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A scalable Newton-Krylov-Schwarz method for the bidomain reaction-diffusion system. (English) Zbl 1205.65261

The paper is an extension of the papers by L. F. Pavarino and S. Scacchi [ibid. 31, 420–445 (2008; Zbl 1185.65179)] and M. Munteanu and L. F. Pavarino [Math. Models Appl. Sci. 19, 1065–1097 (2009; Zbl 1178.65116)].
The authors consider the solution of the bidomain reaction-diffusion system in three dimensions. The bidomain system consists of two degenerate parabolic reaction-diffusion equations coupled with a stiff system of ordinary differential equations. For the discretization in space a finite element method is used. For the discretization in time an implicit scheme is applied which was studied by Munteanu and Pavarino [loc. cit.]. In each time step one has to solve a nonlinear finite element system. Hereby, an outer inexact Newton iteration is used. In each Newton step a Krylov method with a two-level overlapping Schwarz preconditioner is applied. It is shown that the condition number of the preconditioned operator is bounded by \(C(1+H/\delta)\), where \(C\) is a positive constant, \(H\) is the characteristic subdomain size und \(\delta\) the overlap size between the subdomains. The performance of the proposed method is demonstrated by several numerical examples. The computations are performed on parallel systems with up to 2048 processors. The results show the scalability of the presented solver.

MSC:

65M55 Multigrid methods; domain decomposition 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
35K57 Reaction-diffusion equations
65H10 Numerical computation of solutions to systems of equations
65F08 Preconditioners for iterative methods
65Y05 Parallel numerical computation
35K65 Degenerate parabolic equations

Software:

PETSc; NewtonLib
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