# Boundary Element Methods and Their Asymptotic Convergence

• W. L. Wendland
Chapter
Part of the International Centre for Mechanical Sciences book series (CISM, volume 277)

## Abstract

Nowadays the most popular numerical methods for solving elliptic boundary value problems are finite differences, finite elements and, more recently, boundary element methods. The latter are numerical methods for solving integral equations (or their generalizations) on the boundary Γ of the given domain. The reduction of interior or exterior stationary boundary value problems as well as transmission problems to equivalent boundary integral equations is by no means unique, the two most popular reductions are the “direct method” and the “method of potentials”. In all these cases one needs a fundamental solution of the differential equations explicitly since it will be used in numerical computations. This restricts the boundary integral methods to cases of simple computability of a fundamental solution, i.e. essentially to differential equations with constant coefficients. The formulation on the boundary surface F reduces the dimensions of the original problem by one. For the computational treatment the boundary surface is decomposed into a finite number of segments and the boundary functions are approximated by corresponding finite elements, the boundary elements. The appropriately discretized version of the boundary integral equation then provides a finite system of linear approximate equations whose coefficient matrix, the influence matrix is fully distributed.

## Keywords

Boundary Element Method Collocation Method Singular Integral Equation Boundary Integral Equation Pseudodifferential Operator

## References

1. [1]
2. [2]
Agmon, A., Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math. 17 (1964) 35–92.
3. [3]
Agranovich, M.S., Spectral properties of diffraction problems, in The General Method of Natural Vibrations in Diffraction Theory by N.N. Voitovic, B.Z. Katzenellenbaum and A.N. Sivov(Russian) Izdat. Nauka, Moscow 1977.Google Scholar
4. [4]
Ahner, J.F. and Hsiao, G.C., On the two-dimensional exterior boundary-value problems of elasticity, SIAM J. Appl. Math. 31 (1976) 677–685.
5. [5]
Ahner, J.F. and Kleinman, R.E., The exterior Neumann problem for the Helmholtz equation, Arch. Rat. Mech. Anal. 52 (1973) 26–43.
6. [6]
Anselone, P.M., Collectively Compact Operator Approximation Theory, Prentice Hall, London, 1971.
7. [7]
Arnold, D.N. and Wendland, W.L., On the asymptotic convergence of collocation methods, Math. Comp., to appear. (Preprint Nr. 665, Technical Univ. Darmstadt, Dept. Mathematics, D-61 Darmstadt, Fed. Rep. Germany 1982 ).Google Scholar
8. [8]
Arnold, D. and Wendland, W.L., Collocation versus Galerkin proce-dures for boundary integral methods, in Fourth Int. Sem. Boundary Element Methods (ed. C.A. Brebbia), to appear, (Preprint Hr. 671, Technical Univ. Darmstadt, Dept. Mathematics, D-61 Darmstadt 1982 )Google Scholar
9. [9]
Atkinson, K.E., A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, Soc. Ind. Appl. Math. Philadelphia, 1976.Google Scholar
10. [10]
Aubin, J.P,. Approximation of Elliptic Boundary-Value Problems, Wiley-Interscience, New York, 1972.
11. [11]
Aziz, A.K., Dorr, M.R. and Kellogg, R.B., Calculation of electromagnetic scattering by a perfect conductor, Naval Surface Weapons Center Report TR80–245, 1980, Silver Spring, Maryland 20910.Google Scholar
12. [12]
Babuska, I. and Aziz, A.K., Survey lectures on the mathematical foundations of the finite element method, in the Mathematical Foundation of the Finite Element Method with Applications to Partial Differential Equations (ed. A.K. Aziz ), Academic Press, New York (1972) 3–359.Google Scholar
13. [13]
Blue, J., Boundary integral solutions of Laplace’s equation, The Bell System Tech. J. 57 (1978) 2797–2822.
14. [14]
Bolteus, L, and Tullberg, O., BEMSTAT-A new type of boundary element program for two-dimensional elasticity problems, in Boundary Element Methods (ed. C.A. Brebbia ), Springer, Berlin, Heidelberg, New York (1981) 518–537.Google Scholar
15. [15]
Brakhage, H., Über die numerische Behandlung von Integral gleichungen nach der Quadraturformelmethode, Num. Math. 2 (1960) 183–196.
16. [16]
Brakhage, H. and Werner, P., Über das Dirichletsche Außenraumproblem für die Helmholtzsche Schwingungsgleichung, Arch. Math. 16 (1965) 325–329.
17. [17]
Bruhn, G. and Wendland, W.L., Über die näherungsweise Lösung von linearen Funktionalgleichungen, in Funktionalanalysis, Approximationstheorie, Numerische Mathematik (ed. L. Collatz and H. Ehrmann) Intern. Ser. Numer. Math. 7 Birkhäuser Basel (1967) 136–164.Google Scholar
18. [18]
Burago, Y.D., Maz’ja, V.G. and Sapozhnikova, V.D., On the theory of simple and double-layer potentials for domains with irregular boundaries, in Problems in Math. Analysis Vol. 1, Boundary Value Problems and Integral Equations (ed. V.I. Smirnov ), Consultats Bureau, New York (1968) 1–30.Google Scholar
19. [19]
Cea, J., Approximation variationelle des problèmes aux limites, Ann. Inst. Fourier, Grenoble, 14 (1964) 345–444.
20. [20]
Chandler, G.A., Superconvergence of numerical solutions to second kind integral equations, Ph. D. Thesis, Australian National University, 1979.Google Scholar
21. [21]
Ciarlet, P.G., The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, New York, Oxford, 1978.Google Scholar
22. [22]
Colton, D.L., Analytic Theory of Partial Differential Equations, Pitman Publ. London,1980.Google Scholar
23. [23]
Costabel, M. and Stephan, E., Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, Banach Center Publications, Warsaw, to appear (Preprint Nr. 593, Technical Univ. Darmstadt, Dept. Mathematics, D-61 Darmstadt, Fed. Rep. Germany 1981 ).Google Scholar
24. [24]
Costabel, M. and Stephan, E., Curvature terms in the asymptotic expansions for solutions of boundary integral equations on curved polygons, to appear (Preprint Nr. 673, Technical Univ. Darmstadt, Dept. Mathematics, D-61 Darmstadt, Fed. Rep. Germany 1982 ).Google Scholar
25. [25]
Costabel, M., Stephan, E. and Wendland, W.L., On boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain, to appear (Preprint Nr. 670, Technical Univ. Darmstadt, Dept. Mathematics, D-61 Darmstadt, Fed. Rep. Germany 1982 ).Google Scholar
26. [26]
Cruse, T.A., Application of the boundary integral equation method to three-dimensional stress analysis, Comp. Struct. 3 (1973) 309–369.Google Scholar
27. [27]
Djaoua, M., Méthode d’élémnts finis pour la résolution d’un problème extérieur dans IR, Centre de Math. Appl., \$cole Polytechniques, Rapport Int. 3, Palaiseau, France, 1975.Google Scholar
28. [28]
Djaoua, M., A method of calculation of lifting flows around 2-dimensional corner shaped bodies, Centre de Math. Appl, Scole Polytechniques, Rapport Int. 34, Palaiseau, France, 1978.Google Scholar
29. [29]
Durand, M., Diffraction d’ondes acoustiques par un écran mince, Publ. de Math. Appl. Marseille - Toulon 80/3 Université de Provence, France, 1980.Google Scholar
30. [30]
Engels, H., Numerical Quadrature and Cubature, Academic Press, London, New York, Toronto, Sydney, San Francisco, 1980.Google Scholar
31. [31]
Eskin, G.I., Boundary Value Problems for Elliptic Pseudodifferential Equations, AMS, Transi. Math. Mon. 52, Providence, Rhode Island, 1980.Google Scholar
32. [32]
de Figueiredo, D.G., The coerciveness problem for forms over vector valued functions, Comm. Pure Appl. Math. 16 (1963) 63–94.
33. [33]
Filippi, P., Potentiels de couche pour les ondes mécaniques scalaires, Bévue du Cethedec 51 (1977) 121–175.
34. [34]
Filippi, P., Layer potentials and acoustic diffraction, J. Sound and Vibration 54 (1977) 473–500.
35. [35]
Fischer, T., An integral equation procedure for the exterior three-dimensional viscous flow, Integral Equations and Operator Theory 5 (1982) 490–505.
36. [36]
Friedrichs, K.O., Pseudo-Differential Operators, Lecture Notes, Courant Inst. New York University 1970.Google Scholar
37. [37]
Giroire, J., Integral equation methods for exterior problems for the Helmholtz equation, Centre de mathématiques appliquées, Ecole Polytechnique, Palaiseau, France, Rapport interne N2 40 (1978) .Google Scholar
38. [38]
Giroire, J. and Nedelec, J.C., Numerical solution of an exterior Neumann problem using a double layer potential, Math. Comp. 32 (1978) 973–990.
39. [39]
Goldstein, C., Numerical Methods for Helmholtz Type Equations in Unbounded Domains, BNL-26543, Brookhaven Lab., Brookhaven, N.Y. 1979.Google Scholar
40. [40]
Gregoire, J.P., Nedelec, J.C. and Planchard, J., Problèmes relatifs à l’équation d’Helmholtz, Serv. Inf. et Math. Appl. Bull. Direction des Etudes et Recherches, Ser. C, 2 (1974) 15–32.
41. [41]
Günter, N.M., Die Potentialtheorie, Teubner, Leipzig 1957.
42. [42]
Ha Duong, T., La méthode de Schenck pour la résolution numérique du problème de radiation acoustique, E.D.F., Bull. Dir. Etudes Recherches, Ser. C, Math., Informatique, Service Informatique et Mathématiques Appl. 2 (1979) 15–50.Google Scholar
43. [43]
Ha Duong, T., A finite element method for the double-layer potential solutions of the Neumann exterior problem, Math. Meth. Appl. Sci. 2 (1980) 191–208.
44. [44]
Hämmerlin, G. and Schumaker, L.L., Procedures for kernel approximation and solution of Fredholm integral equations of the second kind, Numer. Math. 34 (1980) 125–141.
45. [45]
Hayes, J.K., Kahaner, D.K. and Kellner, R.G., An improved method for numerical conformal mapping, Math. Comp. 26 (1972) 327–334.
46. [46]
Helfrich, H.P., Simultaneous approximation in negative norms of arbitrary order, R.A.I.R.O. Num. Analysis 15 (1981) 231–235.
47. [47]
Hess, J.L., Calculation of acoustic fields about arbitrary three-dimensional bodies by a method of surface source distributions based on certain wave number expansions, Report DAC 66901, Mc Donnell Douglas, 1968.Google Scholar
48. [48]
Hess, J.L. and Smith, A.M.O., Calculation of potential flow about arbitrary bodies, in Progress in Aeronautical Sciences (ed. D. Kuchemann) Pergamon, Oxford 8 (1967) 1–138.Google Scholar
49. [49]
Hildebrandt, St. and Wienholtz, E., Constructive proofs of representation theorems in separable Hilbert space, Comm.Pure Appl. Math. 17 (1964) 369–373.
50. [50]
Hörmander, L., Pseudo-differential operators and non-elliptic boundary problems, Annals Math. 83 (1966) 129–209.
51. [51]
Hoidn, H.-P., Die Kollokationsmethode angewandt auf die Symmsche Integralgleichung, ETH Zürich, Switzerland, in preparation.Google Scholar
52. [52]
Hsiao, G.C., Kopp, P. and Wendland, W.L., A Galerkin collocation method for some integral equations of the first kind, Computing 25 (1980) 89–130.
53. [53]
Hsiao, G.C., Kopp, P. and Wendland, W.L., Some applications of a Galerkin collocation method for integral equations of the first kind, in preparation.Google Scholar
54. [54]
Hsiao, G.C. and Wendland, W.L., A finite element method for some integral equations of the first kind, J. Math. Anal. Appl. 58 (1977) 449–481.
55. [55]
Hsiao, G.C. and Wendland, W.L., The Aubin-Nitsche lemma for integral equations, J. Integral Equations 3 (1981) 299–315.
56. [56]
Jentsch, L., Über stationäre thermoelastische Schwingungen in inhomogenen Körpern, Math. Nachr. 64 (1974) 171–231.
57. [57]
Jentsch, L., Stationäre thermoelastische Schwingungen in stück-weise homogenen Körpern infolge zeitlich periodischer Außen-temperatur, Math. Nachr. 69 (1975) 15–37.
58. [58]
Jones, D.S., Integral equations for the exterior acoustic problem, Quart. J. Mech. Appl. Math. 27 (1974) 129–142.
59. [59]
Kleinman, R.E. and Roach, G.F., Boundary integral equations for the three-dimensional Helmholtz equation, SIAM Review 16 (1974) 214–236.
60. [60]
Kleinman, R. and Wendland, W.L., On Neumann’s method for the exterior Neumann problem for the Helmholtz equation, J. Math. Anal. Appl. 57 (1977) 170–202.
61. [61]
Kohn, J.J. and Nirenberg, L., On the algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965) 269–305.
62. [62]
Kral, J., Integral Operators in Potential Theory, Lecture Notes Math. 823, Springer Berlin, Heidelberg, New York 1980.Google Scholar
63. [63]
Kupradze, W.D., Randwertaufgaben der Schwingungstheorie und Integralgleichungen, Dt. Verlag d. Wissenschaften, Berlin, 1956.Google Scholar
64. [64]
Kupradze, V.D., Potential Methods in the Theory of Elasticity, Israel Program Scientific Transl., Jerusalem, 1965.
65. [65]
Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O., Burchuladze, T.V., Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North Holland, Amsterdam, 1979.Google Scholar
66. [66]
Lamp, U.,Schleicher, T., Stephan, E. and Wendland, W.L., The boundary integral method for a plane mixed boundary value problem, in Advances in Computer Methods for Partial Differential Equations–IV (ed. R. Vichnevetsky and R.S. Stepleman) IMACS Dept. Computer Science, Rutgers Univ. New Brunswick, N.Y. 08903 U.S.A. (1981) 223–229.Google Scholar
67. [67]
Lamp, U., Schleicher, T., Stephan, E. and Wendland, W.L., Theoretical and experimental asymptotic convergence of the boundary integral method for a plane mixed boundary value problem, Proc. Fourth Intern. Seminar on Boundary Element Methods (ed. C.A. Brebbia), Springer, Berlin, Heidelberg, New York, to appear 1982 (Preprint Nr. 667, Technical Univ. Darmstadt, Dept. Mathematics, D-61 Darmstadt Fed. Rep. Germany 1982 ).Google Scholar
68. [68]
Lax, P.D. and Nireonberg, L., On stability for difference schemes; a sharp form of Garding’s inequality, Comm. Pure Appl. Math. 19 (1966) 473–492.
69. [69]
Lehmann, R., Developments at an analytic corner of solutions of elliptic partial differential equations, J. Math. Mech. 8 (1959) 727–760.
70. [70]
MacCamy, R.C., Low frequency acoustic oscillations, Quaterly Appl. Math. 23 (1965) 247–265.
71. [71]
MacCamy, R.C. and Stephan, E., A boundary element method for an exterior problem for three-dimensional Maxwell’s equations, to appear (Preprint Nr. 681, Technical Univ. Darmstadt, Dept. Mathematics, D-61 Darmstadt, Fed. Rep. Germany 1982 ).Google Scholar
72. [72]
Martensen, E., Potentialtheorie, B.G. Teubner, Stuttgart, 1968.Google Scholar
73. [73]
Michlin, S.G., Variationsmethoden der Mathematischen Physik, Akademie-Verlag, Berlin 1962.
74. [74]
Michlin, S.G. and Prössdorf, S., Singuläre Integraloperatoren, Akademie-Verlag, Berlin, 1980.
75. [75]
Müller, C., Foundations of the Mathematical Theory of Electro-magnetic Waves, Springer, Berlin Heidelberg, New York, 1969.Google Scholar
76. [76]
Mustoe, G.G. and Mathews, I.C., Direct boundary integral methods, point collocation and variational procedures, to appear.Google Scholar
77. [77]
Natanson, E., Theory of Functions of a Real Variable, Ungar Publ., New York, 1955.Google Scholar
78. [78]
Nedelec, J.C., Curved finite element methods for the solution of singular integral equations on surfaces in ]R3, Comp. Math. Appl. Mech. Eng. 8 (1976) 61–80.
79. [79]
Nedelec, J.C., Approximation par potentiel de double cuche du problème de Neumann extérieur, C.R. Acad. Sci. Paris, Sér. A 286 (1977) 616–619.Google Scholar
80. [80]
Nedelec, J.C., Formulations variationelles de quelques équations integrales faisant intervenir des parties finies, in Innovative Numerical Analysis for the Engineering Sciences ( R. Shaw ed.), Univ. Press of Virginia, Charlottesville (1980) 517–524.Google Scholar
81. [81]
Nedelec, J.C. and Planchard, J., Une méthode variationnelle d’éléments finis pour la résolution numérique d’un problème extérieur dans 1R3, R.A.I.R.O. 7 (1973) R3, 105–129.
82. [82]
Nitsche, J.A., Zur Konvergenz von Näherungsverfahren bezüglich verschiedener Normen, Num. Math. 15 (1970) 224–228.
83. [83]
Panic, I.J., On the solubility of exterior boundary value problems for the wave equation and for a system of Maxwell’s equations (Russian), Uspehi Mat. Nauk 20 (1965) 221–226.
84. [84]
Petiau, G., La théorie des fonctions de Bessel, Centre National de la Rech. Scientifique, Paris 1955.Google Scholar
85. [85]
Phillips, J.L., The use of the collocation as a projection method for solving linear operator equations, SIAM J. Numer. Anal. 9 (1972) 14–28.
86. [86]
Poggio, A.J. and Miller, E.K., Integral equation solutions of three-dimensional scattering problems, in Computer Techniques for Electromagnetics ( R. Mittra ed.) Pergamon, Oxford, 1973.Google Scholar
87. [87]
Prössdorf, S. and Schmidt, G., A finite element collocation method for singular integral equations, Math. Nachr. 100 (1981) 33–60.
88. [88]
Prössdorf, S. and Schmidt, G., A finite element collocation method for systems of singular integral equations, Preprint P - MATH - 26/81, Akademie d. Wissenschaften DDR, Inst. Math., DDR-1080 Berlin, Mohrenstr. 39, 1981.Google Scholar
89. [89]
Prössdorf, S. and Silbermann, B., Projektionsverfahren und die näherungsweise Lösung singulärer Gleichungen, Teubner, Leipzig, 1977.
90. [90]
Radon, J., Über die Randwertaufgaben beim logarithmischen Potential, Sitz.ber. Akad. Wiss. Wien, Math.-nat. Kl. IIa 128 (1919) 1123–1167.Google Scholar
91. [91]
Rannacher, R. and Wendland, W.L., On the order of pointwise convergence of some boundary element methods, in preparation.Google Scholar
92. [92]
Richter, G.R., Superconvergence of piecewise polynomial Galerkin approximations for Fredholm integral equations of the second kind, Numer. Math. 31 (1978) 63–70.
93. [93]
Le Roux, M.N., Résolution numérique du problème du potential dans le plan part une méthode variationelle d’éléments finis, These, L’Université de Rennes, Ser. A, No. D’ordre 347, No. Ser. 38, 1974.Google Scholar
94. [94]
Ruland, C., Ein Verfahren zur Lösung von in Aussen-gebieten mit Ecken, Appl. Analysis 7 (1978) 69–79.
95. [95]
Saranen, J. and Wendland, W.L., On the asymptotic convergence of collocation methods with spline functions of even degree, to appear (Preprint Nr. 690, Technical Univ. Darmstadt, Dept. Mathematics, D-61 Darmstadt, Fed. Rep. Germany 1982 ).Google Scholar
96. [96]
Schäfer, E., Fehlerabschätzungen für Eigenwertnäherungen nach der Ersatzkernmethode bei Integralgleichungen, Numer. Math. 32 (1979) 281–290.
97. [97]
Schenck, H.A., Improved integral formulation for acoustic radiation problems, J. Acoustic Soc. Amer. 44 (1968) 41–58.
98. [98]
Schmidt, G., On spline collocation for singular integral equations, Math. Nachr., to appear.Google Scholar
99. [99]
Seeley, R., Topics in pseudo-differential operators, in Pseudo-Differential Operators ( L. Nirenberg ed.) C.I.M.E., Cremonese, Roma, 1969.Google Scholar
100. [100]
Sloan, I., Improvement by iteration for compact operator equations, Math. Comp. 30 (1976) 758–764.
101. [101]
Stephan, E., Solution procedures for interface problems in acoustics and electromagnetics, in these Lecture Notes.Google Scholar
102. [102]
Stephan, E. and Wendland, W.L., Remarks to Galerkin and least squares methods with finite elements for general elliptic problems, Springer Lecture Notes Math. 564 (1976) 461–471 and Manuscripta Geodaetica 1 (1976) 93–123.
103. [103]
Stummel, F., Diskrete Konvergenz linearer Operatoren I and II, Math. Zeitschr. 120 (1971) 231–264.
104. [104]
Symm, G.T., Integral equation methods in potential theory, II, Proc. Royal Soc. London 275 (1963) 33–46.
105. [105]
Treves, F., Introduction to Pseudodifferential and Fourier Integral Operators I. Plenum Press, New York and London, 1980.
106. [106]
Ursell, F., On the exterior problems of acoustics, Proc. Camb. Phil. Soc. 74 (1973) 117–125.
107. [107]
Vaillancourt, R., A simple proof of Lax-Nirenberg theorems, Comm. Pure Appl. Math. 23 (1970) 151–163.
108. [108]
Vainikko, G., Funktionalanalysis der Diskretisierungsmethoden, G.B. Teubner, Leipzig, 1976.Google Scholar
109. [109]
Watson, J.0., Advanced implementation of the boundary element method for two- and three-dimensional elastostatics, in “Developments in Boundary Element Methods–1” (ed. P.K. Banerjee and R. Butterfield ), Appl. Science Publ. LTD, London (1979) 31–63.Google Scholar
110. [110]
Wendland, W.L., Die Behandlung von Randwertaufgaben im IR3 mit Hilfe von Einfach- und Doppelschichtpotentialen, Num. Math. 11 (1968) 380–404.
111. [111]
Wendland, W.L., Über Galerkin-Verfahren zur Lösung des Dirichlet Problems in Lp-Räumen für gleichmäßig stark elliptische Differentialgleichungen, Abh. Math. Seminar Univ. Hamburg, 36 (1971) 185–197.
112. [112]
Wendland, W.L., On Galerkin collocation methods for integral equations of elliptic boundary value problems, in Numerical Treatment of Integral Equations (ed. J. Albrecht and L. Collatz), Intern. Ser. Num. Math. 53, Birkhäuser Basel (1980) 244–275.Google Scholar
113. [113]
Wendland, W.L., Asymptotic convergence of boundary element methods, in Lectures on the Numerical Solution of Partial Differential Equations (ed. I. Babus“ka, T.-P. Liu and J. Osborn) Lecture Notes # 20, Univ. of Maryland, College Park Md. (1981) 435–528.Google Scholar
114. [114]
Wendland, W.L., Asymptotic accuracy and convergence, in Progress in Boundary Element Methods (ed. C.A. Brebbia ), Pentech Press, London, Plymouth, 1 (1981) 289–313.
115. [115]
Wendland, W.L., Stephan, E. and Hsiao, G.C., On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Meth. in the Appl. Sci., 1 (1979) 265–321.
116. [116]
Wolfe, P., An integral operator connected with the Helmholtz equation, J. Functional Anal. 36 (1980) 105–113.