ z )^ p (^ r ,^ z ) 2 d ^ rd ^ z bydenition( B–4 ), = Z ^ D 2 ^ 2, ^ pd ^ rd ^ z =^ p ( 0 ) bydenition( B–3b ), = p ( a ), forall p 2 P k ,whichcompletestheproofofItem 1 ToproveItem 2 ,werstshowthat ^ 2, ( 0 ) C min ^y 2 ^ D 2 ( (^ y )) (B–5) Thisistrue,since k ^ 2, k 2L 2 ( ^ D 2 ) =^ 2, ( 0 ) bydenition( B–3b ), = Z ^ D 2 ^ 2 ^ 2, d ^ rd ^ z bydenition( B–3a ), k ^ 2 k L 2 ( ^ D 2 ) k ^ 2, k L 2 ( ^ D 2 ) k ^ 2 k L 2 ( ^ D 2 ) 1 q min ^y 2 ^ D 2 ( (^ y )) k ^ 2, k L 2 ( ^ D 2 ) Thus, k ^ 2, k L 2 ( ^ D 2 ) k ^ 2 k L 2 ( ^ D 2 ) 1 q min ^y 2 ^ D 2 ( (^ y )) andso ^ 2, ( 0 )= k ^ 2, k 2L 2 ( ^ D 2 ) C min ^y 2 ^ D 2 ( (^ y )) 116

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where C = k ^ 2 k 2L 2 ( ^ D 2 ) onlydependsonthexedreferencedomain.Hence, k a k 2L 2r ( D a ) = a ( a ) byItem 1 andsince a 2 P k = 1 2 ^ 2, ( 0 ) bydenition( B–4 ) C 2 min ^y 2 ^ D 2 ( (^ y )) by( B–5 ), = C 2 min y 2 D a ( r ( y )) Thelastequalityholds,since r ( r z ) ismappedto (^ r ,^ z ) .Thiscompletestheproofof Item 2 forcase2. Lastly,weprovethepropositionforcase3.Inthiscase,asi ncase2, a isapoint in R 2+ thatisnotonthe z -axis,butthecorrespondingdomain D a istheopendiskwith center ~ a andradius ,where ~ a =(~ a r ,~ a z ) isobtainedby 0B@ ~ a r ~ a z 1CA = 0B@ a r a z 1CA + 0B@ cos sin sin cos 1CA 0B@ 2 0 1CA forsomexedangle 0 .Considerthemapping 0B@ ^ r ^ z 1CA = 1 0B@ cos sin sin cos 1CA 0B@ r a r z a z 1CA or 0B@ r z 1CA = 0B@ a r a z 1CA + 0B@ cos sin sin cos 1CA 0B@ ^ r ^ z 1CA Then,itisstraightforwardtoshowthat,thismapsendsthed isk D a inthe rz -planetothe disk (^ r 2) 2 +^ z 2 < 1 inthe ^ r ^ z -plane.Wecallthisdisk ^ D 3 .Furthermore,theJacobian arisingfromchangeofvariablesfrom ( r z ) to (^ r ,^ z ) is 2 117

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Letusdene ^ 3 2 ^ P k and ^ 3, 2 ^ P k by Z ^ D 3 ^ 3 ^ pd ^ rd ^ z =^ p (2,0) forall ^ p 2 ^ P k (B–6a) Z ^ D 3 ^ 3, ^ pd ^ rd ^ z =^ p (2,0) forall ^ p 2 ^ P k (B–6b) foranypositiveandboundedfunction (^ r ,^ z ) on ^ D 3 ,where ^ P k isnowthespaceof polynomialsoforderupto k on ^ D 3 .Wethendene a 2 P k by a ( r z )= 1 2 ^ 3, (^ r ,^ z ). (B–7) Then,bydening (^ r ,^ z )= a r + ((cos )^ r (sin )^ z )) ,wehavethat Z D a r a ( r z ) p ( r z ) drdz = Z ^ D 3 (^ r ,^ z ) 1 2 ^ 3, (^ r ,^ z )^ p (^ r ,^ z ) 2 d ^ rd ^ z bydenition( B–7 ), =^ p (2,0) bydenition( B–6b ), = p ( a ). ThiscompletestheproofofItem 1 Next,bythesamewayasincase2,itfollowsthat ^ 3, (2,0) C min ^y 2 ^ D 3 (^ y ) (B–8) andso, k a k 2L 2r ( D a ) = a ( a )= 1 2 ^ 3, (2,0) C 2 min ^y 2 ^ D 3 (^ y ) = C 2 min y 2 D a r ( y ) whichcompletestheproofofItem 2 .Thepropositionholdsforallthreecases1,2,and 3,andthuswearedone. 118

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BIOGRAPHICALSKETCH MinahOhwasborninSeoul,Koreain1981.ShegrewupinbothSe ouland Seattle,WA,andgotherB.S.degreeinAug.2005fromYonseiU niveristy,Seoul,Korea, whereshealsogotamiddle/highschoolteacher'slicensein mathematics.During herundergraduatestudies,shespentayearinSt.OlafColle ge,Northeld,MNasan exchangestudent,andthatiswhenshedecidedtopursuemath ematics.Shewent toUniversityofFloridaforgraduatestudiesinAug.2005an dgotherM.S.degreein mathematicsinMay2007.Shecontinuedtostudynumericalan alysisattheUniversity ofFlorida,niteelementmethodsandmultigridinparticul ar,andgotherPh.D.in mathematicsinMay2010. 122