Optimal convergence rates of an adaptive finite element method for unbounded domains

TL;DR

This paper develops an adaptive finite element method with a residual-based error estimator for unbounded reaction-diffusion problems, achieving optimal convergence rates by adaptively refining the mesh and truncation boundary.

Contribution

It introduces a novel error estimator that accounts for truncation effects and proves the convergence and optimality of the adaptive algorithm for unbounded domains.

Findings

The estimator is reliable and efficient under certain conditions.

The adaptive algorithm converges and attains optimal rates.

Numerical examples confirm theoretical results.

Abstract

We consider linear reaction-diffusion equations posed on unbounded domains, and discretized by adaptive Lagrange finite elements. To obtain finite-dimensional spaces, it is necessary to introduce a truncation boundary, whereby only a bounded computational subdomain is meshed, leading to an approximation of the solution by zero in the remainder of the domain. We propose a residual-based error estimator that accounts for both the standard discretization error as well as the effect of the truncation boundary. This estimator is shown to be reliable and efficient under appropriate assumptions on the triangulation. Based on this estimator, we devise an adaptive algorithm that automatically refines the mesh and pushes the truncation boundary towards infinity. We prove that this algorithm converges and even achieves optimal rates in terms of the number of degrees of freedom. We finally provide numerical examples illustrating our key theoretical findings.