Parallel Schwarz Waveform Relaxation Algorithm for an N-dimensional semilinear heat equation
Tran, Minh-Binh
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014), p. 795-813 / Harvested from Numdam

We present in this paper a proof of well-posedness and convergence for the parallel Schwarz Waveform Relaxation Algorithm adapted to an N-dimensional semilinear heat equation. Since the equation we study is an evolution one, each subproblem at each step has its own local existence time, we then determine a common existence time for every problem in any subdomain at any step. We also introduce a new technique: Exponential Decay Error Estimates, to prove the convergence of the Schwarz Methods, with multisubdomains, and then apply it to our problem.

Publié le : 2014-01-01
DOI : https://doi.org/10.1051/m2an/2013121
Classification:  65M12
@article{M2AN_2014__48_3_795_0,
     author = {Tran, Minh-Binh},
     title = {Parallel Schwarz Waveform Relaxation Algorithm for an N-dimensional semilinear heat equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {48},
     year = {2014},
     pages = {795-813},
     doi = {10.1051/m2an/2013121},
     mrnumber = {3264335},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2014__48_3_795_0}
}
Tran, Minh-Binh. Parallel Schwarz Waveform Relaxation Algorithm for an N-dimensional semilinear heat equation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) pp. 795-813. doi : 10.1051/m2an/2013121. http://gdmltest.u-ga.fr/item/M2AN_2014__48_3_795_0/

[1] J.-D. Benamou and B. Desprès, A domain decomposition method for the Helmholtz equation and related optimal control problems. J. Comput. Phys. 136 (1997) 68-82. | MR 1468624 | Zbl 0884.65118

[2] K. Burrage, C. Dyke and B. Pohl, On the performance of parallel waveform relaxations for differential systems. Appl. Numer. Math. 20 (1996) 39-55. | MR 1385234 | Zbl 0855.65074

[3] Th. Cazenave and A. Harau, An introduction to semilinear evolution equations, vol. 13 of Oxford Lect. Ser. Math. Applic. The Clarendon Press Oxford University Press, New York (1998). Translated from the 1990 French original by Yvan Martel and revised by the authors. | MR 1691574 | Zbl 0926.35049

[4] E.B. Davies, Heat kernels and spectral theory, in vol. 92 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1989). | MR 990239 | Zbl 0699.35006

[5] S. Descombes, V. Dolean and M.J. Gander, Schwarz waveform relaxation methods for systems of semi-linear reaction-diffusion equations, in Domain Decomposition Methods (2009). | Zbl 1217.65187

[6] L.C. Evans, Partial differential equations, Graduate Studies in Mathematics, in vol. 19 of Amer. Math. Soc. Providence, RI (1998). | MR 1625845 | Zbl 0902.35002

[7] A. Friedman, Partial differential equations of parabolic type. Prentice-Hall Inc., Englewood Cliffs, N.J (1964). | MR 181836 | Zbl 0144.34903

[8] M.J. Gander and L. Halpern. Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Numer. Anal. 45 (2007) 666-697. | MR 2300292 | Zbl 1140.65063

[9] M.J. Gander, L. Halpern and F. Nataf, Optimal convergence for overlapping and non-overlapping Schwarz waveform relaxation, in Eleventh International Conference on Domain Decomposition Methods (London, 1998). DDM.org, Augsburg (1999) 27-36. | MR 1827406

[10] M.J. Gander, A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations. Numer. Linear Algebra Appl. 6 (1999) 125-145. Czech-US Workshop in Iterative Methods and Parallel Computing, Part 2 (Milovy 1997). | MR 1695405 | Zbl 0983.65107

[11] M.J. Gander, L. Halpern and F. Nataf, Optimized Schwarz methods. In Domain decomposition methods in sciences and engineering (Chiba, 1999). DDM.org, Augsburg (2001) 15-27. | MR 1827519 | Zbl 1103.65125

[12] M.J. Gander and A.M. Stuart, Space time continuous analysis of waveform relaxation for the heat equation. SIAM J. 19 (1998) 2014-2031. | MR 1638096 | Zbl 0911.65082

[13] M.J. Gander and H. Zhao, Overlapping Schwarz waveform relaxation for the heat equation in n dimensions. BIT 42 (2002) 779-795. | MR 1944537 | Zbl 1022.65112

[14] E. Giladi and H.B. Keller, Space-time domain decomposition for parabolic problems. Numer. Math. 93 (2002) 279-313. | MR 1941398 | Zbl 1019.65076

[15] G.M. Lieberman. Second order parabolic differential equations. World Scientific Publishing Co. Inc., River Edge, NJ (1996). | MR 1465184 | Zbl 0884.35001

[16] P.-L. Lions, On the Schwarz alternating method I. In First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987). SIAM, Philadelphia, PA (1988) 1-42. | MR 972510 | Zbl 0658.65090

[17] P.-L. Lions, On the Schwarz alternating method II. Stochastic interpretation and order properties, in Domain decomposition methods (Los Angeles, CA, 1988). SIAM, Philadelphia, PA (1989) 47-70. | MR 992003 | Zbl 0681.65072

[18] P.-L. Lions, On the Schwarz alternating method III. A variant for nonoverlapping subdomains, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX 1989). SIAM, Philadelphia, PA (1990) 202-223. | MR 1064345 | Zbl 0704.65090

[19] S.H. Lui, On Schwarz methods for monotone elliptic PDEs, in Domain decomposition methods in sciences and engineering (Chiba, 1999). DDM.org, Augsburg (2001) 55-62. | MR 1827522

[20] S.-H. Lui, On monotone iteration and Schwarz methods for nonlinear parabolic PDEs. J. Comput. Appl. Math. 161 (2003) 449-468. | MR 2017025 | Zbl 1038.65090

[21] P. Quittner and Ph. Souplet, Superlinear parabolic problems, Blow-up, global existence and steady states. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel (2007). | MR 2346798 | Zbl 1128.35003

[22] M.-B. Tran, Parallel Schwarz waveform relaxation method for a semilinear heat equation in a cylindrical domain. C. R. Math. Acad. Sci. Paris 348 (2010) 795-799. | MR 2671163 | Zbl 1198.35129