Hostname: page-component-6898cfcb55-kwdml Total loading time: 0 Render date: 2026-11-11T07:43:49.497Z Has data issue: false hasContentIssue false
  • English
  • Français

Parallel Schwarz Waveform Relaxation Algorithm for anN-dimensional semilinear heat equation∗

Published online by Cambridge University Press:  01 April 2014

Minh-Binh Tran
Minh-Binh Tran*
Affiliation:
Basque Center for Applied Mathematics, Alameda de Mazarredo, 14 48009 Bilbao, Basque Country, Spain.. tbinh@bcamath.org

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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

Keywords

Domain decompositionwaveform relaxationSchwarz methodssemilinear heat equation

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014, pp. 795 - 813
Copyright
© EDP Sciences, SMAI 2014

References

Benamou, J.-D. and Desprès, B., A domain decomposition method for the Helmholtz equation and related optimal control problems. J. Comput. Phys. 136 (1997) 68–82. Google Scholar
Burrage, K., Dyke, C. and Pohl, B., On the performance of parallel waveform relaxations for differential systems. Appl. Numer. Math. 20 (1996) 39–55. Google Scholar
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.
E.B. Davies, Heat kernels and spectral theory, in vol. 92 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1989).
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).
L.C. Evans, Partial differential equations, Graduate Studies in Mathematics, in vol. 19 of Amer. Math. Soc. Providence, RI (1998).
A. Friedman, Partial differential equations of parabolic type. Prentice-Hall Inc., Englewood Cliffs, N.J (1964).
Gander, M.J. and Halpern, L.. Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Numer. Anal. 45 (2007) 666–697. Google Scholar
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.
Gander, M.J., 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). Google Scholar
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.
Gander, M.J. and Stuart, A.M., Space time continuous analysis of waveform relaxation for the heat equation. SIAM J. 19 (1998) 2014–2031. Google Scholar
Gander, M.J. and Zhao, H., Overlapping Schwarz waveform relaxation for the heat equation in n dimensions. BIT 42 (2002) 779–795. Google Scholar
Giladi, E. and Keller, H.B., Space-time domain decomposition for parabolic problems. Numer. Math. 93 (2002) 279–313. Google Scholar
G.M. Lieberman. Second order parabolic differential equations. World Scientific Publishing Co. Inc., River Edge, NJ (1996).
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.
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.
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.
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.
Lui, S.-H., On monotone iteration and Schwarz methods for nonlinear parabolic PDEs. J. Comput. Appl. Math. 161 (2003) 449–468. Google Scholar
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).
Tran, M.-B., Parallel Schwarz waveform relaxation method for a semilinear heat equation in a cylindrical domain. C. R. Math. Acad. Sci. Paris 348 (2010) 795–799. Google Scholar