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Picasso, Marco

Adaptive finite elements for a linear parabolic problem. (English) Zbl 0935.65105

Comput. Methods Appl. Mech. Eng. 167, No. 3-4, 223-237 (1998).
Summary: A posteriori error estimates for the heat equation in two space dimensions are presented. A classical discretization is used, Euler backward in time, and continuous, piecewise linear triangular finite elements in space. The error is bounded above and below by an explicit error estimator based on the residual. Numerical results are presented for uniform triangulations and constant time steps. The quality of our error estimator is discussed. An adaptive algorithm is then proposed. Successive Delaunay triangulations are generated, so that the estimated relative error is close to a present tolerance. Again, numerical results demonstrate the efficiency of our approach.

Cited in 1 Review
Cited in 51 Documents

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs

Keywords:

numerical results; error estimates; heat equation; finite elements; adaptive algorithm; Delaunay triangulations

Software:

LAPACK
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\textit{M. Picasso}, Comput. Methods Appl. Mech. Eng. 167, No. 3--4, 223--237 (1998; Zbl 0935.65105)
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References:

[1] Ainsworth, M.; Oden, J.T., A unified approach to a posteriori error estimation using finite element residual methods, Numer. math., 65, 23-50, (1993) · Zbl 0797.65080
[2] Anderson, E.; Bai, Z.; Bischof, C.; Demmal, J.; Dongarra, J.; Du Croz, J.; Greenbaum, A.; Hammarling, S.; McKenney, A.; Ostrou Chov, S.; Soresen, D., (), 19104-2688
[3] Babuska, I.; Duran, R.; Rodriguez, R., Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements, SIAM J. numer. anal., 29, 4, 947-964, (1992) · Zbl 0759.65069
[4] Babuska, I.; Rheinboldt, W.C., Error estimates for adaptive finite element computations, SIAM J. numer. anal., 15, 736-754, (1978) · Zbl 0398.65069
[5] Babuska, I.; Rheinboldt, W.C., A posteriori error estimators in the finite element method, Int. J. numer. methods engrg., 12, 1597-1615, (1978) · Zbl 0396.65068
[6] Babuska, I.; Strouboulis, T.; Upadhyay, C.S., A model study of the quality of a posteriori estimators for linear elliptic problems, (), 307-378, 4
[7] Baranger, J.; El-Amri, H, Estimateurs a posteriori d’erreur pour le calcul adaptatif d’écoulements quasi-newtoniens, Rairo m2an, 25, 1, 31-48, (1991) · Zbl 0712.76068
[8] Bernardi, C., Optimal finite element interpolation on curved domains, SIAM J. numer. anal., 26, 5, 1212-1240, (1989) · Zbl 0678.65003
[9] Caloz, G.; Rappaz, J., Numerical analysis for nonlinear and bifurcation problems, (), 487-638, (Part 2)
[10] Ciarlet, P.G., The finite element method for elliptic problems, (1990), Academic Press London · Zbl 0709.73100
[11] Clément, P., Approximation by finite element functions using local regularization, RAIRO anal. numér., 9, 77-84, (1975) · Zbl 0368.65008
[12] Dautray, R.; Lions, J.-L., Analyse mathématique et calcul numérique pour LES sciences et LES techniques, () · Zbl 0642.35001
[13] Eriksson, K.; Johnson, C., Adaptive finite element methods for parabolic problems I: A linear model problem, SIAM J. numer. anal., 28, 1, 43-77, (1991) · Zbl 0732.65093
[14] Eriksson, K.; Johnson, C., Adaptive finite element methods for parabolic problems IV: nonlinear problems, () · Zbl 0835.65116
[15] Eriksson, K.; Johnson, C., Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems, Math. comput., 60, 201, 167-188, (1993) · Zbl 0795.65074
[16] Johnson, C., Adaptive finite element methods for diffusion and convection problem, Comput. methods appl. mech. engrg., 82, 301-322, (1990) · Zbl 0717.76078
[17] Medina, J.; Picasso, M.; Rappaz, J., Error estimates and adaptive finite elements for nonlinear diffusion-convection problems, Math. models methods appl. sci., 6, 5, 689-712, (1966) · Zbl 0857.65110
[18] Picasso, M., An adaptive finite element algorithm for a two-dimensional stationary Stefan-like problem, Comput. methods appl. mech. engrg., 124, 213-230, (1995) · Zbl 0945.65519
[19] Pousin, J.; Rappaz, J., Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems, Numer. math., 69, 2, 213-232, (1994) · Zbl 0822.65034
[20] Sloan, S.W., A fast algorithm for constructing Delaunay triangulations in the plane, Adv. engrg. software, 9, 1, 34-55, (1987) · Zbl 0628.68044
[21] Verfürth, R., A posteriori error estimators for the Stokes equations, Numer. math., 55, 309-325, (1989) · Zbl 0674.65092
[22] Verfürth, R., A posteriori error estimates for nonlinear problems, finite element discretizations of elliptic equations, Math. comput., 62, 206, 445-475, (1994) · Zbl 0799.65112
[23] Verfürth, R., A posteriori error estimates for nonlinear problems, finite element discretizations of parabolic equations, () · Zbl 0799.65112
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.