WIAS Preprint No. 2251, (2016)

Reliable averaging for the primal variable in the Courant FEM and hierarchical error estimators on red-refined meshes


  • Carstensen, Carsten
  • Eigel, Martin
    ORCID: 0000-0003-2687-4497

2010 Mathematics Subject Classification

  • 35J15 46E22 49Q10 49K20 49K40


  • a posteriori, error analysis, finite element method, averaging, smoothing, hierarchical estimator, adaptivity, mesh refinement, convergence




A hierarchical a posteriori error estimator for the first-order finite element method (FEM) on a red-refined triangular mesh is presented for the 2D Poisson model problem. Reliability and efficiency with some explicit constant is proved for triangulations with inner angles smaller than or equal to pi/2. The error estimator does not rely on any saturation assumption and is valid even in the pre-asymptotic regime on arbitrarily coarse meshes. The evaluation of the estimator is a simple post-processing of the piecewise linear FEM without any extra solve plus a higher-order approximation term. The results also allows the striking observation that arbitrary local averaging of the primal variable leads to a reliable and efficient error estimation. Several numerical experiments illustrate the performance of the proposed a posteriori error estimator for computational benchmarks.

Appeared in

  • Comput. Methods Appl. Math., 16 (2016) pp. 213--230.

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