  The bidomain model is widely used in electro-cardiology to simulate spreading
of excitation in the myocardium and electrocardiograms. It consists of a system
of two parabolic reaction diffusion equations coupled with an ODE system. Its
discretisation displays an ill-conditioned system matrix to be inverted at each
time step: simulations based on the bidomain model therefore are associated
with high computational costs. In this paper we propose a preconditioning for
the bidomain model either for an isolated heart or in an extended framework
including a coupling with the surrounding tissues (the torso). The
preconditioning is based on a formulation of the discrete problem that is shown
to be symmetric positive semi-definite. A block $LU$ decomposition of the
system together with a heuristic approximation (referred to as the monodomain
approximation) are the key ingredients for the preconditioning definition.
Numerical results are provided for two test cases: a 2D test case on a
realistic slice of the thorax based on a segmented heart medical image
geometry, a 3D test case involving a small cubic slab of tissue with
orthotropic anisotropy. The analysis of the resulting computational cost (both
in terms of CPU time and of iteration number) shows an almost linear complexity
with the problem size, i.e. of type $n\log^\alpha(n)$ (for some constant
$\alpha$) which is optimal complexity for such problems.
