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 V_h of continuous piecewise linear functions is not reliable in solving singular perturbation problems. It is also known that the solution of a two-point boundaryvalue 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 V_h and the space spanned by these singular functions. The second method, based on the finite element discretization on a suitably refined mesh, is referred to as mesh refinement method. Both of these methods are proved to be ε-uniformly convergent. Numerical examples which confirm the theory are presented.

Quaestiones Mathematicae 32(2009), 397–413