Error estimates and extrapolation for the numerical solution of Mellin convolution equations

Abstract

In this paper we consider a quadrature method for the numerical solution of a second-kind integral equation over the interval, where the integral operator is a compact perturbation of a Mellin convolution operator. This quadrature method relies upon a singularity subtraction and transformation technique. Stability and convergence order of the approximate solution are well known. We shall derive the first term in the asymptotics of the error which shows that, in the interior of the interval, the approximate solution converges with higher order than over the whole interval. This implies higher orders of convergence for the numerical calculation of smooth functionals to the exact solution. Moreover, the asymptotics allows us to define a new approximate solution extrapolated from the dilated solutions of the quadrature method over meshes with different mesh sizes. This extrapolated solution is designed to improve the low convergence order caused by the non-smoothness of the exact solution even when the transformation technique corresponds to slightly graded meshes. Finally, we discuss the application to the double-layer integral equation over the boundary of polygonal domains and report numerical results.