Ritz–Galerkin Methods for Singular Boundary Value Problems

Abstract

This paper is concerned with the application of the Ritz–Galerkin method to the numerical solution of singular boundary value problems of the type arising when Poisson’s equation on, a domain with cylindrical or spherical symmetry is reduced to a one-dimensional problem. The objective is to derive a priori $L_2 $- and $L_\infty $-norm estimates for the error. The difficulty is that these norms are not natural norms for the reduced problem. With the aid of B-splines we prove some nonstandard approximation-theoretic results and use these to derive the desired error estimates. Some numerical results are presented.