Exponential Convergence of a Generalized FEM for Heterogeneous Reaction-Diffusion Equations
Chupeng Ma, Jens Markus Melenk · Multiscale Modeling and Simulation · 2024
Abstract. A generalized finite element method (FEM) is proposed for solving a heterogeneous reaction-diffusion equation with a singular perturbation parameter [Formula: see text], based on locally approximating the solution on each subdomain by solution of a local reaction-diffusion equation and eigenfunctions of a local eigenproblem. These local problems are posed on some domains slightly larger than the subdomains with oversampling size [Formula: see text]. The method is formulated at the continuous level as a direct discretization of the continuous problem and at the discrete level as a coarse-space approximation for its standard finite element (FE) discretizations. Exponential decay rates for local approximation errors with respect to [Formula: see text] and [Formula: see text] (at the discrete level with [Formula: see text] denoting the fine FE mesh size) and with the local degrees of freedom are established. In particular, it is shown that the method at the continuous level converges uniformly with respect to [Formula: see text] in the standard [Formula: see text] norm, and that if the oversampling size is relatively large with respect to [Formula: see text] and [Formula: see text] (at the discrete level), the solutions of the local reaction-diffusion equations provide good local approximations for the solution and thus the local eigenfunctions are not needed. Numerical results are provided to verify the theoretical results.