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Grosman, Sergey
An equilibrated residual method with a computable error approximation for a singularly perturbed reaction-diffusion problem on anisotropic finite element meshes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, 40 no. 2 (2006), p. 239-267
Texte intégral djvu | pdf | Analyses MR 2241822 | Zbl 1120.65118 | 1 citation dans Numdam
Class. Math.: 65N15, 65N30, 65N50
Mots clés: a posteriori error estimation, singular perturbations, reaction-diffusion problem, robustness, anisotropic solution, stretched elements

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Résumé

Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in a discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both, the perturbation parameters of the problem and the anisotropy of the mesh. The equilibrated residual method has been shown to provide one of the most reliable error estimates for the reaction-diffusion problem. Its modification suggested by Ainsworth and Babuška has been proved to be robust for the case of singular perturbation. In the present work we investigate the modified method on anisotropic meshes. The method in the form of Ainsworth and Babuška is shown here to fail on anisotropic meshes. We suggest a new modification based on the stretching ratios of the mesh elements. The resulting error estimator is equivalent to the equilibrated residual method in the case of isotropic meshes and is proved to be robust on anisotropic meshes as well. Among others, the equilibrated residual method involves the solution of an infinite dimensional local problem on each element. In practical computations an approximate solution to this local problem was successfully computed. Nevertheless, up to now no rigorous analysis has been done showing the appropriateness of any computable approximation. This demands special attention since an improper approximate solution to the local problem can be fatal for the robustness of the whole method. In the present work we provide one of the desired approximations. We prove that the method is not affected by the approximate solution of the local problem.

Bibliographie

[1] M. Ainsworth and I. Babuška, Reliable and robust a posteriori error estimation for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. 36 (1999) 331353 (electronic). See also Corrigendum at http://www.maths.strath.ac.uk/~aas98107/papers.html.  Zbl 0948.65114
[2] M. Ainsworth and J.T. Oden, A unified approach to a posteriori error estimation using element residual methods. Numer. Math. 65 (1993) 2350.
Article |  Zbl 0797.65080
[3] M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis. Wiley (2000).  MR 1885308 |  Zbl 1008.65076
[4] T. Apel, Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements. Computing 60 (1998) 157174.  Zbl 0897.65003
[5] T. Apel, Treatment of boundary layers with anisotropic finite elements. Z. Angew. Math. Mech. (1998).  Zbl 0925.65188
[6] T. Apel, Anisotropic finite elements: local estimates and applications. B.G. Teubner, Stuttgart (1999).  MR 1716824 |  Zbl 0934.65121
[7] T. Apel, S. Grosman, P.K. Jimack and A. Meyer, A new methodology for anisotropic mesh refinement based upon error gradients. Appl. Numer. Math. 50 (2004) 329341.  Zbl 1050.65122
[8] T. Apel and G. Lube, Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem. Appl. Numer. Math. 26 (1998) 415433.  Zbl 0933.65136
[9] I. Babuška and W. Rheinboldt, A posteriori error estimates for the finite element method. Int. J. Numer. Meth. Eng. 12 (1978) 15971615.  Zbl 0396.65068
[10] R. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44 (1985) 283301.  Zbl 0569.65079
[11] H. Bufler and E. Stein, Zur Plattenberechnung mittels finiter Elemente. Ingenier Archiv 39 (1970) 248260.  Zbl 0197.21901
[12] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam. Studies in Mathematics and its Applications, Vol. 4, (1978).  MR 520174 |  Zbl 0383.65058
[13] M. Dobrowolski, S. Gräf and C. Pflaum, On a posteriori error estimators in the infinte element method on anisotropic meshes. Electron. Trans. Numer. Anal. 8 (1999) 3645.  Zbl 0934.65122
[14] S. Grosman, The robustness of the hierarchical a posteriori error estimator for reaction-diffusion equation on anisotropic meshes. SFB393-Preprint 2, Technische Universität Chemnitz, SFB 393 (Germany), (2004).
[15] R. Hagen, S. Roch, and B. Silbermann, C*-algebras and numerical analysis. Marcel Dekker Inc., New York (2001).  MR 1792428 |  Zbl 0964.65055
[16] H. Han and R.B. Kellogg, Differentiability properties of solutions of the equation $-\epsilon ^ 2\delta u+ru=f(x,y)$ in a square. SIAM J. Math. Anal. 21 (1990) 394408.  Zbl 0732.35020
[17] G. Kunert, A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes. Logos Verlag, Berlin, 1999. Also PhD thesis, TU Chemnitz, http://archiv.tu-chemnitz.de/pub/1999/0012/index.html.  Zbl 0919.65066
[18] G. Kunert, An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math. 86 (2000) 471490.  Zbl 0965.65125
[19] G. Kunert, A local problem error estimator for anisotropic tetrahedral finite element meshes. SIAM J. Numer. Anal. 39 (2001) 668689.  Zbl 1004.65112
[20] G. Kunert, Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes. Adv. Comput. Math. 15 (2001) 237259.  Zbl 1049.65121
[21] G. Kunert, Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes. ESAIM: M2AN 35 (2001) 10791109.
Numdam |  Zbl 1041.65072
[22] G. Kunert and R. Verfürth, Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes. Numer. Math. 86 (2000) 283303.  Zbl 0964.65120
[23] P. Ladevèze and D. Leguillon, Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal. 20 (1983) 485509.  Zbl 0582.65078
[24] K.G. Siebert, An a posteriori error estimator for anisotropic refinement. Numer. Math. 73 (1996) 373398.  Zbl 0873.65098
[25] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner Series Advances in Numerical Mathematics. Chichester: John Wiley & Sons. Stuttgart: B.G. Teubner (1996).  Zbl 0853.65108
[26] M. Vogelius and I. Babuška, On a dimensional reduction method. I. The optimal selection of basis functions. Math. Comp. 37 (1981) 3146.  Zbl 0495.65049
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