Scalable Block Preconditioners for the Parabolic-Elliptic Bidomain coupling

Scacchi, Simone

Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 3, pp. 699-714

Résumé

We review some results on parallel multilevel block preconditioners for the Bidomain model of electrocardiology. This model describes the bioelectric activity of the cardiac tissue in terms of the transmembrane electric potential $v$ and the extracellular electric potential $u_{e}$ and it consists of a system of a parabolic non-linear partial differential equation (PDE) for $v$ and an elliptic linear PDE for $u_{e}$. The two PDEs are coupled with a system of ordinary differential equations, modeling the cellular membrane ionic currents. The space and time discretization of the Bidomain system yields at each time step the solution of large and ill-conditioned linear systems. We analyze here the scalability of Multilevel Schwarz Block-Diagonal and Block-Factorized preconditioners for the discrete Bidomain system. New three-dimensional parallel numerical tests on a Linux cluster are performed to compare the Multilevel Schwarz block preconditioners with Block Jacobi (BJ) and Algebraic Multigrid (AMG) preconditioners. The results show that the preconditioners developed are scalable and more efficient than both BJ and AMG preconditioners.