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

TL;DR

The paper develops the Multiscale Spectral Generalized Finite Element Method (MS-GFEM) for solving heterogeneous, singularly perturbed reaction-diffusion equations. It constructs local approximation spaces from eigenfunctions of local spectral problems on oversampling domains and analyzes both continuous and discrete formulations, proving exponential convergence of local errors with respect to oversampling and local degrees of freedom, while achieving uniform convergence in the perturbation parameter $ε$ in the continuous setting. In the preasymptotic discrete regime, the error is governed by the ratio $δ^{*}/h$ rather than $δ^{*}/ε$, with the spectral basis providing further improvements under certain conditions. Numerical experiments corroborate the theory, showing strong parameter-robust performance and substantial reductions in degrees of freedom, validating the method’s practicality for complex, multiscale diffusion with sharp interior or boundary layers.

Abstract

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 $δ^{\ast}$. 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 $δ^{\ast}/\varepsilon$ and $δ^{\ast}/h$ (at the discrete level with $h$ 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 $\varepsilon$ in the standard $H^{1}$ norm, and that if the oversampling size is relatively large with respect to $\varepsilon$ and $h$ (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.

Figures (4)

• Figure 1: Illustration of a subdomain $\omega_i$ that lies within the interior of $\Omega$ (left) and one that intersects the outer boundary $\partial \Omega$ (right) with associated oversampling domains $\omega_i^{\ast}$.
• Figure 1: The diffusion coefficient $A({\bm x})$.
• Figure 2: Plots of $\Vert u_{h}-u_{h}^{G}\Vert_{a,\varepsilon}$ against $n_{\rm loc}$ (the number of local basis functions used per subdomain) for $\varepsilon=10^{-1}$ and $\varepsilon=10^{-4}$.
• Figure 3: The reference solution (left) and the global particular solution (right) for the problem with $\varepsilon=10^{-1}$ (top) and $\varepsilon=10^{-3}$ (down).

Theorems & Definitions (41)

• Theorem 2.2: ma2022novel
• Definition 3.1: Local particular functions
• Theorem 3.3
• Lemma 3.4
• Lemma 3.5
• Proof 1
• Definition 3.6: Local approximation spaces
• Theorem 3.7
• Theorem 3.8
• Lemma 3.9