A virtual element method for a nonlocal Fitzhugh-Nagumo model of cardiac electrophysiology. (English) Zbl 1466.65116
Summary: We present a virtual element method (VEM) for a nonlocal reaction-diffusion system of the cardiac electric field. For this system, we analyze an \(H^1\)-conforming discretization by means of VEM that 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.
MSC:
65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
76Z05 | Physiological flows |
92C35 | Physiological flow |
35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
35Q35 | PDEs in connection with fluid mechanics |