A Solution Method for a Certain Nonlinear Interface Problem in Unbounded Domains

Abstract In this paper, we consider a kind of nonlinear interface problem in unbounded domains. To solve this problem, we discuss a new coupling of finite element and boundary element by adding an auxiliary circle. We first derive the optimal error estimate of finite element approximation to the coupled FEM-BEM problem. Then we introduce a preconditioning steepest descent method for solving the discrete system by constructing a cheap domain decomposition preconditioner. Moreover, we give a complete analysis to the convergence speed of this iterative method.

[1]  Gabriel N. Gatica,et al.  On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R2 , 1992 .

[2]  Salim Meddahi.  An Optimal Iterative Process for the Johnson--Nedelec Method of Coupling Boundary and Finite Elements , 1998 .

[3]  Elijah Polak,et al.  Computational methods in optimization , 1971 .

[4]  Martin Costabel,et al.  Coupling of finite and boundary element methods for an elastoplastic interface problem , 1990 .

[5]  Salim Meddahi,et al.  A dual-dual mixed formulation for nonlinear exterior transmission problems , 2001, Math. Comput..

[6]  Carsten Carstensen,et al.  Interface problem in holonomic elastoplasticity , 1993 .

[7]  Jindřich Nečas,et al.  Introduction to the Theory of Nonlinear Elliptic Equations , 1986 .

[8]  Ernst P. Stephan,et al.  An Adaptive Two-Level Method for the Coupling of Nonlinear FEM-BEM Equations , 1999 .

[9]  Martin Costabel,et al.  Boundary Integral Operators on Lipschitz Domains: Elementary Results , 1988 .

[10]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[11]  J. Pasciak,et al.  The Construction of Preconditioners for Elliptic Problems by Substructuring. , 2010 .

[12]  Martin Costabel,et al.  Symmetric Methods for the Coupling of Finite Elements and Boundary Elements (Invited contribution) , 1987 .

[13]  Gabriel R. Barrenechea,et al.  On the coupling of boundary integral and finite element methods with nonlinear transmission conditions , 1996 .

[14]  R. Glowinski,et al.  Numerical Methods for Nonlinear Variational Problems , 1985 .

[15]  Salim Meddahi,et al.  A FULLY DISCRETE GALERKIN SCHEME FOR A TWO-FOLD SADDLE POINT FORMULATION OF AN EXTERIOR NONLINEAR PROBLEM* , 2001 .

[16]  Ernst P. Stephan,et al.  On the integral equation method for the plane mixed boundary value problem of the Laplacian , 1979 .

[17]  Gabriel N. Gatica,et al.  The Uncoupling of Boundary Integral and Finite Element Methods for Nonlinear Boundary Value Problems , 1995 .

[18]  A. Ženíšek,et al.  Nonlinear elliptic and evolution problems and their finite element approximations , 1990 .