Parallel Multilevel Schwarz and Block Preconditioners for the Bidomain Parabolic-Parabolic and Parabolic-Elliptic Formulations

Abstract

The aim of this work is to develop parallel multilevel and block preconditioners for the Bidomain model of electrocardiology. The Bidomain model describes the electrical activity of the heart tissue and consists of a system of two parabolic nonlinear partial differential equations (PDEs) of reaction-diffusion type (PP formulation) or alternatively of a system of a parabolic nonlinear PDE and an elliptic linear PDE (PE formulation). In both formulations, the PDEs are coupled with a system of ordinary differential equations, modeling the cellular membrane ionic currents. The first goal of the present study is to construct, analyze, and numerically test a multilevel additive Schwarz preconditioner for the PE formulation of the Bidomain model, extending previous results obtained for the PP formulation. Optimal convergence rate estimates are established and confirmed by 3D numerical test on Linux clusters. The second goal of the present study is to analyze the scalability of multilevel Schwarz block-diagonal and block-factorized preconditioners for both PP and PE formulations of the Bidomain model and to compare them with multilevel Schwarz coupled preconditioners. The 3D parallel numerical tests show that block preconditioners for the PP formulation are not scalable, while they are scalable for the PE formulation, but less efficient than the coupled preconditioners.