×

Decoupled Schwarz algorithms for implicit discretizations of nonlinear monodomain and bidomain systems. (English) Zbl 1178.65116

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.

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

Software:

NewtonLib; PETSc
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1090/S0025-5718-98-00930-2 · Zbl 0896.65066 · doi:10.1090/S0025-5718-98-00930-2
[2] DOI: 10.1137/050634785 · Zbl 1114.65110 · doi:10.1137/050634785
[3] DOI: 10.1142/S0218202504003489 · Zbl 1068.92024 · doi:10.1142/S0218202504003489
[4] DOI: 10.1007/88-470-0396-2_6 · Zbl 1387.92056 · doi:10.1007/88-470-0396-2_6
[5] DOI: 10.1007/978-3-0348-8221-7_4 · doi:10.1007/978-3-0348-8221-7_4
[6] Deuflhard P., Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms (2004) · Zbl 1056.65051
[7] DOI: 10.1109/TNSRE.2005.847390 · doi:10.1109/TNSRE.2005.847390
[8] DOI: 10.1137/0915040 · Zbl 0802.65119 · doi:10.1137/0915040
[9] DOI: 10.1007/978-3-642-05221-7 · Zbl 1192.65097 · doi:10.1007/978-3-642-05221-7
[10] DOI: 10.1016/j.cma.2006.03.019 · Zbl 1173.76385 · doi:10.1016/j.cma.2006.03.019
[11] DOI: 10.1016/j.jcp.2003.08.010 · Zbl 1036.65045 · doi:10.1016/j.jcp.2003.08.010
[12] DOI: 10.1023/A:1021900219772 · Zbl 0996.65099 · doi:10.1023/A:1021900219772
[13] LeGrice I. J., Am. J. Physiol. (Heart Circ. Physiol.) 269 pp H571–
[14] DOI: 10.1007/s00205-002-0212-y · Zbl 1055.74041 · doi:10.1007/s00205-002-0212-y
[15] DOI: 10.1002/nme.958 · Zbl 1060.70500 · doi:10.1002/nme.958
[16] DOI: 10.1016/S0167-2789(03)00237-9 · Zbl 1029.92012 · doi:10.1016/S0167-2789(03)00237-9
[17] DOI: 10.1007/BF02241653 · Zbl 0692.65038 · doi:10.1007/BF02241653
[18] DOI: 10.1161/01.RES.68.6.1501 · doi:10.1161/01.RES.68.6.1501
[19] DOI: 10.1142/9789812708229_0021 · doi:10.1142/9789812708229_0021
[20] Munteanu M., Electron. Trans. Number. Anal. 30 pp 359–
[21] DOI: 10.1002/nla.381 · Zbl 1114.65112 · doi:10.1002/nla.381
[22] DOI: 10.1109/10.486288 · doi:10.1109/10.486288
[23] DOI: 10.1002/(SICI)1097-0312(200005)53:5<525::AID-CPA1>3.0.CO;2-M · Zbl 1021.65047 · doi:10.1002/(SICI)1097-0312(200005)53:5<525::AID-CPA1>3.0.CO;2-M
[24] DOI: 10.1137/070706148 · Zbl 1185.65179 · doi:10.1137/070706148
[25] DOI: 10.1137/040615249 · Zbl 1113.35019 · doi:10.1137/040615249
[26] Qu Z., IEEE Trans. Biomed. Engrg. 46 pp 1166–
[27] DOI: 10.1017/S0033583506004227 · doi:10.1017/S0033583506004227
[28] DOI: 10.1002/num.1000 · Zbl 1002.65100 · doi:10.1002/num.1000
[29] DOI: 10.1007/978-3-540-75199-1_79 · doi:10.1007/978-3-540-75199-1_79
[30] DOI: 10.1016/j.mbs.2005.01.001 · Zbl 1063.92018 · doi:10.1016/j.mbs.2005.01.001
[31] DOI: 10.1016/S0306-4522(99)00243-2 · doi:10.1016/S0306-4522(99)00243-2
[32] DOI: 10.1007/b137868 · doi:10.1007/b137868
[33] DOI: 10.1016/j.jcp.2003.11.014 · Zbl 1056.92014 · doi:10.1016/j.jcp.2003.11.014
[34] DOI: 10.1002/mma.740 · Zbl 1108.35090 · doi:10.1002/mma.740
[35] DOI: 10.1016/0010-4809(85)90003-5 · doi:10.1016/0010-4809(85)90003-5
[36] DOI: 10.1109/TBME.2002.804597 · doi:10.1109/TBME.2002.804597
[37] DOI: 10.1109/TBME.2004.834275 · doi:10.1109/TBME.2004.834275
[38] DOI: 10.1109/TBME.2006.879425 · doi:10.1109/TBME.2006.879425
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.