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 approximation to construct projection schemes for a category of variational 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 = for all x €£?an(5'), (b) (u - uo, n X X 4- X^)2^s = ^^^ ^^^ ^ ^^tan(S'), (c) satisfying the projection relations (a) (UAr-Uo,*j)2^5=0, (b) {uN - uo, n X (c) {{uN - uo) + An X {u^j ~ u5), n X ^*j)^^^ = 0, (8.1) for j = 1,2, , A/", anrf A > 0, converges to UQ, i.e \\UN — U0II2 —• 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 It is readily seen that B and J" satisfy the requirements of the discrete approximation theorem and, hence, (a) is proved We consider (b) Define the sesquilinear form B : C^^ni'^) x >Ctan('S') - • Cby (8.2) B ( x , y ) = ( x , n x y - f Ay>2,5 and the linear functional T : ^tan(*5) —^ C by :F{X) (8.3) = (uo, n X X 4- Ax)2 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 -f '{x,Ax)2 (8.4) 2j A m {xyx*j d5 + A / |x|2 dS 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 vectors (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 Consequently, 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 = for all x eC^g^^iS) (8.5) PROJECTION METHODS FOR THE MAXWELL PROBLEM 197 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 = (8.7) Consequently, VLN = 0, and since the system {ii}^^^ is Unearly independent 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 ^ 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 {\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 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\ •^^ ^ " \ U g + A n x u g /nxxi\\* j-l^nxx^j/^^- Then we see that the sequence {t)-t^^my (-) PROJECTION METHODS FOR THE TRANSMISSION PROBLEM 201 satisfying the projection relations // K-u5)+AnxK-u5) \ (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 ( u JV» J • Consequently, the sequence (8.25) satisfying the projection relations f \4 \ / uj + An X uj \ convergesto(^^?2 j = ( ^ ^ | _ ^ ^ ^ ^ ^ § j 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 ...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... £tan('S') -* C defined by and the linear functional f : S^t&ni^) ~* ^ defined by TM-/{ ^o + ^^^Uo •^^ ^ " U g + A n x u g /nxxi\* j-l^nxx^j/^ ^- Then we see that the sequence {t)-t^^my (-) PROJECTION. .. sesquilinear form B : S^l^^iS) x Z^^{S) -> Cby « „ = ( x i , n X yi + Ayi)2 + ( x ^ , n x y^ + Ay^)^^ 200 CHAPTER VIII PROJECTION METHODS IN ELECTROMAGNETICS and the linear