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A model-based block-triangular preconditioner for the bidomain system in electrocardiology. (English) Zbl 1187.92053
Summary: We introduce a preconditioner for the solution of the bidomain system governing the propagation of action potentials in the myocardial tissue. The bidomain model is a degenerate parabolic set of nonlinear reaction-diffusion equations. The nonlinear term describes the ion flux at the cellular level. The degenerate nature of the problem results in a severe ill conditioning of its discretization. Our preconditioning strategy is based on a suitable adaptation of the monodomain model, a simplified version of the bidomain one, which is by far simpler to solve, nevertheless is unable to capture significant features of the action potential propagation. The monodomain preconditioner application to a non-symmetric formulation of the bidomain system results at the algebraic level in a lower block-triangular preconditioner. We prove optimality of the preconditioner with respect to the mesh size, and corroborate our theoretical results with 3D numerical simulations both on idealized and real ventricle geometries.

MSC:
92C50 Medical applications (general)
92C05 Biophysics
35K57 Reaction-diffusion equations
92C55 Biomedical imaging and signal processing
35Q92 PDEs in connection with biology, chemistry and other natural sciences
65T50 Numerical methods for discrete and fast Fourier transforms
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LifeV
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