Hypersingular residuals - a new approach for error estimation in the boundary element method. (English) Zbl 0883.73082
Summary: We present a new approach for a posteriori ‘pointwise’ error estimation in the boundary element method. The estimator relies upon evaluation of the residual of hypersingular integral equations, and is therefore intrinsic to the boundary integral equation approach. A methodology is developed for approximating the error on the boundary as well as in the interior of the domain. Extensive computational experiments have been performed for the two-dimensional Laplace equation, and the numerical results indicate that the error estimates successfully track the form of the exact error curve.
MSC:
| 74S15 | Boundary element methods applied to problems in solid mechanics |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
References:
| [1] | Kita, Adv. Eng. Software 19 pp 21– (1994) · Zbl 0811.65100 · doi:10.1016/0965-9978(94)90043-4 |
| [2] | and (eds.), Adaptive Finite and Boundary Element Methods, Computational Mechanics Publications, Southampton, and Elsevier Applied Science, London, 1993. |
| [3] | and (eds.), Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Wiley, Chichester, 1986. |
| [4] | Noor, Finite Elements Anal. Des. 3 pp 1– (1987) · Zbl 0608.73072 · doi:10.1016/0168-874X(87)90030-8 |
| [5] | and , ’Advances in adaptive improvements–A survey of adaptive finite element methods in computational mechanics’, in and (eds.), State-Of-The-Art Surveys On Computational Mechanics, The American Society of Mechanical Engineers (ASME), New York, Chapter 13, pp. 441-467 (1989) (cites 184 references). |
| [6] | Strouboulis, Comp. Methods Appl. Mech. Eng. 97 pp 399– (1992) · Zbl 0764.65064 · doi:10.1016/0045-7825(92)90053-M |
| [7] | Strouboulis, Comp. Methods Appl. Mech. Eng. 100 pp 359– (1992) · Zbl 0782.65127 · doi:10.1016/0045-7825(92)90090-7 |
| [8] | Babuška, SIAM Rev. 36 pp 578– (1994) · Zbl 0813.65118 · doi:10.1137/1036141 |
| [9] | Mackerle, Finite Elements Anal. Des. 15 pp 177– (1990) · Zbl 0799.73001 · doi:10.1016/0168-874X(93)90064-W |
| [10] | and (eds.), Adaptive Computational Methods for Partial Differential Equations, SIAM (Society for Industrial and Applied Mathematics), Philadelphia, 1983. |
| [11] | Kelly, Int. J. numer. methods eng. 24 pp 1921– (1987) · Zbl 0625.65096 · doi:10.1002/nme.1620241008 |
| [12] | and , The Finite Element Method, Vol. 1: Basic Formulation and Linear Problems, McGraw-Hill, London, 1989. |
| [13] | Zienkiewicz, Int. J. numer. methods. eng. 33 pp 1331– (1992) · Zbl 0769.73084 · doi:10.1002/nme.1620330702 |
| [14] | Zienkiewicz, Int. J. numer. methods eng. 33 pp 1365– (1992) · Zbl 0769.73085 · doi:10.1002/nme.1620330703 |
| [15] | Zienkiewicz, Int. J. numer. methods eng. 37 pp 3195– (1994) · doi:10.1002/nme.1620371812 |
| [16] | and , Finite Element Analysis, Wiley, New York, U.S.A., 1991. |
| [17] | Guiggiani, Int. J. numer. methods eng. 29 pp 1247– (1990) · Zbl 0714.73084 · doi:10.1002/nme.1620290610 |
| [18] | Gray, Eng. Anal. Boundary Elements 6 pp 180– (1989) · doi:10.1016/0955-7997(89)90015-5 |
| [19] | Gray, Int. J. numer. methods eng. 29 pp 1135– (1990) · Zbl 0717.73081 · doi:10.1002/nme.1620290603 |
| [20] | Gray, Eng. Anal. Boundary Elements 11 pp 327– (1993) · doi:10.1016/0955-7997(93)90047-O |
| [21] | and , ’Hypersingular Boundary integral Equations: Their occurrence, interpretation, regularization and computation’, in and (eds.), Developments in Boundary Element Methods, Elsevier Applied Science, London and New York, 1992, Vol. 7, Chapter 6, pp. 207-252. |
| [22] | Guiggiani, ASME J. Appl. Mech. 59 pp 604– (1992) · Zbl 0765.73072 · doi:10.1115/1.2893766 |
| [23] | Rosen, Int. J. numer. methods eng. 35 pp 563– (1992) · Zbl 0768.73088 · doi:10.1002/nme.1620350309 |
| [24] | Rosen, SIAM J. Appl. Math. 53 pp 340– (1993) · Zbl 0771.65076 · doi:10.1137/0153020 |
| [25] | Lutz, Commun. numer. methods eng. 9 pp 909– (1993) · Zbl 0797.65013 · doi:10.1002/cnm.1640091107 |
| [26] | ’Symbolic computation of hypersingular boundary integrals’, in , and (eds.), Advances in Boundary Element Techniques, Springer, Berlin, Heidelberg, 1993, Chapter 8, pp. 157-172. · doi:10.1007/978-3-642-51027-4_8 |
| [27] | Parreira, Adv. Eng., Software 15 pp 249– (1992) · Zbl 0771.65075 · doi:10.1016/0965-9978(92)90107-Q |
| [28] | and , The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton and Elsevier Applied Science, London, 1991. · doi:10.1007/978-94-011-3690-7 |
| [29] | Wei, Int. J. Solids Struct. 31 pp 533– (1994) · Zbl 0946.74563 · doi:10.1016/0020-7683(94)90007-8 |
| [30] | and , Boundary Element Methods in Manufacturing, Oxford University Press, 1996. |
| [31] | and , A Posteriori Pointwise Error Estimates for the Boundary Element Method, Technical Memorandum ORNL/TM-12820, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A. 1994. |
| [32] | Ingber, Comput. Mech. 11 pp 11– (1993) · Zbl 0762.76062 · doi:10.1007/BF00370070 |
| [33] | Hartmann, Ing. Archiv 55 pp 440– (1985) · Zbl 0573.73094 · doi:10.1007/BF00537652 |
| [34] | and , ’On symmetrization in boundary element elastic and elastoplastic analysis’, in and (eds.), Discretization Methods in Structural Mechanics, Springer, Berlin, 1990, pp. 191-200. · doi:10.1007/978-3-642-49373-7_18 |
| [35] | Balakrishna, Comp. Methods Appl. Mech. Eng. 111 pp 335– (1994) · Zbl 0846.73071 · doi:10.1016/0045-7825(94)90138-4 |
| [36] | Yu, Zeitshrift fur angewandte Mathematik und Mechanik 68 pp t435– (1988) |
| [37] | Yu, Comp. Methods Appl. Mech. Eng. 91 pp 1237– (1991) · doi:10.1016/0045-7825(91)90075-H |
| [38] | Wendland, J. Comput. Math. 10 pp 273– (1992) |
| [39] | Rank, Int. J. numer. methods eng. 28 pp 1335– (1989) · Zbl 0705.73260 · doi:10.1002/nme.1620280608 |
| [40] | Sun, Int. J. numer. methods eng. 33 pp 537– (1992) · Zbl 0810.65109 · doi:10.1002/nme.1620330305 |
| [41] | Abe, Adv. Eng. Software 15 pp 231– (1992) · Zbl 0770.65074 · doi:10.1016/0965-9978(92)90105-O |
| [42] | Ingber, Eng. Anal. Boundary Elements 9 pp 13– (1992) · doi:10.1016/0955-7997(92)90120-V |
| [43] | Camp, Math. Comp. Modelling 15 pp 59– (1991) · Zbl 0726.65129 · doi:10.1016/0895-7177(91)90053-A |
| [44] | Krishnasamy, Comput. Mech. 9 pp 267– (1992) · Zbl 0755.65108 · doi:10.1007/BF00370035 |
| [45] | ’Novel formulations of the boundary element method for fracture mechanics and error estimation’, Ph.D. Dissertation, Cornell University, Ithaca, NY, U.S.A., 1995. |
| [46] | and , ’Pointwise error estimates for boundary element calculations’, in and (eds.), Boundary Element Technology IX, Computational Mechanics Publications, Southampton and Boston, 1994, pp. 253-260. |
| [47] | Zhao, Eng. Anal. Boundary Elements 13 pp 201– (1994) · doi:10.1016/0955-7997(94)90022-1 |
| [48] | ’On the convergence of the boundary element method’, Ph.D. Dissertation, Cornell University, Ithaca, NY., U.S.A., 1982. |
| [49] | Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, New York, 1987. |
| [50] | ’Error analysis of boundary integral methods’, Acta Numerica, 287-339 (1992). |
| [51] | Rizzo, Quart. Appl. Math. 25 pp 83– (1967) · Zbl 0158.43406 · doi:10.1090/qam/99907 |
| [52] | Bugeda, Int. J. numer. methods eng. 36 pp 3161– (1993) · Zbl 0780.73049 · doi:10.1002/nme.1620361807 |
| [53] | and , Design Sensitivity Analysis of Structural Systems, Academic Press, Orlando, FL, 1986. · Zbl 0618.73106 |
| [54] | and , Elements of Structural Optimization, 3rd. revised and expanded edition, Kluwer Academic Publishers, Dordrecht, 1992. · doi:10.1007/978-94-011-2550-5 |
| [55] | Jayaswal, Finite Elements Anal. Des. 14 pp 17– (1993) · Zbl 0781.73066 · doi:10.1016/0168-874X(93)90076-3 |
| [56] | Gray, Int. J. numer. methods eng. 36 pp 2357– (1993) · Zbl 0781.73074 · doi:10.1002/nme.1620361404 |
| [57] | Tworzydlo, Comp. Methods Appl. Mech. Eng. 104 pp 87– (1993) · Zbl 0772.73082 · doi:10.1016/0045-7825(93)90208-F |
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