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An equilibrated residual method with a computable error approximation for a singularly perturbed reaction-diffusion problem on anisotropic finite element meshes

  • Sergey Grosman (a1)

Abstract

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.

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[1] Ainsworth, M. and Babuška, I., 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.
[2] Ainsworth, M. and Oden, J.T., A unified approach to a posteriori error estimation using element residual methods. Numer. Math. 65 (1993) 2350.
[3] M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis. Wiley (2000).
[4] Apel, T., Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements. Computing 60 (1998) 157174.
[5] T. Apel, Treatment of boundary layers with anisotropic finite elements. Z. Angew. Math. Mech. (1998).
[6] T. Apel, Anisotropic finite elements: local estimates and applications. B.G. Teubner, Stuttgart (1999).
[7] Apel, T., Grosman, S., Jimack, P.K. and Meyer, A., A new methodology for anisotropic mesh refinement based upon error gradients. Appl. Numer. Math. 50 (2004) 329341.
[8] Apel, T. and Lube, G., Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem. Appl. Numer. Math. 26 (1998) 415433.
[9] Babuška, I. and Rheinboldt, W., A posteriori error estimates for the finite element method. Int. J. Numer. Meth. Eng. 12 (1978) 15971615.
[10] Bank, R. and Weiser, A., Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44 (1985) 283301.
[11] Bufler, H. and Stein, E., Zur Plattenberechnung mittels finiter Elemente. Ingenier Archiv 39 (1970) 248260.
[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).
[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) 36–45.
[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).
[16] Han, H. and Kellogg, R.B., 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.
[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.
[18] Kunert, G., An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math. 86 (2000) 471490.
[19] Kunert, G., A local problem error estimator for anisotropic tetrahedral finite element meshes. SIAM J. Numer. Anal. 39 (2001) 668689.
[20] Kunert, G., Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes. Adv. Comput. Math. 15 (2001) 237259.
[21] Kunert, G., Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes. ESAIM: M2AN 35 (2001) 10791109.
[22] Kunert, G. and Verfürth, R., Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes. Numer. Math. 86 (2000) 283303.
[23] Ladevèze, P. and Leguillon, D., Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal. 20 (1983) 485509.
[24] Siebert, K.G., An a posteriori error estimator for anisotropic refinement. Numer. Math. 73 (1996) 373398.
[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).
[26] Vogelius, M. and Babuška, I., On a dimensional reduction method. I. The optimal selection of basis functions. Math. Comp. 37 (1981) 3146.

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An equilibrated residual method with a computable error approximation for a singularly perturbed reaction-diffusion problem on anisotropic finite element meshes

  • Sergey Grosman (a1)

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