Journal of Integral Equations and Applications

Convergence of adaptive boundary element methods

Carsten Carstensen and Dirk Praetorius

Full-text: Open access

Article information

Source
J. Integral Equations Applications, Volume 24, Number 1 (2012), 1-23.

Dates
First available in Project Euclid: 4 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1333560447

Digital Object Identifier
doi:10.1216/JIE-2012-24-1-1

Mathematical Reviews number (MathSciNet)
MR2911088

Zentralblatt MATH identifier
1238.65124

Subjects
Primary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N38: Boundary element methods 65N50: Mesh generation and refinement 65R20: Integral equations

Keywords
Adaptive boundary element method adaptive algorithm error reduction convergence

Citation

Carstensen, Carsten; Praetorius, Dirk. Convergence of adaptive boundary element methods. J. Integral Equations Applications 24 (2012), no. 1, 1--23. doi:10.1216/JIE-2012-24-1-1. https://projecteuclid.org/euclid.jiea/1333560447


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References

  • J. Bergh and J. Löfström, Interpolation spaces, Grundl. Math. Wissen. 223, Springer Verlag, Berlin, 1976.
  • P. Binev, W. Dahmen and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), 219-268.
  • C. Carstensen, Efficiency of a posteriori BEM error estimates for first kind integral equations on quasi-uniform meshes, Math. Comp. 65 (1996), 69-84.
  • C. Carstensen, An a posteriori error estimate for a first-kind integral equation, Math. Comp. 66 (1997), 139-155.
  • C. Carstensen and 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 (2001), 497-509.
  • C. Carstensen, M. Maischak, D. Praetorius and E. Stephan, Residual-based a posteriori error estimate for hypersingular equation on surfaces, Numer. Math. 97 (2004), 397-425.
  • C. Carstensen, M. Maischak and E. Stephan, A posteriori estimate and h-adaptive algorithm on surfaces for Symm's integral equation, Numer. Math. 90 (2001), 197-213.
  • C. Carstensen and D. Praetorius, Averaging techniques for the effective numerical solution of Symm's integral equation of the first kind, SIAM J. Sci. Comp. 27 (2006), 1226-1260.
  • –––, Averaging techniques for the a posteriori BEM error control for a hypersingular integral equation in two dimensions, SIAM J. Sci. Comp. 29 (2007), 782-810.
  • C. Carstensen and E. Stephan, A posteriori error estimates for boundary element methods, Math. Comp. 64 (1995), 483-500.
  • –––, Adaptive boundary element methods for some first kind integral equations, SIAM J. Numer. Anal. 33 (1996), 2166-2183.
  • J. Cascon, C. Kreuzer, R. Nochetto and K. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008), 2524-2550.
  • A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equations: Convergence rates, Math. Comp. 70 (2001), 27-75.
  • –––, Adaptive wavelet methods II: Beyond the elliptic case, Found. Comput. Math. 2 (2002), 203-245.
  • –––, Adaptive wavelet schemes for nonlinear variational problems, SIAM J. Numer. Anal. 54 (2003), 1785-1823.
  • W. Dahmen, H. Harbrecht and R. Schneider, Adaptive methods for boundary integral equations: Complexity and convergence estimates, Math. Comp. 76 (2007), 1243-1274.
  • W. Doerfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal. 33 (1996), 1106-1124.
  • W. Doerfler and R. Nochetto, Small data oscillation implies the saturation assumption, Numer. Math. 91 (2002), 1-12.
  • C. Erath, S. Ferraz-Leite, S. Funken and D. Praetorius, Energy norm based a posteriori error estimation for boundary element methods in two dimensions, Appl. Numer. Math. 59 (2009), 2713-2734.
  • C. Erath, S. Funken, P. Goldenits and D. Praetorius, Simple error estimators for the Galerkin BEM for some hypersingular integral equation in 2D, Appl. Anal., to appear.
  • V. Ervin and N. Heuer, An adaptive boundary element method for the exterior Stokes problem in three dimensions, IMA J. Numer. Anal. 26 (2006), 337-357.
  • B. Faermann, Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods, Part I: The two-dimensional case, IMA J. Numer. Anal. 20 (2000), 203-234.
  • –––, Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods, Part I: The three-dimensional case, Numer. Math. 92 (2002), 297-325.
  • S. Ferraz-Leite and D. Praetorius, Simple a posteriori error estimators for the $h$-version of the boundary element method, Computing 83 (2008), 135-162.
  • T. Gantumur and R. Stevenson, Computation of singular integral operators in wavelet coordinates, Computing 76 (2006), 77-107.
  • N. Heuer, M. Mellado and E. Stephan, $hp$-adaptive two-level methods for boundary integral equations on curves, Computing 67 (2001), 305-334.
  • M. Maischak, P. Mund and E. Stephan, Adaptive multilevel methods for acoustic scattering, Comput. Meth. Appl. Mech. Eng. 150 (2001), 351-367.
  • W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.
  • P. Morin, R. Nochetto and K. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), 466-488.
  • P. Morin, K. Siebert and A. Veeser, A basic convergence result for conforming adaptive finite elements, Math. Models Meth. Appl. Sci. 18 (2008), 707-737.
  • P. Mund, E. Stephan and J. Weisse, Two-level methods for the single layer potential in $\r^3$, Computing 60 (1998), 243-266.
  • S. Sauter and C. Schwab, Randelemente-Analyse und Implementierung schneller Algorithmen, Teubner, Wiesbaden, 2004.
  • H. Schulz and O. Steinbach, A new a posteriori error estimator in adaptive direct boundary element methods. The Dirichlet problem, Calcolo 37 (2000), 79-96.
  • O. Steinbach, Adaptive boundary element methods based on computational schemes for Sobolev norms, SIAM J. Sci. Comput. 22 (2000), 604-616.
  • R. Stevenson, An optimal adaptive finite element method, SIAM J. Numer. Anal. 42 (2005), 2188-2217.
  • –––, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), 245-269. \noindentstyle