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.