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On the asymptotic exactness of error estimators for linear triangular finite elements

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Summary

This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of elliptic problems. We analyze two estimators based on recovery operators for the gradient of the approximate solution. By using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution.

One of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation. This estimator is equivalent to the error in the energy norm under rather general conditions. However, we show that for the asymptotic exactness, the regularity assumption on the mesh is not merely technical. While doing this, we analyze the relation between superconvergence and asymptotic exactness for some particular examples.

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Author notes

  1. Ricardo Durán

    Present address: Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

  2. Rodolfo Rodríguez

    Present address: Institute for Physical Science and Technology, University of Maryland, 20742, College Park, MD, USA

Authors and Affiliations

  1. Departamento de Matemática, Facultad de Ciencias Exactas, UNLP, C.C. 172, 1900, La Plata, Argentina

    Ricardo Durán, María Amelia Muschietti (Member of CONICET) & Rodolfo Rodríguez

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  1. Ricardo Durán
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  2. María Amelia Muschietti
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  3. Rodolfo Rodríguez
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Durán, R., Muschietti, M.A. & Rodríguez, R. On the asymptotic exactness of error estimators for linear triangular finite elements. Numer. Math. 59, 107–127 (1991). https://doi.org/10.1007/BF01385773

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