Finite Element Methods of High-Order Accuracy for Singular Two-Point Boundary Value Problems with Nonsmooth Solutions

Abstract

In this paper we analyze several finite element methods for singular two-point boundary value problems\[ - (x^\sigma u')' + qu = f,\quad 0 < x < 1.\] Here $\sigma \in [0,1)$, and appropriate boundary conditions are imposed. The solution can be approximated by splines on a nonuniform (“$\beta $ -graded”) mesh. Error bounds of optimal order are proved, and upper and lower bounds on the extent to which the mesh must be graded are obtained. We also consider approximating the solution by functions of the form $x^{ - \sigma } s(x),s(x)$ a spline. Error bounds and numerical results for these “weighted splines” indicate hat they are very efficient. For a third subspace, known error bounds are improved by using a mildly graded mesh.