An adaptive finite element method for linear elliptic problems
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- by Kenneth Eriksson and Claes Johnson PDF
- Math. Comp. 50 (1988), 361-383 Request permission
Abstract:
We propose an adaptive finite element method for linear elliptic problems based on an optimal maximum norm error estimate. The algorithm produces a sequence of successively refined meshes with a final mesh on which a given error tolerance is satisfied. In each step the refinement to be made is determined by locally estimating the size of certain derivatives of the exact solution through computed finite element solutions. We analyze and justify the algorithm in a model case.References
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F. Angrand, V. Billey, J. Periaux, C. Pouletty & J. P. Rosenblum, 2-D and 3-D Euler Computations Around Lifting Bodies on Self Adapted Finite Element Meshes, Sixth International Symposium on Finite Element Methods in Flow Problems, Antibes, 1986.
- I. Babuška, Feedback, adaptivity, and a posteriori estimates in finite elements: aims, theory, and experience, Accuracy estimates and adaptive refinements in finite element computations (Lisbon, 1984) Wiley Ser. Numer. Methods Engrg., Wiley, Chichester, 1986, pp. 3–23. MR 879443 I. Babuška & A. Miller, A Posteriori Error Estimates and Adaptive Techniques for the Finite Element Method, Technical Note BN-968, Univ. of Maryland, 1981. I. Babuška & A. K. Noor, Quality Assessment and Control of Finite Element Solutions, Technical Note BN-1049, Univ. of Maryland, 1986. R. E. Bank, PLTMG Users’ Guide, June, 1981 version, Technical Report, Department of Mathematics, University of California at San Diego, La Jolla.
- L. Demkowicz, Ph. Devloo, and J. T. Oden, On an $h$-type mesh-refinement strategy based on minimization of interpolation errors, Comput. Methods Appl. Mech. Engrg. 53 (1985), no. 1, 67–89. MR 812590, DOI 10.1016/0045-7825(85)90076-3
- Alejandro R. Díaz, Noboru Kikuchi, and John E. Taylor, A method of grid optimization for finite element methods, Comput. Methods Appl. Mech. Engrg. 41 (1983), no. 1, 29–45. MR 723044, DOI 10.1016/0045-7825(83)90051-8 K. Eriksson, "A maximum norm error estimate for the finite element method for linear elliptic problems under weak mesh regularity assumptions." (To appear). K. Eriksson, Adaptive Finite Element Methods Based on Optimal Error Estimates for Linear Elliptic Problems, Tech. Rep. 1987-02, Math. Dept., Chalmers Univ. of Technology, Göteborg, to appear in Math. Comp.
- Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. II. Optimal error estimates in $L_\infty L_2$ and $L_\infty L_\infty$, SIAM J. Numer. Anal. 32 (1995), no. 3, 706–740. MR 1335652, DOI 10.1137/0732033
- Kenneth Eriksson and Claes Johnson, Error estimates and automatic time step control for nonlinear parabolic problems. I, SIAM J. Numer. Anal. 24 (1987), no. 1, 12–23. MR 874731, DOI 10.1137/0724002 K. Eriksson & C. Johnson, An Adaptive Finite Element Method for Linear Advection Problems, Tech. Rep., Math. Dept., Chalmers Univ. of Technology, Göteborg. (To appear).
- I. Babuška, O. C. Zienkiewicz, J. Gago, and E. R. de A. Oliveira (eds.), Accuracy estimates and adaptive refinements in finite element computations, Wiley Series in Numerical Methods in Engineering, John Wiley & Sons, Ltd., Chichester, 1986. Lectures presented at the international conference held in Lisbon, June 1984; A Wiley-Interscience Publication. MR 879442
- Claes Johnson, Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations, SIAM J. Numer. Anal. 25 (1988), no. 4, 908–926. MR 954791, DOI 10.1137/0725051
- Claes Johnson, Yi Yong Nie, and Vidar Thomée, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem, SIAM J. Numer. Anal. 27 (1990), no. 2, 277–291. MR 1043607, DOI 10.1137/0727019
- R. Löhner, K. Morgan, and O. C. Zienkiewicz, An adaptive finite element procedure for compressible high speed flows, Comput. Methods Appl. Mech. Engrg. 51 (1985), no. 1-3, 441–465. FENOMECH ’84, Part I, II (Stuttgart, 1984). MR 822752, DOI 10.1016/0045-7825(85)90042-8
- Rolf Rannacher and Ridgway Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), no. 158, 437–445. MR 645661, DOI 10.1090/S0025-5718-1982-0645661-4
- A. H. Schatz and L. B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. I, Math. Comp. 32 (1978), no. 141, 73–109. MR 502065, DOI 10.1090/S0025-5718-1978-0502065-1
- A. H. Schatz and L. B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. I, Math. Comp. 32 (1978), no. 141, 73–109. MR 502065, DOI 10.1090/S0025-5718-1978-0502065-1
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Math. Comp. 50 (1988), 361-383
- MSC: Primary 65N30; Secondary 65N50
- DOI: https://doi.org/10.1090/S0025-5718-1988-0929542-X
- MathSciNet review: 929542