Isogeometric overlapping Schwarz preconditioners for the Bidomain reaction–diffusion system

Abstract

We present and study an overlapping additive Schwarz preconditioner for the isogeometric discretization of a reaction–diffusion system modeling the heart bioelectrical activity, known as the Bidomain model, consisting of a degenerate system of parabolic and elliptic PDE. This model includes intramural fiber rotation, anisotropic conductivity coefficients and is coupled through the reaction term with a system of ODEs, which models the ionic currents of the cellular membrane. The overlapping Schwarz preconditioner is applied with a PCG accelerator to solve the linear system arising at each time step from the isogeometric discretization in space and a semi-implicit adaptive method in time. A theoretical convergence rate analysis shows that the resulting solver is scalable, optimal in the ratio of subdomain/element size and the convergence rate improves with increasing overlap size. Numerical tests in three-dimensional ellipsoidal domains confirm the theoretical estimates and additionally show the robustness with respect to jump discontinuities of the orthotropic conductivity coefficients.