×

zbMATH — the first resource for mathematics

A posteriori error estimates for boundary element methods. (English) Zbl 0831.65120
The paper presents a new adaptive \(h\)-version of the Galerkin discretization for the boundary element method which is based on a posteriori error estimates. After giving a natural framework for these a posteriori error estimates, three examples are discussed involving the Dirichlet problem, the Neumann problem (for a closed and an open surface), and a transmission problem for the Laplacian.
The approach leads to an upper bound of the global error in energy norms consisting of terms which can be evaluated locally and needs no restriction at all on the mesh for two-dimensional problems. The efficiency of the method is shown by numerical examples which yield almost optimal convergence rates even in the presence of singularities.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65R20 Numerical methods for integral equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. Asadzadeh and K. Eriksson, An adaptive finite element method for a potential problem, Preprint. · Zbl 0814.65134
[2] J. Bergh and J. Löfström, Interpolation spaces, Springer, Berlin, 1976. · Zbl 0344.46071
[3] I. Babuška and A. Miller, A posteriori error estimates and adaptive techniques for the finite element method, Tech. Note BN-968, Institute for Physical Science and Technology, Univ. of Maryland, College Park, MD, 1981.
[4] Martin Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988), no. 3, 613 – 626. · Zbl 0644.35037 · doi:10.1137/0519043 · doi.org
[5] M. Costabel and E. Stephan, The normal derivative of the double layer potential on polygons and Galerkin approximation, Applicable Anal. 16 (1983), no. 3, 205 – 228. · Zbl 0508.31003 · doi:10.1080/00036818308839470 · doi.org
[6] 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. · Zbl 0655.65129
[7] Martin Costabel and Ernst Stephan, A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl. 106 (1985), no. 2, 367 – 413. · Zbl 0597.35021 · doi:10.1016/0022-247X(85)90118-0 · doi.org
[8] Carsten Carstensen and Ernst P. Stephan, Adaptive boundary element methods for some first kind integral equations, SIAM J. Numer. Anal. 33 (1996), no. 6, 2166 – 2183. · Zbl 0863.65073 · doi:10.1137/S0036142993253503 · doi.org
[9] -, Adaptive boundary element methods for transmission problems, Preprint, 1993.
[10] Kenneth Eriksson and Claes Johnson, An adaptive finite element method for linear elliptic problems, Math. Comp. 50 (1988), no. 182, 361 – 383. · Zbl 0644.65080
[11] 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. · Zbl 0732.65093 · doi:10.1137/0728003 · doi.org
[12] V. J. Ervin and E. P. Stephan, A boundary element Galerkin method for a hypersingular integral equation on open surfaces, Math. Methods Appl. Sci. 13 (1990), no. 4, 281 – 289. · Zbl 0717.65092 · doi:10.1002/mma.1670130402 · doi.org
[13] V. J. Ervin, N. Heuer, and E. P. Stephan, On the \?-\? version of the boundary element method for Symm’s integral equation on polygons, Comput. Methods Appl. Mech. Engrg. 110 (1993), no. 1-2, 25 – 38. · Zbl 0842.65076 · doi:10.1016/0045-7825(93)90017-R · doi.org
[14] L. Hörmander, Linear partial differential operators., Springer-Verlag, Berlin-Heidelberg-New York, 1963. · Zbl 0108.09301
[15] 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). · Zbl 0778.73071 · doi:10.1016/0045-7825(92)90020-K · doi.org
[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. · Zbl 0223.35039
[17] J.C. Nedelec, La méthode des éléments finis appliquée aux équations intégrales de la physique, First meeting AFCET-SMF on applied mathematics Palaiseau, Vol. 1, 1978, pp. 181-190. · Zbl 0486.45008
[18] F. V. Postell and E. P. Stephan, On the \?-, \?- and \?-\? versions of the boundary element method — numerical results, Comput. Methods Appl. Mech. Engrg. 83 (1990), no. 1, 69 – 89. · Zbl 0732.65101 · doi:10.1016/0045-7825(90)90125-6 · doi.org
[19] E. Rank, Adaptive boundary element methods (invited contribution), Boundary elements IX, Vol. 1 (Stuttgart, 1987) Comput. Mech., Southampton, 1987, pp. 259 – 278.
[20] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. · Zbl 0867.46001
[21] 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. · Zbl 0799.65115 · doi:10.1080/00036819308840148 · doi.org
[22] 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. , https://doi.org/10.1093/imanum/8.1.105 I. H. Sloan and A. Spence, The Galerkin method for integral equations of the first kind with logarithmic kernel: applications, IMA J. Numer. Anal. 8 (1988), no. 1, 123 – 140. · Zbl 0636.65144 · doi:10.1093/imanum/8.1.123 · doi.org
[23] Ernst P. Stephan, Boundary integral equations for screen problems in \?³, Integral Equations Operator Theory 10 (1987), no. 2, 236 – 257. · Zbl 0653.35016 · doi:10.1007/BF01199079 · doi.org
[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. · Zbl 0657.65138 · doi:10.1007/BF01397551 · doi.org
[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. · Zbl 0758.65072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.