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The preconditioned GMRES method for systems of coupled FEM-BEM equations

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Abstract

We analyze the generalized minimal residual method (GMRES) as a solver for coupled finite element and boundary element equations. To accelerate the convergence of GMRES we apply a hierarchical basis block preconditioner for piecewise linear finite elements and piecewise constant boundary elements. It is shown that the number of iterations which is necessary to reach a given accuracy grows only poly-logarithmically with the number of unknowns.

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References

  1. C. Carstensen, Nonlinear interface problems in solid mechanics - finite element and boundary element coupling, Habilitation, Universität Hannover (1992).

  2. C. Carstensen, M. Kuhn and U. Langer, Fast parallel solvers for symmetric boundary element domain decomposition equations, Institutsbericht 500, Institut für Mathematik, Universität Linz (1996).

  3. M. Costabel, A symmetric method for the coupling of finite elements and boundary elements, in: The Mathematics of Finite Elements and Applications IV, MAFELAP 1987, ed. J.R. Whiteman (Academic Press, London, 1988) pp. 281-288.

    Google Scholar 

  4. S.C. Eisenstat, H.C. Elman and M.H. Schultz, Variational iterative methods for nonsymmetric systems of linear equations, SIAM J. Statist. Comput. 7 (1986) 856-869.

    Article  Google Scholar 

  5. S.A. Funken, Schnelle Lösungsverfahren für FEM-BEM Kopplungsgleichungen, thesis, Universität Hannover (1996).

  6. S.A. Funken and E.P. Stephan, Fast solvers with multigrid preconditioners for linear FEM-BEM coupling, Preprint, Institut für Angewandte Mathematik, Universität Hannover (1995).

  7. M. Hahne, M. Maischak, E.P. Stephan and A. Wathen, Efficient preconditioners for coupled FEM-BEM equations, Numer. Methods Partial Differential Equations (to appear).

  8. M. Hahne, E.P. Stephan and W. Thies, Fast solvers for coupled FEM-BEM equations I, in: Fast Solvers for Flow Problems, eds. W. Hackbusch and G. Wittum, Notes on Numerical Fluid Mechanics 49 (1995) pp. 121-130.

  9. G.C. Hsiao, B.N. Khoromskij and W.L. Wendland, Boundary integral operators and domain decomposition, DFG priority research boundary element methods, Report (1994).

  10. Y. Saad and M.H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Numer. Anal. 20 (1983) 345-357.

    Article  MathSciNet  Google Scholar 

  11. E.P. Stephan and A. Wathen, Convergence of preconditioned minimum residual iteration for coupled finite element/boundary element computations, SIAM J. Numer. Anal. (to appear).

  12. E.P. Stephan and W. Wendland, An augmented Galerkin procedure for the bondary integral method applied to two-dimensional screen and crack problems, Appl. Anal. 18 (1984) 183-219.

    MathSciNet  Google Scholar 

  13. T. Tran and E.P. Stephan, Additive Schwarz methods for the h-version boundary element method, Preprint, Universität Hannover, 1995. Also: Appl. Anal. 60 (1996) 63-84.

    MATH  MathSciNet  Google Scholar 

  14. T. Tran, E.P. Stephan and P. Mund, Hierarchical basis preconditioners for first kind integral equations, Appl. Analysis 65 (1997) 353-372.

    MATH  MathSciNet  Google Scholar 

  15. A. Wathen, B. Fischer and D. Silvester, The convergence rate of the minimum residual method for the Stokes problem, Preprint 76, Universität Hamburg (1993).

  16. Y. Yserentant, On the multi-level splitting of finite element spaces, Numer. Math. 49 (1988) 379-412.

    Article  MathSciNet  Google Scholar 

  17. X. Zhang, Multilevel Schwarz methods, SIAM J. Numer. Anal. 63 (1992) 521-539.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Hannover, Welfengarten 1, D-30167, Hannover, Germany

    Patrick Mund & Ernst P. Stephan

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  1. Patrick Mund
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  2. Ernst P. Stephan
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Mund, P., Stephan, E.P. The preconditioned GMRES method for systems of coupled FEM-BEM equations. Advances in Computational Mathematics 9, 131–144 (1998). https://doi.org/10.1023/A:1018921003084

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