Homogenization and nonlinear CVFE Scheme for a Physiological Anisotropic Model of Electrocardiology

The bidomain and monodomain models are widely used models in simulating cardiac electrical activity. In this talk, we first briefly describe the unfolding homogenization approach to rigorously derive the bidomain equations from a microscopic model with tensorial and space dependent conductivities . Secondly, we present a positive nonlinear control volume finite element (CVFE) scheme, based on Godunov's flux approximation of the diffusion term, for the monodomain model coupled to a physiological ionic model (Beeler-Reuter model) and using an anisotropic diffusion tensor. In this scheme, degrees of freedom are assigned to vertices of a primal triangular mesh, as in finite element methods. The diffusion term which involves an anisotropic tensor is discretized on a dual mesh using the diffusion fluxes provided by the conforming finite element reconstruction on the primal mesh. The scheme ensures the validity of the discrete maximum principle without any restrictions on the transmissibility coefficients. The convergence of the scheme is proved using a compactness argument. Finally, the efficiency of the proposed scheme is illustrated by showing some numerical results.