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Index

aeoustic equation, 38 airfoil equation, 327 algebraic sum of subspaees, 5 arc,48 Arzela lemma, 4

Babuska-Brezzi eondition, 428 Banach theorem, 2 Banach-Steinhaus theorem, 3 Bessel differential equation, 39 Bessel function - of the first kind of order one, 41 - of the first kind of order zero, 39 - of the seeond kind of order zero, 40 - of the third kind, 41 bilinear form, 9 - bounded, 13 - nondegenerate, 9 boundary operator - D 57 - D i 62 , - H, 69 - S,55

Calderon projeetor, 69 eanonieal hypersingular integral, 144,

145 eapacity, 80 Cauehy data, 45 Cauehy integral, 106 Cauchy singular integral equation, 327 Cauehy singular integral operator, 110 CGME,32 CGMR,31 Chebyshev polynomial - of the first kind, 343 - of the seeond kind, 343 classical PPDO, 231 classieal solution of a boundary value

problem, 37 closed graph, 3 eodimension of subspace, 6

eolloeation - a dual method, 433 eolloeation method, 294 eompact embedding, 142 eompaet operator, 3 eompaet set, 3 eomplementable subspace, 5 eonjugate gradients minimizing the

error,32 eonjugate gradients minimizing the

residual, 31 eontinuum, 49 eonvolution operator - nonperiodie, 342 - periodic, 149 eurve,48 C'(il), 36 C' (ii),36 COO(il), 36 C"'(G),105 c+(r) ,C~(r), 116

Dini's eondition, 410 Dirae funetional, 39 Dirae's delta distribution, 134, 436 direet method, 45 direet sum of subspaees, 5 Dirichlet problem, 36 diserete eosine Fourier transform, 246 diserete Fourier eoefficient, 409 diserete Fourier transform, 240, 409 diserete sine Fourier transform, 246 distribution, 134 - periodie, 136 - regular, 134 - singular, 134 double layer operator, 57 double layer potential, 45 dual operator, 9 dual operator in dual system, 10 dual space, 8 dual system, 10

450 Index

Do,57 Dd ,400 D(R), 133 D'(R), 134 D1 (R), 138 D~(R), 138 41 (laplacian), 35 41 (differenee), 200 tSto, 134

eigenelement, 7 eigenvalue, 7 elliptie operator, 224 equilibrium problem of Robin, 79 Euler eonstant, 40 even operator, 333 exterior boundary value problem, 56 exterior of r, 50 E(x),39 Eo(x), 39

fast solver, 272 finite part, 144 Fourier series, 136 Fourier transform, 199 Fredholm alternative, 7 Fredholm operator, 13 fully diserete eolloeation method, 298 fully diserete Galerkin method, 266 fundamental solution, 39 - for -41, 39 - for -~41 + k 2 ), 41 - for 41 ,43 fundamental trigonometrie polynomial,

239 F, F- 1 , 199

Garding's inequality, 229 Galerkin method, 257, 287 - abstract, 426 - spline, 431 Gauß Divergenee Theorem, 37 generalized airfoil equation, 327 generalized eigenveetor, 8 generalized funetion, 134 generalized method of minimal residual,

25 GMRES,25 Green formula, 37

Hölder eontinuous funetion, 105 Hankel function

of order one, 41 - of order zero, 40

harmonie equation, 38 harmonie funetion, 47 Hartogs theorem, 194 Heaviside funetion, 135 Helmholtz equation, 35 Hilbert seale, 161 Hilbert singular integral operator, 152 hypersingular integral, 144, 145 hypersingular operator, 69, 156 Hd1)(r), Hd2)(r), 40 H~I)(r), 41 H).., 141, 316 H)..'I', 168 H;, H;, 244 Hi(R2 ),253 1l~, 194 1lw , llw , 193 1lwl,W2' 1l-w1 ,w2' 194 1l)..lh 195

Wl,W2'

index - of Fredholm operator, 13 - of funetion, 117 - of parametrization, 50 interior of r, 50 Interpolation Theorem, 162 inverse diserete Fourier transform, 240 - of periodie funetion, 410 inverse Fourier transform, 199 inverse inequality - for splines, 404 - for trigonometrie funetions, 247 IS-value, 144

Jordan are, 48 - C k -smooth,49 Jordan eurve, 48 Jordan Theorem, 49 jump relations - for double layer potential, 58 - for normal derivatives of the single

layer potential, 63 Jo(r),40 h(r),41

Kelvin transform, 83 Krylov subspace, 25 Krylov subspace method, 1

Laplace equation, 35 Lax Theorem, 125 Lax-Milgram lemma, 426 layer method, 73, 74

Lippmann-Schwinger equation, 314 locally finite series, 137

Markov's lemma, 209

Neumann function - of order one, 41 - of order zero, 40 Neumann problem, 36 Noether operator, 13 Noether theorem, 124 normal derivative - exterior, 62 - interior, 62 null space, 2 N = {1, 2, ... } (natural numbers), 8 No = NU {O}, 27 n,141 f:n 4i)±,62

odd operator, 333 one-sided inverse, 18 open arc, 325 optimal reduction factor, 26 order - of integral operator, 167 - of PPDO, 203 orientation of a curve, 50 [1-, {}+, 50

parametrix, 224 parametrization - Ck -smooth, 48 - of arc or curve, 48 - periodic, 48 - piecewise smooth, 49 - regular, 48 periodic - analytic function, 193 - distribution, 136 - function, 136 - pseudodifferential operator, 203 - Sobolev space, 141, 168, 253 Petrov-Galerkin method, 433 piecewise smooth boundary, 49 polyhomogenous symbol, 231 positive orientation, 50 positively homogeneous symbol, 231 potential equation, 38 PPDO,203 principal value, 106, 143 projection operator, 5 projector, 5

P+, P-, 183 P+, P_, 116 PD, 251

Index 451

Pn (orthogonal projection to T,.), 241 Pnlon2 ,250

quadrature method, 353 - f,364 - modified, 378 quadrature rule, 353 qualocation method, 433 Qn (interpolation projection to T,.),

242 Qnlon2,250 Q~, Q~, 244 Q~, 365

radiation condition - Rellich formulation, 99 - Sommerfeld formulation, 99 range of an operator, 2 real analytic function, 46 reduced wave equation, 38 regularization method, 286 regularizer , 20 relatively compact set, 3 resolvent set of an operator, 7 Riesz projector, 8 Robin problem, 37 p(T),7 Poo(A), 27

scattering problem, 313 Schauder theorem, 4 simple arc, 48 simple closed curve, 48 single layer operator, 55 single layer potential, 45 singular integral, 143 singular support, 221 smooth spline, 399 Sobolev embedding, 142, 254 Sobolev space - H", 141, 316 - H"'#' 168

" ' " - He, Ho, 244 - Hi(R2 ) 253 - W#,,2(R2), 315 Sokhotski-Plemelj Theorem, 107 Sommerfeld radiation condition - of incoming field, 41 - of outgoing field, 41 spectrum of an operator, 7 spline collocation

452 Index

- mid-point, 412 - nodal point, 412 spline interpolation, 419 stability estimate, 285, 422 stability inequality, 288 strongly elliptie operator, 227 summation by parts, 200 suppf,133 support, 36, 133 symbol - of an operator, 203 - polyhomogenous, 231 - positively homogeneous, 231 Symm's equation, 72 - modified, 181 Symm's operator, 150 S (Cauehy singular operator), 110 S (single layer operator), 55 SN, 137 S(JR) , 199 Sd(.:1), 400 st, 412

test funetion, 133 Three Lines Theorem, 162 time harmonie solution, 36 trapezoidal approximation of an

integral, 358

trapezoidal rule, 355 trigonometrie interpolation, 355 two grid method, 260 two level method, 260 1n,239 T:', T::, 244

unique eontinuation property, 46

V,45

wave equation, 36 weak singularity, 55 winding number - of a function on r, 117 - of a Jordan eurve, 50 - of a periodie funetion, 182 W,45 Wo, 57 WI',2(JR2), 315

YO(T),40 Y1 (T),41

Z={ ... ,-2,-1,0,1,2, ... }, 19 Zn, 239