# Parallel Schwarz Waveform Relaxation Algorithm for an N-dimensional semilinear heat equation

- Volume: 48, Issue: 3, page 795-813
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topTran, 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 48.3 (2014): 795-813. <http://eudml.org/doc/273155>.

@article{Tran2014,

abstract = {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.},

author = {Tran, Minh-Binh},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {domain decomposition; waveform relaxation; Schwarz methods; semilinear heat equation; parallel computation; error estimate; convergence},

language = {eng},

number = {3},

pages = {795-813},

publisher = {EDP-Sciences},

title = {Parallel Schwarz Waveform Relaxation Algorithm for an N-dimensional semilinear heat equation},

url = {http://eudml.org/doc/273155},

volume = {48},

year = {2014},

}

TY - JOUR

AU - Tran, Minh-Binh

TI - Parallel Schwarz Waveform Relaxation Algorithm for an N-dimensional semilinear heat equation

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2014

PB - EDP-Sciences

VL - 48

IS - 3

SP - 795

EP - 813

AB - 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.

LA - eng

KW - domain decomposition; waveform relaxation; Schwarz methods; semilinear heat equation; parallel computation; error estimate; convergence

UR - http://eudml.org/doc/273155

ER -

## References

top- [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. Zbl0884.65118MR1468624
- [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. Zbl0855.65074MR1385234
- [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. Zbl0926.35049MR1691574
- [4] E.B. Davies, Heat kernels and spectral theory, in vol. 92 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1989). Zbl0699.35006MR990239
- [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). Zbl1217.65187
- [6] L.C. Evans, Partial differential equations, Graduate Studies in Mathematics, in vol. 19 of Amer. Math. Soc. Providence, RI (1998). Zbl0902.35002MR1625845
- [7] A. Friedman, Partial differential equations of parabolic type. Prentice-Hall Inc., Englewood Cliffs, N.J (1964). Zbl0144.34903MR181836
- [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. Zbl1140.65063MR2300292
- [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. MR1827406
- [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). Zbl0983.65107MR1695405
- [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. Zbl1103.65125MR1827519
- [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. Zbl0911.65082MR1638096
- [13] M.J. Gander and H. Zhao, Overlapping Schwarz waveform relaxation for the heat equation in n dimensions. BIT42 (2002) 779–795. Zbl1022.65112MR1944537
- [14] E. Giladi and H.B. Keller, Space-time domain decomposition for parabolic problems. Numer. Math.93 (2002) 279–313. Zbl1019.65076MR1941398
- [15] G.M. Lieberman. Second order parabolic differential equations. World Scientific Publishing Co. Inc., River Edge, NJ (1996). Zbl0884.35001MR1465184
- [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. Zbl0658.65090MR972510
- [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. Zbl0681.65072MR992003
- [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. Zbl0704.65090MR1064345
- [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. MR1827522
- [20] S.-H. Lui, On monotone iteration and Schwarz methods for nonlinear parabolic PDEs. J. Comput. Appl. Math.161 (2003) 449–468. Zbl1038.65090MR2017025
- [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). Zbl1128.35003MR2346798
- [22] M.-B. Tran, Parallel Schwarz waveform relaxation method for a semilinear heat equation in a cylindrical domain. C. R. Math. Acad. Sci. Paris348 (2010) 795–799. Zbl1198.35129MR2671163

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.