Acoustic and electromagnetic scattering analysis using discrete sources VIII projection methods in electromagnetic theory

Acoustic and electromagnetic scattering analysis using discrete sources VIII projection methods in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources VIII projection methods in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources VIII projection methods in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources VIII projection methods in electromagnetic theory

VIII

PROJECTION METHODS IN

ELECTROMAGNETIC THEORY

In this section we will use the fundamental theorem of discrete approxima-tion to construct projecapproxima-tion schemes for a category of variaapproxima-tional problems

in the Hilbert spaces £tan(*^) ^^^ '^tanC'^)- ^^ ^he case of the Hilbert space

^tan('^) ^he problems in question read as

(a) (u - uo,x)2,5 = 0 for all x €£?an(5'),

(b) (u - uo, n X X 4- X^)2^s = 0 ^^^ ^^^ ^ ^^tan(S'),

(c) <(u - Uo) + An X (u* - u5), n X x*)2^^ = 0 for all x

eC^^niS)-Here Uo is an element of £tan('^) ^^^ A > 0 In the next sections we will use these results to construct approximate solutions to the boundary-value problems for the Maxwell equations

1 PROJECTION METHODS FOR THE EXTERIOR MAXWELL PROBLEM

Projection methods for solving the exterior Maxwell problem are given by the following theorem

195

196 CHAPTER VIII PROJECTION METHODS IN ELECTROMAGNETICS

THEOREM 1.1: Let {^i}^i be a complete and linearly independent

system of vector functions in C'^si^{S) and UQ E ^tan('S')- Then the sequence

UN = Yli=i ^r*i> satisfying the projection relations

(a) (UAr-Uo,*j)2^5=0,

(b) {uN - uo, n X <ifj -f X^j)2^s = 0> (8.1) (c) {{uN - uo) + An X {u^j ~ u5), n X ^*j)^^^ = 0,

for j = 1,2, , A/", anrf A > 0, converges to UQ, i.e \\UN — U0II2 5 —• 0 as

AT-.00

Proof: To prove (a) we define the sesquilinear form B : /^tanC*^) ^

^tan('S') —* C by iB(x, y) = (x, y}2,5 and the Hnear functional T : ^tanC'S') ~^

C by J^(x) = (uo, x)2 5 It is readily seen that B and J" satisfy the

require-ments of the discrete approximation theorem and, hence, (a) is proved

We consider (b) Define the sesquilinear form B : C^^ni'^) x >Ctan('S') - •

C b y

B ( x , y ) = ( x , n x y - f Ay>2,5

and the linear functional T : ^tan(*5) —^ C by

:F{X) = (uo, n X X 4- Ax)2 5

(8.2)

(8.3)

In this context the conditions of the fundamental discrete approximation

theorem are fulfilled Indeed, J^ is continuous, B is bounded and satisfies

|iB(x,x)| = ( x , n X x4-Ax)2^5 = (x, n x x)2 5 -f '{x,Ax)2 5

2j A m {xyx*j d 5 + A / |x|2 dS

(8.4)

Il2,5 '

where x = x^Cu -h XyBy Here, we used the assumption that if the smooth

surface S has the parametric representation X5= Xs{u,v) at each point on

the surface we can define an orthogonal tangent-normal system of unit

vec-tors (eu,ev,n), where n represents the outward unit normal vector to S

and e^ and e^ are orthogonal unit vectors in the tangent plane of 5

Con-sequently, if { * i } ^ i is a complete system of vector functions in £tan(*5')»

the sequence u/^ = S i = i ^ r * t satisfying the projection relations (8.1b)

converges to the unique solution u = UQ of the variational problem

(u - Uo, n X X -f Ax)2 5 = 0 for all x eC^g^^iS) (8.5)

1 PROJECTION METHODS FOR THE MAXWELL PROBLEM 1 9 7

To show (c) we first restrict attention to the unique solvability of

the corresponding system of equations We will prove that for u^v =

Yli=i ^^^i the system of equations

(uiv -f An X u^, n X ^])^^^ = 0, j = 1,2, , AT, (8.6)

has only the trivial solution Multiply each equation in (8.6) by a^ and

sum the resulting expressions We obtain

(ujv H- An X uj^, n x M)SJ)2^S = 0 (8.7)

Consequently, VLN = 0, and since the system {ii}^^^ is Unearly

indepen-dent on S the conclusion readily follows Let us prove the convergence

The completeness of the system { ^ i } ^ i yields that for uo G Cl^^{S) there

exists a sequence 14^ = Z]i=i ^^^i such that ||€iv||2,5 ^ 0 as iV - • cx),

where €N = UN — UQ represents the error of approximation on S Then, we

have

{{UN - Uo) + An X {W^ - uj) , n X ^])^ ^ = (6iv 4- An x c ^ , n x ^])^ ^

(8.8) for J = 1,2, , iV Subtracting (8.1c) from (8.8) we obtain

{bvLN + An X 6vL)sj,nx ^j^)^ ^ = (ejv + An x €3^,n x **>2^^ (8.9)

for j = 1,2, ,iV Since 6\XN == UN — \IN can be expressed as a linear

combination of ^i, i = 1,2, , iV, we find that

{6\XN -f An X 8\x*N^ n x Sn]^)2^s — (^^ + An x c]^, n x 6x1*1^)2^3 • (8-10)

Therefore, from

A||5uiv||2,5 = USMN 4-An X <5uj;^,n x <5u];^)2^5

= |(€7V + An X €*^, n X 6u*^)r^s\ < (1 + A) ||€Ar||2,5 PuAr||2,5

we obtain

(8.11)

ll^"N|l2,s<^4^INjv|l2,s; (8.12)

and the conclusion follows from the triangle inequality

Let us make some comments

1 The projection relations (8.1b) represents the closeness relations for

the complete system {4»i}^j, with $i = n x * i + A*i The discrete

1 9 8 CHAPTER VIII PROJECTION METHODS IN ELECTROMAGNETICS

approximation theorem can also be applied with the sesquilinear

form B : Cl^{S) x Cl^{S) ^ C defined by

B(x, y) = (x 4- An X X, n X y)2 ^

and the linear functional T : C^^ní^) ~^ ^ defined by

^ ( x ) = (uo -f An X uo, n X 'x.)lg

In this case the sequence u^v = I3t=i ^^"^ii satisfying the

projec-tion relaprojec-tions

((uiv - U o ) - f A n x {UN - u o ) , n X * J ) 2 5 = 0J = l,2, ,iV,

(8.13) converges to UQ Thus, the sequence ú^,

N

ú^ = UÂ -f An X Uiv = ^(^^ (^i -f An X ^ ^ , (8.14)

satisfying the projection relations

(ú^ - u(),n X *,)2^5 = 0 , j = l,2, ,iV, (8.15) converges to UQ = UQ H- An x UQ

2 According to part (c) of theorem 1.1 we see that the sequence ú^y,

N

ú^ = UAT -f An X u;^ = Y^ấifi -f Aaf'*n x ^l (8.16)

1 = 1

satisfying the projection relations

(ú^ - u;,,n X *;)2^^ = 0, j = 1,2, ,7V, (8.17) converges to UQ = UQ + An x UQ

2 PROJECTION METHODS FOR THE TRANSMISSION

PROBLEM

Let us now consider the analog of theorem 1.1 for the product space

£?an(5)

^2 1 f ^ be a complete and linearly

inde-pendent system of vector functions in ii^anC'S')? â^ ( 2 I ^

'^tanC'^)-2 PROJECTION METHODS FOR THE TRANSMISSION PROBLEM 199

Then the sequence

(::f)=|<(ii)'

satisfying the projection relations

<"(("c:^)'(*})L-'

(8.19)

forj = 1,2, , A^, andX > 0, converges to \ 2 ) » ^•^- I 2^ 2 )

\ ^0 / 11 \ ^iv "" "0 /

= 0,

2,5

—^ 0 as N —^ 00

Proof: The proof of (a) follows immediately from the discrete

approxi-mation theorem with the sesquilinear form B : 'Ctan('S') X'Ctan('S') —^ C given

by

and the linear functionary : £?an('5') —^ C given by

Here, and below we use the notations x = I 2 I ^"^^ y = I 2 ) •

For proving (b) we define the sesquilinear form B : S^l^^iS) x Z\^^{S) ->

C b y

= ( x i , n X y i + Ayi)2 5 + ( x ^ , n x y^ + Ay^)^^

2 0 0 CHAPTER VIII PROJECTION METHODS IN ELECTROMAGNETICS

and the linear functional T : ^^J^S) —* C by

•^W - ( ( u l j ' i n x x ^ A x O X , ,

(8.21)

= ( u j , n X x^ + Ax^)*_^ + (ug, n X x^ + Ax^)*^^

Evidently, T \& & continuous functional on i^^^{S) The sesquilinear form

B is bounded and satisfies

| 5 ( x , x ) | = < x ^ n x x l + Ax'>2g + <x2,nxx2^-Ax2>

' 2 , 5

2j A m (xixi*) AS + A / |x»|^ d 5 +

+ 2jjlm{xlxl')dS + xf\x?\' dS

(8.22)

> A {\Wts-^M\l,s)=^\\{i)

>

2,5

with x'^ = x!^eu + xje^ ,fc = 1,2 Therefore, apphcation of the discrete

approximation theorem finish the proof of (b)

The proof of (c) follows in the same manner as in theorem 1.1

We conclude our analysis with some final remarks

1 We may apply the fundamental theorem of discrete approximation

with the sesquihnear form B : S^t&ni^) x £tan('S') -* C defined by

and the linear functional f : S^t&ni^) ~* ^ defined by

TM-/{ ^o + ^^^Uo\ / n x x i \ \ *

•^^ ^ " \ U g + A n x u g j l ^ n x x ^ j / ^ ^ -Then we see that the sequence

{t)-t^^my (-)

2 PROJECTION METHODS FOR THE TRANSMISSION PROBLEM 2 0 1

satisfying the projection relations

/ / K - u 5 ) + A n x K - u 5 ) \ (nx9l\\ _

\ V K - u g j + A n x f e - u g j j ' U x ^ U A s "

(8.24)

CI)-j = 1,2, ,Ar, converges to ( 2 J • Consequently, the sequence

u JV»

(8.25) satisfying the projection relations

convergesto(^^?2 j = ( ^ ^ | _ ^ ^ ^ ^ ^ § j

2 An analogous result can be derived from part (c) of theorem 1.1, i.e the sequence

{ "^N \ _ / uj^ + An X uV* \

(8.27)

satisfying the projection relations

/ < \ / uj + An X uj* \ convergesto(^^2 j = ( ^ ^ | ^ ^ ^ ^ ^ § , j