Exponential convergence of a generalized FEM for heterogeneous reaction-diffusion equations

Understand

A generalized finite element method is proposed for solving a heterogeneous reaction-diffusion equation with a singular perturbation parameter $\varepsilon$, 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 $\delta^{\ast}$.
• 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 FE discretizations.
• Exponential decay rates for local approximation errors with respect to $\delta^{\ast}/\varepsilon$ and $\delta^{\ast}/h$ (at the discrete level with $h$ denoting the fine FE mesh size) and with the local degrees of freedom are established.