Galerkin methods with splines for singular integral equations over (0, 1)

Abstract

Summary. In this paper a convergence analysis of Galerkin methods with splines for strongly elliptic singular integral equations over the interval (0, 1) is given. As trial functions we utilize smoothest polynomial splines on arbitrary meshes and continuous splines on special nonuniform partitions, multiplied by a weight function. Using inequalities of Ggtrding type for singular integral operators in weighted L 2 spaces and the complete asymptotics of solutions at the endpoints, we provide error estimates in certain Sobolev norms.

On spline Galerkin methods for singular integral equations with piecewise continuous coefficients

“…We show for instance, that an asymptotic order of convergence of O(n/-a-l) can be achieved on special nonuniform partitions. This fact was obtained in [28] and [9] by using weighted continuous splines on nonuniform meshes for continuous coefficients a and b and F=(0, 1).…”

“…As a corollary to our proof, which is based on certain localization techniques and on the method of associated operators (see [19]), we obtain that the strong ellipticity of A is sufficient for the stability of the collocation method with piecewise linear splines. Section 3 deals with estimating the error of the Galerkin method (0.3) by means of utilizing the complete asymptotics of the solutions of (0.1) which has been established in [9]. We show for instance, that an asymptotic order of convergence of O(n/-a-l) can be achieved on special nonuniform partitions.…”

“…They were then generalized by Schmidt in [-25]. For F a finite interval, convergence and error analysis for Galerkin's method with (smooth as well as weighted) splines has recently been developed by Elschner [9] for the case of continuous c and d such that Re {c(t)+d(t)} >0 on F. The special case where K is real-valued, c-1 and d is purely imaginary has earlier been considered by Thomas [28].…”

Spline collocation for singular integro-differential equations over (0.1)

“…The obtained numerical results show a high efficiency of these methods, but up to now no convergence results have been proved. A first rigorous analysis for the L2-convergence of Galerkin methods with splines of arbitrary degree was given by Elschner [2]. He proved in particular that Galerkin's method for (1.2) converges in L z if the corresponding operator is invertible and strongly elliptic, i.e., the coefficients satisfy a,,(x)+2bm(x)#O, x~[0,1], 2~ [-1,1].…”

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

“…Spline approximation in weighted Sobolev spaces has been studied by several authors [6], [9], [15], but the kind of result needed here is not available. We assume that u g H~1/2(Y) is the solution of the equation Vu = f with / smooth enough.…”