The finite volume element method for diffusion equations on general triangulations
SIAM Journal on Numerical Analysis
Preconditioning techniques for large linear systems: a survey
Journal of Computational Physics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Adaptivity in Space and Time for Reaction-Diffusion Systems in Electrocardiology
SIAM Journal on Scientific Computing
Parallel black box $$\mathcal {H}$$-LU preconditioning for elliptic boundary value problems
Computing and Visualization in Science
Semi-Implicit Time-Discretization Schemes for the Bidomain Model
SIAM Journal on Numerical Analysis
Multilevel Additive Schwarz Preconditioners for the Bidomain Reaction-Diffusion System
SIAM Journal on Scientific Computing
A model-based block-triangular preconditioner for the Bidomain system in electrocardiology
Journal of Computational Physics
STACOM'11 Proceedings of the Second international conference on Statistical Atlases and Computational Models of the Heart: imaging and modelling challenges
STACOM'12 Proceedings of the third international conference on Statistical Atlases and Computational Models of the Heart: imaging and modelling challenges
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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 nlog^@a(n) (for some constant @a) which is optimal complexity for such problems.