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

Chokri Chniti

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 1, page 23-37
  • ISSN: 0764-583X

Abstract

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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 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.

How to cite

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Chniti, Chokri. "A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 45.1 (2011): 23-37. <http://eudml.org/doc/197476>.

@article{Chniti2011,
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 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. },
author = {Chniti, Chokri},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Corner singularity; domain decomposition method; Kondratiev theory; reaction-diffusion problems; heterogeneous coefficients; corner singularity; domain decomposition; transmission conditions; convergence; mixed boundary value problem; numerical experiments},
language = {eng},
month = {1},
number = {1},
pages = {23-37},
publisher = {EDP Sciences},
title = {A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients},
url = {http://eudml.org/doc/197476},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Chniti, Chokri
TI - A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/1//
PB - EDP Sciences
VL - 45
IS - 1
SP - 23
EP - 37
AB - 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 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.
LA - eng
KW - Corner singularity; domain decomposition method; Kondratiev theory; reaction-diffusion problems; heterogeneous coefficients; corner singularity; domain decomposition; transmission conditions; convergence; mixed boundary value problem; numerical experiments
UR - http://eudml.org/doc/197476
ER -

References

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  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.  Zbl1096.65124
  2. C. Chniti, F. Nataf and F. Nier, Improved interface conditions for 2D domain decomposition with corners: a theoretical determination. CALCOLO45 (2008) 111–147.  Zbl1173.65364
  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.  Zbl1203.65274
  4. O. Dubois, Optimized Schwarz Methods for the Advection-Diffusion Equation and for Problems with Discontinuous Coefficients. Ph.D. Thesis, McGill University, Montréal (2007).  
  5. M.J. Gander, Optimized schwarz methods. SIAM J. Numer. Anal.44 (2006) 699–731.  Zbl1117.65165
  6. P. Grisvard, Singularities in boundary value problems, Research Notes in Applied Mathematics, RMA22. Springer-Verlag (1992).  Zbl0766.35001
  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.  
  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).  Zbl1043.65128
  9. V.A. Kondratiev, Boundary problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obshch.16 (1967) 227–313.  
  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.  
  11. F. Nier, Remarques sur les algorithmes de décomposition de domaines, in Séminaire EDP-École Polytechnique (1998–1999).  

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