×

Goal-oriented error estimation and adaptivity for the finite element method. (English) Zbl 0987.65110

The authors present an analysis for an a posteriori error estimate with respect to a class of elliptic boundary value problems. A new concept of goal oriented adaptivity is introduced and upper and lower bounds are established. Finally, a number of numerical experiments are performed which clearly show the reliability of the bounds in terms of global energy estimates.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Becker, R.; Rannacher, R., Weighted a posteriori error control in FE method, ENUMATH-95, Paris (1995) · Zbl 0968.65083
[2] Becker, R.; Rannacher, R., A feedback approach to error control in finite elements methods: Basic analysis and examples, Institut für Angewandte Mathematik (1996), Universität Heidelberg, Preprint 96-52 · Zbl 0868.65076
[3] Rannacher, R.; Stuttmeier, F. T., A posteriori error control in finite element methods via duality techniques: Application to perfect elasticity (1997), Institut für Angewandte Mathematik: Institut für Angewandte Mathematik Universität Heidelberg, Preprint 97-16
[4] Cirak, F.; Ramm, E., A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem, Comp. Meth. in Appl. Mech. and Eng., 156, 351-362 (1998) · Zbl 0947.74062
[5] Paraschivoiu, M.; Patera, A. T., A hierarchical duality approach to bounds for the outputs of partial differential equations, Comp. Meth. in Appl. Mech. and Eng., 158, 389-407 (1998) · Zbl 0953.76054
[6] Peraire, J.; Patera, A. T., Bounds for linear-functional outputs of coercive partial differential equations: Local indicators and adaptive refinement, (Ladevèze, P.; Oden, J. T., Advances in Adaptive Computational Methods in Mechanics (1998), Elsevier: Elsevier Amsterdam), 199-215
[7] Babuška, I.; Strouboulis, T.; Copps, K.; Gangaraj, S. K.; Upadhyay, C. S., A posteriori error estimation for finite element and generalized finite element method (1998), The University of Texas at Austin, TICAM Report 98-01 · Zbl 0933.74061
[8] S. Prudhomme and J.T. Oden, On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors, Comp. Meth. in Appl. Mech. and Eng. (to appear).; S. Prudhomme and J.T. Oden, On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors, Comp. Meth. in Appl. Mech. and Eng. (to appear). · Zbl 0945.65123
[9] Ainsworth, M.; Oden, J. T., A unified approach to a posteriori error estimation using element residual methods, Numer. Math., 65, 1, 23-50 (1993) · Zbl 0797.65080
[10] Ainsworth, M.; Oden, J. T., A posteriori error estimation in finite element analysis, Comp. Meth. in Appl. Mech. and Eng., 142, 1-88 (1997) · Zbl 0895.76040
[11] Babuška, I.; Rheinboldt, W., A posteriori error estimates for the finite element method, Int. J. Numer. Methods Eng., 12, 1597-1615 (1978) · Zbl 0396.65068
[12] Bank, R. E.; Weiser, A., Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44, 283-301 (1985) · Zbl 0569.65079
[13] Bank, R. E.; Smith, R. K., A posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal., 30, 4, 921-935 (1993) · Zbl 0787.65078
[14] Verfürth, R., A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (1996), Wiley-Teubner: Wiley-Teubner Stuttgart · Zbl 0853.65108
[15] Zienkiewicz, O. C.; Zhu, J. Z., The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Int. J. Numer. Methods Eng., 33, 1331-1364 (1992) · Zbl 0769.73084
[16] Zienkiewicz, O. C.; Zhu, J. Z., The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity, Int. J. Numer. Methods Eng., 33, 1365-1382 (1992) · Zbl 0769.73085
[17] Demkowicz, L.; Oden, J. T.; Rachowicz, W.; Hardy, O., Toward a universal h-p adaptive finite element strategy. Part 1. Constrained approximation and data structure, Comp. Meth. in Appl. Mech. and Eng., 77, 113-180 (1989) · Zbl 0723.73074
[18] Oden, J. T.; Carey, G. F., Finite Elements: Mathematical Aspects, IV (1983), Prentice-Hall · Zbl 0496.65055
[19] Oden, J. T.; Reddy, J. N., An Introduction to the Mathematical Theory of Finite Elements (1976), John Wiley & Sons: John Wiley & Sons New York · Zbl 0336.35001
[20] Prudhomme, S., Adaptive control of error and stability of h-p approximations of the transient Navier-Stokes equations, Ph.D. Thesis (1999), The University of Texas at Austin · Zbl 0965.76046
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.