A matrix–free high–order solver for the numerical solution of cardiac electrophysiology

Abstract

We propose a matrix–free solver for the numerical solution of the cardiac electrophysiology model consisting of the monodomain nonlinear reaction–diffusion equation coupled with a system of ordinary differential equations for the ionic species. Our numerical approximation is based on the high–order Spectral Element Method (SEM) to achieve accurate numerical discretization while employing a much smaller number of Degrees of Freedom than first–order Finite Elements. We combine vectorization with sum–factorization, thus allowing for a very efficient use of high–order polynomials in a high performance computing framework. We validate the effectiveness of our matrix–free solver in a variety of applications and perform different electrophysiological simulations ranging from a simple slab of cardiac tissue to a realistic four–chamber heart geometry. We compare SEM to SEM with Numerical Integration (SEM–NI), showing that they provide comparable results in terms of accuracy and efficiency. In both cases, increasing the local polynomial degree p leads to better numerical results and smaller computational times than reducing the mesh size h. We also implement a matrix–free Geometric Multigrid preconditioner that results in a comparable number of linear solver iterations with respect to a state–of–the–art matrix–based Algebraic Multigrid preconditioner. As a matter of fact, the matrix–free solver proposed here yields up to 45× speed–up with respect to a conventional matrix–based solver.