Munteanu, Marilena; Pavarino, Luca F. Decoupled Schwarz algorithms for implicit discretizations of nonlinear monodomain and bidomain systems. (English) Zbl 1178.65116 Math. Models Methods Appl. Sci. 19, No. 7, 1065-1097 (2009). The authors consider a 3D model for the electric activity of heart tissue, the bidomain system (consisting of 2 diffusion-reaction and several ordinary differential equations) and its simplification, the monodomain system (where the number of diffusion-reaction equations is reduced to 1). For the numerical solution they take the implicit Euler (resp. a third-order Rosenbrock) method for the discretization in time, and trilinear elements in space. This leads to the necessity to solve large nonlinear (resp. linear) systems of algebraic equations. Here, a Newton-Krylov domain decomposition (DD) (resp. a DD) preconditioner is used. The authors prove convergence estimates for the DD preconditioner in both situations and report on a great number of numerical experiments in which, between others, the decoupling of the ordinary differential equations from the diffusion-reaction equations and the use of a time step control shows to be advantageous. Reviewer: Gisbert Stoyan (Budapest) Cited in 1 ReviewCited in 16 Documents MSC: 65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs 35K57 Reaction-diffusion equations 35Q92 PDEs in connection with biology, chemistry and other natural sciences 92C50 Medical applications (general) 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65H10 Numerical computation of solutions to systems of equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65F08 Preconditioners for iterative methods Keywords:nonlinear reaction-diffusion systems; implicit Euler and Rosenbrock methods; domain decomposition; overlapping Schwarz preconditioners; monodomain and bidomain models; Newton-Krylov-Schwarz methods; finite element method; electric activity of heart tissue; convergence; numerical experiments; time step control Software:NewtonLib; PETSc PDFBibTeX XMLCite \textit{M. 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