Parallel multilevel solvers for the cardiac electro-mechanical coupling

Abstract

We develop a parallel solver for the cardiac electro-mechanical coupling. The electric model consists of two non-linear parabolic partial differential equations (PDEs), the so-called Bidomain model, which describes the spread of the electric impulse in the heart muscle. The two PDEs are coupled with a non-linear elastic model, where the myocardium is considered as a nearly-incompressible transversely isotropic hyperelastic material. The discretization of the whole electro-mechanical model is performed by Q1 finite elements in space and a semi-implicit finite difference scheme in time. This approximation strategy yields at each time step the solution of a large scale ill-conditioned linear system deriving from the discretization of the Bidomain model and a non-linear system deriving from the discretization of the finite elasticity model. The parallel solver developed consists of solving the linear system with the Conjugate Gradient method, preconditioned by a Multilevel Schwarz preconditioner, and the non-linear system with a Newton–Krylov-Algebraic Multigrid solver. Three-dimensional parallel numerical tests on a Linux cluster show that the parallel solver proposed is scalable and robust with respect to the domain deformations induced by the cardiac contraction.