This paper reviews the basic mathematical ideas and convergence analysis of domain decomposition methods. These are parallel and scalable iterative methods for the efficient numerical solution of partial differential equations. Two examples are then presented showing the application of domain decomposition methods to large-scale numerical simulations in computational mechanics and electrocardiology.
Questo lavoro illustra le idee principali relative ai metodi di decomposizione dei domini e alla loro analisi di convergenza. Questi algoritmi sono dei metodi paralleli e scalabili per la risoluzione numerica efficiente di equazioni alle derivate parziali. Sono inoltre illustrati due esempi di applicazioni di metodi di decomposizione dei domini a simulazioni numeriche di grande scala in meccanica ed elettrocardiologia computazionale.
@article{BUMI_2005_8_8B_1_21_0,
author = {Luca F. Pavarino},
title = {Domain decomposition methods and scientific computing applications},
journal = {Bollettino dell'Unione Matematica Italiana},
volume = {8-A},
year = {2005},
pages = {21-54},
zbl = {1177.65181},
mrnumber = {2122974},
language = {en},
url = {http://dml.mathdoc.fr/item/BUMI_2005_8_8B_1_21_0}
}
Pavarino, Luca F. Domain decomposition methods and scientific computing applications. Bollettino dell'Unione Matematica Italiana, Tome 8-A (2005) pp. 21-54. http://gdmltest.u-ga.fr/item/BUMI_2005_8_8B_1_21_0/
[1] Special issue on Mapping and control of complex cardiac arrhythmias. Chaos12 (3), 2002.
[2] Special issue on From excitable media to virtual cardiac tissue. Chaos Solit. Frac.13 (8), 2002.
[3] Special issue on Biomedical and bioengineering computing. Comput. Visual. Sci.4 (4), 2003.
[4] et al., High resolution forward and inverse earthquake modeling on terascale computers. In Proceedings of SC2003 (Supercomputing Conference 2003), ACM/IEEE, 2003.
[5] - - , Implicit-explicit methods for timedependent partial differential equations, SIAM J. Numer. Anal., 32 (3) (1995), 797-823. | MR 1335656 | Zbl 0841.65081
[6] , Über Schwarzsche Algorithmen in partiellen Differentialgleichungen der mathematischen Physik, ZAMM, 37 (7/8) (1957), 243-245. | MR 99492 | Zbl 0078.30302
[7] - - - - - , PETSc home page, http://www.mcs.anl.gov/petsc, 2001.
[8] - - - , PETSc users manual, Technical Report ANL-95/11 - Revision 2.1.0, Argonne National Laboratory, 2001.
[9] - , Spectral Methods, in Handbook of Numerical Analysis, Volume V: Techniques of Scientific Computing (Part 2), North-Holland, 1997, 209-485. | MR 1470226
[10] - - - - - , Application of the FETI method to ASCI problems - scalability results on 1000 processors and discussion of highly heterogeneous problems, Int. J. Numer. Meth. Eng., 47 (1-3) (2000), 513-535. | Zbl 0970.74069
[11] - , Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, 15. Springer, New York, 1991. | MR 1115205 | Zbl 0788.73002
[12] - - - , Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988. | MR 917480 | Zbl 0717.76004
[13] - , Domain Decomposition Methods, Acta Numerica, (1994) 61-143. | MR 1288096 | Zbl 0809.65112
[14] - , A parallel solver for reaction-diffusion systems in computational electrocardiology, IMATI-CNR Tech. Rep. 9-PV, 2003, To appear in Math. Mod. Meth. Appl. Sci. | MR 2069498 | Zbl 1068.92024
[15] - , Degenerate evolution systems modeling the cardiac electric field at micro and macroscopic level. In and , Editors, Evolution equations, Semigroups and Functional Analysis, 49-78, Birkhauser, 2002. | MR 1944157 | Zbl 1036.35087
[16] - - , Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal., 31 (1994), 1662-1694. | MR 1302680 | Zbl 0818.65114
[17] - , Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems, Comm. Pure Appl. Math., 48 (1995), 121-155. | MR 1319698 | Zbl 0824.65106
[18] - , Implicit parallel processing in structural mechanics, Comput. Mech. Adv., 2 (1) (1994), 1-124. | MR 1280753 | Zbl 0805.73062
[19] - - - - - - - - , Preventing ventricular fibrillation by flattening cardiac restitution, Proc. Nat. Acad. Sci. USA, 97 (11) (2000), 6061-6066.
[20] - - , Balancing Neumann-Neumann Preconditioners for Mixed Approximations of Heterogeneous Problems in Linear Elasticity, Numer. Math., 95 (2) (2003), 283-324. | MR 2001079 | Zbl 1169.65346
[21] - - - , Editors, Proceedings of the Fourteenth International Conference on Domain Decomposition Methods, UNAM, Mexico City, 2003. | MR 2093729 | Zbl 1103.65004
[22] - , Mathematical Physiology, Springer-Verlag, 1998. | MR 1673204 | Zbl pre05242554
[23] , Terascale implicit methods for partial differential equations. In and , Editors, Recent Advances in Numerical Methods for Partial Differential Equations and Applications, Contemporary Mathematics, 306, AMS (2002), 29-84. | MR 1940622 | Zbl 1018.65121
[24] - - - - - , Editors, Proceedings of the Fifteenth International Conference on Domain Decomposition Methods, Lecture Notes in Computational Science and Engineering, Springer, to appear, 2004. | Zbl 1067.65050
[25] , Domain decomposition methods in computational mechanics, Comput. Mech. Adv., 1 (2) (1994), 121-220. | MR 1263805 | Zbl 0802.73079
[26] , On the Schwarz alternating method. I. In First International Symposium on Domain Decomposition Methods for Partial Differential Equations, et al., Editors, pp. 1-42, SIAM, Philadelphia, 1988. | MR 972510 | Zbl 0658.65090
[27] - , A model of the ventricular cardiac action potential: depolarization, repolarization, and their interaction, Circ. Res., 68 (6) (1991), 1501-1526.
[28] - , Balancing domain decomposition for problems with large jumps in coefficients, Math. Comp., 65 (1996), 1387-1401. | MR 1351204 | Zbl 0853.65129
[29] , An adaptive finite element method for parabolic differential systems: some algorithmic considerations in solving in three space dimensions, SIAM J. Sci. Comput., 21 (4) (2000), 1567-1586. | MR 1756044 | Zbl 0969.65090
[30] - , A polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions, SIAM J. Numer. Anal., 33 (4) (1996), 1303-1335. | MR 1403547 | Zbl 0856.41007
[31] , Neumann-Neumann algorithms for spectral elements in three dimensions, RAIRO M²AN, 31 (1997), pp. 471-493. | MR 1457457 | Zbl 0881.65121
[32] - , Editors, Recent Developments in Domain Decomposition Methods, Lecture Notes in Computational Science and Engineering, vol. 23, Springer-Verlag, 2002. | MR 1962677 | Zbl 0989.00043
[33] - , Iterative substructuring methods for spectral element discretizations of elliptic systems. II: Mixed methods for linear elasticity and Stokes flow, SIAM J. Numer. Anal., 37 (2000), 375-402. | MR 1740774 | Zbl 0951.65123
[34] - , Balancing Neumann-Neumann methods for incompressible Stokes equations, Comm. Pure Appl. Math., 55 (3) (2002), 302-335. | MR 1866366 | Zbl 1024.76025
[35] - , Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, 1994. | MR 1299729 | Zbl 1151.65339
[36] - , Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications, 1999. | MR 1857663 | Zbl 0931.65118
[37] , Gesammelte Mathematische Abhandlungen, volume 2, pp. 133-143, Springer, 1890.
[38] - - , Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, 1996. | MR 1410757 | Zbl 0857.65126
[39] , Lalgorithme de Schwarz dans la théorie de lelasticité, Compt. Rend. Acad. Sci. URSS, IV (XIII 6) (1936), 243-246. | Zbl 0015.40503
[40] - , Domain Decomposition Methods: Algorithms and Theory, Springer-Verlag, 2004. | Zbl 1069.65138
[41] - , Fast parallel direct solvers for coarse grid problems, J. Paral. Distr. Comput., 61 (2) (2001), 151-177. | Zbl 0972.68191
[42] , Electrical turbulence in three-dimensional heart muscle, Science, 266 (5187) (1994), 1003-1006.
[43] , A local space-time adaptive scheme in solving two-dimensional parabolic problems based on domain decomposition methods, SIAM J. Sci. Comput., 23 (1) (2001), 304-322. | MR 1860916 | Zbl 0990.65108