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Adaptive boundary element methods with convergence rates

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Abstract

This paper presents adaptive boundary element methods for positive, negative, as well as zero order operator equations, together with proofs that they converge at certain rates. The convergence rates are quasi-optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods. In particular, no saturation-type assumption is used. The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.

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Notes

  1. In view of (34), one has a computational advantage if \(r=1\), since there would be no fractional norms.

  2. Note that \(p\) and \(\tilde{p}\) are, respectively, \(\tilde{d}\) and \({d}\) as compared to, e.g., [45].

References

  1. Ainsworth, M., McLean, W., Tran, T.: The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling. SIAM J. Numer. Anal. 36(6), 1901–1932 (electronic) 1999. ISSN 0036–1429. doi:10.1137/S0036142997330809

  2. Aurada, M., Feischl, M., Führer, T., Karkulik, M., Praetorius, D.: Efficiency and optimality of some weighted-residual error estimator for adaptive 2d boundary element methods. ASC Report 15/2012. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien (2012a). http://publik.tuwien.ac.at/showentry.php?ID=207951

  3. Aurada, M., Ferraz-Leite, S., Praetorius, D.: Estimator reduction and convergence of adaptive BEM. Appl. Numer. Math. 62(6), 787–801 (2012b). doi:10.1016/j.apnum.2011.06.014

  4. Bernardi, C., Girault, V.: A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35(5), 1893–1916 (1998). doi:10.1137/S0036142995293766

  5. Binev, P., Dahmen, W., DeVore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 97(2), 219–268 (2004). doi:10.1007/s00211-003-0492-7

    Google Scholar 

  6. Carstensen, C., Faermann, B.: Mathematical foundation of a posteriori error estimates and adaptive mesh-refining algorithms for boundary integral equations of the first kind. Eng. Anal. Boundary Elem. 25(7), 497–509 (2001). doi:10.1016/S0955-7997(01)00012-1

  7. Carstensen, C., Praetorius, D.: Convergence of adaptive boundary element methods. J. Integral Equ. Appl. 24(1), 1–23 (2012). doi:10.1216/JIE-2012-24-1-1

    Google Scholar 

  8. Carstensen, C., Maischak, M., Stephan, E.P.: A posteriori error estimate and \(h\)-adaptive algorithm on surfaces for Symm’s integral equation. Numer. Math. 90(2). 197–213 (2001). doi:10.1007/s002110100287

    Google Scholar 

  9. Carstensen, C., Maischak, M., Praetorius, D., Stephan, E.P.: Residual-based a posteriori error estimate for hypersingular equation on surfaces. Numer. Math. 97(3), 397–425 (2004). doi:10.1007/s00211-003-0506-5

    Google Scholar 

  10. Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46(5), 2524–2550 (2008). doi:10.1137/07069047X

    Google Scholar 

  11. Clément, P.P.J.E.: Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. RAIRO Analyse Numérique 9(R–2), 77–84 (1975)

    MATH  Google Scholar 

  12. Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet methods for elliptic operator equations: convergence rates. Math. Comp. 70(233), 27–75 (2001). ISSN 0025–5718. doi:10.1090/S0025-5718-00-01252-7

    Google Scholar 

  13. Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet methods. II. Beyond the elliptic case. Found. Comput. Math. 2(3), 203–245 (2002). ISSN 1615–3375. doi:10.1007/s102080010027

    Google Scholar 

  14. Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19(3), 613–626 (1988). doi:10.1137/0519043

  15. Dahmen, W., Schneider, R.: Wavelets on manifolds. I. Construction and domain decomposition. SIAM J. Math. Anal. 31(1), 184–230 (1999). doi:10.1137/S0036141098333451

    Google Scholar 

  16. Dahmen, W., Stevenson, R.: Element-by-element construction of wavelets satisfying stability and moment conditions. SIAM J. Numer. Anal. 37(1), 319–352 (1999). doi:10.1137/S0036142997330949

    Google Scholar 

  17. Dahmen, W., Faermann, B., Graham, I.G., Hackbusch, W., Sauter, S.A.: Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method. Math. Comput. 73(247), 1107–1138 (2004). doi:10.1090/S0025-5718-03-01583-7

    Google Scholar 

  18. Dahmen, W., Harbrecht, H., Schneider, R.: Compression techniques for boundary integral equations–asymptotically optimal complexity estimates. SIAM J. Numer. Anal. 43(6), 2251–2271 (2006). doi:10.1137/S0036142903428852

    Google Scholar 

  19. Dahmen, W., Harbrecht, H., Schneider, R.: Adaptive methods for boundary integral equations: complexity and convergence estimates. Math. Comput. 76(259), 1243–1274 (2007). doi:10.1090/S0025-5718-07-01970-9

  20. Ding, Z.: A proof of the trace theorem of Sobolev spaces on Lipschitz domains. Proc. Am. Math. Soc. 124(2), 591–600 (1996). doi:10.1090/S0002-9939-96-03132-2

    Google Scholar 

  21. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996). doi:10.1137/0733054

  22. Erath, C., Ferraz-Leite, S., Funken, S.A., Praetorius, D.: Energy norm based a posteriori error estimation for boundary element methods in two dimensions. Appl. Numer. Math. 59(11), 2713–2734 (2009a). doi:10.1016/j.apnum.2008.12.024

  23. Erath, C., Funken, S.A., Goldenits, P., Praetorius, D.: Simple error estimators for the Galerkin BEM for some hypersingular integral equationin 2D. ASC Report 20/2009. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien (2009b). http://publik.tuwien.ac.at/showentry.php?ID=176588

  24. Faermann, B.: Local a-posteriori error indicators for the Galerkin discretization of boundary integral equations. Numer. Math. 79(1), 43–76 (1998). doi:10.1007/s002110050331

  25. Faermann, B.: Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. I. The two-dimensional case. IMA J. Numer. Anal. 20(2), 203–234 (2000). doi:10.1093/imanum/20.2.203

  26. Faermann, B.: Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. II. The three-dimensional case. Numer. Math. 92(3), 467–499 (2002). doi:10.1007/s002110100319

    Google Scholar 

  27. Feischl, M., Karkulik, M., Melenk, J.M., Praetorius, D.: Residual a-posteriori error estimates in BEM: Convergence of h-adaptive algorithms. ASC Report 21/2011. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien (2011a). http://publik.tuwien.ac.at/showentry.php?ID=197313

  28. Feischl, M., Karkulik, M., Melenk, J.M., Praetorius, D.: Quasi-optimal convergence rate for an adaptive boundary element method. ASC Report 28/2011. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien (2011b). http://publik.tuwien.ac.at/showentry.php?ID=198543

  29. Ferraz-Leite, S., Ortner, C., Praetorius, D.: Convergence of simple adaptive Galerkin schemes based on \(h-h/2\) error estimators. Numer. Math. 116(2), 291–316 (2010). doi:10.1007/s00211-010-0292-9

    Google Scholar 

  30. Gantumur, T.: An optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems. J. Comput. Appl. Math. 211(1), 90–102 (2008). doi:10.1016/j.cam.2006.11.013

    Google Scholar 

  31. Gantumur, T., Stevenson, R.: Computation of singular integral operators in wavelet coordinates. Computing 76(1–2), 77–107 (2006). ISSN 0010–485X. doi:10.1007/s00607-005-0135-1

    Google Scholar 

  32. Gantumur, T., Harbrecht, H., Stevenson, R.: An optimal adaptive wavelet method without coarsening of the iterands. Math. Comp. 76(258), 615–629 (2007). ISSN 0025–5718. doi:10.1090/S0025-5718-06-01917-X

    Google Scholar 

  33. Johnen, H., Scherer, K.: On the equivalence of the \(K\)-functional and moduli of continuity and some applications. In: Constructive Theory of Functions of Several Variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976) pp. 119–140. Lecture Notes in Math., vol. 571. Springer, Berlin (1977)

  34. Kim, D.: Trace theorems for Sobolev-Slobodeckij spaces with or without weights. J. Funct. Spaces Appl. 5(3), 243–268 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Marschall, J.: The trace of Sobolev-Slobodeckij spaces on Lipschitz domains. Manuscripta Math. 58(1–2), 47–65 (1987). doi:10.1007/BF01169082

    Google Scholar 

  36. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  37. Mekchay, K., Nochetto, R.H.: Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J. Numer. Anal. 43(5), 1803–1827 (2005). doi:10.1137/04060929X

    Google Scholar 

  38. Morin, P., Nochetto, R.H., Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44(4), 631–658 (2002). doi:10.1137/S0036144502409093 (Revised reprint of “Data oscillation and convergence of adaptive FEM” [SIAM J. Numer. Anal. 38(2), 2000, pp. 466–488])

    Google Scholar 

  39. Nochetto, R.H., von Petersdorff, T., Zhang, C.-S.: A posteriori error analysis for a class of integral equations and variational inequalities. Numer. Math. 116(3), 519–552 (2010). doi:10.1007/s00211-010-0310-y

    Google Scholar 

  40. Oswald, P.: Multilevel finite element approximation. Teubner Skripten zur Numerik. [Teubner Scripts on Numerical Mathematics]. B. G. Teubner, Stuttgart (1994). ISBN 3-519-02719-4. Theory and applications

  41. Sauter, S.A., Schwab, C.: Boundary Element Methods, volume 39 of Springer Series in Computational Mathematics. Springer, Berlin (2011) (Translated and expanded from the 2004 German original)

  42. Schneider, R.: Multiskalen- und Wavelet-Matrixkompression. In: Teubner, B.G. (eds.) Advances in Numerical Mathematics. Stuttgart (1998). Analysisbasierte Methoden zur effizienten Lösung großer vollbesetzter Gleichungssysteme. (Analysis-based methods for the efficient solution of large nonsparse systems of equations)

  43. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990). doi:10.2307/2008497

    Google Scholar 

  44. Stevenson, R.: Locally supported, piecewise polynomial biorthogonal wavelets on nonuniform meshes. Constr. Approx. 19(4), 477–508 (2003). doi:10.1007/s00365-003-0545-2

  45. Stevenson, R.: On the compressibility of operators in wavelet coordinates. SIAM J. Math. Anal. 35(5), 1110–1132 (2004). doi:10.1137/S0036141002411520

    Google Scholar 

  46. Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7(2), 245–269 (2007). doi:10.1007/s10208-005-0183-0

    Google Scholar 

  47. Stevenson, R.: The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77(261), 227–241 (2008). doi:10.1090/S0025-5718-07-01959-X

  48. Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59(3), 572–611 (1984). doi:10.1016/0022-1236(84)90066-1

    Google Scholar 

  49. von Petersdorff, T.: Randwertprobleme der Elastizitätstheorie für Polyeder: Singularitäten und Approximation mit Randelementmethoden. PhD thesis, Technische Hochschule Darmstadt, Switzerland (1989)

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Acknowledgments

I would like to thank Dirk Praetorius for carefully reading an earlier version of this manuscript, and for making several important comments. I thank the anonymous referees for their reviews and suggestions. I also thank Michael Renardy, Nilima Nigam, and Elias Pipping over at mathoverflow.net for pointing out the reference [34], and Doyoon Kim for making his paper available to me. This work is supported by an NSERC Discovery Grant and an FQRNT Nouveaux Chercheurs Grant.

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    Tsogtgerel Gantumur

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Gantumur, T. Adaptive boundary element methods with convergence rates. Numer. Math. 124, 471–516 (2013). https://doi.org/10.1007/s00211-013-0524-x

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