Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

An a posteriori error estimate for a
first-kind integral equation


Author: Carsten Carstensen
Journal: Math. Comp. 66 (1997), 139-155
MSC (1991): Primary 65N38, 65N15, 65R20, 45L10
MathSciNet review: 1372001
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we present a new a posteriori error estimate for the boundary element method applied to an integral equation of the first kind. The estimate is local and sharp for quasi-uniform meshes and so improves earlier work of ours. The mesh-dependence of the constants is analyzed and shown to be weaker than expected from our previous work. Besides the Galerkin boundary element method, the collocation method and the qualocation method are considered. A numerical example is given involving an adaptive feedback algorithm.


References [Enhancements On Off] (What's this?)

  • 1. Douglas N. Arnold and Wolfgang L. Wendland, On the asymptotic convergence of collocation methods, Math. Comp. 41 (1983), no. 164, 349–381. MR 717691, 10.1090/S0025-5718-1983-0717691-6
  • 2. Mohammad Asadzadeh and Kenneth Eriksson, On adaptive finite element methods for Fredholm integral equations of the second kind, SIAM J. Numer. Anal. 31 (1994), no. 3, 831–855. MR 1275116, 10.1137/0731045
  • 3. Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275
  • 4. C. Carstensen: Adaptive boundary element methods and adaptive finite element and boundary element coupling. In Proceedings Boundary Value Problems and Integral Equations on Non-Smooth Domains, eds. M. Costabel, M. Dauge, S. Nicaise. Lecture notes in pure and applied mathematics 167, Marcel Dekker New York 1995, 47-58. CMP 95:03
  • 5. C. Carstensen: Efficiency of a posteriori BEM error estimates for first-kind integral equations on quasi-uniform meshes. Math. Comp. 65 (1996), 69-84. CMP 96:03
  • 6. C. Carstensen: A posteriori error estimate for the symmetric coupling of finite elements and boundary elements, Computing (in press), 1996.
  • 7. Carsten Carstensen and Ernst P. Stephan, A posteriori error estimates for boundary element methods, Math. Comp. 64 (1995), no. 210, 483–500. MR 1277764, 10.1090/S0025-5718-1995-1277764-7
  • 8. C. Carstensen, E.P. Stephan: Adaptive boundary element methods for some first-kind integral equations. SIAM J. Numer. Anal. (1996), to appear.
  • 9. C. Carstensen, E.P. Stephan: Adaptive boundary element methods for transmission problems. J. Austr. Math. Soc. Ser. B (1996), to appear.
  • 10. G. A. Chandler, Discrete norms for the convergence of boundary element methods, Workshop on Theoretical and Numerical Aspects of Geometric Variational Problems (Canberra, 1990) Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 26, Austral. Nat. Univ., Canberra, 1991, pp. 62–78. MR 1139029
  • 11. G. A. Chandler and I. H. Sloan, Spline qualocation methods for boundary integral equations, Numer. Math. 58 (1990), no. 5, 537–567. MR 1080305, 10.1007/BF01385639
  • 12. Martin Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988), no. 3, 613–626. MR 937473, 10.1137/0519043
  • 13. 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
  • 14. B. Faermann: Lokale a-posteriori-Fehlerschätzer bei der Diskretisierung von Randintegralgleichungen. PhD-thesis, University of Kiel, FRG (1993).
  • 15. N. Heuer: hp-Versionen der Randelementemethode. PhD-thesis, University of Hannover, FRG (1992).
  • 16. J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177
  • 17. T. von Petersdorff: Randwertprobleme der Elastizitätstheorie für Polyeder - Singularitäten und Approximation mit Randelementmethoden. PhD-thesis, TH Darmstadt, FRG (1989).
  • 18. E. Rank: Adaptive boundary element methods. in: C.A. Brebbia, W.L. Wendland and G. Kuhn, eds., Boundary Elements 9, Vol. 1, 259-273. Springer Verlag Heidelberg 1987. CMP 21:03
  • 19. 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, 10.1080/00036819308840148
  • 20. 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, 10.1093/imanum/8.1.105
  • 21. E. P. Stephan and M. Suri, The ℎ-𝑝 version of the boundary element method on polygonal domains with quasiuniform meshes, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 6, 783–807 (English, with French summary). MR 1135993
  • 22. E.P. Stephan, W.L. Wendland: Remarks on Galerkin and least squares methods with finite elements for general elliptic problems. Manuscripta Geodaetica 1 (1976) 93-123.
  • 23. 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, 10.1002/mma.1670010302
  • 24. 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, 10.1007/BF01397551
  • 25. 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

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65N38, 65N15, 65R20, 45L10

Retrieve articles in all journals with MSC (1991): 65N38, 65N15, 65R20, 45L10


Additional Information

Carsten Carstensen
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany
Email: cc@numerik.uni-kiel.de

DOI: http://dx.doi.org/10.1090/S0025-5718-97-00790-4
Keywords: Integral equations, boundary element method, a~posteriori error estimate, adaptive algorithm, collocation method, qualocation method
Received by editor(s): February 20, 1995
Received by editor(s) in revised form: November 6, 1995, and January 26, 1996
Article copyright: © Copyright 1997 American Mathematical Society