Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-15T01:29:54.407Z Has data issue: false hasContentIssue false
  • English
  • Français

A matching of singularities in domain decomposition methods for reaction-diffusion problemswith discontinuous coefficients

Published online by Cambridge University Press:  15 April 2010

Chokri Chniti
Chokri Chniti*
Affiliation:
RICAM, Austrian Academy of Sciences, Altenberger strasse 69, 4040 Linz, Austria. [email protected]
Get access

Abstract

In this paper we certify that the same approach proposed in previous works by Chniti et al. [C. R. Acad. Sci.342 (2006) 883–886; CALCOLO45 (2008) 111–147; J. Sci. Comput.38 (2009) 207–228] can be applied to more general operators with strong heterogeneity in the coefficients. We consider here the case of reaction-diffusion problems with piecewise constant coefficients. The problem reduces to determining the coefficients of some transmission conditionsto obtain fast convergence of domain decomposition methods.After explaining the theoretical results, we explicitly compute the coefficients in the transmission boundary conditions. The numerical results presented in this paper confirm the optimality properties.

Keywords

Corner singularitydomain decomposition methodKondratiev theory
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 1 , January 2011 , pp. 23 - 37
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chniti, C., Nataf, F. and Nier, F., Improved interface conditions for the domain decomposition of a non-convex polygonal domain. C. R. Acad. Sci. 342 (2006) 883–886. CrossRef
Chniti, C., Nataf, F. and Nier, F., Improved interface conditions for 2D domain decomposition with corners: a theoretical determination. CALCOLO 45 (2008) 111–147. CrossRef
Chniti, C., Nataf, F. and Nier, F., Improved interface conditions for 2D domain decomposition with corners: Numerical applications. J. Sci. Comput. 38 (2009) 207–228. CrossRef
O. Dubois, Optimized Schwarz Methods for the Advection-Diffusion Equation and for Problems with Discontinuous Coefficients. Ph.D. Thesis, McGill University, Montréal (2007).
Gander, M.J., Optimized schwarz methods. SIAM J. Numer. Anal. 44 (2006) 699–731. CrossRef
P. Grisvard, Singularities in boundary value problems, Research Notes in Applied Mathematics, RMA 22. Springer-Verlag (1992).
C. Japhet and F. Nataf, The Best Interface Conditions for Domain Decomposition Methods: Absorbing Boundary Conditions, in Absorbing Boundaries and Layers, Domain Decomposition Methods – Applications to Large Scale Computation, L. Tourrette and L. Halpern Eds., Nova Science Publishers, Publ. Science (2001) 348–373.
B.N. Khoromskij and G. Wittum, Numerical Solution of Elliptic Differential Equations by Reduction to the Interface, Lect. Notes Comput. Sci. Eng. 36. Springer-Verlag, Berlin (2004).
Kondratiev, V.A., Boundary problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obshch. 16 (1967) 227–313.
P.L. Lions, On the Schwarz Alternating Method III: A variant for Nonoverlapping Subdomains, in Third Internationnal Symposium on Domain Decomposition Methods for Partial Differentiel Equations, held in Houston, Texas, March 20–22, Philadelphia, SIAM (1989) 202–223.
F. Nier, Remarques sur les algorithmes de décomposition de domaines, in Séminaire EDP-École Polytechnique (1998–1999).