Convergence Analysis of a Multiscale Finite Element Method for Singularly Perturbed Problems

Abstract. In this paper we perform an error analysis for a multiscale finite element method for singularly perturbed reaction–diffusion equation. Such method is based on enriching the usual piecewise linear finite element trial spaces with local solutions of the original problem, but do not require these functions to vanish on each element edge. Bubbles are the choice for the test functions allowing static condensation, thus our method is of Petrov–Galerkin type. We perform convergence analysis in different asymptotic regimes, and we show uniform convergence in an appropriate norm with respect to the small parameter. Numerical results show that the new method is able to compute solutions even on coarse meshes.