A direct boundary integral equation method for transmission problems. (English) Zbl 0597.35021

Costabel, Martin; Stephan, Ernst

J. Math. Anal. Appl. 106, 367-413 (1985).

A system of integral equations for the field and its normal derivative on the boundary in acoustic or potential scattering by a penetrable homogeneous object in arbitrary dimensions is presented. The system contains the operators of the single and double layer potentials, of the normal derivative of the single layer, and of the normal derivative of the double layer potential. It defines a strongly elliptic system of pseudodifferential operators. It is shown by the method of Mellin transformation that a corresponding property, namely a Gårding’s inequality in the energy norm, holds also in the case of a polygonal boundary of a plane domain. This yields asymptotic quasioptimal error estimates in Sobolev space for the corresponding Galerkin approximation using finite elements on the boundary only.

Reviewer: R.Kreß

MSC:

35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
49M15	Newton-type methods
65N15	Error bounds for boundary value problems involving PDEs
45A05	Linear integral equations
35A22	Transform methods (e.g., integral transforms) applied to PDEs

Keywords:

system of integral equations; single and double layer potentials; strongly elliptic system of pseudodifferential operators; method of Mellin transformation; Gårding’s inequality in the energy norm; asymptotic quasioptimal error estimates; Galerkin approximation

References:

[1]	Agranovich, M.S, Spectral properties of diffraction problems, (), [Russian] · Zbl 0738.58047
[2]	Aziz, A.K; Dorr, M.R; Kellogg, R.B, A new approximation method for the Helmholtz equation in an exterior domain, SIAM J. numer. anal., 19, 899-908, (1982) · Zbl 0494.65067
[3]	Babuska, I; Aziz, I, Surves lectures on the mathematical foundations of the finite element method, (), 3-359
[4]	Barrar, R.B; Dolph, C.L, On a three-dimensional transmission problem of electromagnetic theory, J. math. mech. anal., 3, 725-743, (1954) · Zbl 0058.42303
[5]	{\scJ. Bielak and R. C. MacCamy}, An exterior interface problem in two-dimensional elastodynamics, J. Math. Anal. Appl., in press. · Zbl 0519.73021
[6]	de Monvel, L.Boutet, Boundary problems for pseudo-differential operators, Acta math., 126, 11-51, (1971) · Zbl 0206.39401
[7]	Chazarain, J; Piriou, A, Introduction à la théorie des equations aux Dérivées partielles linéaires, (1981), Gauthier-Villars Paris · Zbl 0446.35001
[8]	Colton, D; Kress, R, The unique solvability of the null field equations of acoustics, Quart. J. mech. appl. math., 36, 87-95, (1983) · Zbl 0513.73036
[9]	Costabel, M, Boundary integral operators on curved polygons, ann. mat. pura appl. ser. iva, 133, 305-326, (1983) · Zbl 0533.45009
[10]	Costabel, M; Stephan, E, Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, THD-preprint no. 593, (1981), Darmstadt
[11]	Costabel, M; Stephan, E, The boundary integral method for a mixed boundary value problem in a polygonal domain, Adv. comput. methods partial differential equations IV, IMACS, 300-304, (1981)
[12]	Costabel, M; Stephan, E, Curvature terms in the asymptotic expansions for solutions of boundary integral equations on curved polygons, J. integral equations, 5, 353-371, (1983) · Zbl 0538.35022
[13]	Costabel, M; Stephan, E, The normal derivative of the double layer potential on polygons and Galerkin approximation, Applicable anal., 16, 205-228, (1983) · Zbl 0508.31003
[14]	Costabel, M; Stephan, E; Wendland, W.L, On boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain, ann. scuola norm. sup. Pisa, ser. IV, 10, 197-241, (1983) · Zbl 0563.45007
[15]	Dieudonné, J, Eléments d’analyse 8, (1978), Gauthier-Villars Paris
[16]	Dolph, C.L, The integral equation method in scattering theory, (), 201-277 · Zbl 0234.35077
[17]	Durand, M, Layer potentials and boundary value problems for the Helmholtz equation in the complement of a thin obstacle, Math. methods appl. sci., 5, 389-421, (1983) · Zbl 0527.76076
[18]	Eskin, G.I, Boundary value problems for elliptic pseudodifferential equations, (1981), American Mathematical Society Providence, Rhode Island · Zbl 0458.35002
[19]	Giroire, J, Formulation variationelle par équations intégrales de problèmes aux limites extérieurs, ()
[20]	Giroire, J; Nedelec, J.C, Numerical solution of an exterior Neumann problem using a double-layer potential, Math. comp., 22, 973-990, (1978) · Zbl 0405.65060
[21]	Grisvard, P, Boundary value problems in non-smooth domains, () · Zbl 0766.35001
[22]	Günter, N.M, Potential theory and its applications to basic problems of mathematical physics, (1967), Ungar New York · Zbl 0164.41901
[23]	Hariharan, S.I, An integral equation procedure for eddy current problems, () · Zbl 0478.65080
[24]	Hariharan, S.I; Stephan, E, A boundary element method for a two-dimensional interface problem in electromagnetics, Numer. math., 42, 311-322, (1983) · Zbl 0506.65042
[25]	Hildebrandt, S; Wienholtz, E, Constructive proofs of representation theorems in separable Hilbert space, Comm. pure appl. math., 17, 369-373, (1964) · Zbl 0131.13401
[26]	Hsiao, G.C; Mac Camy, R.C, Solution of boundary value problems by integral equations of the first kind, SIAM rev., 15, 687-705, (1973) · Zbl 0235.45006
[27]	Hsiao, G.C; Wendland, W.L, A finite element method for some integral equations of the first kind, J. math. anal. appl., 58, 449-481, (1977) · Zbl 0352.45016
[28]	Hsiao, G.C; Wendland, W.L, The Aubin-Nitsche lemma for integral equations, J. integral equations, 3, 299-315, (1981) · Zbl 0478.45004
[29]	Kellogg, R.B, Singularities in interface problems, (), 351-400 · Zbl 0312.35009
[30]	Kittappa, R; Kleinman, R.E, Acoustic scattering by penetrable homogeneous objects, J. math. phys., 16, 421-432, (1975) · Zbl 0306.76058
[31]	Kleinman, R.E, On the solution of the Helmholtz equation in the neighborhood of a corner, Z. angew. math. mech., 58, 379-381, (1978)
[32]	Kondratiev, V.A, Boundary problems for elliptic equations in domains with conical or angular points, Trans. Moscow math. soc., 16, 227-313, (1967) · Zbl 0194.13405
[33]	Kress, R; Roach, G.F, Transmission problems for the Helmholtz equation, J. math. phys., 19, 1422-1437, (1978) · Zbl 0433.35017
[34]	Kupradze, W.D, Randwertaufgaben der schwingungstheorie und integralgleichungen, (1956), Deutscher Verlag der Wissenschaften Berlin · Zbl 0070.09701
[35]	Lions, J.L; Magenes, E, Non-homogeneous boundary value problems and applications I, (1972), Springer Berlin/Heidelberg/New York · Zbl 0223.35039
[36]	MacCamy, R.C, Variational procedures for a class of exterior interface problems, J. math. anal. appl., 78, 248-266, (1980) · Zbl 0471.65081
[37]	MacCamy, R.C; Marin, S.P, A finite element method for exterior interface problems, Internat. J. math. math. sci., 3, 311-350, (1980) · Zbl 0429.65108
[38]	MacCamy, R.C; Stephan, E, A simple layer potential method for three-dimensional eddy current problems, () · Zbl 0511.35073
[39]	Meyer, W.L; Bell, W.A; Stallybrass, M.P; Zinn, B.T, Boundary integral solutions of three dimensional acoustic radiation problems, J. sound vibration, 59, 245-262, (1978) · Zbl 0391.76052
[40]	Nedelec, J.C, La méthode des éléments finis appliquée aux équations intégrales de la physique, (), 181-190 · Zbl 0486.45008
[41]	Nedelec, J.C; Planchard, J, Une méthode variationnelle d’éléments finis pour la résolution numérique d’un problème extérieur dans \(R\)^{3}, Rairo, 105-129, (1973) · Zbl 0277.65074
[42]	Raugel, G, Résolution numérique de problèmes elliptiques dans des domaines avec coins, ()
[43]	Roach, G.F, On the commutative properties of boundary integral operators, (), 219-227 · Zbl 0368.35028
[44]	Schechter, M, A generalization of the problem of transmission, Ann. scuola norm. sup. Pisa cl. sci., 14, 207-236, (1960) · Zbl 0094.29603
[45]	Stephan, E, Solution procedures for interface problems in acoustics and electromagnetics, (), 291-348
[46]	Stephan, E; Wendland, W.L; Stephan, E; Wendland, W.L, Remarks to Galerkin and least squares methods with finite elements for general elliptic problems, (), 1, 461-471, (1976) · Zbl 0353.65067
[47]	Tikhonov, A.N; Samarskij, A.A, Equations of mathematical physics, (1963), Pergamon Oxford · Zbl 0154.10802
[48]	Weisel, J, (), 1-156, Heft 138
[49]	Wendland, W.L, On Galerkin collocation methods for integral equations of elliptic boundary value problems, (), 224-275
[50]	Wendland, W.L, Asymptotic convergence of boundary element methods, (), 435-528 · Zbl 0561.65086
[51]	Wendland, W.L, (), 281-312
[52]	Wendland, W.L, Boundary element methods and their asymptotic convergence, (), 135-216 · Zbl 0618.65109
[53]	Wendland, W.L, On applications and the convergence of boundary integral methods, (), 463-476
[54]	Wendland, W.L; Stephan, E; Hsiao, G.C, On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. methods appl. sci., 1, 265-321, (1979) · Zbl 0461.65082
[55]	Wendland, W.L, Über potentialtheoretische randwertaufgaben in einem zusammengesetzten gebiet mit übergangsbedingungen and nichtglatten Rändern, Methoden verfahren math. phys., 2, 125-150, (1969)
[56]	Werner, P, Zur mathematischen theorie akustischer wellenfelder, Arch. rat. mech. anal, 6, 231-260, (1960) · Zbl 0097.21103