Anisotropic space-time adaptation for reaction-diffusion problems

TL;DR

This paper introduces an anisotropic space-time adaptive method with a residual error estimator for reaction-diffusion problems, including the monodomain model, demonstrating improved efficiency through numerical verification.

Contribution

It develops a novel anisotropic error estimator and an adaptive algorithm for reaction-diffusion problems, addressing challenges in coupled PDE-ODE systems with minimal regularity assumptions.

Findings

Estimator is reliable under mild assumptions

Adaptive algorithm improves efficiency over uniform meshes

Numerical tests confirm effectiveness of the approach

Abstract

A residual error estimator is proposed for the energy norm of the error for a scalar reaction-diffusion problem and for the monodomain model used in cardiac electrophysiology. The problem is discretized using $P_1$ finite elements in space, and the backward difference formula of second order (BDF2) in time. The estimator for space makes use of anisotropic interpolation estimates, assuming only minimal regularity. Reliability of the estimator is proven under certain mild assumptions on the convergence of the approximate solution. The monodomain model couples a nonlinear parabolic partial differential equation (PDE) with an ordinary differential equation (ODE) and this setting presents challenges theoretically as well as numerically. A space-time adaptation algorithm is proposed to control the global error, using a non-Euclidean metric for mesh adaptation and a simple method to adjust the time step. Numerical examples are used to verify the reliability and efficiency of the estimator, and to test the adaptive algorithm. The potential gains in efficiency of the proposed algorithm compared to methods using uniform meshes is discussed.