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Computing

, Volume 77, Issue 3, pp 253–273 | Cite as

Coarsening of Boundary-element Spaces

  • W. HackbuschEmail author
  • M. Löhndorf
  • S. A. Sauter
Article

Abstract

In this paper, we will present composite boundary elements (CBE) for classical Fredholm boundary integral equations. These new boundary elements allow the low-dimensional discretisation of boundary integral equations where the minimal number of degrees of freedom is independent of the, possibly, huge number of charts which are necessary to describe a complicated surface.

The applications are threefold: (a) The coarse-grid discretisation by composite boundary elements allow the use of multigrid algorithms for solving the fine-grid discretisation independently of the number of patches which are necessary to describe the surface. (b) If the accuracy requirements are moderate, the composite boundary elements allow the low-dimensional discretisation of the integral equation. (c) A posteriori error indicators can be applied already to a low-dimensional discretisation, which do not resolve the domain, to obtain a problem-adapted discretisation.

AMS Subject Classifications

65N38 74S15 78H15 80M15 

Keywords

Boundary-element method multigrid algebraic multigrid 

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References

  1. Bank, R., Smith, R. 2002An algebraic multilevel multigraph algorithmSIAM J. Sci. Comp.2315721592CrossRefMathSciNetGoogle Scholar
  2. Bank, R., Xu, J. 1996An algorithm for coarsening unstructured meshesNumer. Math.73136CrossRefMathSciNetGoogle Scholar
  3. Bank, R. E., Xu, J. 1995A hierarchical basis multi-grid method for unstructured gridsHackbusch, W.Wittum, G. eds. Fast solvers for flow problems. Proc. 10th GAMM-SeminarViewegKielGoogle Scholar
  4. Braess, D. 1995Towards algebraic multigrid for elliptic problems of second orderComputing55379393CrossRefzbMATHMathSciNetGoogle Scholar
  5. Bramble, J.: Multigrid methods. Pitman Research Notes in Mathematics. Longman Scientific & Technical 1993.Google Scholar
  6. Bramble, J., Leyk, Z., Pasciak, J. 1994The analysis of multigrid algorithms for pseudo-differential operators of order minus oneMath. Comp.63461478CrossRefMathSciNetGoogle Scholar
  7. Bramble, J., Pasciak, J., Vassilevksi, P. 1999Computational scales of Sobolev norms with applications to preconditioningMath. Comp.69462480Google Scholar
  8. Brezina, M., Cleary, A., Falgout, R., Henson, V. E., Jones, J., Manteuffel, T., McCormick, S., Ruge, J. 2000Algebraic multigrid based on element interpolation (AMGe)SIAM J. Sci. Comp.2215701592CrossRefMathSciNetGoogle Scholar
  9. Chan, T., Xu, J., Zikatanov, L. 1998An agglomeration multigrid method for unstructured gridsContemp. Math.2186781MathSciNetGoogle Scholar
  10. Chan, T. F., Smith, B. F. 1994Domain decomposition and multigrid algorithms for elliptic problems on unstructured meshesETNA2171182MathSciNetGoogle Scholar
  11. Cleary, A., Falgout, R., Henson, V. E., Jones, J.: Coarse-grid selection for parallel algebraic multigrid. In: Proc. 5th Int. Symp. on Solving Irregularly Structured Problems in Parallel. Lecture Notes in Computer Science. New York: Springer, August 1998. Google Scholar
  12. Cleary, A., Falgout, R., Henson, V. E., Jones, J., Manteuffel, T., McCormick, S., Miranda, J., Ruge, J. 2000Robustness and scalability of algebraic multigridSIAM J. Sci. Comp.2118861908CrossRefMathSciNetGoogle Scholar
  13. Falgout, R. D., Vassilevski, P. 2004On generalizing the algebraic multigrid frameworkSIAM J. Numer. Anal.4216691693CrossRefMathSciNetGoogle Scholar
  14. Feuchter, D., Heppner, I., Sauter, S., Wittum, G. 2003Bridging the gap between geometric and algebraic multi-grid methodsComput. Visual. Sci.6113CrossRefMathSciNetGoogle Scholar
  15. Hackbusch, W. 1985Multi-grid methods and applications2SpringerBerlinGoogle Scholar
  16. Hackbusch, W., Sauter, S. 1997Composite finite elements for problems containing small geometric details. Part II. Implementation and numerical resultsComput. Visual. Sci.11525CrossRefGoogle Scholar
  17. Hackbusch, W., Sauter, S. 1997Composite finite elements for the approximation of pdes on domains with complicated micro-structuresNumer. Math.75447472CrossRefMathSciNetGoogle Scholar
  18. Henson, V. E., Vassilevski, P. 2001Element-free AMGe general algorithms for computing the interpolation weights in AMGSIAM J. Sci. Comp.23629650CrossRefMathSciNetGoogle Scholar
  19. Jones, J., Vassilevski, P. 2001AMGe based on agglomerationSIAM J. Sci. Comp.23109133CrossRefMathSciNetGoogle Scholar
  20. Kornhuber, R., Yserentant, H. 1994Multilevel methods for elliptic problems on domains not resolved by the coarse gridContemp. Math.1804960MathSciNetGoogle Scholar
  21. Langer, U., Pusch, D.: Data-sparse algebraic multigrid methods for large scale boundary-element equation. Appl. Numer. Math. 2004 (online version).Google Scholar
  22. Langer, U., Pusch, D., Reitzinger, S. 2003Efficient preconditioners for boundary-element matrices based on grey-box algebraic multigrid methodsInt. J. Numer. Meth. Eng.5819371953CrossRefMathSciNetGoogle Scholar
  23. Mandel, J., Brezina, M., Vaněk, P. 1999Energy optimization of algebraic multigrid basesComputing62205228CrossRefMathSciNetGoogle Scholar
  24. Rech, M., Sauter, S., Smolianski, A.: Two-scale composite finite element method for the Dirichlet problem on complicated domains). Technical report no. 17-2003, Universität Zürich, 2003.Google Scholar
  25. Reusken, A. 1996A multigrid method based on incomplete Gaussian eliminationNum. Lin. Alg. Appl.3369390CrossRefzbMATHMathSciNetGoogle Scholar
  26. Ruge, J., Stüben, K. 1987Algebraic multigridMcCormick, S. eds. Multigrid methodsSIAMPhiladelphia73130Google Scholar
  27. Sauter, S.: Vergröberung von Finite-Elemente-Räumen. Habilitationsschrift, Universität Kiel, Germany, 1997. Google Scholar
  28. Schmidlin, G., Schwab, C. 2002Wavelet agglomeration on unstructured meshesBarth, T.Chan, T.Haimes, R. eds. Lecture Notes in Computational Science and Engineering, Vol 20SpringerHeidelberg359378Google Scholar
  29. Schöberl, J. 1997NETGEN – An advancing front 2D/3D-mesh generator based on abstract rulesComput. Visual. Sci.14152CrossRefzbMATHGoogle Scholar
  30. Stüben, K.: Algebraicmultigrid (AMG): An introduction with applications.GMD Report 53. 1999. In: Multigrid (U. Trottenberg, C.W. Oosterlee, A. Schüller, eds.). London: Academic Press 2001. Google Scholar
  31. Stüben, K. 2001A review of algebraic multigridJ. Comp. Appl. Math.128281309CrossRefzbMATHGoogle Scholar
  32. Tausch, J., White, J. 2003Multiscale bases for the sparse representation of boundary integral operators on complex geometrySIAM J. Sci. Comp.2516101629CrossRefMathSciNetGoogle Scholar
  33. Vaněk, P., Mandel, J., Brezina, M. 1996Algebraic multigrid by smoothed aggregation for second- and fourth-order elliptic problemsComputing56179196CrossRefMathSciNetGoogle Scholar
  34. Wan, W. L., Chan, T. F., Smith, B. 2000An energy-minimizing interpolation for robust multigrid methodsSIAM J. Sci. Comput.2116321649CrossRefMathSciNetGoogle Scholar
  35. Xu, J. 1996The auxiliary space method and optimal multigrid preconditioning techniques for unstructured gridsComputing56215235CrossRefzbMATHMathSciNetGoogle Scholar
  36. Yserentant, H. 1999Coarse grid spaces for domains with a complicated boundaryNumer. Algorithm21387392CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 2006

Authors and Affiliations

  1. 1.Max-Planck-Institut für Mathematik in den NaturwissenschaftenLeipzigGermany
  2. 2.Advanced Computer Vision GmbH – ACV Tech Gate ViennaWienAustria
  3. 3.Institut für Mathematik, Universität ZürichZürichSwitzerland

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