Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Error analysis of a boundary element collocation method for a screen problem in $ {\bf R}\sp 3$

Authors: M. Costabel, F. Penzel and R. Schneider
Journal: Math. Comp. 58 (1992), 575-586
MSC: Primary 65N38; Secondary 65R20
MathSciNet review: 1122060
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] D. N. Arnold and J. Saranen, On the asymptotic convergence of spline collocation methods for partial differential equations, SIAM J. Numer. Anal. 21 (1984), 459-472. MR 744168 (86g:65221)
  • [2] D. N. Arnold and W. L. Wendland, On the asymptotic convergence of collocation methods, Math. Comp. 41 (1983), 349-381. MR 717691 (85h:65254)
  • [3] J. Aubin, Approximation of elliptic boundary value problems, Wiley-Interscience, New York, 1972.
  • [4] I. Babuška and A. K. Aziz, Survey lectures on the mathematical foundation of the finite element method, The Mathematical Foundation of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, ed.), Academic Press, New York, 1972, pp. 3-359. MR 0421106 (54:9111)
  • [5] M. Dauge, Elliptic boundary value problems in corner domains--smoothness and asymptotics of solutions, Lecture Notes in Math., vol. 1341, Springer-Verlag, Berlin, 1988. MR 961439 (91a:35078)
  • [6] V. J. Ervin and E. P. Stephan, Experimental convergence of boundary element methods for the capacity of the electrified plate, Boundary Elements IX (C. A. Brebbia, W. L. Wendland, and G. Kuhn, eds.), vol. 1, Springer-Verlag, Berlin, 1987, pp. 167-175. MR 965318 (89j:78003)
  • [7] 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 $ {\mathbb{R}^3}$, Math. Meth. Appl. Sci. 13 (1990), 291-303. MR 1074092 (91j:65196)
  • [8] G. I. Èskin, Boundary value problems for elliptic pseudodifferential equations, Transl. Math. Monographs, vol. 52, Amer. Math. Soc., Providence, RI, 1981. MR 623608 (82k:35105)
  • [9] L. S. Frank, Spaces of network functions, Math. USSR-Sb. 15 (1971), 183-226. MR 0290583 (44:7763)
  • [10] R. Hagen and B. Silbermann, On finite element collocation for bisingular integral equations, Appl. Anal. 19 (1985), 117-135. MR 800163 (87a:45014)
  • [11] G. C. Hsiao and S. Prössdorf, On spline collocation for multidimensional singular integral equations (to appear).
  • [12] M. A. Jaswon and G. T. Symm, Integral equation methods in potential theory and elastostatics, Academic Press, London, 1977. MR 0499236 (58:17147)
  • [13] J. L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications, vol. 1, Springer-Verlag, Berlin, 1972.
  • [14] J.-C. Nédélec, Equations intégrales associées aux problèmes aux limites elliptiques dans des domaines de $ {\mathbb{R}^3}$, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques (R. Dautray and J.-L. Lions, eds.), Chapters XI-XIII, Masson, Paris, 1988.
  • [15] F. Penzel, Error estimates for a discretized Galerkin method for a boundary integral equation in two dimensions, THD-Preprint 1276, Technische Hochschule Darmstadt, 1990. MR 1174913 (94c:65142)
  • [16] T. von Petersdorff, Randwertprobleme der Elastizitätstheorie für Polyeder--Singularitäten und Approximation mit Randelementmethoden, Thesis, Technische Hochschule Darmstadt, 1989.
  • [17] S. Prössdorf, Numerische Behandlung singulärer Integralgleichungen, Z. Angew. Math. Mech. 69 (1989), T5-T13. MR 1002327 (90i:65247)
  • [18] S. Prössdorf and A. Rathsfeld, A spline collocation method for singular integral equations with piecewise continuous coefficients, Integral Equations Operator Theory 7 (1984), 536-560. MR 757987 (85h:65276)
  • [19] G. Schmidt, Spline collocation for singular integro-differential equations over (0, 1), Numer. Math. 50 (1987), 337-352. MR 871234 (88d:65183)
  • [20] 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), 145-165. MR 965918 (89i:65121)
  • [21] R. Schneider, Stability of a collocation method for strongly elliptic multidimensional singular integral equations, Numer. Math. 58 (1991), 855-873. MR 1098869 (92a:65354)
  • [22] L. L. Schumaker, Spline functions: Basic theory, Wiley, New York, 1981. MR 606200 (82j:41001)
  • [23] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. MR 0290095 (44:7280)
  • [24] E. P. Stephan, Differenzenapproximationen von Pseudo-Differentialoperatoren, Thesis, Technische Hochschule Darmstadt, 1975.
  • [25] -, Boundary integral equations for screen problems in $ {\mathbb{R}^3}$, Integral Equations Operator Theory 10 (1987), 236-257. MR 878247 (88a:35059)
  • [26] W. L. Wendland, On some mathematical aspects of boundary element methods for elliptic problems, Mathematics of Finite Elements and Applications V (J. Whiteman, ed.), Academic Press, London, 1985, pp. 193-227. MR 811035 (87c:65154)
  • [27] -, Strongly elliptic boundary integral equations, The State of the Art in Numerical Analysis (A. Iserles and M. J. D. Powell, eds.), Clarendon Press, Oxford, 1987, pp. 511-562. MR 921677 (88m:65209)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N38, 65R20

Retrieve articles in all journals with MSC: 65N38, 65R20

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society