Coarsening of Boundary-element Spaces

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.

This is a preview of subscription content, log in to check access.

References

  1. R. Bank R. Smith (2002) ArticleTitleAn algebraic multilevel multigraph algorithm SIAM J. Sci. Comp. 23 1572–1592 Occurrence Handle10.1137/S1064827500381045 Occurrence Handle2002k:65048

    Article  MathSciNet  Google Scholar 

  2. R. Bank J. Xu (1996) ArticleTitleAn algorithm for coarsening unstructured meshes Numer. Math. 73 1–36 Occurrence Handle10.1007/s002110050181 Occurrence Handle97c:65055

    Article  MathSciNet  Google Scholar 

  3. R. E. Bank J. Xu (1995) A hierarchical basis multi-grid method for unstructured grids W. Hackbusch G. Wittum (Eds) Fast solvers for flow problems. Proc. 10th GAMM-Seminar Vieweg Kiel

    Google Scholar 

  4. D. Braess (1995) ArticleTitleTowards algebraic multigrid for elliptic problems of second order Computing 55 379–393 Occurrence Handle10.1007/BF02238488 Occurrence Handle0844.65088 Occurrence Handle97c:65178

    Article  MATH  MathSciNet  Google Scholar 

  5. Bramble, J.: Multigrid methods. Pitman Research Notes in Mathematics. Longman Scientific & Technical 1993.

  6. J. Bramble Z. Leyk J. Pasciak (1994) ArticleTitleThe analysis of multigrid algorithms for pseudo-differential operators of order minus one Math. Comp. 63 461–478 Occurrence Handle10.2307/2153279 Occurrence Handle94m:65184

    Article  MathSciNet  Google Scholar 

  7. J. Bramble J. Pasciak P. Vassilevksi (1999) ArticleTitleComputational scales of Sobolev norms with applications to preconditioning Math. Comp. 69 462–480

    Google Scholar 

  8. M. Brezina A. Cleary R. Falgout V. E. Henson J. Jones T. Manteuffel S. McCormick J. Ruge (2000) ArticleTitleAlgebraic multigrid based on element interpolation (AMGe) SIAM J. Sci. Comp. 22 IssueID5 1570–1592 Occurrence Handle10.1137/S1064827598344303 Occurrence Handle2001k:65184

    Article  MathSciNet  Google Scholar 

  9. T. Chan J. Xu L. Zikatanov (1998) ArticleTitleAn agglomeration multigrid method for unstructured grids Contemp. Math. 218 67–81 Occurrence Handle99h:65205

    MathSciNet  Google Scholar 

  10. T. F. Chan B. F. Smith (1994) ArticleTitleDomain decomposition and multigrid algorithms for elliptic problems on unstructured meshes ETNA 2 171–182 Occurrence Handle95i:65173

    MathSciNet  Google 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.

  12. A. Cleary R. Falgout V. E. Henson J. Jones T. Manteuffel S. McCormick J. Miranda J. Ruge (2000) ArticleTitleRobustness and scalability of algebraic multigrid SIAM J. Sci. Comp. 21 IssueID5 1886–1908 Occurrence Handle10.1137/S1064827598339402 Occurrence Handle2001f:65043

    Article  MathSciNet  Google Scholar 

  13. R. D. Falgout P. Vassilevski (2004) ArticleTitleOn generalizing the algebraic multigrid framework SIAM J. Numer. Anal. 42 1669–1693 Occurrence Handle10.1137/S0036142903429742 Occurrence Handle2005i:65047

    Article  MathSciNet  Google Scholar 

  14. D. Feuchter I. Heppner S. Sauter G. Wittum (2003) ArticleTitleBridging the gap between geometric and algebraic multi-grid methods Comput. Visual. Sci. 6 IssueID1 1–13 Occurrence Handle10.1007/s00791-003-0102-3 Occurrence Handle2004d:65040

    Article  MathSciNet  Google Scholar 

  15. W. Hackbusch (1985) Multi-grid methods and applications EditionNumber2 Springer Berlin

    Google Scholar 

  16. W. Hackbusch S. Sauter (1997) ArticleTitleComposite finite elements for problems containing small geometric details. Part II. Implementation and numerical results Comput. Visual. Sci. 1 15–25 Occurrence Handle10.1007/s007910050002

    Article  Google Scholar 

  17. W. Hackbusch S. Sauter (1997) ArticleTitleComposite finite elements for the approximation of pdes on domains with complicated micro-structures Numer. Math. 75 447–472 Occurrence Handle10.1007/s002110050248 Occurrence Handle97k:65251

    Article  MathSciNet  Google Scholar 

  18. V. E. Henson P. Vassilevski (2001) ArticleTitleElement-free AMGe general algorithms for computing the interpolation weights in AMG SIAM J. Sci. Comp. 23 629–650 Occurrence Handle10.1137/S1064827500372997 Occurrence Handle2002j:65040

    Article  MathSciNet  Google Scholar 

  19. J. Jones P. Vassilevski (2001) ArticleTitleAMGe based on agglomeration SIAM J. Sci. Comp. 23 109–133 Occurrence Handle10.1137/S1064827599361047 Occurrence Handle2002g:65035

    Article  MathSciNet  Google Scholar 

  20. R. Kornhuber H. Yserentant (1994) ArticleTitleMultilevel methods for elliptic problems on domains not resolved by the coarse grid Contemp. Math. 180 49–60 Occurrence Handle95j:65165

    MathSciNet  Google Scholar 

  21. Langer, U., Pusch, D.: Data-sparse algebraic multigrid methods for large scale boundary-element equation. Appl. Numer. Math. 2004 (online version).

  22. U. Langer D. Pusch S. Reitzinger (2003) ArticleTitleEfficient preconditioners for boundary-element matrices based on grey-box algebraic multigrid methods Int. J. Numer. Meth. Eng. 58 1937–1953 Occurrence Handle10.1002/nme.839 Occurrence Handle2004j:65199

    Article  MathSciNet  Google Scholar 

  23. J. Mandel M. Brezina P. Vaněk (1999) ArticleTitleEnergy optimization of algebraic multigrid bases Computing 62 205–228 Occurrence Handle10.1007/s006070050022 Occurrence Handle2000j:65124

    Article  MathSciNet  Google 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.

  25. A. Reusken (1996) ArticleTitleA multigrid method based on incomplete Gaussian elimination Num. Lin. Alg. Appl. 3 369–390 Occurrence Handle10.1002/(SICI)1099-1506(199609/10)3:5<369::AID-NLA89>3.0.CO;2-M Occurrence Handle0906.65118 Occurrence Handle98a:65137

    Article  MATH  MathSciNet  Google Scholar 

  26. J. Ruge K. Stüben (1987) Algebraic multigrid S. McCormick (Eds) Multigrid methods SIAM Philadelphia 73–130

    Google Scholar 

  27. Sauter, S.: Vergröberung von Finite-Elemente-Räumen. Habilitationsschrift, Universität Kiel, Germany, 1997.

  28. G. Schmidlin C. Schwab (2002) Wavelet agglomeration on unstructured meshes T. Barth T. Chan R. Haimes (Eds) Lecture Notes in Computational Science and Engineering, Vol 20 Springer Heidelberg 359–378

    Google Scholar 

  29. J. Schöberl (1997) ArticleTitleNETGEN – An advancing front 2D/3D-mesh generator based on abstract rules Comput. Visual. Sci. 1 41–52 Occurrence Handle10.1007/s007910050004 Occurrence Handle0883.68130

    Article  MATH  Google 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.

  31. K. Stüben (2001) ArticleTitleA review of algebraic multigrid J. Comp. Appl. Math. 128 IssueID1-2 281–309 Occurrence Handle10.1016/S0377-0427(00)00516-1 Occurrence Handle0979.65111

    Article  MATH  Google Scholar 

  32. J. Tausch J. White (2003) ArticleTitleMultiscale bases for the sparse representation of boundary integral operators on complex geometry SIAM J. Sci. Comp. 25 1610–1629 Occurrence Handle10.1137/S1064827500369451 Occurrence Handle2004d:65151

    Article  MathSciNet  Google Scholar 

  33. P. Vaněk J. Mandel M. Brezina (1996) ArticleTitleAlgebraic multigrid by smoothed aggregation for second- and fourth-order elliptic problems Computing 56 179–196 Occurrence Handle10.1007/BF02238511 Occurrence Handle97c:65207

    Article  MathSciNet  Google Scholar 

  34. W. L. Wan T. F. Chan B. Smith (2000) ArticleTitleAn energy-minimizing interpolation for robust multigrid methods SIAM J. Sci. Comput. 21 IssueID4 1632–1649 Occurrence Handle10.1137/S1064827598334277 Occurrence Handle2001a:65162

    Article  MathSciNet  Google Scholar 

  35. J. Xu (1996) ArticleTitleThe auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids Computing 56 215–235 Occurrence Handle10.1007/BF02238513 Occurrence Handle0857.65129 Occurrence Handle97d:65062

    Article  MATH  MathSciNet  Google Scholar 

  36. H. Yserentant (1999) ArticleTitleCoarse grid spaces for domains with a complicated boundary Numer. Algorithm 21 387–392 Occurrence Handle10.1023/A:1019157313043 Occurrence Handle0939.65134 Occurrence Handle1725737

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to W. Hackbusch.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hackbusch, W., Löhndorf, M. & Sauter, S.A. Coarsening of Boundary-element Spaces. Computing 77, 253–273 (2006). https://doi.org/10.1007/s00607-005-0160-0

Download citation

AMS Subject Classifications

  • 65N38
  • 74S15
  • 78H15
  • 80M15

Keywords

  • Boundary-element method
  • multigrid
  • algebraic multigrid