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.
Referenze Bibliografiche
[1] Special issue on Mapping and control of complex cardiac arrhythmias. Chaos 12 (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] V. AKCELIK et al., High resolution forward and inverse earthquake modeling on terascale computers. In Proceedings of SC2003 (Supercomputing Conference 2003), ACM/IEEE, 2003.
[5]
O. M. ASCHER -
S. J. RUUTH -
B. T. R. WETTON,
Implicit-explicit methods for timedependent partial differential equations,
SIAM J. Numer. Anal.,
32 (3) (
1995), 797-823. |
MR 1335656 |
Zbl 0841.65081[6]
I. BABUŠKA,
Über Schwarzsche Algorithmen in partiellen Differentialgleichungen der mathematischen Physik,
ZAMM,
37 (7/8) (
1957), 243-245. |
MR 99492 |
Zbl 0078.30302[7] S. BALAY - K. BUSCHELMAN - W. D. GROPP - D. KAUSHIK - L. CURFMAN MCINNES - B. F. SMITH, PETSc home page, http://www.mcs.anl.gov/petsc, 2001.
[8] S. BALAY - W. D. GROPP - L. CURFMAN MCINNES - B. F. SMITH, PETSc users manual, Technical Report ANL-95/11 - Revision 2.1.0, Argonne National Laboratory, 2001.
[9]
C. BERNARDI -
Y. MADAY,
Spectral Methods, in
Handbook of Numerical Analysis, Volume V: Techniques of Scientific Computing (Part 2),
North-Holland,
1997, 209-485. |
MR 1470226[10]
M. BHARDWAJ -
D. DAY -
C. FARHAT -
M. LEISONNE -
K. PIERSON -
D. RIXEN,
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]
F. BREZZI -
M. FORTIN,
Mixed and Hybrid Finite Element Methods,
Springer Series in Computational Mathematics,
15.
Springer, New York,
1991. |
MR 1115205 |
Zbl 0788.73002[12]
C. CANUTO -
M. Y. HUSSAINI -
A. QUARTERONI -
T. A. ZANG,
Spectral Methods in Fluid Dynamics,
Springer-Verlag, Berlin,
1988. |
MR 917480 |
Zbl 0717.76004[13]
T. F. CHAN -
T. P. MATHEW,
Domain Decomposition Methods,
Acta Numerica, (
1994) 61-143. |
MR 1288096 |
Zbl 0809.65112[14]
P. COLLI FRANZONE -
L. F. PAVARINO,
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]
P. COLLI FRANZONE -
G. SAVARÉ,
Degenerate evolution systems modeling the cardiac electric field at micro and macroscopic level. In
A.Lorenzi and
B.Ruf, Editors,
Evolution equations, Semigroups and Functional Analysis, 49-78,
Birkhauser,
2002. |
MR 1944157 |
Zbl 1036.35087[16]
M. DRYJA -
B. F. SMITH -
O. B. WIDLUND,
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]
M. DRYJA -
O. B. WIDLUND,
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]
C. FARHAT -
F.-X. ROUX,
Implicit parallel processing in structural mechanics,
Comput. Mech. Adv.,
2 (1) (
1994), 1-124. |
MR 1280753 |
Zbl 0805.73062[19] A. GARFINKEL - Y.-H. KIM - O. VOROSHILOVSKY - Z. QU - J. R. KIL - M.-H. LEE - H. S. KARAGUEUZIAN - J. N. WEISS - P.-S. CHEN, Preventing ventricular fibrillation by flattening cardiac restitution, Proc. Nat. Acad. Sci. USA, 97 (11) (2000), 6061-6066.
[20]
P. GOLDFELD -
L. F. PAVARINO -
O. B. WIDLUND,
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]
I.HERRERA -
D.KEYES -
O.WIDLUND -
R.YATES, Editors,
Proceedings of the Fourteenth International Conference on Domain Decomposition Methods, UNAM, Mexico City,
2003. |
MR 2093729 |
Zbl 1103.65004[23]
D. E. KEYES,
Terascale implicit methods for partial differential equations. In
X.Feng and
T. P.Schulze, 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]
R.KORNHUBER -
R. H. W.HOPPE -
D. E.KEYES -
J.PERIAUX -
O.PIRONNEAU -
J.XU, 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]
P. LE TALLEC,
Domain decomposition methods in computational mechanics,
Comput. Mech. Adv.,
1 (2) (
1994), 121-220. |
MR 1263805 |
Zbl 0802.73079[26]
P.-L. LIONS,
On the Schwarz alternating method. I. In
First International Symposium on Domain Decomposition Methods for Partial Differential Equations,
R.Glowinski et al., Editors, pp. 1-42, SIAM, Philadelphia,
1988. |
MR 972510 |
Zbl 0658.65090[27] C. LUO - Y. RUDY, A model of the ventricular cardiac action potential: depolarization, repolarization, and their interaction, Circ. Res., 68 (6) (1991), 1501-1526.
[28]
J. MANDEL -
M. BREZINA,
Balancing domain decomposition for problems with large jumps in coefficients,
Math. Comp.,
65 (
1996), 1387-1401. |
MR 1351204 |
Zbl 0853.65129[29]
P. K. MOORE,
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]
L. F. PAVARINO -
O. B. WIDLUND,
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[32]
L. F.PAVARINO -
A.TOSELLI, 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]
L. F. PAVARINO -
O. B. WIDLUND,
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]
L. F. PAVARINO -
O. B. WIDLUND,
Balancing Neumann-Neumann methods for incompressible Stokes equations,
Comm. Pure Appl. Math.,
55 (3) (
2002), 302-335. |
MR 1866366 |
Zbl 1024.76025[35]
A. QUARTERONI -
A. VALLI,
Numerical Approximation of Partial Differential Equations,
Springer-Verlag, Berlin,
1994. |
MR 1299729 |
Zbl 1151.65339[36]
A. QUARTERONI -
A. VALLI,
Domain Decomposition Methods for Partial Differential Equations,
Oxford Science Publications,
1999. |
MR 1857663 |
Zbl 0931.65118[37] H. A. SCHWARZ, Gesammelte Mathematische Abhandlungen, volume 2, pp. 133-143, Springer, 1890.
[38]
B. F. SMITH -
P. BJøRSTAD -
W. D. GROPP,
Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations,
Cambridge University Press,
1996. |
MR 1410757 |
Zbl 0857.65126[39]
S. L. SOBOLEV,
Lalgorithme de Schwarz dans la théorie de lelasticité,
Compt. Rend. Acad. Sci. URSS,
IV (XIII 6) (
1936), 243-246. |
Zbl 0015.40503[40]
A. TOSELLI -
O. B. WIDLUND,
Domain Decomposition Methods: Algorithms and Theory,
Springer-Verlag,
2004. |
Zbl 1069.65138[41]
H. M. TUFO -
P. F. FISCHER,
Fast parallel direct solvers for coarse grid problems,
J. Paral. Distr. Comput.,
61 (2) (
2001), 151-177. |
Zbl 0972.68191[42] A. T. WINFREE, Electrical turbulence in three-dimensional heart muscle, Science, 266 (5187) (1994), 1003-1006.
[43]
H. YU,
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