Non-symmetric Algebraic Multigrid Preconditioners for the Bidomain Reaction–Diffusion system

We deal with the efficient solution of the so-called bidomain system which is possibly the most complete model for the cardiac bioelectric activity. We study the performance of a non-symmetric structured algebraic multigrid preconditioner on the formulation generally used of the bidomain model, i.e. the one characterized by a parabolic equation coupled with an elliptic one. Our numerical results show that, for this formulation, the non-symmetric preconditioner provides the best overall performance. 1 The Bidomain Model The excitation process in the myocardium is a complex phenomenon characterized by rapid ionic fluxes through the cellular membrane separating the intracellular and the interstitial fluid in the myocardium [7]. The bidomain is the most complete model for the cardiac bioelectric activity, and it consists of a non-linear Reaction-Diffusion (R-D) system of equations for the intraand extracellular potential ui and ue, coupled through the transmembrane potential v := ui−ue [9]. The nonlinearity arises through the current-voltage relationship across the membrane which is described by a set of nonlinear ODEs (see [7]). The anisotropic properties of the media are modeled by the intraand extracellular conductivity tensors Mi = Mi(x) and Me = Me(x) that satisfy a uniform ellipticity condition, see [10]. Micol Pennacchio Istituto di Matematica Applicata e Tecnologie Informatiche del CNR, via Ferrata, 1, I-27100 Pavia, Italy, e-mail: [email protected] Valeria Simoncini Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5, I-40127 Bologna, Italy; CIRSA, Ravenna and IMATI-CNR, Pavia, Italy e-mail: [email protected]