Improved interface conditions for 2D domain decomposition with corners: a theoretical determination

This article deals with a local improvement of domain decomposition methods for 2-dimensional elliptic problems for which either the geometry or the domain decomposition presents conical singularities. The problem amounts to determining the coefficients of interface boundary conditions so that the domain decomposition algorithm has rapid convergence. Specific problems occur in the presence of conical singularities. Starting from the method used for regular interfaces, we derive a local improvement by matching the singularities, that is, the initial terms of the asymptotic expansion arond the corner, provided by Kondratiev theory. This theoretical approach leads to the explicit computation of coefficients in the interface boundary conditions, which have been tested numerically. This final numerical step is presented in a companion article [4]. This article focuses on the method used to compute these coefficients and provides detailed examples for a model problem.

[1]  Frédéric Nataf,et al.  Improved Interface Conditions for 2D Domain Decomposition with Corners: Numerical Applications , 2009, J. Sci. Comput..

[2]  Frédéric Nataf,et al.  Convergence rate of some domain decomposition methods for overlapping and nonoverlapping subdomains , 1997 .

[3]  M. Dauge.  Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions , 1988 .

[4]  R. Glowinski,et al.  Third International Symposium on Domain Decomposition Methods for Partial Differential Equations , 1990 .

[5]  Thomas Hagstrom,et al.  Numerical Experiments on a Domain Decomposition Algorithm for Nonlinear Elliptic Boundary Value Problems , 1988 .

[6]  Frédéric Nataf,et al.  Convergence of domain decomposition methods via semi-classical calculus semiclassical calculus , 1998 .

[7]  Frédéric Nataf,et al.  Domain Decomposition Methods for Fluid Dynamics , 1995 .

[8]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[9]  Francis Nier,et al.  Remarques sur les algorithmes de décomposition de domaines , 1999 .

[10]  Frédéric Nataf,et al.  Méthode de décomposition de domaine pour l'équation d'advection-diffusion , 1991 .

[11]  R. Noyé,et al.  Numerical Solutions of Partial Differential Equations , 1983 .

[12]  J. Hopcroft,et al.  Modeling, mesh generation, and adaptive numerical methods for partial differential equations , 1995 .

[13]  P. Grisvard,et al.  BEHAVIOR OF THE SOLUTIONS OF AN ELLIPTIC BOUNDARY VALUE PROBLEM IN A POLYGONAL OR POLYHEDRAL DOMAIN , 1976 .

[14]  Alfio Quarteroni,et al.  Adaptive Domain Decomposition Methods for Advection-Diffusion Problems , 1995 .

[15]  Séminaire sur les équations aux dérivées partielles,et al.  Séminaire equations aux dérivées partielles , 1988 .

[16]  W. Rudin.  Real and complex analysis , 1968 .

[17]  T. Chan,et al.  Domain decomposition algorithms , 1994, Acta Numerica.

[18]  V. A. Kondrat'ev,et al.  Boundary problems for elliptic equations in domains with conical or angular points , 1967 .

[19]  A. Majda,et al.  Radiation boundary conditions for acoustic and elastic wave calculations , 1979 .

[20]  Alfio Quarteroni,et al.  Domain Decomposition Methods for Partial Differential Equations , 1999 .