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An adaptive finite element method for linear elliptic problems. (English) Zbl 0644.65080

A practical adaptive finite element method is developed for a Poisson equation with Dirichlet or Neumann boundary conditions when the exact solution contains a singularity dependent on \(r^{\beta}\) where \(\beta\) lies between 1 and 2 and where r is the distance from the singularity. Criteria are developed for estimating the error in the solution and gradient and techniques given for minimal refinements of the mesh based on the local error estimates. Two numerical examples are given for sectors with internal angles \(3\pi\) /4 and \(\pi\).
Reviewer: K.E.Barrett

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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[1] 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.
[2] 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.
[3] I. Babuška & A. Miller, A Posteriori Error Estimates and Adaptive Techniques for the Finite Element Method, Technical Note BN-968, Univ. of Maryland, 1981.
[4] I. Babuška & A. K. Noor, Quality Assessment and Control of Finite Element Solutions, Technical Note BN-1049, Univ. of Maryland, 1986.
[5] R. E. Bank, PLTMG Users’ Guide, June, 1981 version, Technical Report, Department of Mathematics, University of California at San Diego, La Jolla.
[6] L. Demkowicz, Ph. Devloo, and J. T. Oden, On an \?-type mesh-refinement strategy based on minimization of interpolation errors, Comput. Methods Appl. Mech. Engrg. 53 (1985), no. 1, 67 – 89. · Zbl 0556.73081
[7] 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. · Zbl 0509.73071
[8] K. Eriksson, ”A maximum norm error estimate for the finite element method for linear elliptic problems under weak mesh regularity assumptions.” (To appear).
[9] 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.
[10] Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. II. Optimal error estimates in \?_{\infty }\?\(_{2}\) and \?_{\infty }\?_{\infty }, SIAM J. Numer. Anal. 32 (1995), no. 3, 706 – 740. · Zbl 0830.65094
[11] 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. · Zbl 0618.65104
[12] K. Eriksson & C. Johnson, An Adaptive Finite Element Method for Linear Advection Problems, Tech. Rep., Math. Dept., Chalmers Univ. of Technology, Göteborg. (To appear). · Zbl 0732.65093
[13] I. Babuška, O. C. Zienkiewicz, J. Gago, and E. R. de A. Oliveira , 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. · Zbl 0663.65001
[14] 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. · Zbl 0661.65076
[15] 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. · Zbl 0701.65063
[16] 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). · Zbl 0568.76074
[17] Rolf Rannacher and Ridgway Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), no. 158, 437 – 445. · Zbl 0483.65007
[18] 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. , https://doi.org/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. II. Refinements, Math. Comp. 33 (1979), no. 146, 465 – 492. · Zbl 0382.65058
[19] 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. , https://doi.org/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. II. Refinements, Math. Comp. 33 (1979), no. 146, 465 – 492. · Zbl 0382.65058
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