A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 1, p. 23-37

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; CALCOLO 45 (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 conditions to 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.

DOI : https://doi.org/10.1051/m2an/2010031
Classification:  65N55,  35J05,  65N30,  35J25,  35R05
Keywords: Corner singularity, domain decomposition method, Kondratiev theory
@article{M2AN_2011__45_1_23_0,
     author = {Chniti, Chokri},
     title = {A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {1},
     year = {2011},
     pages = {23-37},
     doi = {10.1051/m2an/2010031},
     zbl = {1270.65074},
     mrnumber = {2781130},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2011__45_1_23_0}
}
A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 1, pp. 23-37. doi : 10.1051/m2an/2010031. http://www.numdam.org/item/M2AN_2011__45_1_23_0/

[1] C. Chniti, F. Nataf and F. Nier, Improved interface conditions for the domain decomposition of a non-convex polygonal domain. C. R. Acad. Sci. 342 (2006) 883-886. | MR 2224641 | Zbl 1096.65124

[2] C. Chniti, F. Nataf and F. Nier, Improved interface conditions for 2D domain decomposition with corners: a theoretical determination. CALCOLO 45 (2008) 111-147. | MR 2424651 | Zbl 1173.65364

[3] C. Chniti, F. Nataf and F. Nier, Improved interface conditions for 2D domain decomposition with corners: Numerical applications. J. Sci. Comput. 38 (2009) 207-228. | MR 2471014 | Zbl 1203.65274

[4] O. Dubois, Optimized Schwarz Methods for the Advection-Diffusion Equation and for Problems with Discontinuous Coefficients. Ph.D. Thesis, McGill University, Montréal (2007). | MR 2711738

[5] M.J. Gander, Optimized schwarz methods. SIAM J. Numer. Anal. 44 (2006) 699-731. | MR 2218966 | Zbl 1117.65165

[6] P. Grisvard, Singularities in boundary value problems, Research Notes in Applied Mathematics, RMA 22. Springer-Verlag (1992). | MR 1173209 | Zbl 0766.35001

[7] 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. | MR 2039948

[8] 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). | MR 2045003 | Zbl 1043.65128

[9] V.A. Kondratiev, Boundary problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obshch. 16 (1967) 227-313. | MR 226187 | Zbl 0194.13405

[10] 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. | MR 1064345 | Zbl 0704.65090

[11] F. Nier, Remarques sur les algorithmes de décomposition de domaines, in Séminaire EDP-École Polytechnique (1998-1999). | Numdam | MR 1721327 | Zbl 1058.65514