Efficiency of a posteriori BEM–error estimates for first-kind integral equations on quasi–uniform meshes

Carstensen, Carsten

doi:10.1090/S0025-5718-96-00671-0

Efficiency of a posteriori BEM–error estimates for first-kind integral equations on quasi–uniform meshes
HTML articles powered by AMS MathViewer

by Carsten Carstensen PDF

Math. Comp. 65 (1996), 69-84 Request permission

Abstract:

In the numerical treatment of integral equations of the first kind using boundary element methods (BEM), the author and E. P. Stephan have derived a posteriori error estimates as tools for both reliable computation and self-adaptive mesh refinement. So far, efficiency of those a posteriori error estimates has been indicated by numerical examples in model situations only. This work affirms efficiency by proving the reverse inequality. Based on best approximation, on inverse inequalities and on stability of the discretization, and complementary to our previous work, an abstract approach yields a converse estimate. This estimate proves efficiency of an a posteriori error estimate in the BEM on quasi–uniform meshes for Symm’s integral equation, for a hypersingular equation, and for a transmission problem.

References

I. Babuška and A. Miller, A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator, Comput. Methods Appl. Mech. Engrg. 61 (1987), no. 1, 1–40. MR 880421, DOI 10.1016/0045-7825(87)90114-9
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
C. Carstensen, Adaptive boundary element methods and adaptive finite element and boundary element coupling, Boundary Value Problems and Integral Equations in Nonsmooth Domains (M. Costabel, M. Dauge, S. Nicaise, eds.), Proc. Conf. at the CIRM, Luminy, Lecture Notes in Pure and Appl. Math. vol. 167, Marcel Dekker, 1995, pp. 47–58.
Carsten Carstensen and Ernst P. Stephan, A posteriori error estimates for boundary element methods, Math. Comp. 64 (1995), no. 210, 483–500. MR 1277764, DOI 10.1090/S0025-5718-1995-1277764-7
—, Adaptive boundary element methods for some first kind integral equations, SIAM J. Numer. Anal. (accepted for publication 1994).
—, Adaptive boundary element methods for transmission problems, J. Austral. Math. Soc. Ser. B (accepted for publication 1995).
—, Adaptive coupling of boundary elements and finite elements, Math. Modelling Numer. Anal. (accepted for publication 1995).
C. Carstensen, S. Funken, and E. P. Stephan, On the adaptive coupling of FEM and BEM in 2–d–elasticity, Preprint, Institut für Angewandte Mathematik, Universität Hannover, 1993.
Martin Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988), no. 3, 613–626. MR 937473, DOI 10.1137/0519043
Martin Costabel and Ernst Stephan, Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, Mathematical models and methods in mechanics, Banach Center Publ., vol. 15, PWN, Warsaw, 1985, pp. 175–251. MR 874845
Martin Costabel and Ernst Stephan, A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl. 106 (1985), no. 2, 367–413. MR 782799, DOI 10.1016/0022-247X(85)90118-0
M. Crouzeix and V. Thomée, The stability in $L_p$ and $W^1_p$ of the $L_2$-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521–532. MR 878688, DOI 10.1090/S0025-5718-1987-0878688-2
Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635, DOI 10.1007/978-3-662-02888-9
Kenneth Eriksson and Claes Johnson, An adaptive finite element method for linear elliptic problems, Math. Comp. 50 (1988), no. 182, 361–383. MR 929542, DOI 10.1090/S0025-5718-1988-0929542-X
Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal. 28 (1991), no. 1, 43–77. MR 1083324, DOI 10.1137/0728003
B. Faehrmann, Lokale a–posteriori–Fehlerschätzer bei der Diskretisierung von Randintegralgleichungen, Ph.D. thesis, University of Kiel, FRG, 1993.
N. Heuer, hp–Versionen der Randelementemethode, Ph.D. thesis, University of Hannover, FRG, 1992.
W. Gui and I. Babuška, The $h,\;p$ and $h$-$p$ versions of the finite element method in $1$ dimension. I. The error analysis of the $p$-version, Numer. Math. 49 (1986), no. 6, 577–612. MR 861522, DOI 10.1007/BF01389733
G. C. Hsiao and W. L. Wendland, The Aubin-Nitsche lemma for integral equations, J. Integral Equations 3 (1981), no. 4, 299–315. MR 634453
Claes Johnson and Peter Hansbo, Adaptive finite element methods in computational mechanics, Comput. Methods Appl. Mech. Engrg. 101 (1992), no. 1-3, 143–181. Reliability in computational mechanics (Kraków, 1991). MR 1195583, DOI 10.1016/0045-7825(92)90020-K
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350177
J. C. Nedelec, La méthode des élements finis appliquée aux equations intégrales de la physique, First meeting AFCET–SMF on applied mathematics Palaiseau, Vol. 1, 1978, pp. 181–190.
E. Rank, Adaptive boundary element methods, Boundary Elements 9, Vol. 1 (C. A. Brebbia, W. L. Wendland and G. Kuhn, eds.), Springer-Verlag, Heidelberg, 1987, pp. 259—278.
J. Saranen and W. L. Wendland, Local residual-type error estimates for adaptive boundary element methods on closed curves, Appl. Anal. 48 (1993), no. 1-4, 37–50. MR 1278122, DOI 10.1080/00036819308840148
I. H. Sloan and A. Spence, The Galerkin method for integral equations of the first kind with logarithmic kernel: theory, IMA J. Numer. Anal. 8 (1988), no. 1, 105–122. MR 967846, DOI 10.1093/imanum/8.1.105
Ernst Stephan and Wolfgang Wendland, Remarks to Galerkin and least squares methods with finite elements for general elliptic problems, Ordinary and partial differential equations (Proc. Fourth Conf., Univ. Dundee, Dundee, 1976) Lecture Notes in Math., Vol. 564, Springer, Berlin, 1976, pp. 461–471. MR 0520343
W. L. Wendland, E. Stephan, and G. C. Hsiao, On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Methods Appl. Sci. 1 (1979), no. 3, 265–321. MR 548943, DOI 10.1002/mma.1670010302
R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques, Preprint, 1993.
W. L. Wendland and De Hao Yu, Adaptive boundary element methods for strongly elliptic integral equations, Numer. Math. 53 (1988), no. 5, 539–558. MR 954769, DOI 10.1007/BF01397551
W. L. Wendland and De Hao Yu, A posteriori local error estimates of boundary element methods with some pseudo-differential equations on closed curves, J. Comput. Math. 10 (1992), no. 3, 273–289. MR 1167929