A Virtual Element Method for a Nonlocal FitzHugh-Nagumo Model of Cardiac Electrophysiology

TL;DR

This paper develops a Virtual Element Method for simulating cardiac electrophysiology using a nonlocal FitzHugh-Nagumo model, enabling accurate solutions on complex polygonal meshes with proven convergence and error estimates.

Contribution

It introduces a novel VEM discretization for a nonlocal cardiac model, with rigorous convergence analysis and optimal error estimates on polygonal meshes.

Findings

Convergence of the VEM discretization is established.

Optimal order space-time error estimates are derived.

Numerical tests confirm theoretical results.

Abstract

We present a Virtual Element Method (VEM) for a nonlocal reaction-diffusion system of the cardiac electric field. To this system, we analyze an $H^1(\Omega)$-conforming discretization by means of VEM which can make use of general polygonal meshes. Under standard assumptions on the computational domain, we establish the convergence of the discrete solution by considering a series of a priori estimates and by using a general $L^p$ compactness criterion. Moreover, we obtain optimal order space-time error estimates in the $L^2$ norm. Finally, we report some numerical tests supporting the theoretical results.