Citation

## Material Information

Title:
Multiscale Analysis of Partial Differential Equations Modeling Voltage Potential
Creator:
BHAT, YERMAL SUJEET ( Author, Primary )
2008

## Subjects

Subjects / Keywords:
Approximation ( jstor )
Boundary conditions ( jstor )
Boundary value problems ( jstor )
Electric potential ( jstor )
Embeddings ( jstor )
Finite element method ( jstor )
Inhomogeneity ( jstor )
Mathematics ( jstor )
Uniqueness ( jstor )
Zinc ( jstor )

## Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Yermal Sujeet Bhat. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
7/24/2006
Resource Identifier:
496613763 ( OCLC )

Full Text

MULTISCALE ANALYSIS
OF PARTIAL DIFFERENTIAL EQUATIONS
MODELING VOLTAGE POTENTIAL

By

YERMAL SUJEET BHAT

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2006

by

Yermal Sujeet Bhat

To Y.L. Bhat, 1936-1994.

ACKNOWLEDGMENTS

My deep gratitude to Y.L.B., L.L.B., V.L.B., and S.L. Moskow and the

University of Florida Mathematics Dept. I would also like to thank all my family

and friends for their help and support.

page

ACKNOWLEDGMENTS ................... ....... iv

LIST OF TABLES ........... ...... ........ ...... vii

LIST OF FIGURES ............. .... ............ viii

ABSTRACT ........... .. ............... .... x

CHAPTER

1 INTRODUCTION .............................. 1

2 THE ELECTROLYTIC VOLTAGE POTENTIAL MODEL ....... 5

2.1 Butler-Volmer Boundary Conditions ................ 5
2.2 Existence and Uniqueness .......... ............ 8
2.3 Regularity . . . . . . . 15

3 A CORRECTOR BASED ON NEUMANN BOUNDARY DATA ..... 18

3.1 A Neumann Boundary Condition ........ .......... 18
3.2 Finite Element Method Implementation ..... ... ........ 24
3.3 Numerical Results .. .............. ........ 31

4 A CORRECTOR BASED ON ROBIN BOUNDARY DATA ....... 42

4.1 A Robin Boundary Condition ........ ......... .. 42
4.2 Numerical Results ........................... 46

5 SHIFTING MATERIAL BOUNDARIES ............ .... 58

5.1 The Electrostatic Conductivity Model ............ 58
5.2 Estimating the H'(Q) norm of Uo uh ............ .. 59
5.3 Formal Asymptotics ........... ............. 63

6 CONCLUSION ...................... ........ 69

REFERENCES ...... ...................... ...... 71

BIOGRAPHICAL SKETCH .......

LIST OF TABLES
Table page

3-1 Table of estimates over 0 and convergence rates . . .... 32

3-2 Table of estimates over F and estimates of the gradient over .... 32

4-1 Table of estimates .................. .......... 47

LIST OF FIGURES

Zinc loses electrons to silver . ............

The base is a heterogeneous surface . ........

2-2 Perimeter increases while anodic area fraction stays constant.

Two-dimensional analogue .. ........

Limiting behaviour of u, on F as e approaches
(b) = 1/11, (c) = 1/25, (d) = 1/40

u,, = 1/5 . . . . .

uo + EU('), e = 1/5 ...........

u, E = 1/11.....................

Uo + C', E = 1/11 .. ..........

The potential on the boundary F, e = 1/5

The potential on the boundary F, e = 1/11

The potential on the boundary F, e = 1/25

The potential on the boundary F, e = 1/40

L norm of Vu, on F as e approaches 0..

u e = 1/5 . . . . .

uo + e U), e = 1/5 .. ..........

u 6, e = 1/11 ......

0 + E ', e 1/11 . . ....

Graph of u, (above) and uo + u1) (below) on

The approximation and the original, e = 1/5

4-7 Graph of u, (above) and uo + u1) (below)

on

zero

for: (a) e

the boundary

the boundary

4-8 The approximation and the original, e = 1/11 . .

Figure

1-1

2-1

page

3

6

1/5,

= 1/5

= 1/11

r,

P, <

. -. 53

4-9 Graph of u, on the boundary F, e = 1/25

4-10 Graph of uo + u1) on the boundary F, e =

4-11 Graph of u, on the boundary F, e = 1/40

4-12 Graph of uo + ui) on the boundary P, e =

5-1 Perturbation due to shifting between two

1/25.........

= 1/40 . . .

dielectrics a1 and 2.....

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MULTISCALE ANALYSIS
OF PARTIAL DIFFERENTIAL EQUATIONS
MODELING VOLTAGE POTENTIAL

By

Yermal Sujeet Bhat

May 2006

Chair: Shari Moskow
Major Department: Mathematics

We study a nonlinear elliptic boundary value problem arising from electro-

chemistry. The boundary condition is of an exponential type. We examine the

questions of existence and uniqueness of solutions to this boundary value problem.

We then treat the problem from the point of view of homogenization theory. The

boundary condition has a periodic structure. We find a limiting or effective prob-

lem as the period approaches zero, along with a correction term and convergence

estimates. This correction term satisfies a boundary value problem with Neumann

boundary conditions. We do numerical experiments to investigate the behavior

of galvanic currents near the boundary as the period approaches zero. We then

consider a correction term which satisfies a boundary value problem with a Robin

boundary condition. We do numerical experiments to investigate our approxi-

mation based on this corrector term. We then use asymptotics to analyze the

behaviour of the steady state voltage potential of a conductor with a region of in-

homogeneity. The boundary of the inhomogeneity shifts slightly, we do asymptotics

utilizing the small shift.

CHAPTER 1
INTRODUCTION

Modern perturbation theory is primarily concerned with constructing ap-

proximations of solutions to mathematical models that have a parameter which is

approaching zero. One such class of models are boundary value problems in which

the domain has a periodic structure. In this case the period size is the small scale

parameter. These types of boundary value problems often arise in the study of,

for example, composite materials, macroscopic parameters of crystalline struc-

tures, fluid mechanics and aerodynamics. Perturbation theory is an example of

an analytical approximation as opposed to a numerical approximation. There are

many techniques to formulate these analytical approximations. One fundamental

technique is through the use of a multiple scales asymptotic expansion. The foun-

dational ideas for this approach appear in the early 1800s. In 1812 Laplace used

asymptotic series to analyze some special functions. In 1823, Poisson constructed

an expansion of a Bessel function. In 1886 Poincar6 used asymptotic expansions to

study solutions of differential equations. The idea behind the asymptotic approx-

imations is that the solution can expressed as a sum of terms of different orders

of magnitude with respect to the small scale. For example if e,(x) is the solution

to some boundary value problem with small scale parameter E then we begin by

assuming the solution 0, has an asymptotic expansion of the form

o 0o + E0!1) + E20!2) + .

The general procedure is to then substitute the above expansion back into

the original boundary value problem to determine associated boundary value

problems for o, 01), 02 ), ... We try to find simpler equations that describe the

behaviour of the solution on various orders of e. Here we wish to use multiple scales

analysis to develop an asymptotic expansion of the solution to some phenomena

related to conductivity. We first use perturbation theory to approximate the

solution of a nonlinear boundary value problem which models galvanic corrosion.

As a second application of asymptotics we wish to describe the conductivity

properties of a material with a shifting dielectric boundary

In the electrochemistry community there is much interest in the study of

galvanic interactions on heterogeneous surfaces [12], [13]. When two different

metals in electrical contact, referred to as anode and cathode, are immersed in an

electrolytic solution, the difference in rest potential generates an electron flow. This

electron flow is called a galvanic current and may lead to a deterioration (corrosion)

of the anode.

In Figure 1-1 a strip of silver (Ag) and a strip of zinc (Zn) have been im-

mersed in a saltwater solution. The zinc strip gives up electrons to the silver strip.

The silver strip is said to be cathodic, and reduction takes place (Ag gains elec-

trons.) Simultaneously oxidation takes place at the zinc strip, zinc loses electrons,

and is said to be anodic. Zinc dissolves into the solution, the zinc electrode is being

corroded and the electron flow is known as galvanic current. The driving force

of the electron transport process is the difference in potential of the two metals

involved. See Newman [15] for a complete introduction to the subject.

Here we study the electrostatic problem on a surface where anodes are

arranged periodically in a cathodic matrix. Mathematically the potential is

modeled as a function, 0, over a Euclidean domain Q. Part of the boundary of Q

is electrochemically active while the rest of the boundary is inert. It is the active

region of the boundary that is made up of anodic and cathodic portions. The

potential over both of these regions satisfies an exponential boundary condition

of Butler and Volmer, but with different material parameters on each portion.

Ag Zn
hydrogen
gas
NaCI

^H ^ e- ^

Figure 1-1: Zinc loses electrons to silver

In Morris and Smyrl [12] the authors study such a problem numerically, using

finite elements. Additionally various interesting aspects of the two-dimensional,

homogeneous model with the Butler-Volmer condition have been analyzed in Bryan

and Vogelius [5], Turner and Hou [9], and Vogelius and Xu [19]. To the best of

our knowledge, however, studies coming from the applied mathematics community

have been restricted to two dimensions. The main reason for this is that one can

bound exponentials of the two-dimensional weak solution on the boundary by

using an Orlicz estimate [18], [19]. Such an estimate would require more than H1

regularity in higher dimensions. In this paper, we attempt to treat a periodically

heterogeneous problem, in two and three dimensions, from the point of view of

homogenization theory.

Our second application of asymptotics is to a problem pertaining to electro-

static conductivity. We consider a PDE which models the steady state voltage

potential of a metal with a small inhomogeneity. There is a jump in the conduc-

tivity across the boundary of the inhomogeneous region. The boundary of the

inhomogeneous region shifts by some small amount due to some type of physical

stress. We now wish to describe the voltage potential of the conductor with shifted

boundary as a perturbation of the voltage potential of the original conductor. We

do asymptotics to establish an estimate.

In Chapter 2 we formally present the electrolytic voltage potential model.

We tackle the issue of existence and uniqueness of the solution to the model and

then discuss the issue of regularity. In Chapter 3 we construct an asymptotic ap-

proximation of the solution to the original problem. Here the second term of the

approximation satisfies a linear boundary value problem with Neumann data. We

then establish some convergence estimates and do numerical implementation. Using

a finite element method approach we implement and test our asymptotic approx-

imation and convergence estimates. In Chapter 4 we propose an approxiamation

in which the second term satisfies a boundary value problem with Robin boundary

data. We then do numerical implementation and testing of this approximation.

In Chapter 5 we perform asymptotics on the "shifting boundary" problem. We

establish some convergence estimates and do some formal asymptotics. In Chapter

6 we discuss future work to be done.

CHAPTER 2
THE ELECTROLYTIC VOLTAGE POTENTIAL MODEL

2.1 Butler-Volmer Boundary Conditions

Now we formally present the three-dimensional model for electrolytic voltage

potential on a heterogeneous surface. The domain 0 is of cylindrical shape with

base some two-dimensional domain. The bottom base is assumed to contain a

periodic arrangement of islands (anodes). We call this collection of islands 8QA and

the remainder of the bottom of the base tQc cathodicc plane). The electrolytic

voltage potential, 0, satisfies the following nonlinear elliptic boundary value

problem,

A0 = 0 in Q
JA [eas(4-V) -V e-aac(4-A)] on aA (2.1)
8n
S= Jc[ea"(-vc) e -a -vc)] on aQc
on
= 0 on 9 \ {tQA U 8Qc}
8n

where aaai, c a, ce, are the transfer coefficients and it is assumed that the sums

(aaa + aac) and (ac. + ace) are equal to one. The positive constants JA, JC are
the anodic and cathodic polarization parameters and VA, Vc are the anodic and

cathodic rest potentials respectively. Note that V4 represents galvanic current.

These boundary conditions are the so-called the Butler-Volmer exponential

boundary conditions.

In the numerical studies of [12], the authors observed that for fixed ratios of

anodic to cathodic areas on the bottom base, the resulting current increased ap-

proximately linearly with the length of the perimeter between the two regions, and

Figure 2-1: The base is a heterogeneous surface

they hypothesized that it is the ratio of anodic area to perimeter that determines

the size of the resulting current.

Figure 2-2: Perimeter increases while anodic area fraction stays constant.

As a special case of increasing perimeter with approximately fixed area

fraction, we consider a periodic structure with period approaching zero. Our goal is

to expand the solution asymptotically with respect to the period size. Convergence

results involving these approximations could provide insight into the behavior of

the current for small period size; and possibly lead to techniques for computing

approximate solutions to (2.1).

We model the periodic structure by letting

f (y, v) = A(y)[e"(y)(v-v(y)) e-(1-"(Y))(v-v(y))]

for any v R and y E Y, the boundary period cell, which for simplicity we take

to be the unit square; Y = [0, 1] x [0, 1]. Here A, a, and V are all piecewise smooth

Y-periodic functions. We also assume there exist constants Ao, A0, ao, Ao and Vo

such that:

0 < Ao < A(y) < Ao, (2.2)

0 < ao < a(y) < Ao < 1,

(2.3)

and

V(y)I < Vo. (2.4)

See [5] and [19] for an analysis of when A < 0.

Consider the problem

Au = 0 in Q
S= f(x/,,u,) on r (2.5)
On
= 0 on 0Q \r.
On

As is typical in homogenization problems, one expects that as c -- 0, the solutions

will converge in some sense to a solution of a problem with an averaged boundary

condition. Define fo(v) to be a cell average of f(y,v), that is,

fo(v) = f(y,v)dy.

Consider the candidate for the homogenized problem

Auo = 0 in Q
Buo
=u fo(uo) onr (2.6)
an
Buo
= 0 on dQ \r.
On

Remark If, as is the case in [12], Y = Y UY2 and the functions A,a,V are

piecewise constant, each taking on the values A,, ai, V respectively in Y5, then

fo(v) = I1 jI[eal(-1-) e-(l-")(v-vi)] + YA2 [ 2(V-V2) e-(1 -2)(-V2)].

That is, the above homogenized boundary condition would depend on the volume

fraction of anodic to cathodic regions.

2.2 Existence and Uniqueness
In this section we show that the energy minimization forms of the nonlinear
problem (2.5) and (2.6) have unique solutions in H1(Q) in any dimension. Some
elements of the proof are similar to those in [9] and [19]. For a given E, define the
following energy functional,

,(v) = Vv 2 dx + F( v)da (2.7)

where,
F(y, v) = A(y) eY)(v-V(y)) + ) e-(1-a(y))(v-V(y))
a(y) 1 a(y)
We show the existence and uniqueness of a minimizer of (2.7). Formally, we show
the existence of a function u, E H1(V) such that

E,(u) = min E(u). (2.8)

Note that E, is not necessarily bounded on all of H1'() (unless n = 2 for which
we can use an Orlicz estimate). However this does not pose a problem. We set E,
equal to (2.7) where it is well defined and to +oo where it is not, as in [7], p.444.
In the two-dimensional case of the model, due to the boundedness of E, on Hl(S),
direct calculation shows u, satisfies the variational form of (2.5),

SVu Vv dx = f(x/e, u)v dc for any v c H1'(). (2.9)

In the three-dimensional case, if u, is an energy minimizer, we will have that

/ F(x/e,u,) da < oo, (2.10)

and hence by the positivity of each term of F(x/e, u,), we have that each term is
separately in L1(Q). Therefore,

E,(ue + tv) < 00,

(2.11)

for any t E R and for any v which is smooth on P. Standard arguments then show
that u, satisfies,

/ Vu. Vv dx f(x/e, ue)v doa for any v E C0(Q).

Additionally, if we knew that u, E CO(Q) then f(x/e,ue) is bounded and hence
clearly in H-1/2(P). So by the density of C"(0) functions in HI(Q), u, in this case
would satisfy

/ Vue Vv dx f(x/e, u)v doa for any v E H (). (2.12)

Consider also the functional

Eo(v) = V12 dx + Fo(v) do. (2.13)

where,

Fo(v) = Fj F(y,v)dy.

Here again the energy Eo is not necessarily bounded but as before, we set Eo equal
to (2.13) where it is well defined and to +oo where it is not. Direct calculations
show that a minimizer uo of (2.13) will satisfy,

/ Vu o Vv dx = f(uo)v dao for any v E H1'(), (2.14)

assuming u0 is continuous (actually we will see that uo is a constant.)
Theorem 2.2.1 (Existence and Uniqueness of the Minimizer). Let E, be defined
by (2.7), where X,a, and V satisfy (2.2)-(2.4). Then there exists a unique function
u, E H1(Q) satisfying
E,(ue) = min E,(u).
neH' ()
Proof. Note that

2 F(y,v) = A(y)a(y)ea(Y)(v-V(y)) + A(y)(l a(y)) e-(1-a())(v-V(y))
(9v2

since A > 0, a > 0, and 1 a > 0 we have that 92F > 0. Clearly the partial

derivative is bounded below. That is, there exists a constant co, independent of y

and v such that,
92
2 F(y, v) > co > 0.

Since F is smooth in the second variable, for any v,w E H1(Q) and for any y, there

exists some ( between v + w and v w such that

a02
F(y, v + w) + F(y, v w) 2F(y, v) = Fly, J)w2

which from the lower bound yields

F(, v + w) + F(, v w) 2F(X,v) > c0w2
6 E E

whence

E,(v + w) + E,(v w) 2E,(v) > +f Iw 2 dz + co0 w2cdr

> Wco2H111 ) (2.15)

where the last inequality follows by a variant of Poincar6. Now let {u'}'l be a

minimizing sequence, that is

E,(u) -> inf E,(u) as n oo.
ueHi(n)

Since all the terms of (2.7) are nonnegative, clearly

inf E,(u) > oo.
ueHl(2)

Note that without loss of generality we can choose the minimizing sequence so that

all terms have finite energy (since infu Hi() E,(u) < E(0) and E(0) is bounded

independently of E.) Let
un + u2
V--
2

and
-n U
W --u
2

Then v + w = un and v w = u, so (2.15) implies

E(v + w) + E,(v w) 2E,(v) > 4||u U7||n1

which implies,

E(t) + E,(u) 2 inf E,(v) > u u"Jn)
veH1 (c) 4

Now if we let m, n -- oc, we see that {uf}n is a Cauchy sequence in the Hilbert

Space H'(Q). Define u, to be its limit in H1'(). Then we have

S- u, in H1(Q)

which by the Trace Theorem implies,

u u, in L2(P)

which implies ([17],p.68) there exists a subsequence {u~k}k, which we label {zu}k

such that

u -> us a.e. in P.

Since F is smooth in the second variable, and u. -> u a.e. in P we have that

F(X,u6) = lim F( ,u,) a.e..
E k-o0o E

Now note that clearly F(Q, u) > 0 for any k. So, by Fatou's Lemma we have,

SF(, u~)da, < lim inf F( zu)du,.

Thus from this and the fact that u. -- u, in H1(Q), we can conclude that,

E,(u) < liminfke, EE(u)

Hence,

E,(u)= inf E (u).

So we have shown the existence of a minimizer.

Suppose u, and w, are both minimizers of the energy functional, i.e.

E,(u) = inf E,(u)=
ueHI(n)

E, (w,).

Now if we let

v = (u, + w,)/2

w = (Ue w,)/2

then substituting v and w into (2.15) yields,

Ue + w, jo 2
E,(uE) + E(w) 2E,( 2 >) W> H 'I i(n).

However, this implies,

u wll (n) < E,(ue) + E(w) 2 infuEHi(a) E,(u) = 0.

Hence u, = wE in H'(Q). Thus we have shown the uniqueness of the minimizer. O

Note that this argument can be generalized to address the n-dimensional

problem, i.e. the case in which we have f C R",r C R"-1 with boundary period

cell Y = [0, 1]"-1. The existence and uniqueness of a minimizer uo of Eo follows

from the same proof.

limk_,, E(uk)

inf.ueH(n) E,(u).

Corollary 2.2.2. There exists a constant C, depending on Ao, ao,Ao and Vo but
independent of e such that,

IIU'IIHl() < c

where u, is a weak solution to (2.5).

Proof. Consider the function v 0. Then

E,(v) == E(0) = F(, o)d < M

for M independent of e (but depending on Ao, o, Ao and Vo ). Then since u, is a
minimizer,
E,(u,) < E,(0) < M.

Since both terms in E, are positive,

IIVuE 2(| ) < M.

We also have that
SF(x,u,)da,
By examining the form of F(y,v), we see that there exists some constant d,
depending on Ao, ao, and Ao but independent of e and x such that

duE- V(x )
Hence,

Su V(x) d\x < M/d,

which by the boundedness of V implies that

SIU, do, < M

where M is independent of E. One variant of the Poincar6 inequality says that there
exists C such that

\\u, jueda\x\L2(Q)
Finally the reverse triangle inequality yields,

IIIIL2() < C IIVUEIL2(Q) + M,

which proves the corollary. W

We conclude this section by establishing the fact that the solution to the
homogenized problem (2.6) is in a fact a constant. This fact follows easily once we
have established the following lemma.
Lemma 2.2.3. There exists a constant K such that fo(K) = 0.

Proof. Recall that

fo() = j f(y,v)dy (2.16)

where

f(y, v) = A(y)[e -v()) e--"))-v())]

for any v E R and y E Y, where Y = [0, 1] x [0, 1]. Also recall that A, a, and
V are all piecewise smooth Y-periodic functions for which there exist constants

Ao, Ao, ao, Ao and Vo such that:

0 < Ao < A(y) < Ao, (2.17)

0 < co < a(y) < Ao < 1, (2.18)

and

IV(y)I < Vo. (2.19)

Now if v > Vo then (2.19) and (2.18) imply

(1 a(y))(v V(y)) < 0 for all y Y.

a(y)(v V(y)) > 0 and

So we have
e-(-a(y))(v-V(y)) < ea(y)(v-V(y)) for all y E Y.

Thus (2.17),(2.18), and (2.19) imply that if v > Vo then f(y,v) > 0 for all y E Y.
Similarly if v < -Vo then f(y,v) < 0 for ally E Y. Now note that (2.16)
implies that if v > Vo then fo(v) > 0 and if v < -Vo then fo(v) < 0. Now since

fo(v) is continuous by the Intermediate Value Theorem there exists a constant
K E (-Vo, Vo) such that fo(K) = 0. Thus the lemma is established. W

Theorem 2.2.4. Let uo be a minimizer of (2.6) then uo is a constant.

Proof. Suppose K is a constant such that fo(K) = 0. Such a constant exists by
Lemma 2.2.3. Then clearly u0 = K is a strong solution of (2.6). Note that this
argument and Lemma 2.2.3 hold in any dimension. O

2.3 Regularity

We conclude this chapter with a short discussion of the regularity of the solu-
tions u, and u0 for the two and three-dimensional case. For the two-dimensional
case of this problem, i.e. when the medium is layered as in Turner and Hou [9], and
Vogelius and Xu [19] (see Figure 2-3), using imbeddings of Sobolev Spaces (W"'p)
into Orlicz Spaces (L') we can show that f(x2/e, U) and fo(uo) are bounded in
L2(F) independently of c.
We first give a general definition of an Orlicz Space before presenting the
imbedding result. See [2] or [16] for a thorough discussion of Orlicz Spaces. Let Q2
be a bounded domain in R". Let 4 be a Young function, i.e. a real-valued, convex
function such that 4(x) = 4(-x), 4(0) = 0, and 4 / oo as x / oo. Then the
Orlicz space LQ(0) is the set of all measurable functions f on 2 such that

4 44(af) dx < o for some a > 0.
oo soeo

X2

1

Qr

x,
1

Figure 2-3: Two-dimensional analogue.

Now define the norm

I|| f |L() =inf k> : j )( ) dx < l

then L2(Q) becomes a Banach space. For example when 4(x) = IxlP then L' = LP.
We now state the imbedding result established by [18] and [2].
Theorem 2.3.1 (Trudinger's Theorem). Let Q be a bounded domain in R"
satisfying the cone condition. Let mp = n and p > 1. Set

a(x) = eCPp-1) 1.

Then there exists the imbedding

Wm,P() -- Ln(Q).

In the two-dimensional case of (2.5) we can conclude from Trudinger's Theo-
rem [18], [19] that there exists a constant C such that for any v H'I() and any
real / we have,
Jefll~dx2 < ecf2(jlVil(nl')( r + 1).

Then from standard elliptic regularity theory this implies that u, and uo are in
H3/2(Q), with the norm bounded independently of e. By the trace theorem we

then obtain bounds for u, and uo in H'(F). Since P is one-dimensional it follows

that u, and u0 are continuous on F and bounded pointwise, and their tangential

derivatives are bounded in L2(F) For the homogenized solution we have much

more regularity, u0 is in fact the constant that satisfies fo(uo) = 0. For nonzero

boundary conditions on the inactive region, uo would still be a smooth bounded

function. So for the two-dimensional version of this problem we have the following

lemma:

Lemma 2.3.2. If Q C R2 is a rectangle and F is an edge, then u, e C(Q) where

u, is a weak solution of (2.5). Furthermore, there exists a constant D, the value of

which does not depend on e, such that,

I6U(x) lc(n) < D.

In the three-dimensional case of (2.5) since u, E H'I() implies m = 1, so

mp \$ n and thus Trudinger's Theorem does not apply. In fact there is a large

class of Sobolev imbedding theorems pertaining to the case mp = n, however

there are no imbeddings of HI(Q) for a cyindrically shaped domain 2 C R3

that are applicable to (2.5). In general, there has been no rigorous analysis of

the regularity of three-dimensional solutions of elliptic boundary value problems

with L1 Neumann boundary data. Now if we assume that (2.5) models electrolytic

voltage potential then it is physically reasonable to make some assumptions about

the regularity of u, in the three-dimensional case. In the next chapter we show that

if these physically justifiable assumptions are made we can establish convergence

estimates for our asymptotic approximation for the three-dimensional case.

CHAPTER 3
A CORRECTOR BASED ON NEUMANN BOUNDARY DATA

3.1 A Neumann Boundary Condition

To show u, converges to uo we will add a correction term and prove estimates

in terms of powers of E. The convergence of u, to uo when n = 2 will then easily

follow from this. The same estimate holds when n = 3 if we know that the

solutions are continuous and uniformly bounded. When n = 2 we will see that the

convergence is strong in HI(Q) and of the order of vE.

Let u0 be a minimizer of (2.13) and define the correction ut4' to satisfy,

Au)) = 0 in 0
a() 1 I
--u = -(f(,uo) -fo(uo)) + e, on (3.1)
An 6 E
uc 1)
6 = 0on 80 \r
On
I U1) d3 = 0 (3.2)

where,

ee = (fo(uo) f(x/,uo))dax.

Hence e, is chosen such that the solution always exists and the condition (3.2)

guarantees this solution is unique. We note that given uo, this is a linear problem.

Now if u, and uo are in L(P), let

De =max l ||IU|L -(), '0 o L-(r)} (3.3)

and let,

M= sup (y, w). (3.4)
(y,w)EYX[-De,De] ov

The next estimate holds for dimension n = 2 or 3 but depends on the constant Me.

We do not know a prior that D, is finite in general when n = 3. However, such an

assumption seems to be physically reasonable and known to be the case when the

medium is layered.

Proposition 3.1.1. Let n = 2 or 3 and let u,, Uo be minimizers of (2.7),(2.13)

respectively, and let u(') be the solution to (3.1). Assume also that u, E Co(Q).

Then there exists constants C and D independent of e such that

U6 Uo eu |Hi() < Ce(M, + D),

where Me is defined by (3.4). Furthermore, there exists constants C1, and C2

independent of E such that,

IU)||L2:(r) < C1 and |e6j < C2.

Proof. Let

ZC = UE UO ECU

since u, is continuous, by (2.12), we have that for any v E H1(Q),

Vze Vv dx= Vu Vv dx V Vv dx e Vu Vv dx

j f(-,uI)vdux + f( ,uo)vdorx +e evdox.

So,

Vze Vv dx + [ ( -, e) f- (x,o)]vdox c ecvdo =0.

Now note that Uo and u, are defined pointwise on P. So, by the Mean Value

Theorem, for each fixed c and x E F there exists between uo(x) and u,(x) such

that,
xx f x .
f ( ,u) f( ,uo) = (u, uo) ( ,(X ).
E E a2) E

By subtracting and adding Eu() we have,

f (,) f o) = Z (, + E ( af

which, if we pick v = z,, yields,

SVz|VI dx+jf z29f(x, j ()dd d=-c M 4)( C)dX + zd .

Since f > co, this implies

C~I J() < 4 z62dx+jzfi(,da

= M-e af (8 (X)z dax + ee zcd&e .

So by applying Hilders Inequality and then the Trace Theorem we have,

C 2 f< 1 )1/2
_< ll (t ,( e ) ll ( ) l (l ) L2() + l f i l/2) ~ L2
0 Io| | I r, 1|1/2) 1 H
(9~I~v E~~~lO~) U' L() e r/)Zlx~,

Thus, we can write,

tif x
||Hll'Z(| ) < C ( vL() E 1).

Now recall for any v we have,

(f(y,v)- fo(v))dy= 0

so there exists a continuous Y-periodic function r(y,v) such that

Ary(y,v) = f(y,v) fo(v),Vv R.

So we have,

(3.5)

(3.6)

ee = (fo(uo)- f(-,uo))dar

= Ayr(X, uo)dox
p Jr 6

= Vyr(,uo) v dsx
ar C

where the last equality is arrived at using integration by parts and the fact that the
8r dr
chain rule implies -(x/, Uo) = (x/e, Uo). Note that the differential operators
ay ax
Vy and A. are with respect to y E Y, that is, they are surface operators. Now

since u0 is bounded pointwise on F and since r(y, v) is a continuously differentiable

Y-periodic function we have,

ee < C (3.7)

where C is bounded independent of c. Now we show that I.I')lIL2(r) is similarly

bounded. Let w, E H1(Q) satisfy,

Aw = 0 in 0

=- uld( on r (3.8)
On
Owo
= 0 on 9Q \ r
8n
SWe do, = 0

then,

(1) wd j=z = V Vw dx wdax
fE (t 9n C2 e an C

where the last two qualities follow from integration by parts. Now, since u(1'

satisfies (3.1), we have

f au ', ff(x/e, uo) fo(Uo) d
weda = f + e[ w6d
aQ tn E' I X
l= Ayr(x/e,uo)wedcx e, w'da

1 Ayr(x/e, Uo)wdax
e Jr

where the second equality follows from (3.6) and the last equality holds since
/ wda, = 0. Now using the chain rule we can write

Ayr(x/e, uo) = 2 Ar(x/E, o),

where As is a surface Laplacian on F. Thus, we have,

j(u())2dox = Ac Ar(x/e,uo)w da
r Jr
= e V7r(x/C,Uo)Vwdu-E WEdsx, (3.9)

where v is the outward unit normal to &F. Note that when n = 2, we use the last
integral to represent endpoint evaluation. So, by Hilders Inequality,

ef Vjr(xa/e,Uo)Vwcedax -JE r wds, < Ell Vc r L2(r) VWE L2(r)

+E IL2(ar)WllWll2(ar). (3.10)

Then by the Trace Theorem we have,

IIW6e| 2(ar) < C1 IWe |IH (r) < C2||w~ e|| H3/2(). (3.11)

Similarly,
IIVW IL2() < C311 W6 Hi(rF) < CllEllH3/2(). (3.12)

Then (3.9), (3.10), (3.11) and (3.12) imply,

|u 2() < c ( VXr|L2(P) + | L2(P)) W H| /2().

Now since we satisfies (3.8) we have from standard elliptic regularity theory [10],

I HWll3,/2(n) < CU 1) llL2(r)

where C is independent of e and so we can write,

I LJ() JI2(r) < Ce | V r(x/e,U o) -2(rr) /E,~O IL2(

= ( ||Vr(x/e, uo) L2() + 11r uo) L2( ,

where the last equality follows from the chain rule. Consequently, since we have
that u0 is continuous on P and bounded pointwise and since r(y,v) is a continu-
ously differentiable Y-periodic function we can conclude that

I4u) L2(r) < D (3.13)

where D is bounded independently of e. Then (3.5), (3.7) and (3.13) imply the
main result of the proposition:

Of x
KZ H'U() CE(Il e LP) l U L2(Plr) + e1 ) < Ce(M+ D),

where M, is defined by (3.4). O

Note that in light of Lemma 2.3.2, we can easily establish the following corollaries:
Corollary 3.1.2. When n = 2, i.e. for the case in which f c R2, F c R with
boundary period cell Y = [0, 1] there exists a constant C independent of e such that,

Iu, Uo eu,1) Hi(n) < Ce. (3.14)

Corollary 3.1.3. When n = 2, for u, the weak solution of (2.5), and uo the weak
solution of (2.6), there exists a constant C independent of E such that,

|IUE uo \Hi(n) < C (3.15)

Estimate (3.15) follows from the fact that,

U("n) < C ll HH-/2() < CC-1/2,

where the last inequality follows by interpolating between L2(P) and Hl(P)

(see [11], Section 11.5) and then using duality (as in [14]). Finally note that

estimate (3.14) also holds for n = 3 if we know that D, defined by (3.3) is

uniformly bounded.

3.2 Finite Element Method Implementation

We wish to numerically observe the behaviour of the homogenized boundary

value problems as a way to describe the behaviour of the current near the bound-

ary. We use a finite element method approach to the 2-D problem. For the 2-D

problem the domain 0 is a unit square and the boundary P is the left side of the

unit square, that is

r { (x21,2) E 1 =: 1}

(see Figure 2-3). In this case we impose a grid of points(called nodes) on the

unit square and triangulate the domain, then introduce a finite set of piecewise

continuous basis functions. Here we use standard finite element method "tent"

functions. We impose a grid of node points on the unit square which are evenly

spaced both on the x and y-axes. We label the nodes starting at the origin and

moving to the right. We label the nodes Pi, i = 1,..., m, where P1 = (0, 0) and

P, = (1,1). Furthermore, if there are N nodes on the axis (i.e. if the axis is

divided into N 1 pieces) then m = N2 and PN-1 = (1 1/N, 0), PN = (1, 0),

PN+I = (0,1/N), PN+2 = (1/N, 1/N) etc. Once the node points have been

established we triangulate the domain in a predetermined fashion. Note that the

node points P1, P2, PN+I and PN+2 form a square. We form one triangle by using

as vertices the node points P1, P2 and PN+I and another triangle by the node

points P2, PN+I and PN+2 Proceeding in this fashion we can form yet one more

triangle by using as vertices the node points P2, P3 and PN+2 and yet another by

using P3, PN+2 and PN+s. The entire domain can be triangulated in this fashion

using the node points as vertices. Once the domain has been triangulated we

introduce basis functions. Note that in the original problem (2.5) we attempt to

minimize the energy functional

E,(v)= j Vv2dx + F(X, v)dOx

over the space H1(Q). However, numerically we minimize the energy functional

over the space Vh where,

Vh = {v E C(Q) : v is a linear function when restricted to each triangle in Q}.

Now we introduce a set of basis functions bi(xi,X2), i = 1,... ,m. We use the

standard finite element method "tent" function, that is for each i define bi E Vh by,

bi(Pj)= I ifi= for j= 1,...,m.
0 if i j

m
So any function v E Vh can be written as v(x) = vjjbj(x), where q = v(Pj).
j=1
Furthermore, we can reformulate the energy minimization problem in the following

way, if we denote the minimizer of the energy functional as uh then

m
E,(uh) = min E,(v) = min E(E rjbj(x)).
veVh i1,-,/rm L
j=1
m
So if (1,..., m minimizes the energy functional then we can write uh Z= ( jbj(x).
j=1
Thus, by triangulating the domain and introducing basis functions of Vh we are

able to discretize the problem. Now we wish to solve the energy minimization
m
problem. Note that if we let v(x) = y rl2jb(x) then the energy functional has the
j=1
form,
E( T m
EE() = A J+ F(x, 7bC(x))riT.
j=1

where A = (aij), 1 < i, j < m such that aij = f Vbi Vbj di and ry is the m x 1

vector 7r = (qi,... ,rm)T. Hence we wish to minimize the energy EE(r7) over all

] E R". Since we plan to use a gradient descent based optimization method to

minimize the energy we are required to calculate the gradient of the energy as well.

For the homogenized problem (2.6) we wish to minimize the functional

Eo(v) = I Vv2dx + Fo(v)d
2 JQ Jr

over the space H1(Q). Thus numerically we wish to minimize
m
Eo(0) = A + Fo( yjbj(x))dax.
j=1

over all 97 E R". To conclude this section, as an example, we include some detailed

calculations to determine the boundary integral I Fo(uh)dx2 explicitly as a

function of (. Then we find the gradient of the boundary integral with respect to (.

Note that

SN-l 1
Fo(uh)dx2 = f Fo(uh)dx where
i=1 r

P i-

< x2 < N

and,

J Fo(uh)dx2 = ji F(v, h)dy dx2

In the two-dimensional case Y = Y1 U Y2 =

A1,
(2 =
A2,

if Ye

if Y2

, a(y)

{ 1,

a2,

S ea(h-V) + -(1-a)(-V)dy dx2
[, k] U (k, 1] and furthermore,
[0, 1/k] U (I/k, 1] and furthermore,

if y E Y

if y Y2

,and V(y)= ,
V2,

if ye Yi

if E Y2

Hence,

F Fo(uh)dx2
. r

A lea,(uh-Vi) eA -(1-ac)(uh-V1)
kr ka k(1 a- )
(k 1)A2 e"2((-2) + (k 1)A2 e 2)(Uh-v2) dx2.
ka2 k(1 a2)

Now, note that Uh

N2
S(p and uhF
j=i

N
SiNYPiN. Also note that
i=1

(N 1)x2 + on Fi

(N r = IN(x2)= {

and for i = 2,..., N 1 we have

PiN r = OivN(2)=

I 1

(N 1)x2 -(i- 2),

-(N 1)x2 + i,

on Fr_1

on ri

otherwise

1)2 (N 2), on rN-1
otherwise

This implies uh ri

(iNV)iN + (i+l)N z(i+l)N for i

1,...,N -1. Whence,

Uh rI = p iN1iN + (i+1)NN)(i+1)N = {iN(-(N 1)Z2 + i) + i(i+l)N((N 1)x2

= ((i+I)N iN)((N 1)x2 i) + (i+ )N = Gi(x2).

(Q(i+l)N- iN)((N- 1)2 -i) + (i+l)N, if iN # C(iS+)N
' (+i)N, if JiN = '(iS+)N

Thus,

SFo(uh)dx2
Jr

ir kVl k(1- )
(k 1)A2ea2(Gi(X2)2) + (k 1)A2 -(1-2)(Gi(X2)-V2) dx2
ka2 k(l a2)

otherwise

and finally

(i 1))

So, if iN ~(i+t)N then,

F (uh)dx2 = l i ea1([(( -+t)N -CN)((N-1)x2-i)+ i+tN]-vl- )
1 A e-(1-a )([((i+ )N IN)((N-1)x2-i)+2 ( i+)oN]-V)
i J 1h ka2

k (1 o1) )
+ (k ka1)A ea2([((i-+l,),N n )((N-1)2-i)+(E+l-)N] V 2)

(k 1) A2 e_(I- )([(12 (i+ )-INg )((N-_I) 2-i)+ (i+I)N]-V2) dx2,
k(1 a2)
and, if iN = (i+t)N then,

SFo(uh)dX2 ( ) A1 --(1-at)((i+1)N-Vi)}d2
rr ka, k(1 l)
S (kr )A2 e2 + 2) ( 1)A2 e-(1-a2)((j+)-Nv2) }d2
r ka2 k(1 a2)
Recall Pi = {x rl i < X2 < }, so if iN # -(i+I)N then for i = 1,...,N- 1

we have,

f A1 ead+t-w ) eatiW- v)
SFo(u)dZ2= 2 AT --- -- c--
r ka(N 1) (i+1)- iN
A1 e-(1-at)((i+1)N-vi) e-(1-a ()(iN -vi)
k(l ai)2(N 1) (+il)N iN
(k 1)A2 e2((i+l)N-VV2) ea2(iN-V2)
ka2(N 1) (i+i)N (iN
(k 1)A2 e-(1-02)(C( +)-V2) e-(1-a2)(jN-V2)
k(1 a2)2(N 1) (i+1)N iN

and, if iNv = (i+)N then for i = 1,..., N 1 we have,

fI Ui2 17 A eA((i+(1)-V) A -(-a)(i+) i)
F, kl(N 1) k(1 a1)(N 1)
(k A 1))A2) ( i1A2, e- 1- a)(g e)- V2)
ka(N 1) k(1 2)(N 1)

N2
Recall that Uh = (jp so that Eo(uh) = E0(1,... ,N2). But uhl
j=1
iNV)iN + (i+i)NVb(i+1)N for i = 1,..., N 1 implies that the boundary integral
is a function of N,(2N,*... (N-1)N,(N2 only, i.e. we can write j Fo(uh) =

g9(N, 2N, >(N-1)N,N2). So we see that the gradient of the boundary integral
with respect to ( is

VC Fo(Uh) = (0,...,0, 0,...,0, 0,......,0, ).

Now note that,

g ar f
8g a

(9iN -9- iN L _
89 8 iN{ Jr.

and -g {
(9&2 ( N9 &

Thus if &N # (2N then,

89 { Fo((

A1 ea 1(2N-
ka (N- 1)

Fo(&N, 2N)dX2)},

;N
i)

Fo({(i-1)N, iN)dX2 + I Fo(iN, (i+I)N)dx2} for i = 2,..., N 1,

Fo(%(N-1)N, AN2)d2 }.

, 2N)d2 =
- eal(EN-V) al(2N &N)eal(EN-Vi)
(b2N N)2

A1 g-(1--a)(g2N-V) -(1- a)(N-Vt) + (1 a )(2N N)e-(1-at)(N-V1)
k(l a)2(N 1) (2N N)2
(k 1)A2 e"2(2N-V2) e2(C-V2) 2(N N)e2(1-V2)
ka2(N 1) (J2N (N)2
(k 1)A2 e-(1-2)(E2N-V2) e-(1-a2)(N--V2) + (1 -_ )(2N N)e1- 2)(CN-V2)
k(1 aa2(N 1) (2N 2N)2
and if (N-1)N # 4N2 then,

N zag N(9 Fo((N-1)N',N2)dx2} =

A1 eaI(t(N-1)N -V) eai(N 2-Vi) + a 1(N2 (N_1))eat(N2-Vt)
k(N 1) (N2 (N -1))2
AI e-(1-c)(e(N- N-V) e-(l-ac)(E2 -V) (1 ac)(CN2 i(N-1)N)e-(1-a1)(12-V1)
k(1 )2(N 1) (NN2 (N_-)N)2

-1
V--

(k 1)A2 e"2((N- 1N-V2) ea62(e2-V) + 02(N2 (N_-)N)ea2(N-V2)
ka2(N 1) (N2 (Ni))2
(k 1)A2 e-(1-"2)((1)-V2) e-(1-02)(jN2-V2) (1 Q2)(N2 \$(N-1)N)- (1- 2)(e2-2i)
k(1 a2)2(N 1) (N2 (N-1)N)2

Now if JN = (2N then we have,

g9 A1 (ea(2n-Vi) -(1-a)(2N-Vi)) (k 1)A2 (ea2(2y-V2) -(1-a)(21-V2))
aN 2k(N 1) 2k(N 1)

and if (N-1)N = N2 then,

ag A1
9 A ,l (eci*(w- v-i) e-(1-a1)(E(jv-1)Nv-v)
8(92 2k(N 1)
(k -( 1)A2 (ea2( (N i)N-V2) -(1-02)((Ni)N_-V2)
2k(N 1)

Now recall that for i = 2,..., N 1 we have,

ag a Fo((i-l)N, iN)d2 + Fo(iN, (i+l)N)dX2}.
a-iN -iN i-_ Ji

So if (i-i)N iN then for i= 2,..., N 1 we have,

a{ f Fo(4(i-1)N,\$ iN)dx} =
A, e i((-I)N-V i) ei (g Vi) + a1(iN (i-1)N)ea(-NVi)
kac(N 1) (S (_i-1)N)2
1 e-(l-a)(( _-)N-Vi) -_ e-(1-ai)(aEN-V) -_ (1 _- aQ)(SiN (i-1)N)e-(l-a2 )(lN-V2)
k(1 I)2( 1) iN (i-1)N )2
(k 1)A2 ea2((-_).-V2) ea2(N-V2) + 2(iN (i-1)N)e'2(eN-V2)
+kac(N 1) (6N (-i1)N)2
(k 1)A2 e-(1-ag)(-)N-V2) e-(1-a)(N (1 -V2) 2) (2iN (i-1)N)e-(1-a2)(CN-V2)
k(1 2)2 (N 1) (N (-1)N)2
and, if 'JN \$(iS+i)N then for i = 2,..., N 1 we have,

{ Fo (iN, (i+1)N)dx2 }
A1 eai((.+i)N -Vi) eai(~t -Vi) a ((i+i)N iN)eaL(EN -Vi)
ka(N- ) ((i+l)N iN)
1 e-(1-a)((+i)N- V) e-(1i -)(-_ Vi) + a1 (i+)N iN)e-(1-ai)(- Vi)
k(1 i)2(N 1) ((+l)N -iN )2

(k 1)A2 ea2(f(c+l,)-v2) ea2(EN -V2) 02((i+l)N \$iN)ea2('N -V2)
ka(N 1) ((i)+)N Si)2
(k 1)A2 e-(1-a2)((+1)N-V2) e-(1-a2)(N-V2) + (1 Q)((i+l)N iN)e-(1-a2)( -V2)
k(1 c2)2(N 1) (\$(i+l)N iN)2

Finally, if (i-1)N = tiN then for i = 2,..., N 1 we have

a Fo((i-1)N, AiN)dx2 1 (e
XNi. 2k(N 1)
(k 1)A2 (ei2(@i-1)N-V2)
2k(N 1)

and if AiN = k(i+1)N then for i = 2,..., N 1 we have,

o iN FO(iN, (i+l)N)dx2}
(9tiN 'i

- A, (eai(g(i+i)N--V1)
2k(N- 1) (e
+ (k 1) A2 (ea2((i+1)N-2)
2k(N 1)

Once we are able to determine the energy and the gradient of the energy we

can implement the optimization strategy developed in [8]. Coding was done in

FORTRAN and the results of our implementation are discussed in the next section.

3.3 Numerical Results

Here we will both test the accuracy of our asymptotic expansion and observe

the behavior of the current by performing numerical experiments in two dimen-

sions. Note that for the two-dimensional problem the domain Q is a unit square

and the boundary F is the right side of the unit square, that is

r = {(Xl,X2) E : Xi = 1},

(see Figure 2-3). To compute solutions u uo, and u0 we use piecewise linear
finite elements on a regular mesh. To avoid singularities within elements, we chose

a grid which conforms to the medium. To perform the nonlinear minimization

e (1 a2)(C(~frt)N 1/2)).

(when solving for u,), we use a conjugate gradient descent based algorithm devel-

oped by Hager and Zhang, [8]. Note that the homogenized solution uo is simply a

constant value here, which we can find by Newton's Method. The correction, u(1',

we compute using standard finite elements for a linear problem, again conforming

to the media.

We perform these computations for c = 1/5, e = 1/11, e = 1/25 and c = 1/40.

We use the following parameter values for our simulation: JA = 1, Jc = 10,

VA = 0.5, Vc = 1.0, a, = 0.5, a, = 0.85, and Y = YA U Yc where YA = [0, 1/3]

and Yc = [1/3, 1]. Note that for the parameter values used in this implementation,

we have u0 = 0.9758. We have analytically shown that the estimates below hold for

the case of layered media and wish to numerically verify these estimates:

ilUE Uo EU 1) ll(n) < C1E

|| UO H'o (Q) < C2v

The results are summarized in Table 3-1. The estimates above are all bounded

by a term of the form CCe. We estimate this exponent a in the table below. Note

that the numerical results in Table 3-1 are in compliance with the given estimates.
Table 3-1: Table of estimates over Q and convergence rates
E 1/5 1/11 1/25 1/40 ce
I|ur (uo + eu'))lHi(n) .0189 .0090 .0040 .0025 .9699 .9843 .9913
IU, uollHH(n) .0537 .0360 .0238 .0188 .5057 .5061 .5060
IUe uo LL2(n .0063 .0027 .0011 .0007 1.0808 1.0722 1.0676

Table 3-2: Table of estimates over F and estimates of the gradient over F
E 1/5 1/11 1/25 1/40
Iu (uo + i-())lL2(r) 0.0108 0.0050 0.0022 0.0014
Iu Uol\L2(r) 0.0128 0.0057 0.0025 0.0015
IIVU V(uo + ,)) L2(r) 0.1027 0.0710 0.0475 0.0377
IIVu, Vuo|L2(r) 0.1235 0.0817 0.0536 0.0422

In Figure 3-2 and Figure 3-3 we plot the "correct" and asymptotic approxi-

mation of the potential on Q when e = 1/5. We see that the macroscopic behavior

is captured by the expansion. Figure 3-4 and Figure 3-5 show the same for

e = 1/11. In Figures 3-1(a)-3-1(d) we can view the limiting behavior of u, on F as

E approaches 0. To examine the influence of the corrector term more closely, in Fig-

ures 3-6-3-9 we graph both the "correct" solution and the asymptotic expansion

over P with material regions indicated. Note that the asymptotic approximation is

not exact and in fact is slightly skewed. This is probably due to the linearization of

the corrector term. In Figure 3-10 we graph the L"-norm of Vu, on the boundary

for various values of e. We see that according to our simulations of the layered me-

dia case, the current remains bounded as the perimeter becomes arbitrarily large,

suggesting that the linear relation between current and perimeter observed in [12]

may not hold for all geometries. Our results, however, do not directly contradict

the observations made in [12], where the computations were done for a fixed num-

ber of anodes with a varying geometry. Furthermore, since the estimates here are

merely in H1(Q), pointwise estimates for the gradient (current) on the boundary do

not follow.

0.99

- 0.98
C-

S0.97

. 0.96
-5

0 0.2 0.4 0.6 0.8 1
x2

1

0.99

0.98

0.97

0.96

0.95

0.94

0.93
0

..... uO

--- u
s

0.2 0.4 0.6 0.8 1

o 0.97
CL
01
S0.96

0.95

0.94

0.93 -
0

1

0.99

0.98

o 0.97
0.

S0.96

0.95

0.94

0.93
0

---.... UO

SU

0.2 0.4 0.6 0.8 1

U
0.2 0.4 0.6 0 u.8 1

0.2 0.4 0.6 0.8 1

Figure 3-1: Limiting behaviour of u, on r

e = 1/11, (c) e = 1/25, (d) e = 1/40

as c approaches zero for: (a) e = 1/5, (b)

-

1

0.99

0.98

0.97

0.96

0.95

0.94-

0.93

0.8 .. .
0.6 ... ... "
0.4 0.6
0.4
0.2 0.2
x2 0 0
x2 xl

Figure 3-2: u,, = 1/5

1.02

1.-

0.96 ...

0.94

0.92

1 1

S0 0 x1
X2 -_

Figure 3-3: Uo + Eu ', = 1/5

0.9856

0.955,,. ,
0.975 .

0 .4 .. .." :.:. .B
0 W,8

097

S039,
0 97.
2 oe

Figure 3-4:0, u = 1/11

0.99

0.95 .......

Figure 3-5: Uo + cu), e = 1/11

1.02
--U
1.01 .. u
UO+EUr /
U+ E u(I1)
1 Region A /
Region C I
0.99 /

50uo- 0.97578
-.-.-.- - -.---.-- -. ..-..-- - - -- - -

0.97
t \t t \', \
/ /
0.96

0.95

0.94

0.93
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
X2

Figure 3-6: The potential on the boundary F, e = 1/5

U
0.995

0.99
Region A

0.985 Region C

0.98

0.975

0.97

0.965

0.96

0.955 I

0.95
0 0.1 0.2 0.3 0.4 0.5
x2

0.6 0.7 0.8 0.9 1

Figure 3-7: The potential on the boundary F, e = 1/11

S 098 I I I\ 1 '

0.97

0.965

0.96
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2

Figure 3-8: The potential on the boundary F, E = 1/25

--us
U

SRegion A
Region C

** n J (, 1 1'

0.1 0.2

0.4 0.5 0.6 0.7 0.8 0.9
x2

Figure 3-9: The potential on the boundary F, e

0.984

0.982

0.98

H 0.978

, 0.976

> 0.974

0.9721

III.

0 97d

0 968

0.966 -
0

I j -M 1.I.-I m ---. m -- I r, i i i r i 1 1 r l n l l -- -- r -- I I I --I. .

1/40

0.8

0.78

0.76

0.74- e=1/5

e=1/11 0

"o 0-
0.72 E=1/40

S0.7
o

0.68

0.66

0.64

0.62

0.6 I
0 0.05 0.1 0.15 0.2 0.25

Figure 3-10: L" norm of Vu, on P as E approaches 0.

CHAPTER 4
A CORRECTOR BASED ON ROBIN BOUNDARY DATA

4.1 A Robin Boundary Condition

Note that from Figures 3-6 through 3-9 we see that the corrector developed

in the Chapter 3 is slightly shifted away from the original. While the shape of this

approximation is good, the overall location of the approximation is poor. In this

section we attempt to improve the model used to determine the corrector term u').

Thus instead of using (3.1), suppose the correction u') satisfies the Robin boundary

condition problem

Au' = 0 in 0
_() 1 x ()f
-- (f( ,uo) fo(uo)) + v (x/e,uo) on r (4.1)
8n E v
0 on 8Q \r
8n

Before we develop rigorous estimates or provide numerical data to verify that this

approximation will be more accurate let us first intuitively motivate our reason for

proposing (4.1). The original model utilized in Chapter 3 to determine u(1) requires

only Neumann boundary data. By adding Dirichlet boundary data to the Neumann

boundary data we hope that the resulting approximation will have good shape as

well as location. Note that it is not a priori obvious that the term u\$'-f(x/e, o)

should be added to the boundary condition.

Assuming u, u0o + cu'1 implies u( ) (u, Uo)/e which motivates the

boundary condition (f(x/e, u) fo(uo))/c. Using f(x/e, u,) f(x/e, uo) motivates

the Neumann boundary conditioned used in Chapter 3. Here using the Taylor

Series expansion of f(y,v) in the variable v about uo, i.e. using

af
f(x/e,u,) f(x/e,uo) + a(x/e,Uo)(U, uo)

with Eu') z (u, u,) then yields the Robin boundary condition (4.1).

Proposition 4.1.1. Let n = 2 and let u,, Uo be minimizers of (2.7),(2.13)

respectively, and let uit) be the solution to (4.1). Then there exists a constant C

independent of e such that

IIUC Uo CM) II(D) C2.
\\uUo-eu4\\ H'i(Q) < CE2.

Proof. Let

ZC = U UO CUM

since u, is continuous, by (2.12), we have that for any v E H1(Q),

/ Vz Vv dx = '. Vv dx Vuo Vv dx e Vuz Vv dx
j f(x,u)yvdc, + f( ,uo)vdca, + af

So,

Vze Vv dx + [f ( ,ue) f (,Uo)]vdox, c (x/c,Uo)U'vd = 0.

Now note that uo and u, are defined pointwise on P. So, by Taylors Theorem, using

Lagrange's form of the remainder term we have that for each fixed E and x E r

there exists (f between uo(x) and u,(x) such that,

af x I a2f (x
f(X, U') f(X, uo) = f(l,o)(U, Uo) + lJ2f (,9 )( uo)2.
E E 9v e 2V2 v2 E

By subtracting and adding eiu4 within the parentheses of the first term on the

right hand side we have,

(X,) x fo(x 8 2(fx o 1 82f xo
f(X,ue) f(,uo)= ( ,Uo)z, + ( ,UO)U1 + (, )(U -UO)2
c 6 (9v E av E E 2 (9V2 E

Thus making the above substitution yields
z f^ ^af 1 / 2
jVz Vv dx j+ -(x/e,uo)zevda= = j ({(x/, )(U )2vdaU .

Now if we pick v = z, this yields
/I f 1x af : d g 1 X2f .x
SVzdx + d7;u U (2 dux
I + v C 2 8v2 E
Since tf > co, using a variant of Poincar6 yields

Co <1() j< IVZ,2dx+ --j ,uo)zd

(a 02f (X, (U 2.xo)2 ,dor
= 2 j Ea C

Now note that

2 y, ) = A(y) [(y)2e(y)(v-V(y)) )2e-(-a(y))(v-V())
< A(y)[a(y)2e"(y)(v"-(y)) + (1 a(y))2e-(1-(y))(v-V(Y))]
af
< (y,v)
8v
where the last inequality follows from the fact that 0 < a(y) < 1 for all y E Y. So

collzE(ll ) < 2 )- I U uO)21z da. (4.2)

Now note that when n = 2

I|iU Uo11 L-(r) < C l||1 -U 1o 11(,) < C2 1'E Uo1H/2 ) (4.3)

where the first inequality follows from the Sobolev Imbedding Theorem [2] and
the second inequality follows from the Trace Theorem. Now from standard elliptic
regularity theory [10] we have

lue Uo11x3/2( ) <_C a (u- UO) L2(r). (4.4)

By the Mean Value Theorem, for each fixed e and x E P there exists rvf between
uo(x) and u,(x) such that,

8f x
f(X, UE) f (, uo) = (E UO) (f,

Now recall for any v we have,

/y(f(y, v) fo(v))dy = 0

so there exists a continuous Y-periodic function g(y, v) such that

(y, v) = f(y, v) fo(v), Vv E R.
aR

f(x/,, ) f(x/E, o) + f(x/,,uo) fo(uo)
(f Xg
(U o) (x ,)+ Ea (x/E, Uo).

This implies

f(x/e, U) fo(Uo)

an (u, Uno) L = | f(x/, U') fo(UO) L2(r)
an
fx 8g
= |(u- no) av + 6 ax (x/Ce,uo) L2(r)
af x g
< \\(ue Uo) ( ,x)\ L2(r) + (x/e, uo) L2(r)
8g
< Me6IIUE- UIL2(r) + 'll (x/6co) IL2(r)

where Me is defined by (3.4). Note that if u' is a weak solution to (3.1) then

IIE U0 L2(r) = IIU-U CU + UI2(r)

< i||U UO 6UIL2(r) + EIU L2(r)
< C(M, + D) + EC

(4.5)

where the last inequality follows from Proposition 3.1.1. So in the two-dimensional
case we have

ilue 01L2(,r) < C (4.6)

for some constant C independent of e. Thus in the two-dimensional case (4.3),

(4.4), (4.5) and (4.6) imply that

IUe uoILs-(r) < Ce (4.7)

for some constant C independent of e. Then clearly

(uE uo)2 < IU, U I 0 o() < (CE)2.

So applying the above estimate to (4.2) yields,

Co 2 ((Ce)2 x ,
C0 o Z, ) < J2 11lP) L1()

< c8, lz6L2 P)

< C0E2 11 !^:)I

where the last inequality follows from the Trace Theorem. Thus, we can write,

I|z|i
where C is independent of c and so the proposition is proved. [

4.2 Numerical Results

We use the same parameter values as before and utilize the same finite
element based approach as outline in Chapter 3 to discretize the model. The
linear problem (4.1) can be solved directly in MATLAB. We provide graphs of
the approximation as a three-dimensional function in Figures 4-2 and 4-4. In
Figures 4-5, 4-7, 4-10 and 4-12 we graph the new approximation on the active
boundary. In Figures 4-6, and 4-8 we graph both the approximation and the

47

"correct" solution on the boundary. As we see from the graphs and Table 4-1 this

approximation is much more accurate than the approximation used in Chapter 3.

Using a corrector based on Robin boundary data yields a substantially more

accurate approximation without becoming numerically cumbersome. The new

corrector is a substantial improvement over the corrector used in Chapter 3.

Table 4-1: Table of estimates
6 1/2 1/3 1/4 1/5 1/6
Ilut (uo + eut llIoo ) .0019 .0013 .0009 .0007 .0004
IIu (no + U7 )1ll2 r) .5. 29e-4 .2908e-4 .1i91e-4 .1212e-4 .0736e-4
IIu, (uo + Eu1 )IIL2 ( .5254e-4 .2427e-4 .1290e-4 .0720e-4 .0417e-4
lu 0 (uo + eu1))\Hi(q) .8406e-4 .4420e-4 .2877e-4 2124e-4 ,2104e-4

u ,e=15

0.99

0.98-

0.97

0.96

0.95-

0.94 .

0.93 ..... .. .. .. .

0.6 0.2 0.8
0.9.
0.2
x2 0 0

Figure 4-1: u, 6 = 1/5

u e (1), =1/5

0.99 :

0.98

0.97 *

0.96

0.95.
: '" "- "

0.94.

0.93 .

0. 6 8" ... . .. 1

0.2 0 1

x2 0 0 0

Figure 4-2: uo + cu('), E = 1/5

49

u ,e.1/11

0.99

0.985

20 .
0. 975

0.965

0.96

0.955

x2 0 0 x

Figure 4-3: ua, c 1/11

uo+e u(1), Coll1

0 .98 .. : ... .

0.985
0.97 -

0.965-

0.96.

0.4 0:.6

0.2 2 0.4

X2 0 0x

Figure 4-4: uo + eu) e = 1/11

1 -

0.99

0.98

0.97

0.96

0.95

0.94

0.93
0

1

0.99

0.98

0.97

0.96

0.95

0.94

0.93
0

0.2 0.4 0.6 0.8 1

Figure 4-5: Graph of u, (above) and u0 + u1) (below) on the boundary F, E = 1/5

I I I I I I 1 I I

I I I I I I I I I

44444
4" *4

e
c
4

4 *

44

4 4

* 4

4 4

4 4

44

S
I I "B-I

0 0.1 02 03 0.4 05 B. 0.7 08
X2

Figure 4-6: The approximation and the original, E = 1/5

os09-

094

0. 1

I I I I I I I I I

0.99

0.985

0.98

0.975

0.97

0.965

0.96

0.955
0

0.99

0.985

0.98

0.975

0.97

0.965

0.96

0.955
0

0.2 0.4 0.6 0.8 1

Figure 4-7: Graph of u, (above) and uo + u'l) (below) on the boundary F, e = 1/11

0.2 0.4 0.6 0.8 1

448b
4 4 4

4 4 4 4

44 44

44
Vo

44 4 4
4 4 4 4

4 4 4 4

V V

0.1 0.2 0.3 0.4 0.5
X2

0.8 0.7 0.8 09

Figure 4-8: The approximation and the original, c = 1/11

0 OP

0.s95

0.975

0.97

e 4

4 4

4g

44
4
4
4

f
4
4
4

4

44
4 4

4 4

a
4
4r

4 4
4

0965s .

44p

44

44
Vg

I Ilo

098

0 972

0966

0904 -
0 01 02 03 04 05 06 07 08 09
X2

Figure 4-9: Graph of u, on the boundary F, e = 1/25

0984

S|Region C

098

0978

0976

09 74

0972

097

0968

0966

0 01 02 03 04 05 06 07 08 09
X2

Figure 4-10: Graph of Uo+ u1) on the boundary r, ( = 1/25

0982

098

0978

0976

10974

0972

097

01 02 03 04 05
X2

06 07 08 09

Figure 4-11: Graph of u, on the boundary F, e = 1/40

0982

H Region A
09 Region C

0978

0976

0974

0972

097

01 02 03 04 05
X2

06 07 08 09

Figure 4-12: Graph of Uo + u1) on the boundary F, c = 1/40

CHAPTER 5
SHIFTING MATERIAL BOUNDARIES

5.1 The Electrostatic Conductivity Model

We wish to utilize asymptotic expansions to approximate the solution of a

linear elliptic boundary value problem over a two-dimensional domain with shifting

material boundaries. This is a problem that pertains to electrostatic conductivity

and has applications to photonic bandgap (PBG) optical materials. We consider

a PDE which models the steady state voltage potential of a conductor with a

small inhomogeneity in which there is a discontinuity in the conductivity across

the boundary of the inhomogeneity. The boundary of the inhomogeneity shifts by

some small amount h (Figure 5-1). The shift results in a new steady state voltage

potential for the conductor. Let uo be the solution to the boundary value problem

V ((oVuo) = 0 on Q
Buo
an
'0 o = g on 0 (5.1)

where 2 C R2, and g E L2(Q) and

ao(x)= { a(x), if x ED
oa2(x), if x G \D

where D is the region of the small inhomogeneity and a~, a2 > 0. Let uh be the

solution to the problem with shifted boundaries, that is uh is a solution to

V (ahVuh) = 0 on Q
tuh
ah = g on 90. (5.2)
Here,
Here,

ah(x) l(x), ifx c Dh
a2(x), ifx cf Q\Dh
where Dh is the region of the small inhomogeneity but with shifted boundary.
Here we assume D C Dh. The steady state voltage potential of the conductor
with shifted inhomogeneity is viewed as a perturbation of the original steady state
voltage potential. We wish to establish an estimate and do formal asymptotics.
X,

r,,- ,D
1X

Figure 5-1: Perturbation due to shifting between two dielectrics al and a2.

5.2 Estimating the H'I() norm of uo uh

We conclude this chapter with an estimate. We use energy methods to
rigorously develop an estimate characterizing the limiting behaviour of uh. First we
must establish a lemma.
Lemma 5.2.1. Let C = 1/ min{il, a2} then for e > C/4 we have

4Ce2(eg a1)2
| |V( Uh) L2() < e ) VU IIL2(Dh\D)*

Proof. For v E H'l(), let Eh(v) be the energy defined by

Eh(v) = f I Vv 2 dx gv dx,
2 C2 Jan

then

Eh(uo) = Vul 2 dx j 9 dx.
L Jn Jac

Thus we have

/sh |V(oUo Uh)|2 dx = h VUo 02 dx 2 j h (VuLh Vuo)dxI + ) jh VUh2 dx
= h | Vuo 2 dx 2 9uo dx+ f hVUh2 d
J- Jan Jn
= 2Eh(O)+ gIh Vuhu2 dx (5.3)

where we used the fact that the variational form of (5.2) implies

4 ah(VUh. Vo)dx = I guo dx.

Now note that the variational form of (5.1) implies that

f gud da = o f VUa 12 dx

and the definition of Ch implies

/ s"hlVuol2 dx =\D U VoU2 dx + fD O- Vuo2 dx + fh\D aO Vuo 2 dx
J JO\Dh JD JDh\D
and similarly the definition of ao implies

/ ao VIuo 2 dx= 2 Vuo 2 dx + Vu 2 dx + \ 2 V 2 dx. (5.4)
JQ Jn\Dh JD JDh\D

S ohlVUol2 dx )O 4 Vuol2 dx

( \h 2Vaol2 dx + fD (Uo2 dx+ f\D aUo2 dx)
(f\Dh IVUo2 d + DVo2 dx + Dh\D 2IVU 2 dx)

2 2\1VUo12 dx- 1 7lVuol2 dx + aC2) Vu0o2 dx.
SOJn\Df z D hDh\D z

Eh (Uo)

Equation (5.4) then implies

Eh(,u) = o- o Vo 2 dx + 2 Vo0 2 dx + -2) VU0a12 dx
Souo2dx+ h\D aia uodx.
I romVUO 2 d + 2) dx.
2 Jn f Dh\D 2

So

2Eh(uo) + h |VUh 2 dx = o7h VUh 2 dx ao Vuo0 2 dx
nlo Jla Jn
+ fD\ (i -- )uo 2 dx.
JDh\D
Now the variational form of (5.1) and (5.2) imply

C/ao VU02dx = ah(Vuh Vuo) dx,

and

/ ah Vuh2dx 0o (VUo VUh)dx
J O JOi

(5.5)

so we have

/ (ah Vuh |2 _o I0 VU2)d = f (2 V- oi)VUo V
J O JDh\D

So by (5.3), (5.5) and (5.6) we have that

j ah V(uo uh)|2 dx 2Eh(uo) + h VUha 2 dx

= (02 Or)Vuo Vuhdx + f (\
J D \D JfDh\D

Thus

/ ohIV(UO h)d2 dx
JO

JDh\D 2

JDh\D

'Uhdx.

(5.6)

01 2) IVU1o2 dx.

a1) Vu Vuah dx h\(2 )- V1)0 Uo2 dx
JD(\D
O1) (VUh Vuo) V80 dx.

So, we have

= / ((2 (1i) V(Uh Un ) Vt

< IDKD4 u.)12dx+j
< V(U Uo)|2dx + D
J4c fDh\D

0o dx

(\D
E (

6(72 1,)21Vuo02 dx

2 I)21Vuo0 2 dx

thus C = 1/ min{ai, T2} implies

I V(uo- u 2 dx < C h V(uo- Uh)2 dx

< -C V(U -uo) dx+ Ce(x2 a 1l)2 VU02 dx
4c D h\D

Hence

(1 -) V(uo U) 1 dx< C'e(2 ,T)2 Vol2 dx
46 f Dh\D

which implies

IV(o h)12 dx < (4 Ce(a2 1)2f u 2 dx
2n (4E -G C)Dh\D

and so
V(U4 Uh) C2(2 a1)2
||V(o -U)||Ln) 4 -L2(Dh\D)
and thus the lemma is proved. E

We now establish an estimate for the H1'() norm of uo uh.
Proposition 5.2.2. Let C = 1/ min{(a,a2} and suppose e > C/4 then there exists
a constant K(T1, (2), independent of h, such that

(| U U 12 + IV(Uo ut)12) dx < K(7i1,a2)h.

Proof. By a variant of Poincar6 we have

So Uh12 dx < Ci( V(uo -U,)2 dx + | Uo-h 2 d)

Ja/ hV(uo Uh) I dx

= 1 IV(u, u)l2 dx (5.7)

where the last equality follows from the fact that u, = uh on 80. Thus it suffices to
show

S|IV(uo- u)|2 dx< 2h.

Now by Lemma 5.2.1 we have that

4CG2(a:2 01)2
IIV(uo UI)IL2() L < 4CC (2- l u)2 V L2(D\D). (5.8)

Note that due to elliptic regularity IVuo| is uniformly bounded on Dh \ D thus

IVUo IL2(Dh\D) < C!3Dh \ D 1/2

and note that there exists some positive real number a, independent of h, such that

IDh \ D < ah for all h as h 0. So it follows that

IIVolIL2(Dh\D)
Define
2(^4\E2 ,( 42 U2 71)2
K(a-,0-2) = (C' + 1)30 4c' 1
4E C
then (5.7), (5.8), and (5.9) imply that

/ (ou u2 + I V(0- h)|12) dx < K(a, 2)h

and thus the proposition is proved. O

5.3 Formal Asymptotics
We conclude this chapter with some formal asymptotics. We attempt to
characterize the new steady state voltage potential by developing an asymptotic
expansion in terms of the shift h using integral equation asymptotics and Green's
function. Note that the shifting boundary also causes a perturbation in the
conductivity of the conductor. The asymptotic expansion must be developed in

such a way as to address this discontinuity. Let Jz(x) be the Dirac delta function
centered at z, and let N(x, z) be the solution to

-V, *.go(x)VNN(x, z) = z(x) in 0
8N(x,z) 1
2 =on 8Q.

Then

uo(z) = uo(x)6r(x) dx

= o() (Vx. -ao(x)VxN(x, z)) dx

= Vauo(x) go(o(x) VN(x,z) dx- 2uo 8 dga
2 fan anx
= Vxuo(x) *.o(x)VxN(x,z) dx

where the last inequality follows from the fact that uo E H'I() and we assume
/ no d(x = 0. Now using integration by parts yields
Jan

uo(z) = (V~ .c o(x)Vuo(x))N(x,z) dx+ j a N(x,z) dgU.

So (5.1) implies

uo(z) = ~~2 auoN(x, z) da
JOfan xn
= 29g(x)N(x,z) dau. (5.10)
Jan
Similarly we have

Uh(Z) = Uh(x)6z(x) dx

= j (x)(Vx .-o(x)VxN(x,z)) dx

= VxUh(x) ao(x)VxN(x,z) dx 2Uh- dar
/n 1 2dnx

where the last inequality follows by integration by parts. Now assuming

/ Uh dox = 0
an
yields

Uh(z) = f (o(x)VxUh(x) V-N(x, z) dx

= j ao(x)Vuh(x) -VxN(x, z) dx + ao(x)VUh(x) VN(x,z) dx.
Jn\Dh J Dh
Then D C Dh and integration by parts in the first integral implies

Uh(z) = oo(x)-N(x,z) do, + o ,(x)Vuh(x) VN(x,z) dx
J(n\Dh) anx D
+ f a2(x)VUh(x) VN(x,z) dx. (5.11)
J D \D
Note that Dh C 2 implies

oo(x) N(x,) d N(x,z) d
.a(n\Dh) an, fan anx
X) a2x) N(x, z) dox
and using integration by parts on the last two integrals appearing in the right hand
side of (5.11) yields

Uh(z) = -(x)l N(x, z) dao,- a2(x) ( 1 N(x,z) dcx,
S i(x) N(x, z) do + 2(x) N(x, z) da.
'D an, 9(Dh\D) an.
Now recall that D C Dh and let 3D = F1 and let r2 and P3 be such that r3 C F1
and aDh = (rl \ r3) U2 (Figure 5-1). Equation (5.10) implies

Uh() = uo(Z) \ 2(X) (-2 ) N xz) d
U(r (\r3)Urh,2 ( ))
+ 1(x) UhN(x, z) dua, + a2 (x)aUhN(x,z) dat.
ri x 13rsU F2 an,

Note we have that

/1a '2 (7Ua on F2
S8n, }n.
and since duh/dn, is continuous across F3 we have

a h = (a0 + on F3.
K ian,,) K Z on an.
Thus

Uh(Z) = uo(z) (-i(X) 2(x))(ah ) N(x,z) dao

+ (Ci() U2(X)) alh N(x, z, ) da,

and so

Uh(z) uO(z) (Ol(X) "2(X)) 2( J }N(zx, z) dc

/ -Uh N(x, z) dux]

Let
(s) = (x1(s), 2(s)), s

be a parametric equation for the curve OD and let

q(s) = (x1(s), i2(s)), sel

be a parametric equation for the perturbed boundary 8Dh where

(i1(S),C2(S)) = (X1(S), 2(S)) + hO(S)v(S)

and where 0(s) is a positive real, function and v is the outward pointing normal
vector field on D. In particular let f(s) = (x1(s),X(s)), s E [a,b] be a parametriza-
tion of F3 and let q(s) = (1l(s), 2(s)), s E [a,b] be a parametrization of F2.

Then

2 (an N(x, z) deo- = (i(s), Z2(s))N((i,22 i), (z, 2))q'(s) ds
JIdKa 2 V7 d a, p Uh<
and

(ah N(x, z) dc- = (2X(S), 2(s))N(( 2) (Zl,Z2)) ds.
NUh -
Now we use a first order Taylor Series expansion of (1, i2) and N((1,2), 2 (1, 2))
about the point (xl, x2) E F3. Recall that the first order Taylor Series expansion of
f(xl, x2) about the point (xl, 2) is given by
8 f (,, Z2 ,xf(xl, x2)
f/(ix, 2) = f((l,2) + (l ( 1 l) + ()2 X2).

To approximate Iq'(s)l we use a first order Taylor series expansion of g(yl,y2)
v1+ y about the point (xl, 2) since I'(s)l = g(1,2). Note that for any
s G [a,b] we have 2(s) x(s) = h4(s)v(s) and thus

Uh(z) = uo(z) + h(-2 Ca1)I

where

I,= [ (s) (V( i(s))N(x l(s), X2(S), )'(S)

+ (s) (VN(x (s),x2(s),z) V(S)) l'(s)

a( )-Uh)n,/i)( v')\1
-+ ) N(xl(s), X2(S),Z) ) ds.
\ V(i) + (X2 2)
Now if we assume OD and aDh are parametrized by arclength then I|'(s)l = 1 and
thus

Uh(z) = uo(z) + h(-2 71)12

where

I2,= [b )(v(U) (s))N(xi(s),2),)

+ Uh () (VN(xi(s), 2(S), Z) V())

+-( ) N(xil(s),x2(s),z)(P'(s) ('/u + Ow/))] ds.

Note that to complete the formal asymptotics we need to show

8uh Quo
as h 0.
On on

We leave this as the topic of future work.

CHAPTER 6
CONCLUSION

We have analyzed a Butler-Volmer type model which describes the potential

distribution in a system of anodic islands in a coplanar cathodic matrix with

a periodic structure. By using a mutli-scale approach we have determined the

limiting problem for the boundary value problem (2.5) as the period approaches

zero. Furthermore, by introducing a linear correction, we have developed an

asymptotic expansion which closely estimates the solution of the original boundary

value problem. Essentially, we have taken a nonlinear heterogeneous problem and

decomposed it, in a sense, into a nonlinear homogeneous problem and a linear

heterogeneous problem.

Hence the homogenization approach to this problem gives insight into the

behaviour of the solution while also providing an efficient computational technique.

The corrector term, although inhomogeneous, solves a linear problem, and was

therefore not difficult to compute in our experiments. However, in higher dimen-

sions or for very small scale problems, one may want to homogenize the corrector

term itself. This could perhaps be done by solving a cell problem or looking at

the tail behaviour, as in Achdou et al. [1] or Allaire and Amar [3]. In this paper

we have used the language and terminology of galvanic corrosion but this analysis

could also carry over to a more general class of elliptic problems with nonlinear

boundary conditions having periodic structure (assuming the appropriate convexity

conditions.) Future work must address the continuity and boundedness issues

of the three-dimensional problem, i.e. the lack of an applicable Orlicz estimate

must be resolved. We wish to do three-dimensional numerics and we also wish to

consider the model for the case A < 0.

With respect to the corrector introduced in Chapter 4 we wish to develop a

convergence estimate for the three-dimensional case. In addition to implementing

thrree-dimensional numerical simulations we wish to use the multiscale analysis

developed in Chapter 2 and Chapter 4 to construct a first order approximation of

solutions to other nonlinear PDE.

With regards to the electrostatic voltage potential model of Chapter 5, we wish

to complete the asymptotic analysis presented there. We also wish to do numerics

simulating electrostatic voltage potential. The end goal is to work up to a three-

dimensional time harmonic Maxwell's equation so that we may model propagation

phenomena and apply this research to PBG structures.

REFERENCES

[1] Y. ACHDOU, O. PIRONNEAU, AND F. VALENTIN, Effective boundary
conditions for laminar flows over periodic rough boundaries, J. Comput.
Phys., 147, No.l, (1998), pp.187-218.

[2] R.A. ADAMS AND J.J. FOURNIER, Sobolev Spaces, 2nd edition, Elsevier,
Oxford, U.K., 2003.

[3] G. ALLAIRE AND M. AMAR, Boundary layer tails in periodic homogenization,
ESAIM Control Optim. Cale. Var., Vol.4, (1999), pp.209-243.

[4] Y.S. BHAT AND S. MOSKOW, Homogenization of a nonlinear elliptic bound-
ary value problem modeling galvanic currents, Multiscale Model. Simul., 5
(2006), no.l, pp.149-169.

[5] K. BRYAN AND M. VOGELIUS, Singular solutions to a nonlinear elliptic
boundary value problem originating from corrosion modeling, Quart. Appl.
Math, Vol.60, No.4, (2002), pp.675-694.

[6] D.J. CEDIO-FENGYA, S. MOSKOW AND M. VOGELIUS, Identification of
conductivity imperfections of small diameter by boundary measurements,
University of Minnesota, IMA Preprint Series, No.1502, (1997).

[7] L.C. EVANS, Partial Differential Equations, American Mathematical Society,
Providence, Rhode Island, 1998.

[8] W.W. HAGER AND H. ZHANG CG DESCENT, A conjugate gradient method
with guaranteed descent, ACM Transactions on Mathematical Software, (2005),
accepted.

[9] L.S. HOU AND J.C. TURNER, Analysis and finite element approximation of
an optimal control problem in electrochemistry with current density controls,
Numer. Math., Vol.71, No.3, (1995), pp.289-315.

[10] D. JERISON AND C.E. KENIG, The Neumann problem on Lipschitz domains,
Bull. Amer. Math. Soc., Vol.4, (1981), pp.203-207.

[11] J.L. LIONS AND E. MAGENES, Non-Homogeneous Boundary Value Problems
and Applications, Vol.1, Springer-Verlag, New York, 1972.

[12] R. MORRIS AND W. SMYRL, Galvanic Interactions on periodically regular
heterogeneous surfaces, AIChE Journal, Vol.34, (1988), pp.723-732.

[13] R. MORRIS AND W. SMYRL, Galvanic Interactions on random heterogeneous
surfaces, J. Electrochem. Soc, No.ll, (1989), pp.3237-3248.

[14] S. MOSKOW AND M. VOGELIUS, First order corrections to the homogenized
eigenvalues of a periodic composite medium: The case of Neumann boundary
conditions, Indiana Journal, accepted.

[15] J.S. NEWMAN, Electrochemical Systems, Prentice-Hall, Englewood Cliffs, New
Jersey, 1973.

[16] M.M. RAO AND Z.D. REN, Theory of Orlicz Spaces, Marcel Dekker, New
York, 1991.

[17] W. RUDIN, Real and Complex Analysis, McGraw-Hill, New York, 1966.

[18] N.S. TRUDINGER, On imbeddings into Orlicz Spaces and some applications, J.
Math. Mech., Vol.17, (1967), pp.473-483.

[19] M. VOGELIUS AND J.M. Xu, A nonlinear elliptic boundary value problem
related to corrosion modeling, Quart. Appl. Math., Vol.56, No.3, (1998),
pp.479-505.

BIOGRAPHICAL SKETCH

Sujeet Bhat was born in Bangalore, India in 1972. He lived in Malaysia, In-

donesia and the Philippines from 1973 to 1990. In 1990 he graduated from the

International School, Manila. Sujeet obtained a bachelors degree in mathematics

from the University of Florida in 1995, and a master's degree in applied mathe-

matics from the University of Texas at Dallas in May, 1998. He began his study

towards a doctoral degree under the supervision of Professor Shari Moskow in 2003

and received his doctorate in mathematics from the University of Florida in May,

2006. He accepted a two year Industrial Postdoctoral Fellowship from the Institute

for Mathematics and its Applications (IMA) at the University of Minnesota in

April, 2006.

Full Text

PAGE 1

PAGE 2

PAGE 3

ToY.L.Bhat,1936-1994.

PAGE 4

ACKNOWLEDGMENTS MydeepgratitudetoY.L.B.,L.L.B.,V.L.B.,andS.L.Moskow andthe UniversityofFloridaMathematicsDept.Iwouldalsoliketo thankallmyfamily andfriendsfortheirhelpandsupport. iv

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS.............................iv LISTOFTABLES.................................vii LISTOFFIGURES................................viii ABSTRACT....................................x CHAPTER 1INTRODUCTION..............................1 2THEELECTROLYTICVOLTAGEPOTENTIALMODEL.......5 2.1Butler{VolmerBoundaryConditions................5 2.2ExistenceandUniqueness......................8 2.3Regularity...............................15 3ACORRECTORBASEDONNEUMANNBOUNDARYDATA.....18 3.1ANeumannBoundaryCondition..................18 3.2FiniteElementMethodImplementation............... 24 3.3NumericalResults...........................31 4ACORRECTORBASEDONROBINBOUNDARYDATA.......42 4.1ARobinBoundaryCondition....................42 4.2NumericalResults...........................46 5SHIFTINGMATERIALBOUNDARIES..................58 5.1TheElectrostaticConductivityModel............... .58 5.2Estimatingthe H 1 (n)normof u 0 )Tj/T1_1 11.955 Tf11.88 0 Td(u h ...............59 5.3FormalAsymptotics..........................63 6CONCLUSION................................69 REFERENCES...................................71 v

PAGE 6

BIOGRAPHICALSKETCH............................73 vi

PAGE 7

LISTOFTABLES Table page 3{1Tableofestimatesovernandconvergencerates........ ....32 3{2Tableofestimatesover)-322(andestimatesofthegradientov er)-292(.....32 4{1Tableofestimates.............................47 vii

PAGE 8

LISTOFFIGURES Figure page 1{1Zincloseselectronstosilver...................... .3 2{1Thebaseisaheterogeneoussurface.................. .6 2{2Perimeterincreaseswhileanodicareafractionstaysco nstant......6 2{3Two-dimensionalanalogue......................... 16 3{1Limitingbehaviourof u on)-322(as approacheszerofor:(a) =1 = 5, (b) =1 = 11,(c) =1 = 25,(d) =1 = 40...............34 3{2 u =1 = 5.................................35 3{3 u 0 + u (1) =1 = 5.............................35 3{4 u =1 = 11................................36 3{5 u 0 + u (1) =1 = 11............................36 3{6Thepotentialontheboundary, =1 = 5...............37 3{7Thepotentialontheboundary, =1 = 11...............38 3{8Thepotentialontheboundary, =1 = 25...............39 3{9Thepotentialontheboundary, =1 = 40...............40 3{10 L 1 normof r u on)-322(as approaches0.................41 4{1 u =1 = 5.................................48 4{2 u 0 + u (1) =1 = 5.............................48 4{3 u =1 = 11................................49 4{4 u 0 + u (1) =1 = 11............................49 4{5Graphof u (above)and u 0 + u (1) (below)ontheboundary, =1 = 550 4{6Theapproximationandtheoriginal, =1 = 5..............51 4{7Graphof u (above)and u 0 + u (1) (below)ontheboundary, =1 = 1152 4{8Theapproximationandtheoriginal, =1 = 11.............53 viii

PAGE 9

4{9Graphof u ontheboundary, =1 = 25................54 4{10Graphof u 0 + u (1) ontheboundary, =1 = 25............55 4{11Graphof u ontheboundary, =1 = 40................56 4{12Graphof u 0 + u (1) ontheboundary, =1 = 40............57 5{1Perturbationduetoshiftingbetweentwodielectrics 1 and 2 .....59 ix

PAGE 10

AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy MULTISCALEANALYSIS OFPARTIALDIFFERENTIALEQUATIONS MODELINGVOLTAGEPOTENTIAL By YermalSujeetBhat May2006 Chair:ShariMoskowMajorDepartment:Mathematics Westudyanonlinearellipticboundaryvalueproblemarisin gfromelectrochemistry.Theboundaryconditionisofanexponentialtype .Weexaminethe questionsofexistenceanduniquenessofsolutionstothisb oundaryvalueproblem. Wethentreattheproblemfromthepointofviewofhomogeniza tiontheory.The boundaryconditionhasaperiodicstructure.Wendalimiti ngoreectiveproblemastheperiodapproacheszero,alongwithacorrectionte rmandconvergence estimates.Thiscorrectiontermsatisesaboundaryvaluep roblemwithNeumann boundarycondidtions.Wedonumericalexperimentstoinves tigatethebehavior ofgalvaniccurrentsneartheboundaryastheperiodapproac heszero.Wethen consideracorrectiontermwhichsatisesaboundaryvaluep roblemwithaRobin boundarycondition.Wedonumericalexperimentstoinvesti gateourapproximationbasedonthiscorrectorterm.Wethenuseasymptotics toanalyzethe behaviourofthesteadystatevoltagepotentialofaconduct orwitharegionofinhomogeneity.Theboundaryoftheinhomogeneityshiftsslig htly,wedoasymptotics utilizingthesmallshift. x

PAGE 11

CHAPTER1 INTRODUCTION Modernperturbationtheoryisprimarilyconcernedwithcon structingapproximationsofsolutionstomathematicalmodelsthathave aparameterwhichis approachingzero.Onesuchclassofmodelsareboundaryvalu eproblemsinwhich thedomainhasaperiodicstructure.Inthiscasetheperiods izeisthesmallscale parameter.Thesetypesofboundaryvalueproblemsoftenari seinthestudyof, forexample,compositematerials,macroscopicparameters ofcrystallinestructures,ruidmechanicsandaerodynamics.Perturbationtheo ryisanexampleof an analytical approximationasopposedtoa numerical approximation.Thereare manytechniquestoformulatetheseanalyticalapproximati ons.Onefundamental techniqueisthroughtheuseofamultiplescalesasymptotic expansion.Thefoundationalideasforthisapproachappearintheearly1800s.I n1812Laplaceused asymptoticseriestoanalyzesomespecialfunctions.In182 3,Poissonconstructed anexpansionofaBesselfunction.In1886Poincareusedasy mptoticexpansionsto studysolutionsofdierentialequations.Theideabehindt heasymptoticapproximationsisthatthesolutioncanexpressedasasumoftermso fdierentorders ofmagnitudewithrespecttothesmallscale.Forexampleif ( x )isthesolution tosomeboundaryvalueproblemwithsmallscaleparameter thenwebeginby assumingthesolution hasanasymptoticexpansionoftheform 0 + (1) + 2 (2) + : Thegeneralprocedureistothensubstitutetheaboveexpans ionbackinto theoriginalboundaryvalueproblemtodetermineassociate dboundaryvalue problemsfor 0 ; (1) ; (2) ;::: .Wetrytondsimplerequationsthatdescribethe 1

PAGE 12

2 behaviourofthesolutiononvariousordersof .Herewewishtousemultiplescales analysistodevelopanasymptoticexpansionofthesolution tosomephenomena relatedtoconductivity.Werstuseperturbationtheoryto approximatethe solutionofanonlinearboundaryvalueproblemwhichmodels galvaniccorrosion. Asasecondapplicationofasymptoticswewishtodescribeth econductivity propertiesofamaterialwithashiftingdielectricboundar y Intheelectrochemistrycommunitythereismuchinterestin thestudyof galvanicinteractionsonheterogeneoussurfaces[12],[13 ].Whentwodierent metalsinelectricalcontact,referredtoasanodeandcatho de,areimmersedinan electrolyticsolution,thedierenceinrestpotentialgen eratesanelectronrow.This electronrowiscalledagalvaniccurrentandmayleadtoadet erioration(corrosion) oftheanode. InFigure1{1astripofsilver(Ag)andastripofzinc(Zn)hav ebeenimmersedinasaltwatersolution.Thezincstripgivesupelect ronstothesilverstrip. Thesilverstripissaidtobe cathodic ,and reduction takesplace(Aggainselectrons.)Simultaneously oxidation takesplaceatthezincstrip,zincloseselectrons, andissaidtobe anodic .Zincdissolvesintothesolution,thezincelectrodeisbei ng corrodedandtheelectronrowisknownasgalvaniccurrent.T hedrivingforce oftheelectrontransportprocessisthedierenceinpotent ialofthetwometals involved.SeeNewman[15]foracompleteintroductiontothe subject. Herewestudytheelectrostaticproblemonasurfacewherean odesare arrangedperiodicallyinacathodicmatrix.Mathematicall ythepotentialis modeledasafunction, ,overaEuclideandomainn.Partoftheboundaryofn iselectrochemicallyactivewhiletherestoftheboundaryi sinert.Itistheactive regionoftheboundarythatismadeupofanodicandcathodicp ortions.The potentialoverbothoftheseregionssatisesanexponentia lboundarycondition ofButlerandVolmer,butwithdierentmaterialparameters oneachportion.

PAGE 13

3 Figure1{1:Zincloseselectronstosilver InMorrisandSmyrl[12]theauthorsstudysuchaproblemnume rically,using niteelements.Additionallyvariousinterestingaspects ofthetwo-dimensional, homogeneousmodelwiththeButler{Volmerconditionhavebe enanalyzedinBryan andVogelius[5],TurnerandHou[9],andVogeliusandXu[19] .Tothebestof ourknowledge,however,studiescomingfromtheappliedmat hematicscommunity havebeenrestrictedtotwodimensions.Themainreasonfort hisisthatonecan boundexponentialsofthetwo-dimensionalweaksolutionon theboundaryby usinganOrliczestimate[18],[19].Suchanestimatewouldr equiremorethan H 1 regularityinhigherdimensions.Inthispaper,weattemptt otreataperiodically heterogeneousproblem,intwoandthreedimensions,fromth epointofviewof homogenizationtheory. Oursecondapplicationofasymptoticsistoaproblempertai ningtoelectrostaticconductivity.WeconsideraPDEwhichmodelsthestea dystatevoltage potentialofametalwithasmallinhomogeneity.Thereisaju mpintheconductivityacrosstheboundaryoftheinhomogeneousregion.The boundaryofthe inhomogeneousregionshiftsbysomesmallamountduetosome typeofphysical stress.Wenowwishtodescribethevoltagepotentialofthec onductorwithshifted boundaryasaperturbationofthevoltagepotentialoftheor iginalconductor.We doasymptoticstoestablishanestimate.

PAGE 14

4 InChapter2weformallypresenttheelectrolyticvoltagepo tentialmodel. Wetackletheissueofexistenceanduniquenessofthesoluti ontothemodeland thendiscusstheissueofregularity.InChapter3weconstru ctanasymptoticapproximationofthesolutiontotheoriginalproblem.Hereth esecondtermofthe approximationsatisesalinearboundaryvalueproblemwit hNeumanndata.We thenestablishsomeconvergenceestimatesanddonumerical implementation.Using aniteelementmethodapproachweimplementandtestourasy mptoticapproximationandconvergenceestimates.InChapter4weproposea napproxiamation inwhichthesecondtermsatisesaboundaryvalueproblemwi thRobinboundary data.Wethendonumericalimplementationandtestingofthi sapproximation. InChapter5weperformasymptoticsonthe\shiftingboundar y"problem.We establishsomeconvergenceestimatesanddosomeformalasy mptotics.InChapter 6wediscussfutureworktobedone.

PAGE 15

CHAPTER2 THEELECTROLYTICVOLTAGEPOTENTIALMODEL 2.1 Butler{Volmer Boundary Conditions Nowweformallypresentthethree-dimensionalmodelforele ctrolyticvoltage potentialonaheterogeneoussurface.Thedomainnisofcyli ndricalshapewith basesometwo-dimensionaldomain.Thebottombaseisassume dtocontaina periodicarrangementofislands(anodes).Wecallthiscoll ectionofislands @ n A and theremainderofthebottomofthebase @ n C (cathodicplane).Theelectrolytic voltagepotential, ,satisesthefollowingnonlinearellipticboundaryvalue problem, =0inn @ @n = J A [ e aa ( V A ) e ac ( V A ) ]on @ n A (2.1) @ @n = J C [ e ca ( V C ) e cc ( V C ) ]on @ n C @ @n =0on @ n nf @ n A [ @ n C g where aa ; ac ; ca ; cc arethetransfercoecientsanditisassumedthatthesums ( aa + ac )and( ca + cc )areequaltoone.Thepositiveconstants J A ;J C are theanodicandcathodicpolarizationparametersand V A ;V C aretheanodicand cathodicrestpotentialsrespectively.Notethat r representsgalvaniccurrent. Theseboundaryconditionsaretheso-calledtheButler{Vol merexponential boundaryconditions. Inthenumericalstudiesof[12],theauthorsobservedthatf orxedratiosof anodictocathodicareasonthebottombase,theresultingcu rrentincreasedapproximatelylinearlywiththelengthoftheperimeterbetwe enthetworegions,and 5

PAGE 16

6 Figure2{1:Thebaseisaheterogeneoussurface theyhypothesizedthatitistheratioofanodicareatoperim eterthatdetermines thesizeoftheresultingcurrent. Figure2{2:Perimeterincreaseswhileanodicareafraction staysconstant. Asaspecialcaseofincreasingperimeterwithapproximatel yxedarea fraction,weconsideraperiodicstructurewithperiodappr oachingzero.Ourgoalis toexpandthesolutionasymptoticallywithrespecttothepe riodsize.Convergence resultsinvolvingtheseapproximationscouldprovideinsi ghtintothebehaviorof thecurrentforsmallperiodsize;andpossiblyleadtotechn iquesforcomputing approximatesolutionsto(2.1). Wemodeltheperiodicstructurebyletting f ( y;v )= ( y )[ e ( y )( v V ( y )) e (1 ( y ))( v V ( y )) ] forany v 2 R and y 2 Y ,theboundaryperiodcell,whichforsimplicitywetake tobetheunitsquare; Y =[0 ; 1] [0 ; 1].Here ;; and V areallpiecewisesmooth Y -periodicfunctions.Wealsoassumethereexistconstants 0 ; 0 ; 0 ;A 0 and V 0 suchthat: 0 < 0 ( y ) 0 ; (2.2) 0 < 0 ( y ) A 0 < 1 ; (2.3)

PAGE 17

7 and j V ( y ) j V 0 : (2.4) See[5]and[19]forananalysisofwhen < 0. Considertheproblem u =0inn @u @n = f ( x=;u )on(2.5) @u @n =0on @ n n : Asistypicalinhomogenizationproblems,oneexpectsthata s 0,thesolutions willconvergeinsomesensetoasolutionofaproblemwithana veragedboundary condition.Dene f 0 ( v )tobeacellaverageof f ( y;v ),thatis, f 0 ( v )= Z Y f ( y;v ) dy: Considerthecandidateforthehomogenizedproblem u 0 =0inn @u 0 @n = f 0 ( u 0 )on(2.6) @u 0 @n =0on @ n n : Remark If,asisthecasein[12], Y = Y 1 S Y 2 andthefunctions ;;V are piecewiseconstant,eachtakingonthevalues i ; i ;V i respectivelyin Y i ,then f 0 ( v )= j Y 1 j 1 e 1 ( v V 1 ) e (1 1 )( v V 1 ) + j Y 2 j 2 e 2 ( v V 2 ) e (1 2 )( v V 2 ) : Thatis,theabovehomogenizedboundaryconditionwoulddep endonthevolume fractionofanodictocathodicregions.

PAGE 18

8 2.2 Existence and Uniqueness Inthissectionweshowthattheenergyminimizationformsof thenonlinear problem(2.5)and(2.6)haveuniquesolutionsin H 1 (n)inanydimension.Some elementsoftheproofaresimilartothosein[9]and[19].For agiven ,denethe followingenergyfunctional, E ( v )= 1 2 Z n jr v j 2 dx + Z F ( x ;v ) d x (2.7) where, F ( y;v )= ( y ) ( y ) e ( y )( v V ( y )) + ( y ) 1 ( y ) e (1 ( y ))( v V ( y )) : Weshowtheexistenceanduniquenessofaminimizerof(2.7). Formally,weshow theexistenceofafunction u 2 H 1 (n)suchthat E ( u )=min u 2 H 1 (n) E ( u ) : (2.8) Notethat E isnotnecessarilyboundedonallof H 1 (n)(unless n =2forwhich wecanuseanOrliczestimate).Howeverthisdoesnotposeapr oblem.Weset E equalto(2.7)whereitiswelldenedandto+ 1 whereitisnot,asin[7],p.444. Inthetwo-dimensionalcaseofthemodel,duetotheboundedn essof E on H 1 (n), directcalculationshows u satisesthevariationalformof(2.5), Z n r u r vdx = Z f ( x=;u ) vd x forany v 2 H 1 (n) : (2.9) Inthethree-dimensionalcase,if u isanenergyminimizer,wewillhavethat Z F ( x=;u ) d x < 1 ; (2.10) andhencebythepositivityofeachtermof F ( x=;u ),wehavethateachtermis separatelyin L 1 (n).Therefore, E ( u + tv ) < 1 ; (2.11)

PAGE 19

9 forany t 2 R andforany v whichissmoothon.Standardargumentsthenshow that u satises, Z n r u r vdx = Z f ( x=;u ) vd x forany v 2 C 1 ( n) : Additionally,ifweknewthat u 2 C 0 ( n)then f ( x=;u )isboundedandhence clearlyin H 1 = 2 ().Sobythedensityof C 1 ( n)functionsin H 1 (n), u inthiscase wouldsatisfy Z n r u r vdx = Z f ( x=;u ) vd x forany v 2 H 1 (n) : (2.12) Consideralsothefunctional E 0 ( v )= 1 2 Z n jr v j 2 dx + Z F 0 ( v ) d x (2.13) where, F 0 ( v )= Z Y F ( y;v ) dy: Hereagaintheenergy E 0 isnotnecessarilyboundedbutasbefore,weset E 0 equal to(2.13)whereitiswelldenedandto+ 1 whereitisnot.Directcalculations showthataminimizer u 0 of(2.13)willsatisfy, Z n r u 0 r vdx = Z f ( u 0 ) vd x forany v 2 H 1 (n) ; (2.14) assuming u 0 iscontinuous(actuallywewillseethat u 0 isaconstant.) Theorem2.2.1 (ExistenceandUniquenessoftheMinimizer) Let E bedened by(2.7),where ,and V satisfy(2.2)-(2.4).Thenthereexistsauniquefunction u 2 H 1 (n) satisfying E ( u )=min u 2 H 1 (n) E ( u ) : Proof. Notethat @ 2 @v 2 F ( y;v )= ( y ) ( y ) e ( y )( v V ( y )) + ( y )(1 ( y )) e (1 ( y ))( v V ( y )) ;

PAGE 20

10 since > 0 ;> 0 ; and1 > 0wehavethat @ 2 @v 2 F> 0 : Clearlythepartial derivativeisboundedbelow.Thatis,thereexistsaconstan t c 0 ,independentof y and v suchthat, @ 2 @v 2 F ( y;v ) c 0 > 0 : Since F issmoothinthesecondvariable,forany v;w 2 H 1 (n)andforany y ,there existssome between v + w and v w suchthat F ( y;v + w )+ F ( y;v w ) 2 F ( y;v )= @ 2 @v 2 F ( y; ) w 2 whichfromthelowerboundyields F ( x ;v + w )+ F ( x ;v w ) 2 F ( x ;v ) c 0 w 2 whence E ( v + w )+ E ( v w ) 2 E ( v ) R n jr w j 2 dx + c 0 R w 2 d x ~ c 0 k w k 2H 1 (n) (2.15) wherethelastinequalityfollowsbyavariantofPoincare. Nowlet f u n g 1n =1 bea minimizingsequence,thatis E ( u n ) inf u 2 H 1 (n) E ( u )as n !1 : Sinceallthetermsof(2.7)arenonnegative,clearly inf u 2 H 1 (n) E ( u ) > 1 : Notethatwithoutlossofgeneralitywecanchoosetheminimi zingsequencesothat alltermshaveniteenergy(sinceinf u 2 H 1 (n) E ( u ) E (0)and E (0)isbounded independentlyof .)Let v = u n + u m 2

PAGE 21

11 and w = u n u m 2 : Then v + w = u n and v w = u m ,so(2.15)implies E ( v + w )+ E ( v w ) 2 E ( v ) ~ c 0 4 k u n u m k 2H 1 (n) whichimplies, E ( u n )+ E ( u m ) 2inf v 2 H 1 (n) E ( v ) ~ c 0 4 k u n u m k 2H 1 (n) : Nowifwelet m;n !1 ,weseethat f u n g n isaCauchysequenceintheHilbert Space H 1 (n).Dene u tobeitslimitin H 1 (n).Thenwehave u n u in H 1 (n) whichbytheTraceTheoremimplies, u n u in L 2 () whichimplies([17],p.68)thereexistsasubsequence f u n k g k ,whichwelabel f u k g k suchthat u k u a.e.in : SinceFissmoothinthesecondvariable,and u k u a.e.inwehavethat F ( x ;u )=lim k !1 F ( x ;u k )a.e. : Nownotethatclearly F ( x ;u k ) > 0forany k .So,byFatou'sLemmawehave, Z F ( x ;u ) d x liminf k !1 Z F ( x ;u k ) d x : Thusfromthisandthefactthat u k u in H 1 (n),wecanconcludethat, E ( u ) liminf k !1 E ( u k )

PAGE 22

12 =lim k !1 E ( u k ) =inf u 2 H 1 (n) E ( u ) : Hence, E ( u )=inf u 2 H 1 (n) E ( u ) : Sowehaveshowntheexistenceofaminimizer. Suppose u and w arebothminimizersoftheenergyfunctional,i.e. E ( u )=inf u 2 H 1 (n) E ( u )= E ( w ) : Nowifwelet v =( u + w ) = 2 and w =( u w ) = 2 thensubstituting v and w into(2.15)yields, E ( u )+ E ( w ) 2 E ( u + w 2 ) ~ c 0 4 k u w k 2H 1 (n) : However,thisimplies, ~ c 0 4 k u w k 2H 1 (n) E ( u )+ E ( w ) 2inf u 2 H 1 (n) E ( u )=0 : Hence u = w in H 1 (n).Thuswehaveshowntheuniquenessoftheminimizer. Notethatthisargumentcanbegeneralizedtoaddressthe n -dimensional problem, i.e. thecaseinwhichwehaven R n ; R n 1 withboundaryperiod cell Y =[0 ; 1] n 1 .Theexistenceanduniquenessofaminimizer u 0 of E 0 follows fromthesameproof.

PAGE 23

13 Corollary2.2.2. Thereexistsaconstant C ,dependingon 0 ;a 0 ;A 0 and V 0 but independentof suchthat, k u k H 1 (n) C where u isaweaksolutionto(2.5). Proof. Considerthefunction v 0.Then E ( v )= E (0)= Z F ( x ; 0) d x M for M independentof (butdependingon 0 ;a 0 ;A 0 and V 0 ).Thensince u isa minimizer, E ( u ) E (0) M: Sincebothtermsin E arepositive, kr u k 2L 2 (n) M: Wealsohavethat Z F ( x ;u ) d x M: Byexaminingtheformof F ( y;v ),weseethatthereexistssomeconstant d dependingon 0 ;a 0 ; and A 0 butindependentof and x suchthat d j u V ( x ) j F ( x ;u ) : Hence, Z j u V ( x ) j d x M=d; whichbytheboundednessof V impliesthat Z j u j d x ~ M

PAGE 24

14 where ~ M isindependentof .OnevariantofthePoincareinequalitysaysthatthere exists ^ C suchthat k u Z u d x k L 2 (n) ^ C kr u k L 2 (n) : Finallythereversetriangleinequalityyields, k u k L 2 (n) ^ C kr u k L 2 (n) + ~ M; whichprovesthecorollary. Weconcludethissectionbyestablishingthefactthattheso lutiontothe homogenizedproblem(2.6)isinafactaconstant.Thisfactf ollowseasilyoncewe haveestablishedthefollowinglemma.Lemma2.2.3. Thereexistsaconstant K suchthat f 0 ( K )=0 Proof. Recallthat f 0 ( v )= Z Y f ( y;v ) dy (2.16) where f ( y;v )= ( y )[ e ( y )( v V ( y )) e (1 ( y ))( v V ( y )) ] forany v 2 R and y 2 Y ,where Y =[0 ; 1] [0 ; 1].Alsorecallthat ;; and V areallpiecewisesmooth Y -periodicfunctionsforwhichthereexistconstants 0 ; 0 ; 0 ;A 0 and V 0 suchthat: 0 < 0 ( y ) 0 ; (2.17) 0 < 0 ( y ) A 0 < 1 ; (2.18) and j V ( y ) j V 0 : (2.19) Nowif v>V 0 then(2.19)and(2.18)imply ( y )( v V ( y )) > 0and (1 ( y ))( v V ( y )) < 0forall y 2 Y

PAGE 25

15 Sowehave e (1 ( y ))( v V ( y )) V 0 then f ( y;v ) > 0forall y 2 Y Similarlyif v< V 0 then f ( y;v ) < 0forall y 2 Y .Nownotethat(2.16) impliesthatif v>V 0 then f 0 ( v ) > 0andif v< V 0 then f 0 ( v ) < 0.Nowsince f 0 ( v )iscontinuousbytheIntermediateValueTheoremthereexis tsaconstant K 2 ( V 0 ;V 0 )suchthat f 0 ( K )=0.Thusthelemmaisestablished. Theorem2.2.4. Let u 0 beaminimizerof(2.6)then u 0 isaconstant. Proof. Suppose K isaconstantsuchthat f 0 ( K )=0.Suchaconstantexistsby Lemma2.2.3.Thenclearly u 0 = K isastrongsolutionof(2.6).Notethatthis argumentandLemma2.2.3holdinanydimension. 2.3 Regularity Weconcludethischapterwithashortdiscussionoftheregul arityofthesolutions u and u 0 forthetwoandthree-dimensionalcase.Forthetwo-dimensi onal caseofthisproblem, i.e. whenthemediumislayeredasinTurnerandHou[9],and VogeliusandXu[19](seeFigure2{3),usingimbeddingsofSo bolevSpaces( W m;p ) intoOrliczSpaces( L )wecanshowthat f ( x 2 =;u )and f 0 ( u 0 )areboundedin L 2 ()independentlyof WerstgiveageneraldenitionofanOrliczSpacebeforepre sentingthe imbeddingresult.See[2]or[16]forathoroughdiscussiono fOrliczSpaces.Letn beaboundeddomainin R n .LetbeaYoungfunction, i.e. areal-valued,convex functionsuchthat( x )=( x ),(0)=0,and %1 as x %1 .Thenthe Orliczspace L (n)isthesetofallmeasurablefunctions f onnsuchthat Z n ( f ) dx< 1 forsome > 0 :

PAGE 26

16 Figure2{3:Two-dimensionalanalogue. Nowdenethenorm k f k L (n) =inf k> 0: Z n ( f k ) dx< 1 then L (n)becomesaBanachspace.Forexamplewhen( x )= j x j p then L = L p Wenowstatetheimbeddingresultestablishedby[18]and[2] Theorem2.3.1 ( Trudinger'sTheorem). Let n beaboundeddomainin R n satisfyingtheconecondition.Let mp = n and p> 1 .Set ( x )= e p= ( p 1) 1 : Thenthereexiststheimbedding W m;p (n) L (n) : Inthetwo-dimensionalcaseof(2.5)wecanconcludefromTru dinger'sTheorem[18],[19]thatthereexistsaconstant C suchthatforany v 2 H 1 (n)andany real wehave, Z e j v j dx 2 e C 2 ( k v k H 1 (n) +1) ( j j +1) : Thenfromstandardellipticregularitytheorythisimplies that u and u 0 arein H 3 = 2 (n),withthenormboundedindependentlyof .Bythetracetheoremwe

PAGE 27

17 thenobtainboundsfor u and u 0 in H 1 ().Sinceisone-dimensionalitfollows that u and u 0 arecontinuousonandboundedpointwise,andtheirtangent ial derivativesareboundedin L 2 ().Forthehomogenizedsolutionwehavemuch moreregularity, u 0 isinfacttheconstantthatsatises f 0 ( u 0 )=0.Fornonzero boundaryconditionsontheinactiveregion, u 0 wouldstillbeasmoothbounded function.Soforthetwo-dimensionalversionofthisproble mwehavethefollowing lemma:Lemma2.3.2. If n R 2 isarectangleand isanedge,then u 2 C ( n) where u isaweaksolutionof(2.5).Furthermore,thereexistsacons tant D ,thevalueof whichdoesnotdependon ,suchthat, k u ( x ) k C ( n) D: Inthethree-dimensionalcaseof(2.5)since u 2 H 1 (n)implies m =1,so mp 6 = n andthusTrudinger'sTheoremdoesnotapply.Infactthereis alarge classofSobolevimbeddingtheoremspertainingtothecase mp = n ,however therearenoimbeddingsof H 1 (n)foracyindricallyshapeddomainn R 3 thatareapplicableto(2.5).Ingeneral,therehasbeennori gorousanalysisof theregularityofthree-dimensionalsolutionsofelliptic boundaryvalueproblems with L 1 Neumannboundarydata.Nowifweassumethat(2.5)modelsele ctrolytic voltagepotentialthenitisphysicallyreasonabletomakes omeassumptionsabout theregularityof u inthethree-dimensionalcase.Inthenextchapterweshowth at ifthesephysicallyjustiableassumptionsaremadewecane stablishconvergence estimatesforourasymptoticapproximationforthethree-d imensionalcase.

PAGE 28

CHAPTER3 ACORRECTORBASEDONNEUMANNBOUNDARYDATA 3.1 A Neumann Boundary Condition Toshow u convergesto u 0 wewilladdacorrectiontermandproveestimates intermsofpowersof .Theconvergenceof u to u 0 when n =2willtheneasily followfromthis.Thesameestimateholdswhen n =3ifweknowthatthe solutionsarecontinuousanduniformlybounded.When n =2wewillseethatthe convergenceisstrongin H 1 (n)andoftheorderof p Let u 0 beaminimizerof(2.13)anddenethecorrection u (1) tosatisfy, u (1) =0inn @u (1) @n = 1 ( f ( x ;u 0 ) f 0 ( u 0 ))+ e on(3.1) @u (1) @n =0on @ n n Z u (1) d x =0 (3.2) where, e = 1 Z ( f 0 ( u 0 ) f ( x=;u 0 )) d x : Hence e ischosensuchthatthesolutionalwaysexistsandthecondit ion(3.2) guaranteesthissolutionisunique.Wenotethatgiven u 0 ,thisisa linear problem. Nowif u and u 0 arein L 1 (),let D =max n k u k L 1 () ; k u 0 k L 1 () o (3.3) andlet, M =sup ( y;w ) 2 Y [ D ;D ] @f @v ( y;w ) : (3.4) 18

PAGE 29

19 Thenextestimateholdsfordimension n =2or3butdependsontheconstant M Wedo not know apriori that D isniteingeneralwhen n =3.However,suchan assumptionseemstobephysicallyreasonableandknowntobe thecasewhenthe mediumislayered.Proposition3.1.1. Let n =2 or 3 andlet u u 0 beminimizersof(2.7),(2.13) respectively,andlet u (1) bethesolutionto(3.1).Assumealsothat u 2 C 0 ( n) Thenthereexistsconstants C and D independentof suchthat k u u 0 u (1) k H 1 (n) C ( M + D ) ; where M isdenedby(3.4).Furthermore,thereexistsconstants C 1 ,and C 2 independentof suchthat, k u (1) k L 2 () C 1 and j e j C 2 : Proof. Let z = u u 0 u (1) ; since u iscontinuous,by(2.12),wehavethatforany v 2 H 1 (n), Z n r z r vdx = Z n r u r vdx Z n r u 0 r vdx Z n r u (1) r vdx = Z f ( x ;u ) vd x + Z f ( x ;u 0 ) vd x + Z e vd x : So, Z n r z r vdx + Z [ f ( x ;u ) f ( x ;u 0 )] vd x Z e vd x =0 : Nownotethat u 0 and u aredenedpointwiseon.So,bytheMeanValue Theorem,foreachxed and x 2 thereexists x between u 0 ( x )and u ( x )such that, f ( x ;u ) f ( x ;u 0 )=( u u 0 ) @f @v ( x ; x ) :

PAGE 30

20 Bysubtractingandadding u (1) wehave, f ( x ;u ) f ( x ;u 0 )= z @f @v ( x ; x )+ u (1) @f @v ( x ; x ) which,ifwepick v = z ,yields, Z n jr z j 2 dx + Z z 2 @f @v ( x ; x ) d x = Z u (1) @f @v ( x ; x ) z d x + e Z z d x : Since @f @v c 0 ,thisimplies ~ c 0 k z k 2H 1 (n) Z n jr z j 2 dx + Z z 2 @f @v ( x ; x ) d x = Z u (1) @f @v ( x ; x ) z d x + e Z z d x : SobyapplyingHoldersInequalityandthentheTraceTheore mwehave, ~ c 0 k z k 2H 1 (n) k @f @v ( x ; x ) k L 1 () k u (1) k L 2 () k z k L 2 () + j e jj j 1 = 2 k z k L 2 () ( k @f @v ( x ; x ) k L 1 () k u (1) k L 2 () + j e jj j 1 = 2 ) k z k H 1 (n) : Thus,wecanwrite, k z k H 1 (n) C ( k @f @v ( x ; x ) k L 1 () k u (1) k L 2 () + j e j ) : (3.5) Nowrecallforany v wehave, Z Y ( f ( y;v ) f 0 ( v )) dy =0 sothereexistsacontinuous Y -periodicfunction r ( y;v )suchthat y r ( y;v )= f ( y;v ) f 0 ( v ) ; 8 v 2 R: (3.6) Sowehave, e = 1 Z ( f 0 ( u 0 ) f ( x ;u 0 )) d x = 1 Z y r ( x ;u 0 ) d x

PAGE 31

21 = Z @ r y r ( x ;u 0 ) ds x wherethelastequalityisarrivedatusingintegrationbypa rtsandthefactthatthe chainruleimplies @r @y ( x=;u 0 )= @r @x ( x=;u 0 ).Notethatthedierentialoperators r y and y arewithrespectto y 2 Y ,thatis,theyaresurfaceoperators.Now since u 0 isboundedpointwiseonandsince r ( y;v )isacontinuouslydierentiable Y -periodicfunctionwehave, e C (3.7) where C isboundedindependentof .Nowweshowthat k u (1) k L 2 () issimilarly bounded.Let w 2 H 1 (n)satisfy, w =0inn @w @n = u (1) on(3.8) @w @n =0on @ n n Z w d x =0 then, Z ( u (1) ) 2 d x = Z u (1) @w @n d x = Z n r u (1) r w dx = Z @ n @u (1) @n w d x wherethelasttwoequalitiesfollowfromintegrationbypar ts.Now,since u (1) satises(3.1),wehave Z @ n @u (1) @n w d x = Z f ( x=;u 0 ) f 0 ( u 0 ) + e w d x = 1 Z y r ( x=;u 0 ) w d x e Z w d x = 1 Z y r ( x=;u 0 ) w d x

PAGE 32

22 wherethesecondequalityfollowsfrom(3.6)andthelastequ alityholdssince Z w d x =0.Nowusingthechainrulewecanwrite y r ( x=;u 0 )= 2 x r ( x=;u 0 ) ; where x isasurfaceLaplacianon.Thus,wehave, Z ( u (1) ) 2 d x = Z x r ( x=;u 0 ) w d x = Z r x r ( x=;u 0 ) r w d x Z @ @ x r @ w ds x ; (3.9) where istheoutwardunitnormalto @ .Notethatwhen n =2,weusethelast integraltorepresentendpointevaluation.So,byHolders Inequality, Z r x r ( x=;u 0 ) r w d x Z @ @ x r @ w ds x kr x r k L 2 () kr w k L 2 () + k @ x r @ k L 2 ( @ ) k w k L 2 ( @ ) : (3.10) ThenbytheTraceTheoremwehave, k w k L 2 ( @ ) C 1 k w k H 1 () C 2 k w k H 3 = 2 (n) : (3.11) Similarly, kr w k L 2 () C 3 k w k H 1 () C 4 k w k H 3 = 2 (n) : (3.12) Then(3.9),(3.10),(3.11)and(3.12)imply, k u (1) k 2L 2 () kr x r k L 2 () + k @ x r @ k L 2 ( @ ) k w k H 3 = 2 (n) : Nowsince w satises(3.8)wehavefromstandardellipticregularityth eory[10], k w k H 3 = 2 (n) C k u (1) k L 2 () where C isindependentof andsowecanwrite, k u (1) k L 2 () C kr x r ( x=;u 0 ) k L 2 () + k @ x r ( x=;u 0 ) @ k L 2 ( @ )

PAGE 33

23 = C kr y r ( x=;u 0 ) k L 2 () + k @ y r ( x=;u 0 ) @ k L 2 ( @ ) ; wherethelastequalityfollowsfromthechainrule.Consequ ently,sincewehave that u 0 iscontinuousonandboundedpointwiseandsince r ( y;v )isacontinuouslydierentiable Y -periodicfunctionwecanconcludethat k u (1) k L 2 () D (3.13) where D isboundedindependentlyof .Then(3.5),(3.7)and(3.13)implythe mainresultoftheproposition: k z k H 1 (n) C ( k @f @v ( x ; x ) k L 1 () k u (1) k L 2 () + j e j ) ~ C ( M + ~ D ) ; where M isdenedby(3.4). NotethatinlightofLemma2.3.2,wecaneasilyestablishthe followingcorollaries: Corollary3.1.2. When n =2 i.e. forthecaseinwhich n R 2 ; R with boundaryperiodcell Y =[0 ; 1] thereexistsaconstant C independentof suchthat, k u u 0 u (1) k H 1 (n) C: (3.14) Corollary3.1.3. When n =2 ,for u theweaksolutionof(2.5),and u 0 theweak solutionof(2.6),thereexistsaconstant C independentof suchthat, k u u 0 k H 1 (n) C p : (3.15) Estimate(3.15)followsfromthefactthat, k u (1) k H 1 (n) C k @u (1) @n k H 1 = 2 () C 1 = 2 ; wherethelastinequalityfollowsbyinterpolatingbetween L 2 ()and H 1 () (see[11],Section11.5)andthenusingduality(asin[14]). Finallynotethat

PAGE 34

24 estimate(3.14)alsoholdsfor n =3ifweknowthat D denedby(3.3)is uniformlybounded. 3.2 Finite Element Method Implementation Wewishtonumericallyobservethebehaviourofthehomogeni zedboundary valueproblemsasawaytodescribethebehaviourofthecurre ntneartheboundary.Weuseaniteelementmethodapproachtothe2-Dproblem .Forthe2-D problemthedomainnisaunitsquareandtheboundaryisthel eftsideofthe unitsquare,thatis = f ( x 1 ;x 2 ) 2 n: x 1 =1 g (seeFigure2{3).Inthiscaseweimposeagridofpoints(call ednodes)onthe unitsquareandtriangulatethedomain,thenintroduceani tesetofpiecewise continuousbasisfunctions.Hereweusestandardniteelem entmethod\tent" functions.Weimposeagridofnodepointsontheunitsquarew hichareevenly spacedbothonthe x and y -axes.Welabelthenodesstartingattheoriginand movingtotheright.Welabelthenodes P i ;i =1 ;:::;m ,where P 1 =(0 ; 0)and P m =(1 ; 1).Furthermore,ifthereare N nodesontheaxis( i.e. iftheaxisis dividedinto N 1pieces)then m = N 2 and P N 1 =(1 1 =N; 0), P N =(1 ; 0), P N +1 =(0 ; 1 =N ), P N +2 =(1 =N; 1 =N )etc.Oncethenodepointshavebeen establishedwetriangulatethedomaininapredeterminedfa shion.Notethatthe nodepoints P 1 ;P 2 ;P N +1 and P N +2 formasquare.Weformonetrianglebyusing asverticesthenodepoints P 1 ;P 2 and P N +1 andanothertrianglebythenode points P 2 ;P N +1 and P N +2 .Proceedinginthisfashionwecanformyetonemore trianglebyusingasverticesthenodepoints P 2 ;P 3 and P N +2 andyetanotherby using P 3 ;P N +2 and P N +3 .Theentiredomaincanbetriangulatedinthisfashion usingthenodepointsasvertices.Oncethedomainhasbeentr iangulatedwe introducebasisfunctions.Notethatintheoriginalproble m(2.5)weattemptto

PAGE 35

25 minimizetheenergyfunctional E ( v )= 1 2 Z n jr v j 2 dx + Z F ( x ;v ) d x overthespace H 1 (n).However,numericallyweminimizetheenergyfunctiona l overthespace V h where, V h = f v 2 C (n): v isalinearfunctionwhenrestrictedtoeachtriangleinn g : Nowweintroduceasetofbasisfunctions b i ( x 1 ;x 2 ) ;i =1 ;:::;m .Weusethe standardniteelementmethod\tent"function,thatisfore ach i dene b i 2 V h by, b i ( P j )= 8><>: 1if i = j 0if i 6 = j for j =1 ;:::;m: Soanyfunction v 2 V h canbewrittenas v ( x )= m X j =1 j b j ( x ),where j = v ( P j ). Furthermore,wecanreformulatetheenergyminimizationpr obleminthefollowing way,ifwedenotetheminimizeroftheenergyfunctionalas u h then E ( u h )=min v 2 V h E ( v )=min 1 ;:::; m E ( m X j =1 j b j ( x )) : Soif 1 ;:::; m minimizestheenergyfunctionalthenwecanwrite u h = m X j =1 j b j ( x ). Thus,bytriangulatingthedomainandintroducingbasisfun ctionsof V h weare abletodiscretizetheproblem.Nowwewishtosolvetheenerg yminimization problem.Notethatifwelet v ( x )= m X j =1 j b j ( x )thentheenergyfunctionalhasthe form, E ( )= 1 2 T A + Z F ( x ; m X j =1 j b j ( x )) d x : where A =( a ij ) ; 1 i;j m suchthat a ij = R n r b i r b j d ~ x and isthe m 1 vector = h 1 ;:::; m i T .Hencewewishtominimizetheenergy E ( )overall

PAGE 36

26 2 R m .Sinceweplantouseagradientdescentbasedoptimizationm ethodto minimizetheenergywearerequiredtocalculatethegradien toftheenergyaswell. Forthehomogenizedproblem(2.6)wewishtominimizethefun ctional E 0 ( v )= 1 2 Z n jr v j 2 dx + Z F 0 ( v ) d x overthespace H 1 (n).Thusnumericallywewishtominimize E 0 ( )= 1 2 T A + Z F 0 ( m X j =1 j b j ( x )) d x : overall 2 R m .Toconcludethissection,asanexample,weincludesomedet ailed calculationstodeterminetheboundaryintegral Z F 0 ( u h ) dx 2 explicitlyasa functionof .Thenwendthegradientoftheboundaryintegralwithrespe ctto Notethat Z F 0 ( u h ) dx 2 = N 1 X i =1 Z i F 0 ( u h ) dx 2 where i = x 2 2 i 1 N 1 x 2 i N 1 and,Z i F 0 ( u h ) dx 2 = Z i Z Y F ( y;u h ) dydx 2 = Z i Z Y e ( u h V ) + 1 e (1 )( u h V ) dy dx 2 : Inthetwo-dimensionalcase Y = Y 1 [ Y 2 =[0 ; 1 =k ] [ (1 =k; 1]andfurthermore, ( y )= 8><>: 1 ; if y 2 Y 1 2 ; if y 2 Y 2 ; ( y )= 8><>: 1 ; if y 2 Y 1 2 ; if y 2 Y 2 ; and V ( y )= 8><>: V 1 ; if y 2 Y 1 V 2 ; if y 2 Y 2 : Hence, Z i F 0 ( u h ) dx 2 = Z i n 1 k 1 e 1 ( u h V 1 ) + 1 k (1 1 ) e (1 1 )( u h V 1 ) + ( k 1) 2 k 2 e 2 ( u h V 2 ) + ( k 1) 2 k (1 2 ) e (1 2 )( u h V 2 ) o dx 2 :

PAGE 37

27 Now,notethat u h = N 2 X j =1 j j and u h = N X i =1 iN iN .Alsonotethat N = N ( x 2 )= 8><>: ( N 1) x 2 +1 ; on 1 0 ; otherwise ; andfor i =2 ;:::;N 1wehave iN = iN ( x 2 )= 8>>>><>>>>: ( N 1) x 2 ( i 2) ; on i 1 ( N 1) x 2 + i; on i 0 ; otherwise andnally N 2 = N 2 ( x 2 )= 8><>: ( N 1) x 2 ( N 2) ; on N 1 0 ; otherwise : Thisimplies u h i = iN iN + ( i +1) N ( i +1) N for i =1 ;:::;N 1.Whence, u h i = iN iN + ( i +1) N ( i +1) N = iN ( ( N 1) x 2 + i )+ ( i +1) N (( N 1) x 2 ( i 1)) =( ( i +1) N iN )(( N 1) x 2 i )+ ( i +1) N = G i ( x 2 ) : So, u h i = 8><>: ( ( i +1) N iN )(( N 1) x 2 i )+ ( i +1) N ; if iN 6 = ( i +1) N ( i +1) N ; if iN = ( i +1) N : Thus, Z i F 0 ( u h ) dx 2 = Z i n 1 k 1 e 1 ( G i ( x 2 ) V 1 ) + 1 k (1 1 ) e (1 1 )( G i ( x 2 ) V 1 ) + ( k 1) 2 k 2 e 2 ( G i ( x 2 ) V 2 ) + ( k 1) 2 k (1 2 ) e (1 2 )( G i ( x 2 ) V 2 ) o dx 2 :

PAGE 38

28 So,if iN 6 = ( i +1) N then, Z i F 0 ( u h ) dx 2 = Z i n 1 k 1 e 1 ([( ( i +1) N iN )(( N 1) x 2 i )+ ( i +1) N ] V 1 ) + 1 k (1 1 ) e (1 1 )([( ( i +1) N iN )(( N 1) x 2 i )+ ( i +1) N ] V 1 ) o dx 2 + Z i n ( k 1) 2 k 2 e 2 ([( ( i +1) N iN )(( N 1) x 2 i )+ ( i +1) N ] V 2 ) + ( k 1) 2 k (1 2 ) e (1 2 )([( ( i +1) N iN )(( N 1) x 2 i )+ ( i +1) N ] V 2 ) o dx 2 ; and,if iN = ( i +1) N then, Z i F 0 ( u h ) dx 2 = Z i n 1 k 1 e 1 ( ( i +1) N V 1 ) + 1 k (1 1 ) e (1 1 )( ( i +1) N V 1 ) o dx 2 + Z i n ( k 1) 2 k 2 e 2 ( ( i +1) N V 2 ) + ( k 1) 2 k (1 2 ) e (1 2 )( ( i +1) N V 2 ) o dx 2 : Recall i = x 2 2 i 1 N 1 x 2 i N 1 ,soif iN 6 = ( i +1) N thenfor i =1 ;:::;N 1 wehave, Z i F 0 ( u h ) dx 2 = 1 k 2 1 ( N 1) e 1 ( ( i +1) N V 1 ) e 1 ( iN V 1 ) ( i +1) N iN 1 k (1 1 ) 2 ( N 1) e (1 1 )( ( i +1) N V 1 ) e (1 1 )( iN V 1 ) ( i +1) N iN + ( k 1) 2 k 2 2 ( N 1) e 2 ( ( i +1) N V 2 ) e 2 ( iN V 2 ) ( i +1) N iN ( k 1) 2 k (1 2 ) 2 ( N 1) e (1 2 )( ( i +1) N V 2 ) e (1 2 )( iN V 2 ) ( i +1) N iN ; and,if iN = ( i +1) N thenfor i =1 ;:::;N 1wehave, Z i F 0 ( u h ) dx 2 = 1 k 1 ( N 1) e 1 ( ( i +1) N V 1 ) + 1 k (1 1 )( N 1) e (1 1 )( ( i +1) N V 1 ) + ( k 1) 2 k 2 ( N 1) e 2 ( ( i +1) N V 2 ) + ( k 1) 2 k (1 2 )( N 1) e (1 2 )( ( i +1) N V 2 ) :

PAGE 39

29 Recallthat u h = N 2 X j =1 j j sothat E 0 ( u h )= E 0 ( 1 ;:::; N 2 ).But u h i = iN iN + ( i +1) N ( i +1) N for i =1 ;:::;N 1impliesthattheboundaryintegral isafunctionof N ; 2 N ;:::; ( N 1) N ; N 2 only,i.e.wecanwrite Z F 0 ( u h )= g ( N ; 2 N ;:::; ( N 1) N ; N 2 ).Soweseethatthegradientoftheboundaryintegral withrespectto is r Z F 0 ( u h )= h 0 ;:::; 0 ; @g @ N ; 0 ;:::; 0 ; @g @ 2 N ; 0 ;::::::; 0 ; @g @ N 2 i : Nownotethat, @g @ N = @ @ N Z 1 F 0 ( N ; 2 N ) dx 2 ; @g @ iN = @ @ iN Z i 1 F 0 ( ( i 1) N ; iN ) dx 2 + Z i F 0 ( iN ; ( i +1) N ) dx 2 for i =2 ;:::;N 1 ; and @g @ N 2 = @ @ N 2 Z N 1 F 0 ( ( N 1) N ; N 2 ) dx 2 : Thusif N 6 = 2 N then, @g @ N = @ @ N Z 1 F 0 ( N ; 2 N ) dx 2 = 1 k 2 1 ( N 1) e 1 ( 2 N V 1 ) e 1 ( N V 1 ) 1 ( 2 N N ) e 1 ( N V 1 ) ( 2 N N ) 2 1 k (1 1 ) 2 ( N 1) e (1 1 )( 2 N V 1 ) e (1 1 )( N V 1 ) +(1 1 )( 2 N N ) e (1 1 )( N V 1 ) ( 2 N N ) 2 + ( k 1) 2 k 2 2 ( N 1) e 2 ( 2 N V 2 ) e 2 ( N V 2 ) 2 ( 2 N N ) e 2 ( N V 2 ) ( 2 N N ) 2 ( k 1) 2 k (1 2 ) 2 ( N 1) e (1 2 )( 2 N V 2 ) e (1 2 )( N V 2 ) +(1 2 )( 2 N N ) e (1 2 )( N V 2 ) ( 2 N N ) 2 ; andif ( N 1) N 6 = N 2 then, @g @ N 2 = @ @ N 2 Z N 1 F 0 ( ( N 1) N ; N 2 ) dx 2 = 1 k 21 ( N 1) e 1 ( ( N 1) N V 1 ) e 1 ( N 2 V 1 ) + 1 ( N 2 ( N 1) N ) e 1 ( N 2 V 1 ) ( N 2 ( N 1) N ) 2 1 k (1 1 ) 2 ( N 1) e (1 1 )( ( N 1) N V 1 ) e (1 1 )( N 2 V 1 ) (1 1 )( N 2 ( N 1) N ) e (1 1 )( N 2 V 1 ) ( N 2 ( N 1) N ) 2

PAGE 40

30 + ( k 1) 2 k 22 ( N 1) e 2 ( ( N 1) N V 2 ) e 2 ( N 2 V 2 ) + 2 ( N 2 ( N 1) N ) e 2 ( N 2 V 2 ) ( N 2 ( N 1) N ) 2 ( k 1) 2 k (1 2 ) 2 ( N 1) e (1 2 )( ( N 1) N V 2 ) e (1 2 )( N 2 V 2 ) (1 2 )( N 2 ( N 1) N ) e (1 2 )( N 2 V 2 ) ( N 2 ( N 1) N ) 2 : Nowif N = 2 N thenwehave, @g @ N = 1 2 k ( N 1) ( e 1 ( 2 N V 1 ) e (1 1 )( 2 N V 1 ) )+ ( k 1) 2 2 k ( N 1) ( e 2 ( 2 N V 2 ) e (1 2 )( 2 N V 2 ) ) ; andif ( N 1) N = N 2 then, @g @ N 2 = 1 2 k ( N 1) ( e 1 ( ( N 1) N V 1 ) e (1 1 )( ( N 1) N V 1 ) ) + ( k 1) 2 2 k ( N 1) ( e 2 ( ( N 1) N V 2 ) e (1 2 )( ( N 1) N V 2 ) ) : Nowrecallthatfor i =2 ;:::;N 1wehave, @g @ iN = @ @ iN Z i 1 F 0 ( ( i 1) N ; iN ) dx 2 + Z i F 0 ( iN ; ( i +1) N ) dx 2 : Soif ( i 1) N 6 = iN thenfor i =2 ;:::;N 1wehave, @ @ iN Z i 1 F 0 ( ( i 1) N ; iN ) dx 2 = 1 k 21 ( N 1) e 1 ( ( i 1) N V 1 ) e 1 ( iN V 1 ) + 1 ( iN ( i 1) N ) e 1 ( iN V 1 ) ( iN ( i 1) N ) 2 1 k (1 1 ) 2 ( N 1) e (1 1 )( ( i 1) N V 1 ) e (1 1 )( iN V 1 ) (1 1 )( iN ( i 1) N ) e (1 1 )( iN V 1 ) ( iN ( i 1) N ) 2 + ( k 1) 2 k 22 ( N 1) e 2 ( ( i 1) N V 2 ) e 2 ( iN V 2 ) + 2 ( iN ( i 1) N ) e 2 ( iN V 2 ) ( iN ( i 1) N ) 2 ( k 1) 2 k (1 2 ) 2 ( N 1) e (1 2 )( ( i 1) N V 2 ) e (1 2 )( iN V 2 ) (1 2 )( iN ( i 1) N ) e (1 2 )( iN V 2 ) ( iN ( i 1) N ) 2 ; and,if iN 6 = ( i +1) N thenfor i =2 ;:::;N 1wehave, @ @ iN Z i F 0 ( iN ; ( i +1) N ) dx 2 = 1 k 21 ( N 1) e 1 ( ( i +1) N V 1 ) e 1 ( iN V 1 ) 1 ( ( i +1) N iN ) e 1 ( iN V 1 ) ( ( i +1) N iN ) 2 1 k (1 1 ) 2 ( N 1) e (1 1 )( ( i +1) N V 1 ) e (1 1 )( iN V 1 ) +(1 1 )( ( i +1) N iN ) e (1 1 )( iN V 1 ) ( ( i +1) N iN ) 2

PAGE 41

31 + ( k 1) 2 k 22 ( N 1) e 2 ( ( i +1) N V 2 ) e 2 ( iN V 2 ) 2 ( ( i +1) N iN ) e 2 ( iN V 2 ) ( ( i +1) N iN ) 2 ( k 1) 2 k (1 2 ) 2 ( N 1) e (1 2 )( ( i +1) N V 2 ) e (1 2 )( iN V 2 ) +(1 2 )( ( i +1) N iN ) e (1 2 )( iN V 2 ) ( ( i +1) N iN ) 2 : Finally,if ( i 1) N = iN thenfor i =2 ;:::;N 1wehave @ @ iN Z i 1 F 0 ( ( i 1) N ; iN ) dx 2 = 1 2 k ( N 1) ( e 1 ( ( i 1) N V 1 ) e (1 1 )( ( i 1) N V 1 ) ) + ( k 1) 2 2 k ( N 1) ( e 2 ( ( i 1) N V 2 ) e (1 2 )( ( i 1) N V 2 ) ) ; andif iN = ( i +1) N thenfor i =2 ;:::;N 1wehave, @ @ iN Z i F 0 ( iN ; ( i +1) N ) dx 2 = 1 2 k ( N 1) ( e 1 ( ( i +1) N V 1 ) e (1 1 )( ( i +1) N V 1 ) ) + ( k 1) 2 2 k ( N 1) ( e 2 ( ( i +1) N V 2 ) e (1 2 )( ( i +1) N V 2 ) ) : Onceweareabletodeterminetheenergyandthegradientofth eenergywe canimplementtheoptimizationstrategydevelopedin[8].C odingwasdonein FORTRANandtheresultsofourimplementationarediscussed inthenextsection. 3.3 Numerical Results Herewewillbothtesttheaccuracyofourasymptoticexpansi onandobserve thebehaviorofthecurrentbyperformingnumericalexperim entsintwodimensions.Notethatforthetwo-dimensionalproblemthedomain nisaunitsquare andtheboundaryistherightsideoftheunitsquare,thatis = f ( x 1 ;x 2 ) 2 n: x 1 =1 g ; (seeFigure2{3).Tocomputesolutions u u 0 ,and u (1) ,weusepiecewiselinear niteelementsonaregularmesh.Toavoidsingularitieswit hinelements,wechose agridwhichconformstothemedium.Toperformthenonlinear minimization

PAGE 42

32 (whensolvingfor u ),weuseaconjugategradientdescentbasedalgorithmdevel opedbyHagerandZhang,[8].Notethatthehomogenizedsolut ion u 0 issimplya constantvaluehere,whichwecanndbyNewton'sMethod.The correction, u (1) wecomputeusingstandardniteelementsforalinearproble m,againconforming tothemedia. Weperformthesecomputationsfor =1 = 5, =1 = 11, =1 = 25and =1 = 40. Weusethefollowingparametervaluesforoursimulation: J A =1, J C =10, V A =0 : 5, V C =1 : 0, aa =0 : 5, ca =0 : 85,and Y = Y A S Y C where Y A =[0 ; 1 = 3] and Y C =[1 = 3 ; 1].Notethatfortheparametervaluesusedinthisimplement ation, wehave u 0 =0 : 9758.Wehaveanalyticallyshownthattheestimatesbelowho ldfor thecaseoflayeredmediaandwishtonumericallyverifythes eestimates: k u u 0 u (1) k H 1 (n) C 1 k u u 0 k H 1 (n) C 2 p TheresultsaresummarizedinTable3{1.Theestimatesabove areallbounded byatermoftheform C .Weestimatethisexponent inthetablebelow.Note thatthenumericalresultsinTable3{1areincompliancewit hthegivenestimates. Table3{1:Tableofestimatesovernandconvergencerates 1/5 1/11 1/25 1/40 k u ( u 0 + u (1) ) k H 1 (n) .0189 .0090 .0040 .0025 .9699 .9843 .9913 k u u 0 k H 1 (n) .0537 .0360 .0238 .0188 .5057 .5061 .5060 k u u 0 k L 2 (n) .0063 .0027 .0011 .0007 1.0808 1.0722 1.0676 Table3{2:Tableofestimatesoverandestimatesofthegrad ientover 1/5 1/11 1/25 1/40 k u ( u 0 + u (1) ) k L 2 () 0.0108 0.0050 0.0022 0.0014 k u u 0 k L 2 () 0.0128 0.0057 0.0025 0.0015 kr u r ( u 0 + u (1) ) k L 2 () 0.1027 0.0710 0.0475 0.0377 kr u r u 0 k L 2 () 0.1235 0.0817 0.0536 0.0422 InFigure3{2andFigure3{3weplotthe\correct"andasympto ticapproximationofthepotentialonnwhen =1 = 5.Weseethatthemacroscopicbehavior

PAGE 43

33 iscapturedbytheexpansion.Figure3{4andFigure3{5showt hesamefor =1 = 11.InFigures3{1(a)-3{1(d)wecanviewthelimitingbehavi orof u onas approaches0.Toexaminetheinruenceofthecorrectortermm oreclosely,inFigures3{6{3{9wegraphboththe\correct"solutionandtheasy mptoticexpansion overwithmaterialregionsindicated.Notethattheasympt oticapproximationis notexactandinfactisslightlyskewed.Thisisprobablydue tothelinearizationof thecorrectorterm.InFigure3{10wegraphthe L 1 -normof r u ontheboundary forvariousvaluesof .Weseethataccordingtooursimulationsofthelayeredmediacase,thecurrentremainsboundedastheperimeterbecom esarbitrarilylarge, suggestingthatthelinearrelationbetweencurrentandper imeterobservedin[12] maynotholdforallgeometries.Ourresults,however,donot directlycontradict theobservationsmadein[12],wherethecomputationswered oneforaxednumberofanodeswithavaryinggeometry.Furthermore,sinceth eestimateshereare merelyin H 1 (n),pointwiseestimatesforthegradient(current)ontheb oundarydo notfollow.

PAGE 44

34 Figure3{1:Limitingbehaviourof u onas approacheszerofor:(a) =1 = 5,(b) =1 = 11,(c) =1 = 25,(d) =1 = 40

PAGE 45

35 Figure3{2: u =1 = 5 Figure3{3: u 0 + u (1) =1 = 5

PAGE 46

36 Figure3{4: u =1 = 11 Figure3{5: u 0 + u (1) =1 = 11

PAGE 47

37 Figure3{6:Thepotentialontheboundary, =1 = 5

PAGE 48

38 Figure3{7:Thepotentialontheboundary, =1 = 11

PAGE 49

39 Figure3{8:Thepotentialontheboundary, =1 = 25

PAGE 50

40 Figure3{9:Thepotentialontheboundary, =1 = 40

PAGE 51

41 Figure3{10: L 1 normof r u onas approaches0.

PAGE 52

CHAPTER4 ACORRECTORBASEDONROBINBOUNDARYDATA 4.1 A Robin Boundary Condition NotethatfromFigures3{6through3{9weseethatthecorrect ordeveloped intheChapter3isslightlyshiftedawayfromtheoriginal.W hiletheshapeofthis approximationisgood,theoveralllocationoftheapproxim ationispoor.Inthis sectionweattempttoimprovethemodelusedtodeterminethe correctorterm u (1) Thusinsteadofusing(3.1),supposethecorrection u (1) satisestheRobinboundary conditionproblem u (1) =0inn @u (1) @n = 1 ( f ( x ;u 0 ) f 0 ( u 0 ))+ u (1) @f @v ( x=;u 0 )on(4.1) @u (1) @n =0on @ n n Beforewedeveloprigorousestimatesorprovidenumericald atatoverifythatthis approximationwillbemoreaccurateletusrstintuitively motivateourreasonfor proposing(4.1).TheoriginalmodelutilizedinChapter3to determine u (1) requires onlyNeumannboundarydata.ByaddingDirichletboundaryda tatotheNeumann boundarydatawehopethattheresultingapproximationwill havegoodshapeas wellaslocation.Notethatitisnotaprioriobviousthatthe term u (1) @f @v ( x=;u 0 ) shouldbeaddedtotheboundarycondition. Assuming u u 0 + u (1) implies u (1) ( u u 0 ) = whichmotivatesthe boundarycondition( f ( x=;u ) f 0 ( u 0 )) = .Using f ( x=;u ) f ( x=;u 0 )motivates theNeumannboundaryconditionedusedinChapter3.Hereusi ngtheTaylor 42

PAGE 53

43 Seriesexpansionof f ( y;v )inthevariable v about u 0 i.e. using f ( x=;u ) f ( x=;u 0 )+ @f @v ( x=;u 0 )( u u 0 ) with u (1) ( u u 0 )thenyieldstheRobinboundarycondition(4.1). Proposition4.1.1. Let n =2 andlet u u 0 beminimizersof(2.7),(2.13) respectively,andlet u (1) bethesolutionto(4.1).Thenthereexistsaconstant C independentof suchthat k u u 0 u (1) k H 1 (n) C 2 : Proof. Let z = u u 0 u (1) ; since u iscontinuous,by(2.12),wehavethatforany v 2 H 1 (n), Z n r z r vdx = Z n r u r vdx Z n r u 0 r vdx Z n r u (1) r vdx = Z f ( x ;u ) vd x + Z f ( x ;u 0 ) vd x + Z @f @v ( x=;u 0 ) u (1) vd x : So, Z n r z r vdx + Z [ f ( x ;u ) f ( x ;u 0 )] vd x Z @f @v ( x=;u 0 ) u (1) vd x =0 : Nownotethat u 0 and u aredenedpointwiseon.So,byTaylorsTheorem,using Lagrange'sformoftheremaindertermwehavethatforeachx ed and x 2 thereexists x between u 0 ( x )and u ( x )suchthat, f ( x ;u ) f ( x ;u 0 )= @f @v ( x ;u 0 )( u u 0 )+ 1 2 @ 2 f @v 2 ( x ; x )( u u 0 ) 2 : Bysubtractingandadding u (1) withintheparenthesesofthersttermonthe righthandsidewehave, f ( x ;u ) f ( x ;u 0 )= @f @v ( x ;u 0 ) z + @f @v ( x ;u 0 ) u (1) + 1 2 @ 2 f @v 2 ( x ; x )( u u 0 ) 2 :

PAGE 54

44 Thusmakingtheabovesubstitutionyields Z n r z r vdx + Z @f @v ( x=;u 0 ) z vd x = 1 2 Z @ 2 f @v 2 ( x=; x )( u u 0 ) 2 vd x : Nowifwepick v = z ,thisyields Z n jr z j 2 dx + Z @f @v ( x ; x ) z 2 d x = 1 2 Z @ 2 f @v 2 ( x ; x )( u u 0 ) 2 z d x : Since @f @v c 0 ,usingavariantofPoincareyields ~ c 0 k z k 2H 1 (n) Z n jr z j 2 dx + Z @f @v ( x ;u 0 ) z 2 d x = 1 2 Z @ 2 f @v 2 ( x ; x )( u u 0 ) 2 z d x : Nownotethat @ 2 f @v 2 ( y;v ) = ( y )[ ( y ) 2 e ( y )( v V ( y )) (1 ( y )) 2 e (1 ( y ))( v V ( y )) ] ( y )[ ( y ) 2 e ( y )( v V ( y )) +(1 ( y )) 2 e (1 ( y ))( v V ( y )) ] @f @v ( y;v ) wherethelastinequalityfollowsfromthefactthat0 < ( y ) < 1forall y 2 Y .So ~ c 0 k z k 2H 1 (n) 1 2 k @f @v ( x ; x ) k L 1 () Z ( u u 0 ) 2 j z j d x : (4.2) Nownotethatwhen n =2 k u u 0 k L 1 () C 1 k u u 0 k H 1 () C 2 k u u 0 k H 3 = 2 (n) (4.3) wheretherstinequalityfollowsfromtheSobolevImbeddin gTheorem[2]and thesecondinequalityfollowsfromtheTraceTheorem.Nowfr omstandardelliptic regularitytheory[10]wehave k u u 0 k H 3 = 2 (n) C k @ @n ( u u 0 ) k L 2 () : (4.4)

PAGE 55

45 BytheMeanValueTheorem,foreachxed and x 2 thereexists x between u 0 ( x )and u ( x )suchthat, f ( x ;u ) f ( x ;u 0 )=( u u 0 ) @f @v ( x ; x ) : Nowrecallforany v wehave, Z Y ( f ( y;v ) f 0 ( v )) dy =0 sothereexistsacontinuous Y -periodicfunction g ( y;v )suchthat @g @v ( y;v )= f ( y;v ) f 0 ( v ) ; 8 v 2 R: Thisimplies f ( x=;u ) f 0 ( u 0 )= f ( x=;u ) f ( x=;u 0 )+ f ( x=;u 0 ) f 0 ( u 0 ) =( u u 0 ) @f @v ( x ; x )+ @g @x ( x=;u 0 ) : So, k @ @n ( u u 0 ) k L 2 () = k f ( x=;u ) f 0 ( u 0 ) k L 2 () = k ( u u 0 ) @f @v ( x ; x )+ @g @x ( x=;u 0 ) k L 2 () k ( u u 0 ) @f @v ( x ; x ) k L 2 () + k @g @x ( x=;u 0 ) k L 2 () M k u u 0 k L 2 () + k @g @x ( x=;u 0 ) k L 2 () : (4.5) where M isdenedby(3.4).Notethatif~ u isaweaksolutionto(3.1)then k u u 0 k L 2 () = k u u 0 ~ u + ~ u k L 2 () k u u 0 ~ u k L 2 () + k ~ u k L 2 () C ( M + D )+ C 1

PAGE 56

46 wherethelastinequalityfollowsfromProposition3.1.1.S ointhetwo-dimensional casewehave k u u 0 k L 2 () ~ C (4.6) forsomeconstant ~ C independentof .Thusinthetwo-dimensionalcase(4.3), (4.4),(4.5)and(4.6)implythat k u u 0 k L 1 () C (4.7) forsomeconstant C independentof .Thenclearly ( u u 0 ) 2 k u u 0 k 2L 1 () ( C ) 2 : Soapplyingtheaboveestimateto(4.2)yields, ~ c 0 k z k 2H 1 (n) ( C ) 2 2 k @f @v ( x ; x ) k L 1 () k z k L 1 () ^ C 2 k z k L 2 () ~ C 2 k z k H 1 (n) : wherethelastinequalityfollowsfromtheTraceTheorem.Th us,wecanwrite, k z k H 1 (n) C 2 where C isindependentof andsothepropositionisproved. 4.2 Numerical Results Weusethesameparametervaluesasbeforeandutilizethesam enite elementbasedapproachasoutlineinChapter3todiscretize themodel.The linearproblem(4.1)canbesolveddirectlyinMATLAB.Wepro videgraphsof theapproximationasathree-dimensionalfunctioninFigur es4{2and4{4.In Figures4{5,4{7,4{10and4{12wegraphthenewapproximatio nontheactive boundary.InFigures4{6,and4{8wegraphboththeapproxima tionandthe

PAGE 57

47 \correct"solutionontheboundary.Asweseefromthegraphs andTable4{1this approximationismuchmoreaccuratethantheapproximation usedinChapter3. UsingacorrectorbasedonRobinboundarydatayieldsasubst antiallymore accurateapproximationwithoutbecomingnumericallycumb ersome.Thenew correctorisasubstantialimprovementoverthecorrectoru sedinChapter3. Table4{1:Tableofestimates 1/2 1/3 1/4 1/5 1/6 k u ( u 0 + u (1) ) k L 1 (n) .0019 .0013 .0009 .0007 .0004 k u ( u 0 + u (1) ) k L 2 () .5729e-4 .2908e-4 .1791e-4 .1212e-4 .0736e-4 k u ( u 0 + u (1) ) k L 2 (n) .5254e-4 .2427e-4 .1290e-4 .0720e-4 .0417e-4 k u ( u 0 + u (1) ) k H 1 (n) .8406e-4 .4420e-4 .2877e-4 .2124e-4 .2104e-4

PAGE 58

48 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 x 1 u e e =1/5 x 2 x 3 Figure4{1: u =1 = 5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 x 1 u 0 + e u e (1) e =1/5 x 2 x 3 Figure4{2: u 0 + u (1) =1 = 5

PAGE 59

49 Figure4{3: u =1 = 11 Figure4{4: u 0 + u (1) =1 = 11

PAGE 60

50 Figure4{5:Graphof u (above)and u 0 + u (1) (below)ontheboundary, =1 = 5

PAGE 61

51 Figure4{6:Theapproximationandtheoriginal, =1 = 5

PAGE 62

52 Figure4{7:Graphof u (above)and u 0 + u (1) (below)ontheboundary, =1 = 11

PAGE 63

53 Figure4{8:Theapproximationandtheoriginal, =1 = 11

PAGE 64

54 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.964 0.966 0.968 0.97 0.972 0.974 0.976 0.978 0.98 0.982 0.984 x 2Voltage Potential u 0 u e Region A Region C Figure4{9:Graphof u ontheboundary, =1 = 25

PAGE 65

55 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.964 0.966 0.968 0.97 0.972 0.974 0.976 0.978 0.98 0.982 0.984 x 2Voltage Potential u 0 u 0 + e u (1)e Region A Region C Figure4{10:Graphof u 0 + u (1) ontheboundary, =1 = 25

PAGE 66

56 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.968 0.97 0.972 0.974 0.976 0.978 0.98 0.982 x 2Voltage Potential u 0 u e Region A Region C Figure4{11:Graphof u ontheboundary, =1 = 40

PAGE 67

57 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.968 0.97 0.972 0.974 0.976 0.978 0.98 0.982 x 2Voltage Potential u 0 u 0 + e u (1)e Region A Region C Figure4{12:Graphof u 0 + u (1) ontheboundary, =1 = 40

PAGE 68

CHAPTER5 SHIFTINGMATERIALBOUNDARIES 5.1 The Electrostatic Conductivity Model Wewishtoutilizeasymptoticexpansionstoapproximatethe solutionofa linearellipticboundaryvalueproblemoveratwo-dimensio naldomainwithshifting materialboundaries.Thisisaproblemthatpertainstoelec trostaticconductivity andhasapplicationstophotonicbandgap(PBG)opticalmate rials.Weconsider aPDEwhichmodelsthesteadystatevoltagepotentialofacon ductorwitha smallinhomogeneityinwhichthereisadiscontinuityinthe conductivityacross theboundaryoftheinhomogeneity.Theboundaryoftheinhom ogeneityshiftsby somesmallamount h (Figure5{1).Theshiftresultsinanewsteadystatevoltage potentialfortheconductor.Let u 0 bethesolutiontotheboundaryvalueproblem r ( 0 r u 0 )=0onn 0 @u 0 @n = g on @ n(5.1) wheren R 2 ,and g 2 L 2 (n)and 0 ( x )= 8><>: 1 ( x ) ; if x 2 D 2 ( x ) ; if x 2 n n D where D istheregionofthesmallinhomogeneityand 1 ; 2 > 0.Let u h bethe solutiontotheproblemwithshiftedboundaries,thatis u h isasolutionto r ( h r u h )=0onn h @u h @n = g on @ n : (5.2) Here, 58

PAGE 69

59 h ( x )= 8><>: 1 ( x ) ; if x 2 D h 2 ( x ) ; if x 2 n n D h where D h istheregionofthesmallinhomogeneitybutwithshiftedbou ndary. Hereweassume D D h .Thesteadystatevoltagepotentialoftheconductor withshiftedinhomogeneityisviewedasaperturbationofth eoriginalsteadystate voltagepotential.Wewishtoestablishanestimateanddofo rmalasymptotics. s 1 s 2 G 2 G 3 G 1 = x 1 x 26D6D h h Figure5{1:Perturbationduetoshiftingbetweentwodielec trics 1 and 2 5.2 Estimating the H 1 (n) norm of u 0 u h Weconcludethischapterwithanestimate.Weuseenergymeth odsto rigorouslydevelopanestimatecharacterizingthelimitin gbehaviourof u h .Firstwe mustestablishalemma.Lemma5.2.1. Let C =1 = min f 1 ; 2 g thenfor >C= 4 wehave kr ( u 0 u h ) k L 2 (n) r 4 C 2 ( 2 1 ) 2 4 C kr u 0 k L 2 ( D h n D ) : Proof. For v 2 H 1 (n),let E h ( v )betheenergydenedby E h ( v )= 1 2 Z n h jr v j 2 dx Z @ n gvdx;

PAGE 70

60 then E h ( u 0 )= 1 2 Z n h jr u 0 j 2 dx Z @ n gu 0 dx: ThuswehaveZ n h jr ( u 0 u h ) j 2 dx = Z n h jr u 0 j 2 dx 2 Z n h r u h r u 0 dx + Z n h jr u h j 2 dx = Z n h jr u 0 j 2 dx 2 Z @ n gu 0 dx + Z n h jr u h j 2 dx =2 E h ( u 0 )+ Z n h jr u h j 2 dx (5.3) whereweusedthefactthatthevariationalformof(5.2)impl ies Z n h r u h r u 0 dx = Z @ n gu 0 dx: Nownotethatthevariationalformof(5.1)impliesthat Z @ n gu 0 d = Z n 0 jr u 0 j 2 dx andthedenitionof h implies Z n h jr u 0 j 2 dx = Z n n D h 2 jr u 0 j 2 dx + Z D 1 jr u 0 j 2 dx + Z D h n D 1 jr u 0 j 2 dx andsimilarlythedenitionof 0 implies Z n 0 jr u 0 j 2 dx = Z n n D h 2 jr u 0 j 2 dx + Z D 1 jr u 0 j 2 dx + Z D h n D 2 jr u 0 j 2 dx: (5.4) So E h ( u 0 )= 1 2 Z n h jr u 0 j 2 dx Z n 0 jr u 0 j 2 dx = 1 2 Z n n D h 2 jr u 0 j 2 dx + Z D 1 jr u 0 j 2 dx + Z D h n D 1 jr u 0 j 2 dx Z n n D h 2 jr u 0 j 2 dx + Z D 1 jr u 0 j 2 dx + Z D h n D 2 jr u 0 j 2 dx = 1 2 Z n n D h 2 jr u 0 j 2 dx 1 2 Z D 1 jr u 0 j 2 dx + Z D h n D ( 1 2 1 2 ) jr u 0 j 2 dx:

PAGE 71

61 Equation(5.4)thenimplies E h ( u 0 )= 1 2 Z n 0 jr u 0 j 2 dx + 1 2 Z D h n D 2 jr u 0 j 2 dx + Z D h n D ( 1 2 1 2 ) jr u 0 j 2 dx = 1 2 Z n 0 jr u 0 j 2 dx + Z D h n D 1 2 ( 1 2 ) jr u 0 j 2 dx: So 2 E h ( u 0 )+ Z n h jr u h j 2 dx = Z n h jr u h j 2 dx Z n 0 jr u 0 j 2 dx + Z D h n D ( 1 2 ) jr u 0 j 2 dx: (5.5) Nowthevariationalformof(5.1)and(5.2)imply Z n 0 jr u 0 j 2 dx = Z n h r u h r u 0 dx; and Z n h jr u h j 2 dx = Z n 0 r u 0 r u h dx sowehave Z n h jr u h j 2 0 jr u 0 j 2 dx = Z D h n D ( 2 1 ) r u 0 r u h dx: (5.6) Soby(5.3),(5.5)and(5.6)wehavethat Z n h jr ( u 0 u h ) j 2 dx =2 E h ( u 0 )+ Z n h jr u h j 2 dx = Z D h n D ( 2 1 ) r u 0 r u h dx + Z D h n D ( 1 2 ) jr u 0 j 2 dx: Thus Z n h jr ( u 0 u h ) j 2 dx = Z D h n D ( 2 1 ) r u 0 r u h dx Z D h n D ( 2 1 ) jr u 0 j 2 dx = Z D h n D ( 2 1 )( r u h r u 0 ) r u 0 dx:

PAGE 72

62 So,wehave Z n h jr ( u 0 u h ) j 2 dx = Z D h n D ( 2 1 ) r ( u h u 0 ) r u 0 dx Z D h n D 1 4 jr ( u h u 0 ) j 2 dx + Z D h n D ( 2 1 ) 2 jr u 0 j 2 dx Z n 1 4 jr ( u h u 0 ) j 2 dx + Z D h n D ( 2 1 ) 2 jr u 0 j 2 dx thus C =1 = min f 1 ; 2 g implies Z n jr ( u 0 u h ) j 2 dx C Z n h jr ( u 0 u h ) j 2 dx Z n C 4 jr ( u h u 0 ) j 2 dx + Z D h n D C ( 2 1 ) 2 jr u 0 j 2 dx: Hence (1 C 4 ) Z n jr ( u 0 u h ) j 2 dx Z D h n D C ( 2 1 ) 2 jr u 0 j 2 dx whichimplies Z n jr ( u 0 u h ) j 2 dx 4 4 C C ( 2 1 ) 2 Z D h n D jr u 0 j 2 dx andso kr ( u 0 u h ) k L 2 (n) r 4 C 2 ( 2 1 ) 2 4 C kr u 0 k L 2 ( D h n D ) andthusthelemmaisproved. Wenowestablishanestimateforthe H 1 (n)normof u 0 u h Proposition5.2.2. Let C =1 = min f 1 ; 2 g andsuppose >C= 4 thenthereexists aconstant K ( 1 ; 2 ) ,independentof h ,suchthat Z n j u 0 u h j 2 + jr ( u 0 u h ) j 2 dx K ( 1 ; 2 ) h: Proof. ByavariantofPoincarewehave Z n j u 0 u h j 2 dx ~ C 1 Z n jr ( u 0 u h ) j 2 dx + Z @ n j u 0 u h j 2 d

PAGE 73

63 = ~ C 1 Z n jr ( u 0 u h ) j 2 dx (5.7) wherethelastequalityfollowsfromthefactthat u 0 = u h on @ n.Thusitsucesto show Z n jr ( u 0 u h ) j 2 dx ~ C 2 h: NowbyLemma5.2.1wehavethat kr ( u 0 u h ) k L 2 (n) r 4 C 2 ( 2 1 ) 2 4 C kr u 0 k L 2 ( D h n D ) : (5.8) Notethatduetoellipticregularity jr u 0 j isuniformlyboundedon D h n D thus kr u 0 k L 2 ( D h n D ) ~ C 3 j D h n D j 1 = 2 andnotethatthereexistssomepositiverealnumber ,independentof h ,suchthat j D h n D j h forall h as h 0.Soitfollowsthat kr u 0 k L 2 ( D h n D ) ~ C 3 p h: (5.9) Dene K ( 1 ; 2 )=( ~ C 1 +1) ~ C 3 2 4 C 2 ( 2 1 ) 2 4 C then(5.7),(5.8),and(5.9)implythat Z n j u 0 u h j 2 + jr ( u 0 u h ) j 2 dx K ( 1 ; 2 ) h andthusthepropositionisproved. 5.3 Formal Asymptotics Weconcludethischapterwithsomeformalasymptotics.Weat temptto characterizethenewsteadystatevoltagepotentialbydeve lopinganasymptotic expansionintermsoftheshift h usingintegralequationasymptoticsandGreen's function.Notethattheshiftingboundaryalsocausesapert urbationinthe conductivityoftheconductor.Theasymptoticexpansionmu stbedevelopedin

PAGE 74

64 suchawayastoaddressthisdiscontinuity.Let z ( x )betheDiracdeltafunction centeredat z ,andlet N ( x;z )bethesolutionto r x 0 ( x ) r x N ( x;z )= z ( x )inn 2 @N ( x;z ) @n x = 1 j @ n j on @ n : Then u 0 ( z )= Z n u 0 ( x ) z ( x ) dx = Z n u 0 ( x ) r x 0 ( x ) r x N ( x;z ) dx = Z n r x u 0 ( x ) 0 ( x ) r x N ( x;z ) dx Z @ n 2 u 0 @N @n x d x = Z n r x u 0 ( x ) 0 ( x ) r x N ( x;z ) dx wherethelastinequalityfollowsfromthefactthat u 0 2 H 1 (n)andweassume Z @ n u 0 d x =0.Nowusingintegrationbypartsyields u 0 ( z )= Z n r x 0 ( x ) r x u 0 ( x ) N ( x;z ) dx + Z @ n 2 @u 0 @n x N ( x;z ) d x : So(5.1)implies u 0 ( z )= Z @ n 2 @u 0 @n x N ( x;z ) d x = Z @ n 2 g ( x ) N ( x;z ) d x : (5.10) Similarlywehave u h ( z )= Z n u h ( x ) z ( x ) dx = Z n u h ( x ) r x 0 ( x ) r x N ( x;z ) dx = Z n r x u h ( x ) 0 ( x ) r x N ( x;z ) dx Z @ n 2 u h @N @n x d x

PAGE 75

65 wherethelastinequalityfollowsbyintegrationbyparts.N owassuming Z @ n u h d x =0 yields u h ( z )= Z n 0 ( x ) r x u h ( x ) r x N ( x;z ) dx = Z n n D h 0 ( x ) r x u h ( x ) r x N ( x;z ) dx + Z D h 0 ( x ) r x u h ( x ) r x N ( x;z ) dx: Then D D h andintegrationbypartsintherstintegralimplies u h ( z )= Z @ (n n D h ) 0 ( x ) @u h @n x N ( x;z ) d x + Z D 1 ( x ) r x u h ( x ) r x N ( x;z ) dx + Z D h n D 2 ( x ) r x u h ( x ) r x N ( x;z ) dx: (5.11) Notethat D h nimplies Z @ (n n D h ) 0 ( x ) @u h @n x N ( x;z ) d x = Z @ n 2 ( x ) @u h @n x N ( x;z ) d x Z @D h 2 ( x ) @u h @n x + N ( x;z ) d x andusingintegrationbypartsonthelasttwointegralsappe aringintherighthand sideof(5.11)yields u h ( z )= Z @ n 2 ( x ) @u h @n x N ( x;z ) d x Z @D h 2 ( x ) @u h @n x + N ( x;z ) d x + Z @D 1 ( x ) @u h @n x N ( x;z ) d x + Z @ ( D h n D ) 2 ( x ) @u h @n x N ( x;z ) d x : Nowrecallthat D D h andlet @D = 1 andlet 2 and 3 besuchthat 3 1 and @D h =( 1 n 3 ) S 2 (Figure5{1).Equation(5.10)implies u h ( z )= u 0 ( z ) Z ( 1 n 3 ) S 2 2 ( x ) @u h @n x + N ( x;z ) d x + Z 1 1 ( x ) @u h @n x N ( x;z ) d x + Z 3 S 2 2 ( x ) @u h @n x N ( x;z ) d x :

PAGE 76

66 Notewehavethat 1 @u h @n x = 2 @u h @n x + on 2 andsince @u h =@n x iscontinuousacross 3 wehave @u h @n x = @u h @n x + on 3 : Thus u h ( z )= u 0 ( z ) Z 2 1 ( x ) 2 ( x ) @u h @n x N ( x;z ) d x + Z 3 1 ( x ) 2 ( x ) @u h @n x N ( x;z ) d x ; andso u h ( z )= u 0 ( z ) 1 ( x ) 2 ( x ) Z 2 @u h @n x N ( x;z ) d x Z 3 @u h @n x N ( x;z ) d x : Let ~p ( s )= h x 1 ( s ) ;x 2 ( s ) i ;s 2 I beaparametricequationforthecurve @D andlet ~q ( s )= h ~ x 1 ( s ) ; ~ x 2 ( s ) i ;s 2 I beaparametricequationfortheperturbedboundary @D h where h ~ x 1 ( s ) ; ~ x 2 ( s ) i = h x 1 ( s ) ;x 2 ( s ) i + h ( s ) ( s ) andwhere ( s )isapositivereal,functionand istheoutwardpointingnormal vectoreldon D .Inparticularlet ~p ( s )= h x 1 ( s ) ;x 2 ( s ) i s 2 [ a;b ]beaparametrizationof 3 andlet ~q ( s )= h ~ x 1 ( s ) ; ~ x 2 ( s ) i s 2 [ a;b ]beaparametrizationof 2

PAGE 77

67 Then Z 2 @u h @n x N ( x;z ) d x = Z b a @u h @n x (~ x 1 ( s ) ; ~ x 2 ( s )) N (~ x 1 ; ~ x 2 ) ; ( z 1 ;z 2 ) j ~q 0 ( s ) j ds and Z 3 @u h @n x N ( x;z ) d x = Z b a @u h @n x ( x 1 ( s ) ;x 2 ( s )) N ( x 1 ;x 2 ) ; ( z 1 ;z 2 ) j ~p 0 ( s ) j ds: NowweusearstorderTaylorSeriesexpansionof @u h @n x (~ x 1 ; ~ x 2 )and N (~ x 1 ; ~ x 2 ) ; ( z 1 ;z 2 ) aboutthepoint( x 1 ;x 2 ) 2 3 .RecallthattherstorderTaylorSeriesexpansionof f (~ x 1 ; ~ x 2 )aboutthepoint( x 1 ;x 2 )isgivenby f (~ x 1 ; ~ x 2 )= f ( x 1 ;x 2 )+ @f ( x 1 ;x 2 ) @ ~ x 1 (~ x 1 x 1 )+ @f ( x 1 ;x 2 ) @ ~ x 2 (~ x 2 x 2 ) : Toapproximate j ~q 0 ( s ) j weusearstorderTaylorseriesexpansionof g ( y 1 ;y 2 )= p y 2 1 + y 2 2 aboutthepoint( x 1 ;x 2 )since j ~q 0 ( s ) j = g (~ x 01 ; ~ x 02 ).Notethatforany s 2 [ a;b ]wehave~ x ( s ) x ( s )= h ( s ) ( s )andthus u h ( z )= u 0 ( z )+ h ( 2 1 ) I 1 where I 1 = Z b a ( s ) r @u h @n x ( s ) N ( x 1 ( s ) ;x 2 ( s ) ;z ) j ~p 0 ( s ) j + @u h @n x ( s ) r N ( x 1 ( s ) ;x 2 ( s ) ;z ) ( s ) j ~p 0 ( s ) j + @u h @n x N ( x 1 ( s ) ;x 2 ( s ) ;z ) ~p 0 ( s ) ( 0 + 0 ) p ( x 01 ) 2 +( x 02 ) 2 ds: Nowifweassume @D and @D h areparametrizedbyarclengththen j ~p 0 ( s ) j =1and thus u h ( z )= u 0 ( z )+ h ( 2 1 ) I 2

PAGE 78

68 where I 2 = Z b a ( s ) r @u h @n x ( s ) N ( x 1 ( s ) ;x 2 ( s ) ;z ) + @u h @n x ( s ) r N ( x 1 ( s ) ;x 2 ( s ) ;z ) ( s ) + @u h @n x N ( x 1 ( s ) ;x 2 ( s ) ;z )( ~p 0 ( s ) ( 0 + 0 )) ds: Notethattocompletetheformalasymptoticsweneedtoshow @u h @n @u 0 @n as h 0 : Weleavethisasthetopicoffuturework.

PAGE 79

CHAPTER6 CONCLUSION WehaveanalyzedaButler{Volmertypemodelwhichdescribes thepotential distributioninasystemofanodicislandsinacoplanarcath odicmatrixwith aperiodicstructure.Byusingamutli-scaleapproachwehav edeterminedthe limitingproblemfortheboundaryvalueproblem(2.5)asthe periodapproaches zero.Furthermore,byintroducingalinearcorrection,weh avedevelopedan asymptoticexpansionwhichcloselyestimatesthesolution oftheoriginalboundary valueproblem.Essentially,wehavetakena nonlinearheterogeneous problemand decomposedit,inasense,intoa nonlinearhomogeneous problemanda linear heterogeneous problem. Hencethehomogenizationapproachtothisproblemgivesins ightintothe behaviourofthesolutionwhilealsoprovidinganecientco mputationaltechnique. Thecorrectorterm,althoughinhomogeneous,solvesalinea rproblem,andwas thereforenotdiculttocomputeinourexperiments.Howeve r,inhigherdimensionsorforverysmallscaleproblems,onemaywanttohomoge nizethecorrector termitself.Thiscouldperhapsbedonebysolvingacellprob lemorlookingat thetailbehaviour,asinAchdouetal.[1]orAllaireandAmar [3].Inthispaper wehaveusedthelanguageandterminologyofgalvaniccorros ionbutthisanalysis couldalsocarryovertoamoregeneralclassofellipticprob lemswithnonlinear boundaryconditionshavingperiodicstructure(assumingt heappropriateconvexity conditions.)Futureworkmustaddressthecontinuityandbo undednessissues ofthethree-dimensionalproblem, i.e. thelackofanapplicableOrliczestimate mustberesolved.Wewishtodothree-dimensionalnumericsa ndwealsowishto considerthemodelforthecase < 0. 69

PAGE 80

70 WithrespecttothecorrectorintroducedinChapter4wewish todevelopa convergenceestimateforthethree-dimensionalcase.Inad ditiontoimplementing thrree-dimensionalnumericalsimulationswewishtouseth emultiscaleanalysis developedinChapter2andChapter4toconstructarstorder approximationof solutionstoothernonlinearPDE. Withregardstotheelectrostaticvoltagepotentialmodelo fChapter5,wewish tocompletetheasymptoticanalysispresentedthere.Weals owishtodonumerics simulatingelectrostaticvoltagepotential.Theendgoali stoworkuptoathreedimensionaltimeharmonicMaxwell'sequationsothatwemay modelpropagation phenomenaandapplythisresearchtoPBGstructures.

PAGE 81

REFERENCES [1] Y.Achdou,O.Pironneau,andF.Valentin Eectiveboundary conditionsforlaminarrowsoverperiodicroughboundaries ,J.Comput. Phys.,147,No.1,(1998),pp.187-218. [2] R.A.AdamsandJ.J.Fournier SobolevSpaces ,2ndedition,Elsevier, Oxford,U.K.,2003. [3] G.AllaireandM.Amar Boundarylayertailsinperiodichomogenization ESAIMControlOptim.Calc.Var.,Vol.4,(1999),pp.209-243 [4] Y.S.BhatandS.Moskow Homogenizationofanonlinearellipticboundaryvalueproblemmodelinggalvaniccurrents ,MultiscaleModel.Simul.,5 (2006),no.1,pp.149-169. [5] K.BryanandM.Vogelius Singularsolutionstoanonlinearelliptic boundaryvalueproblemoriginatingfromcorrosionmodelin g ,Quart.Appl. Math,Vol.60,No.4,(2002),pp.675-694. [6] D.J.Cedio-Fengya,S.MoskowandM.Vogelius Identicationof conductivityimperfectionsofsmalldiameterbyboundarym easurements UniversityofMinnesota,IMAPreprintSeries,No.1502,(19 97). [7] L.C.Evans PartialDierentialEquations ,AmericanMathematicalSociety, Providence,RhodeIsland,1998. [8] W.W.HagerandH.Zhang CGDESCENT,Aconjugategradientmethod withguaranteeddescent ,ACMTransactionsonMathematicalSoftware,(2005), accepted. [9] L.S.HouandJ.C.Turner Analysisandniteelementapproximationof anoptimalcontrolprobleminelectrochemistrywithcurren tdensitycontrols Numer.Math.,Vol.71,No.3,(1995),pp.289-315. [10] D.JerisonandC.E.Kenig TheNeumannproblemonLipschitzdomains Bull.Amer.Math.Soc.,Vol.4,(1981),pp.203-207. [11] J.L.LionsandE.Magenes Non-HomogeneousBoundaryValueProblems andApplications,Vol.I ,Springer-Verlag,NewYork,1972. [12] R.MorrisandW.Smyrl GalvanicInteractionsonperiodicallyregular heterogeneoussurfaces ,AIChEJournal,Vol.34,(1988),pp.723-732. 71

PAGE 82

72 [13] R.MorrisandW.Smyrl GalvanicInteractionsonrandomheterogeneous surfaces ,J.Electrochem.Soc,No.11,(1989),pp.3237-3248. [14] S.MoskowandM.Vogelius Firstordercorrectionstothehomogenized eigenvaluesofaperiodiccompositemedium:ThecaseofNeum annboundary conditions ,IndianaJournal,accepted. [15] J.S.Newman ElectrochemicalSystems ,Prentice-Hall,EnglewoodClis,New Jersey,1973. [16] M.M.RaoandZ.D.Ren TheoryofOrliczSpaces ,MarcelDekker,New York,1991. [17] W.Rudin RealandComplexAnalysis ,McGraw-Hill,NewYork,1966. [18] N.S.Trudinger OnimbeddingsintoOrliczSpacesandsomeapplications ,J. Math.Mech.,Vol.17,(1967),pp.473-483. [19] M.VogeliusandJ.M.Xu Anonlinearellipticboundaryvalueproblem relatedtocorrosionmodeling ,Quart.Appl.Math.,Vol.56,No.3,(1998), pp.479-505.

PAGE 83

BIOGRAPHICALSKETCH SujeetBhatwasborninBangalore,Indiain1972.HelivedinM alaysia,IndonesiaandthePhilippinesfrom1973to1990.In1990hegrad uatedfromthe InternationalSchool,Manila.Sujeetobtainedabachelors degreeinmathematics fromtheUniversityofFloridain1995,andamaster'sdegree inappliedmathematicsfromtheUniversityofTexasatDallasinMay,1998.He beganhisstudy towardsadoctoraldegreeunderthesupervisionofProfesso rShariMoskowin2003 andreceivedhisdoctorateinmathematicsfromtheUniversi tyofFloridainMay, 2006.HeacceptedatwoyearIndustrialPostdoctoralFellow shipfromtheInstitute forMathematicsanditsApplications(IMA)attheUniversit yofMinnesotain April,2006. 73