Coupling of mixed finite elements and boundary elements

The symmetric coupling of mixed finite element and boundary element methods is analysed for a model interface problem with the Laplacian. The coupling involves a further continuous ansatz function on the interface to link the discontinuous displacement field to the necessarily continuous boundary ansatz function. Quasi-optimal a priori error estimates and sharp a posteriori error estimates are established which justify adaptive mesh-refining algorithms. Numerical experiments prove the adaptive coupling as an efficient tool for the numerical treatment of transmission problems.

[1]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[2]  E. Stein,et al.  Symmetric coupling of boundary elements and Raviart–Thomas-type mixed finite elements in elastostatics , 1996 .

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

[4]  Irene A. Stegun,et al.  Pocketbook of mathematical functions , 1984 .

[5]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[6]  Carsten Carstensen,et al.  Weighted Clement-Type Interpolation And A Posteriori Analysis For FEM , 1997 .

[7]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[8]  A. Alonso.  Error estimators for a mixed method , 1996 .

[9]  Martin Costabel,et al.  Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation , 1985 .

[10]  Ian H. Sloan,et al.  The Galerkin Method for Integral Equations of the First Kind with Logarithmic Kernel: Theory , 1988 .

[11]  Carsten Carstensen,et al.  A posteriori error estimate for the mixed finite element method , 1997, Math. Comput..

[12]  Salim Meddahi,et al.  An a-posteriori error estimate for the coupling of BEM and mixed-FEM , 1999 .

[13]  Carsten Carstensen,et al.  Adaptive coupling of boundary elements and finite elements , 1995 .

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

[15]  Wolfgang L. Wendland,et al.  On Asymptotic Error Estimates for Combined Bem and Fem , 1988 .

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

[17]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[18]  Carsten Carstensen,et al.  An a posteriori error estimate for a first-kind integral equation , 1997, Math. Comput..

[19]  Carsten Carstensen,et al.  Efficiency of a posteriori BEM-error estimates for first-kind integral equations on quasi-uniform meshes , 1996, Math. Comput..

[20]  F. Thomasset.  Finite element methods for Navier-Stokes equations , 1980 .

[21]  Dieter Gaier,et al.  Integralgleichungen erster Art und konforme Abbildung , 1976 .

[22]  Gabriel N. Gatica,et al.  Boundary-Field Equation Methods for a Class of Nonlinear Problems , 1995 .

[23]  Carsten Carstensen,et al.  Coupling of Nonconforming Finite Elements and Boundary Elements I: A Priori Estimates , 1999, Computing.

[24]  Ernst P. Stephan,et al.  Remarks to Galerkin and least squares methods with finite elements for general elliptic problems , 1976 .

[25]  Coupling of adaptively refined dual mixed finite elements and boundary elements in linear elasticity , 1995 .

[26]  Carsten Carstensen,et al.  Coupling of Nonconforming Finite Elements and Boundary Elements II: A Posteriori Estimates and Adaptive Mesh-Refinement , 1999, Computing.

[27]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

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

[29]  R. Verfürth,et al.  Edge Residuals Dominate A Posteriori Error Estimates for Low Order Finite Element Methods , 1999 .

[30]  Dietrich Braess,et al.  A Posteriori Error Estimators for the Raviart--Thomas Element , 1996 .