The cardiac Bidomain model consists in a reaction-diffusion system of PDEs for the intra- and extra-cellular cardiac potentials coupled with a nonlinear system of ODEs accounting for the cellular model of ionic currents. Fully implicit methods in time have been considered in a few studies, see e.g. [16] and references therein. As in most of previous work (see [18] for a review), in this study we consider an Implicit-Explicit operator splitting technique in order to separate the part of the system of PDEs describing diffusion of cardiac potentials from the large and stiff nonlinear system of ODEs accounting for the reaction terms. The resulting space-time discretization of the so-called parabolic-parabolic Bidomain operator leads to a large, sparse, symmetric positive semidefinite linear system which must be solved at each time step of a cardiac beat simulation using a Krylov subspace method. Given a component by component finite element discretization of the cardiac potentials, the coefficient matrix of the linear system to be solved is K=[A_i 0 0 A_e]+X/(δ_t)[(M-M/(-M M))] (1) where δ_t is the value of the time step and X the membrane capacitance per unit volume; M and A_(i,e) are the mass and stiffness matrices with entries {M}_(rs)=∫_Ω Φ_h~rΦ_h~s, {A_(i,e)}_(rs)=∫_ΩD_(i,e)▽_(Φ_h~r)·▽_(Φ_h~s), where for sake of simplicity the same finite element basis {Φ_h~j} is considered for each cardiac potential. Anisotropic conductivity tensors D_i(x) and D_e(x) model propagation of electrical signals with orthotropic anisotropy D_(i,e)(x)=3 ∑ j-1σ_i~(i,e)(x)a_j(x)a_j(x)~T, with σ_j~(i,e)(x) ＞ 0 the conductivity coefficient of the intra- and extra-cellular media measured along the orthonormal triplet {a_j(x)}_(j=1)~3, describing cardiac fiber rotation. For additional details on the operator splitting technique adopted and the diffusion tensors, see.
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