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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Error analysis of a boundary element collocation method for a screen problem in $\textbf {R}^ 3$
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by M. Costabel, F. Penzel and R. Schneider PDF
Math. Comp. 58 (1992), 575-586 Request permission

Abstract:

We examine the numerical approximation of the first-kind integral equation on a plane rectangle defined by the single-layer potential of the three-dimensional Laplacian. The solution is approximated by nodal collocation with piecewise bilinear trial functions on a rectangular grid. We prove stability and convergence of this method in the Sobolev space ${\tilde H^{ - 1/2}}$. A key ingredient in the proof is the observation that the collocation equations define symmetric positive definite Toeplitz matrices.
References
  • Douglas N. Arnold and Jukka Saranen, On the asymptotic convergence of spline collocation methods for partial differential equations, SIAM J. Numer. Anal. 21 (1984), no. 3, 459–472. MR 744168, DOI 10.1137/0721034
  • Douglas N. Arnold and Wolfgang L. Wendland, On the asymptotic convergence of collocation methods, Math. Comp. 41 (1983), no. 164, 349–381. MR 717691, DOI 10.1090/S0025-5718-1983-0717691-6
  • J. Aubin, Approximation of elliptic boundary value problems, Wiley-Interscience, New York, 1972.
  • Ivo Babuška and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR 0421106
  • Monique Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR 961439, DOI 10.1007/BFb0086682
  • V. J. Ervin and E. P. Stephan, Experimental convergence of boundary element methods for the capacity of the electrified square plate, Boundary elements IX, Vol. 1 (Stuttgart, 1987) Comput. Mech., Southampton, 1987, pp. 167–175. MR 965318
  • V. J. Ervin, E. P. Stephan, and S. Abou El-Seoud, An improved boundary element method for the charge density of a thin electrified plate in $\textbf {R}^3$, Math. Methods Appl. Sci. 13 (1990), no. 4, 291–303. MR 1074092, DOI 10.1002/mma.1670130403
  • G. I. Eskin, Boundary value problems for elliptic pseudodifferential equations, Translations of Mathematical Monographs, vol. 52, American Mathematical Society, Providence, R.I., 1981. Translated from the Russian by S. Smith. MR 623608
  • L. S. Frank, Spaces of network functions, Mat. Sb. (N.S.) 86 (128) (1971), 187–233 (Russian). MR 0290583
  • Roland Hagen and Bernd Silbermann, A finite element collocation method for bisingular integral equations, Applicable Anal. 19 (1985), no. 2-3, 117–135. MR 800163, DOI 10.1080/00036818508839538
  • G. C. Hsiao and S. Prössdorf, On spline collocation for multidimensional singular integral equations (to appear).
  • M. A. Jaswon and G. T. Symm, Integral equation methods in potential theory and elastostatics, Computational Mathematics and Applications, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1977. MR 0499236
  • J. L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications, vol. 1, Springer-Verlag, Berlin, 1972. J.-C. NĂ©dĂ©lec, Equations intĂ©grales associĂ©es aux problèmes aux limites elliptiques dans des domaines de ${\mathbb {R}^3}$, Analyse MathĂ©matique et Calcul NumĂ©rique pour les Sciences et les Techniques (R. Dautray and J.-L. Lions, eds.), Chapters XI-XIII, Masson, Paris, 1988.
  • F. Penzel, Error estimates for a discretized Galerkin method for a boundary integral equation in two dimensions, Numer. Methods Partial Differential Equations 8 (1992), no. 5, 405–421. MR 1174913, DOI 10.1002/num.1690080502
  • T. von Petersdorff, Randwertprobleme der Elastizitätstheorie fĂĽr Polyeder—Singularitäten und Approximation mit Randelementmethoden, Thesis, Technische Hochschule Darmstadt, 1989.
  • S. Prössdorf, Numerische Behandlung singulärer Integralgleichungen, Proceedings of the Annual Scientific Meeting of the GAMM (Vienna, 1988), 1989, pp. T5–T13 (German). MR 1002327
  • Siegfried Prössdorf and Andreas Rathsfeld, A spline collocation method for singular integral equations with piecewise continuous coefficients, Integral Equations Operator Theory 7 (1984), no. 4, 536–560. MR 757987, DOI 10.1007/BF01238865
  • Gunther Schmidt, Spline collocation for singular integro-differential equations over $(0,1)$, Numer. Math. 50 (1987), no. 3, 337–352. MR 871234, DOI 10.1007/BF01390710
  • G. Schmidt and H. Strese, The convergence of a direct BEM for the plane mixed boundary value problem of the Laplacian, Numer. Math. 54 (1988), no. 2, 145–165. MR 965918, DOI 10.1007/BF01396971
  • R. Schneider, Stability of a spline collocation method for strongly elliptic multidimensional singular integral equations, Numer. Math. 58 (1991), no. 8, 855–873. MR 1098869, DOI 10.1007/BF01385658
  • Larry L. Schumaker, Spline functions: basic theory, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. MR 606200
  • Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • E. P. Stephan, Differenzenapproximationen von Pseudo-Differentialoperatoren, Thesis, Technische Hochschule Darmstadt, 1975.
  • Ernst P. Stephan, Boundary integral equations for screen problems in $\textbf {R}^3$, Integral Equations Operator Theory 10 (1987), no. 2, 236–257. MR 878247, DOI 10.1007/BF01199079
  • W. L. Wendland, On some mathematical aspects of boundary element methods for elliptic problems, The mathematics of finite elements and applications, V (Uxbridge, 1984) Academic Press, London, 1985, pp. 193–227. MR 811035
  • W. L. Wendland, Strongly elliptic boundary integral equations, The state of the art in numerical analysis (Birmingham, 1986) Inst. Math. Appl. Conf. Ser. New Ser., vol. 9, Oxford Univ. Press, New York, 1987, pp. 511–562. MR 921677
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Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Math. Comp. 58 (1992), 575-586
  • MSC: Primary 65N38; Secondary 65R20
  • DOI: https://doi.org/10.1090/S0025-5718-1992-1122060-9
  • MathSciNet review: 1122060