Reliable finite element methods for self-adjoint singular perturbation problems

Abstract

It is well known that the standard finite element method based on the space Vh of continuous piecewise linear functions is not reliable in solving singular perturbation problems. It is also known that the solution of a two-point boundary-value singular perturbation problem admits a decomposition into a regular part and a finite linear combination of explicit singular functions. Taking into account this decomposition, first, we design a finite element method (which we call Singular Function Method) where the space of trial and test functions is the direct sum of Vh and the space spanned by these singular functions. The second method, based on the finite element discretization on a suitably redefined mesh, is referred to as Mesh Refinement Method. Both of these methods are proved to be e-uniformly convergent. Numerical examples which confirm the theory are presented.