On the Convergence of Collocation Methods for Boundary Integral Equations on Polygons

The integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions. Convergence proofs and error estimates for these approximation methods have been only available in the following cases: Fredholm integral equations of the second kind (4), (7), one-dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves (2), (3), (17), (26)-(29), and some special results on the classical Neumann integral equation of potential theory for polygonal plane domains (5), (8), (9). Here we give convergence proofs for collocation with piecewise linear trial functions for Neumann's integral equation and Symm's integral equation on plane curves with corners. We derive asymptotic error estimates in Sobolev norms and analyze the effect of graded meshes. 0. Introduction. In this paper we give convergence proofs and asymptotic error estimates in Sobolev norms for collocation with piecewise linear spline trial func- tions applied to two basic integral equations of potential theory on plane polygons, namely the integral equation of the second kind with the double layer potential ("Neumann's integral equation"), and the integral equation of the first kind with the simple layer potential ("Symm's integral equation"). We use an idea of Arnold and Wendland (2), namely considering Dirac delta functions (the "test functions" in the collocation method) as second derivatives of piecewise linear functions. Therefore, similar results as presented here should be possible for splines of higher odd order. Corresponding results for even-order splines are not yet available. Thus, for one of the simplest methods of numerically solving Dirichlet's problem on a plane domain with corners, the midpoint collocation with piecewise constant trial functions for the first-kind integral equation with the simple layer potential, convergence is still an open problem. The method of Fourier series that yields the convergence proof in the case of a smooth boundary (27) cannot be applied in the presence of corners. We apply the method of local Mellin transformation that has previously been used to derive error estimates for Galerkin methods for a wide class of operators, including those occurring in boundary element methods in acoustics, electromag-

[1]  Martin Costabel,et al.  On boundary integral equations of the first kind for the bi-laplacian in a polygonal plane domain , 1983 .

[2]  Boundary integral operators on curved polygons , 1983 .

[3]  J. Elschner.  Galerkin methods with splines for singular integral equations over (0, 1) , 1984 .

[4]  Siegfried Prössdorf,et al.  Ein Lokalisierungsprinzip in der Theorie der Spline‐Approximationen und einige Anwendungen , 1984 .

[5]  W. Wendland,et al.  Über die Näherungsweise Lösung von Linearen Funktionalgleichungen , 1967 .

[6]  D. Arnold,et al.  The convergence of spline collocation for strongly elliptic equations on curves , 1985 .

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

[8]  R. Taylor,et al.  The Numerical Treatment of Integral Equations , 1978 .

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

[10]  W. Wendland.  Boundary Element Methods and Their Asymptotic Convergence , 1983 .

[11]  P. Anselone,et al.  Collectively Compact Operator Approximation Theory and Applications to Integral Equations , 1971 .

[12]  Stefan Hildebrandt,et al.  Constructive proofs of representation theorems in separable Hilbert space , 1964 .

[13]  Martin Costabel,et al.  A direct boundary integral equation method for transmission problems , 1985 .

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

[15]  K. Atkinson,et al.  A survey of numerical methods for the solution of Fredholm integral equations of the second kind , 1977 .

[16]  S. Prössdorf,et al.  Finite element methods for singular integral equations on an interval , 1984 .

[17]  G. Schmidt.  On Spline Collocation for Singular Integral Equations , 1983 .

[18]  Ivo Babugka,et al.  A Finite Element Scheme for Domains with Corners , 2022 .

[19]  M. A. Krasnoselʹskii,et al.  Approximate Solution of Operator Equations , 1972 .

[20]  Ivan G. Graham,et al.  Product integration-collocation methods for noncompact integral operator equations , 1988 .

[21]  Siegfried Prössdorf,et al.  Projektionsverfahren und die näherungsweise Lösung singulärer Gleichungen , 1977 .

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

[23]  Martin Costabel,et al.  Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation , 1985 .

[24]  I. Gohberg,et al.  Convolution Equations and Projection Methods for Their Solution , 1974 .

[25]  Wolfgang L. Wendland,et al.  On the asymptotic convergence of collocation methods with spline functions of even degree , 1985 .