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A-BEM

  • Joachim Gwinner
  • Ernst Peter Stephan
Chapter
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 52)

Abstract

First in this chapter we give a general framework of adaptive Petrov–Galerkin methods for the solution of operator equations in Banach spaces. This approach is made precise in the application to Symm’s integral equation. Then we present more general adaptive BEM. Here we use the residual error estimator and prove reliability and efficiency in 2D. Finally we analyze the hierarchical error estimator and demonstrate its applicability in two-level adaptive BEM for scalar and vector boundary value problems. Special emphasis is given to the 3D case for the weakly singular integral equation (Sect. 10.3) and for the hyper singular integral equation (Sect. 10.4). In Sect. 10.5 we present a two-level adaptive BEM for the weakly singular operator and the h-version on surface pieces. In Sect. 10.6 based on a two-level subspace decomposition for the p-version BEM we give hierarchical error estimators for the hypersingular integral operator on curves. Finally recent developments on the convergence of the adaptive BEM for the h-version are given in Sect. 10.7.

References

  1. 8.
    M. Aurada, M. Feischl, Th. Führer, M. Karkulik, J.M. Melenk, D. Praetorius, Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity. Comput. Mech. 51, 399–419 (2014)MathSciNetCrossRefGoogle Scholar
  2. 9.
    M. Aurada, M. Feischl, Th. Führer, M. Karkulik, J.M. Melenk, D. Praetorius, Local inverse estimates for non-local boundary integral operators. Math. Comput. 86, 2651–2686 (2017)MathSciNetCrossRefGoogle Scholar
  3. 10.
    M. Aurada, M. Feischl, Th. Führer, M. Karkulik, D. Praetorius, Efficiency and optimality of some weighted-residual error estimator for adaptive 2D boundary element methods. Comput. Methods Appl. Math. 13, 305–332 (2013)MathSciNetCrossRefGoogle Scholar
  4. 11.
    M. Aurada, M. Feischl, Th. Führer, M. Karkulik, D. Praetorius, Energy norm based error estimators for adaptive BEM for hypersingular integral equations. Appl. Numer. Math. 95, 15–35 (2015)MathSciNetCrossRefGoogle Scholar
  5. 12.
    M. Aurada, M. Feischl, M. Karkulik, D. Praetorius, A posteriori error estimates for the Johnson-Nédélec FEM-BEM coupling. Eng. Anal. Bound. Elem., 255–266 (2012)MathSciNetCrossRefGoogle Scholar
  6. 13.
    M. Aurada, M. Feischl, D. Praetorius, Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems. ESAIM Math. Model. Numer. Anal. 46, 1147–1173 (2012)MathSciNetCrossRefGoogle Scholar
  7. 15.
    I. Babuška, A. Craig, J. Mandel, J. Pitkäranta, Efficient preconditioning for the p-version finite element method in two dimensions. SIAM J. Numer. Anal. 28, 624–661 (1991)MathSciNetCrossRefGoogle Scholar
  8. 26.
    I. Babuška, M. Vogelius, Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44, 75–102 (1984)MathSciNetCrossRefGoogle Scholar
  9. 32.
    R.E. Bank, R.K. Smith, A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30, 921–935 (1993)MathSciNetCrossRefGoogle Scholar
  10. 44.
    J. Bergh, J. Löfström, Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, No. 223 (Springer, Berlin-New York, 1976)Google Scholar
  11. 74.
    C. Carstensen, Efficiency of a posteriori BEM-error estimates for first-kind integral equations on quasi-uniform meshes. Math. Comput. 65, 69–84 (1996)MathSciNetCrossRefGoogle Scholar
  12. 76.
    C. Carstensen, An a posteriori error estimate for a first-kind integral equation. Math. Comput. 66, 139–155 (1997)MathSciNetCrossRefGoogle Scholar
  13. 77.
    C. Carstensen, S. Bartels, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming, and mixed FEM. Math. Comput. 71, 945–969 (2002)MathSciNetCrossRefGoogle Scholar
  14. 78.
    C. Carstensen, B. Faermann, Mathematical foundation of a posteriori error estimates and adaptive mesh-refining algorithms for boundary integral equations of the first kind. Eng. Anal. Bound. Elem. 25, 497–509 (2001)CrossRefGoogle Scholar
  15. 79.
    C. Carstensen, M. Feischl, M. Page, D. Praetorius, Axioms of adaptivity. Comput. Math. Appl. 67, 1195–1253 (2016)MathSciNetCrossRefGoogle Scholar
  16. 82.
    C. Carstensen, S.A. Funken, E.P. Stephan, A posteriori error estimates for hp-boundary element methods. Appl. Anal. 61, 233–253 (1996)MathSciNetCrossRefGoogle Scholar
  17. 84.
    C. Carstensen, D. Gallistl, J. Gedicke, Justification of the saturation assumption. Numer. Math. 134, 1–25 (2016)MathSciNetCrossRefGoogle Scholar
  18. 86.
    C. Carstensen, M. Maischak, D. Praetorius, E.P. Stephan, Residual-based a posteriori error estimate for hypersingular equation on surfaces. Numer. Math. 97, 397–425 (2004)MathSciNetCrossRefGoogle Scholar
  19. 87.
    C. Carstensen, M. Maischak, E.P. Stephan, A posteriori error estimate and h-adaptive algorithm on surfaces for Symm’s integral equation. Numer. Math. 90, 197–213 (2001)MathSciNetCrossRefGoogle Scholar
  20. 88.
    C. Carstensen, D. Praetorius, Averaging techniques for the effective numerical solution of Symm’s integral equation of the first kind. SIAM J. Sci. Comput. 27, 1226–1260 (2006)MathSciNetCrossRefGoogle Scholar
  21. 89.
    C. Carstensen, D. Praetorius, Convergence of adaptive boundary element methods. J. Integr. Equ. Appl. 24, 1–23 (2012)MathSciNetCrossRefGoogle Scholar
  22. 91.
    C. Carstensen, E.P. Stephan, Adaptive coupling of boundary elements and finite elements. RAIRO Modél. Math. Anal. Numér. 29, 779–817 (1995)MathSciNetCrossRefGoogle Scholar
  23. 92.
    C. Carstensen, E.P. Stephan, A posteriori error estimates for boundary element methods. Math. Comput. 64, 483–500 (1995)MathSciNetCrossRefGoogle Scholar
  24. 93.
    C. Carstensen, E.P. Stephan, Adaptive boundary element methods for some first kind integral equations. SIAM J. Numer. Anal. 33, 2166–2183 (1996)MathSciNetCrossRefGoogle Scholar
  25. 143.
    W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)MathSciNetCrossRefGoogle Scholar
  26. 144.
    W. Dörfler, R.H. Nochetto, Small data oscillation implies the saturation assumption. Numer. Math. 91, 1–12 (2002)MathSciNetCrossRefGoogle Scholar
  27. 161.
    C. Erath, S. Ferraz-Leite, S. Funken, D. Praetorius, Energy norm based a posteriori error estimation for boundary element methods in two dimensions. Appl. Numer. Math. 59, 2713–2734 (2009)MathSciNetCrossRefGoogle Scholar
  28. 162.
    C. Erath, S. Funken, P. Goldenits, D. Praetorius, Simple error estimators for the Galerkin BEM for some hypersingular integral equation in 2D. Appl. Anal. 92, 1194–1216 (2013)MathSciNetCrossRefGoogle Scholar
  29. 164.
    V.J. Ervin, N. Heuer, An adaptive boundary element method for the exterior Stokes problem in three dimensions. IMA J. Numer. Anal. 26, 297–325 (2006)MathSciNetCrossRefGoogle Scholar
  30. 170.
    B. Faermann, Local a-posteriori error indicators for the Galerkin discretization of boundary integral equations. Numer. Math. 79, 43–76 (1998)MathSciNetCrossRefGoogle Scholar
  31. 171.
    B. Faermann, Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. I. The two-dimensional case. IMA J. Numer. Anal. 20, 203–234 (2000)MathSciNetCrossRefGoogle Scholar
  32. 172.
    B. Faermann, Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. II. The three-dimensional case. Numer. Math. 92, 467–499 (2002)MathSciNetzbMATHGoogle Scholar
  33. 174.
    M. Feischl, Th. Führer, N. Heuer, M. Karkulik, D. Praetorius, Adaptive boundary element methods. Arch. Comput. Methods Eng. 22, 309–389 (2015)MathSciNetCrossRefGoogle Scholar
  34. 175.
    M. Feischl, Th. Führer, M. Karkulik, J.M. Melenk, D. Praetorius, Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, part I: Weakly-singular integral equation. Calcolo 51, 531–562 (2014)MathSciNetCrossRefGoogle Scholar
  35. 176.
    M. Feischl, Th. Führer, G. Mitscha-Eibl, D. Praetorius, E.P. Stephan, Convergence of adaptive BEM and adaptive FEM-BEM coupling for estimators without h-weighting factor. Comput. Methods Appl. Math. 14, 485–508 (2014)MathSciNetCrossRefGoogle Scholar
  36. 177.
    M. Feischl, M. Karkulik, J.M. Melenk, D. Praetorius, Quasi-optimal convergence rate for an adaptive boundary element method. SIAM J. Numer. Anal. 51, 1327–1348 (2013)MathSciNetCrossRefGoogle Scholar
  37. 178.
    S. Ferraz-Leite, D. Praetorius, Simple a posteriori error estimators for the h-version of the boundary element method. Computing 83, 135–162 (2008)MathSciNetCrossRefGoogle Scholar
  38. 198.
    M. Gläfke, M. Maischak, E.P. Stephan, Coupling of FEM and BEM for a transmission problem with nonlinear interface conditions. Hierarchical and residual error indicators. Appl. Numer. Math. 62, 736–753 (2012)CrossRefGoogle Scholar
  39. 233.
    N. Heuer, Additive Schwarz method for the p-version of the boundary element method for the single layer potential operator on a plane screen. Numer. Math. 88, 485–511 (2001)MathSciNetCrossRefGoogle Scholar
  40. 234.
    N. Heuer, An hp-adaptive refinement strategy for hypersingular operators on surfaces. Numer. Methods Partial Differ. Equ. 18, 396–419 (2002)MathSciNetCrossRefGoogle Scholar
  41. 238.
    N. Heuer, M.E. Mellado, E.P. Stephan, hp-adaptive two-level methods for boundary integral equations on curves. Computing 67, 305–334 (2001)MathSciNetCrossRefGoogle Scholar
  42. 239.
    N. Heuer, M.E. Mellado, E.P. Stephan, A p-adaptive algorithm for the BEM with the hypersingular operator on the plane screen. Int. J. Numer. Methods Eng. 53, 85–104 (2002)MathSciNetCrossRefGoogle Scholar
  43. 265.
    M. Karkulik, J.M. Melenk, Local high-order regularization and applications to hp-methods. Comput. Math. Appl. 70, 1606–1639 (2015)MathSciNetCrossRefGoogle Scholar
  44. 274.
    A. Krebs, M. Maischak, E.P. Stephan, Adaptive FEM-BEM coupling with a Schur complement error indicator. Appl. Numer. Math. 60, 798–808 (2010)MathSciNetCrossRefGoogle Scholar
  45. 284.
    J.-L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I. Grundlehren der mathematischen Wissenschaften, vol. 181 (Springer, New York-Heidelberg, 1972). Translated from the French by P. KennethGoogle Scholar
  46. 293.
    M. Maischak, Manual of the Software Package Maiprogs, version 3.7.1 edn., 2012Google Scholar
  47. 294.
    M. Maischak, P. Mund, E.P. Stephan, Adaptive multilevel BEM for acoustic scattering. Comput. Methods Appl. Mech. Eng. 150, 351–367 (1997)MathSciNetCrossRefGoogle Scholar
  48. 308.
    P. Morin, K.G. Siebert, A. Veeser, A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18, 707–737 (2008)MathSciNetCrossRefGoogle Scholar
  49. 312.
    P. Mund, E.P. Stephan, An adaptive two-level method for the coupling of nonlinear FEM-BEM equations. SIAM J. Numer. Anal. 36, 1001–1021 (1999)MathSciNetCrossRefGoogle Scholar
  50. 313.
    P. Mund, E.P. Stephan, An adaptive two-level method for hypersingular integral equations in R 3. In: Proceedings of the 1999 International Conference on Computational Techniques and Applications (Canberra), vol. 42, 2000, pp. C1019–C1033Google Scholar
  51. 314.
    P. Mund, E.P. Stephan, J. Weisse, Two-level methods for the single layer potential in R 3. Computing 60, 243–266 (1998)MathSciNetCrossRefGoogle Scholar
  52. 337.
    L.F. Pavarino, Additive Schwarz methods for the p-version finite element method. Numer. Math. 66, 493–515 (1994)MathSciNetCrossRefGoogle Scholar
  53. 348.
    E. Rank, A posteriori error estimates and adaptive refinement for some boundary integral element method. In: Proceedings Int. Conf. on Accuracy Estimates and Adaptive Refinements in FE Computations (ARFEC, Lisbon, 1984), pp. 55–64Google Scholar
  54. 369.
    H. Schulz, O. Steinbach, A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem. Calcolo 37, 79–96 (2000)MathSciNetCrossRefGoogle Scholar
  55. 371.
    H. Schulz, O. Steinbach, W.L. Wendland, On Adaptivity in Boundary Element Methods. Aspects of the Boundary Element Method (Kluwer, 2001). In: IUTAM/IACM/IABEM Symposium on Advanced Mathematical and Computational Mechanics, pp. 315–325CrossRefGoogle Scholar
  56. 375.
    Ch. Schwab, W.L. Wendland, On the extraction technique in boundary integral equations. Math. Comput. 68, 91–122 (1999)MathSciNetCrossRefGoogle Scholar
  57. 377.
    K.G. Siebert, A convergence proof for adaptive finite elements without lower bound. IMA J. Numer. Anal. 31, 947–970 (2011)MathSciNetCrossRefGoogle Scholar
  58. 389.
    O. Steinbach, Adaptive finite element-boundary element solution of boundary value problems. J. Comput. Appl. Math. 106, 307–316 (1999)MathSciNetCrossRefGoogle Scholar
  59. 390.
    O. Steinbach, Adaptive boundary element methods based on computational schemes for Sobolev norms. SIAM J. Sci. Comput. 22, 604–616 (2000)MathSciNetCrossRefGoogle Scholar
  60. 396.
    E.P. Stephan, A boundary integral equation method for three-dimensional crack problems in elasticity. Math. Methods Appl. Sci. 8, 609–623 (1986)MathSciNetCrossRefGoogle Scholar
  61. 398.
    E.P. Stephan, Boundary integral equations for screen problems in R 3. Integr. Equ. Oper. Theory 10, 236–257 (1987)MathSciNetCrossRefGoogle Scholar
  62. 405.
    E.P. Stephan, M. Suri, The h-p version of the boundary element method on polygonal domains with quasiuniform meshes. RAIRO Modél. Math. Anal. Numér. 25, 783–807 (1991)MathSciNetCrossRefGoogle Scholar
  63. 423.
    T. von Petersdorff, Randwertprobleme der Elastizitätstheorie für Polyeder - Singularitäten und Approximation mit Randelementmethoden, Ph.D. thesis, TU Darmstadt, 1989Google Scholar
  64. 434.
    W.L. Wendland, D.H. Yu, Adaptive boundary element methods for strongly elliptic integral equations. Numer. Math. 53, 539–558 (1988)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Joachim Gwinner
    • 1
  • Ernst Peter Stephan
    • 2
  1. 1.Fakultät für Luft- und RaumfahrttechnikUniversität der Bundeswehr MünchenNeubiberg/MünchenGermany
  2. 2.Institut für Angewandte MathematikLeibniz Universität HannoverHannoverGermany

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