Mardal, Kent-Andre; Nielsen, Bjørn Fredrik; Cai, Xing; Tveito, Aslak An order optimal solver for the discretized bidomain equations. (English) Zbl 1199.65111 Numer. Linear Algebra Appl. 14, No. 2, 83-98 (2007). Summary: The electrical activity in the heart is governed by the bidomain equations. In this paper, we analyse an order optimal method for the algebraic equations arising from the discretization of this model. Our scheme is defined in terms of block Jacobi or block symmetric Gauss-Seidel preconditioners. Furthermore, each block in these methods is based on standard preconditioners for scalar elliptic or parabolic partial differential equations. Such preconditioners can be realized in terms of multigrid or domain decomposition schemes, and are thus readily available by applying ‘off-the-shelves’ software. Finally, our theoretical findings are illuminated by a series of numerical experiments. Cited in 16 Documents MSC: 65F10 Iterative numerical methods for linear systems 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 92C05 Biophysics 65F08 Preconditioners for iterative methods 35Q92 PDEs in connection with biology, chemistry and other natural sciences Keywords:bidomain model; preconditioning; multigrid; bidomain equations; block Jacobi; block symmetric Gauss-Seidel; domain decomposition; numerical experiments PDFBibTeX XMLCite \textit{K.-A. Mardal} et al., Numer. Linear Algebra Appl. 14, No. 2, 83--98 (2007; Zbl 1199.65111) Full Text: DOI References: [1] . Mathematical Physiology. Springer: New York, 1998. · Zbl 0913.92009 [2] http://www.cellml.org/ [3] , , , , . Computing the Electrical Activity in the Human Heart, Volume 1, Monographs in Computational Science and Engineering. Springer: Berlin, 2004. [4] Qu, IEEE Transactions on Biomedical Engineering 46 pp 1166– (1999) [5] Sundnes, Computer Methods in Biomechanics and Biomedical Engineering 5 pp 397– (2002) [6] Winslow, Physica D 64 pp 281– (1993) [7] , . A second order scheme for solving the coupled bidomain and forward problem. 2002. [8] Bank, Mathematics of Computation 36 pp 35– (1981) [9] Multigrid Methods. Pitman Research Notes in Mathematical Sciences, vol. 294. Longman Scientific & Technical: Essex, England, 1993. [10] Olshanskii, Computing 65 pp 193– (2000) [11] Cai, Numerische Mathematik 60 pp 41– (1991) [12] . Domain decomposition algorithms. Acta Numerica 1994; 61–143. · Zbl 0809.65112 [13] Arnold, Mathematical Modelling and Numerical Analysis 31 pp 517– (1997) [14] Elman, International Journal for Numerical Methods in Fluids 40 pp 333– (2002) [15] Mardal, Numerische Mathematik 98 pp 305– (2004) [16] Rusten, SIAM Journal on Matrix Analysis and Applications 13 pp 887– (1992) [17] Bai, Advances in Computational Mathematics 10 pp 169– (1999) [18] Iterative Solution of Large Sparse Systems of Equations. Springer: New York, 1994. · doi:10.1007/978-1-4612-4288-8 [19] Computer Solution of Large Linear Systems. Elsevier: Amsterdam, 1999. [20] Hooke, Mathematical Biosciences 120 pp 127– (1994) [21] A Survey of Preconditioned Iterative Methods. Addison-Wesley/Pitman: Reading/London, 1995. · Zbl 0834.65014 [22] Xu, SIAM Review 34 pp 581– (1992) [23] Iterative Methods for Sparse Linear Systems. PWS Publishing Company: Massachusetts, 1996. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.