A Space–Time Finite Element Method for the Linear Bidomain Equations

Abstract

In this work, we study a Galerkin–Petrov space–time finite element method for a linear system of parabolic–elliptic equations with in general anisotropic conductivity matrices , which may be considered as a simplified version of the nonlinear bidomain equations . The discretization is based on a stable space–time variational formulation employing continuous and piecewise linear finite elements in both spatial and temporal directions simultaneously. We show stability of the space–time formulation on both the continuous and discrete level for such a coupled problem under a rather general condition on the conductivity matrices. We further discuss the construction of a monolithic algebraic multigrid (AMG) method for solving the coupled linear system of algebraic equations globally. Numerical experiments are performed to demonstrate the convergence of the space–time finite element approximations, and the performance of the AMG method with respect to the mesh discretization parameter. Finally, we apply the space–time finite element method to the nonlinear bidomain equations in order to show the applicability of the proposed approach.