The K-Operator and the Galerkin Method for Strongly Elliptic Equations on Smooth Curves: Local Estimates

Abstract

Superconvergence in the L2-norm for the Galerkin approximation of the integral equation Lu = f is studied, where I is a strongly elliptic pseudodifferential operator on a smooth, closed or open curve. Let Uf, be the Galerkin approximation to u . By using the ^-operator, an operator that averages the values of uh , we will construct a better approximation than uh itself. That better approximation is a legacy of the highest order of convergence in negative norms. For Symm's equation on a slit the same order of convergence can be recovered if the mesh is suitably graded. In this paper we shall study a way of increasing the order of local convergence in the L2-norm for the Galerkin approximation to the solution of strongly ellip- tic pseudodifferential equations on a smooth, closed or open curve in R2. This kind of integral equation is of importance in solving interior or exterior bound- ary value problems of potential theory. The most common example is Symm's first-kind integral equation with logarithmic kernel (see (6, 7, 12)). Other appli- cations are hypersingular integral equations and singular integral equations of Cauchy type, which occur, e.g., in elasticity (see (18)). For general Petrov-Galerkin methods when smoothest splines are used as trial and test functions, local error estimates were proved in (11) for smooth closed curves and in (16) for smooth open curves. Consider, for example, Symm's equation. With piecewise constant functions used as trial and test functions, it was proved that the local L2-error converges with order 0(h) in the case of smooth closed curves (11) and with order 0(hxi2) in the case of smooth open curves (16). However, it is well known that the highest orders of global convergence achieved (in negative norms) are 0(h3) for the closed smooth case (5) and 0(h) for the open smooth case (4, 13). The purpose of this article is to construct, from the Galerkin solution, a better approximate solution which inherits the highest possible orders of global convergence to give best local con- vergence in the L2-norm (e.g., in the example mentioned above, order 0(hy) for the closed case and 0(h) for the open case can be achieved locally in the L2-norm). That better approximation is constructed by averaging the values of the Galerkin solution, using the K-operator.