Error analysis of a boundary element collocation method for a screen problem in

We examine the numerical approximation of the first-kind integral equation on a plane rectangle defined by the single-layer potential of the threedimensional 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 j1/2. A key ingredient in the proof is the observation that the collocation equations define symmetric positive definite Toeplitz matrices.

[1]  Ernst P. Stephan,et al.  An improved boundary element method for the charge density of a thin electrified plate in ℝ3 , 1990 .

[2]  B. Silbermann,et al.  A Finite Element Collocation Method for Bisingular Integral Equations , 1985 .

[3]  G. Schmidt,et al.  The convergence of a direct BEM for the plane mixed boundary value problem of the Laplacian , 1988 .

[4]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .

[5]  F. Penzel.  Error estimates for a discretized Galerkin method for a boundary integral equation in two dimensions , 1992 .

[6]  S. Prössdorf.  Numerische Behandlung singulärer Integralgleichungen , 1989 .

[7]  S. Prössdorf,et al.  A spline collocation method for singular integral equations with piecewise continuous coefficients , 1984 .

[8]  M. Dauge.  Elliptic boundary value problems on corner domains , 1988 .

[9]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[10]  S. Prössdorf,et al.  On the stability of the spline collocation method for a class of integral equations of the first kind , 1988 .

[11]  A. Aziz.  The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations , 1972 .

[12]  D. Arnold,et al.  On the Asymptotic Convergence of Spline Collocation Methods for Partial Differential Equations , 1984 .

[13]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[14]  Experimental Convergence of Boundary Element Methods for the Capacity of the Electrified Square Plate , 1987 .

[15]  M. A. Jaswon,et al.  Integral equation methods in potential theory and elastostatics , 1977 .

[16]  Douglas N. Arnold,et al.  On the asymptotic convergence of collocation methods , 1983 .

[17]  L. S. Frank.  SPACES OF NETWORK FUNCTIONS , 1971 .

[18]  Wolfgang L. Wendland,et al.  ON SOME MATHEMATICAL ASPECTS OF BOUNDARY ELEMENT METHODS FOR ELLIPTIC PROBLEMS , 1985 .

[19]  Reinhold Schneider,et al.  Stability of a spline collocation method for strongly elliptic multidimensional singular integral equations , 1990 .

[20]  J. Aubin.  Approximation of Elliptic Boundary-Value Problems , 1980 .

[21]  G. Schmidt.  Spline collocation for singular integro-differential equations over (0.1) , 1986 .

[22]  Ernst P. Stephan,et al.  Boundary integral equations for screen problems in IR3 , 1987 .