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Robust Equilibrated Error Estimator for Diffusion Problems: Mixed Finite Elements in Two Dimensions

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Abstract

This paper introduces and analyzes an equilibrated a posteriori error estimator for mixed finite element approximations to the diffusion problem in two dimensions. The estimator, which is a generalization of those in Braess and Schöberl (Math Comput 77:651–672, 2008) and Cai and Zhang (SIAM J Numer Anal 50(1):151–170, 2012), is based on the Prager–Synge identity and on a local recovery of a gradient in the curl free subspace of the \(H(\text {curl})\)-confirming finite element spaces. The resulting estimator admits guaranteed reliability, and its robust local efficiency is proved under the quasi-monotonicity condition of the diffusion coefficient. Numerical experiments are given to confirm the theoretical results.

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Authors and Affiliations

  1. Department of Mathematics, Emory University, 30322, Atlanta, Georgia

    Difeng Cai

  2. Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, 47907-2067, USA

    Zhiqiang Cai

  3. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong SAR, China

    Shun Zhang

Authors
  1. Difeng Cai
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  2. Zhiqiang Cai
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  3. Shun Zhang
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Correspondence to Zhiqiang Cai.

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Z. Cai: This work was supported in part by the National Science Foundation under Grant DMS-1522707. S. Zhang: This work was supported in part by Research Grants Council of the Hong Kong SAR, China under the GRF Grant Project No. 11305319, CityU.

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Cai, D., Cai, Z. & Zhang, S. Robust Equilibrated Error Estimator for Diffusion Problems: Mixed Finite Elements in Two Dimensions. J Sci Comput 83, 22 (2020). https://doi.org/10.1007/s10915-020-01199-9

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  • DOI: https://doi.org/10.1007/s10915-020-01199-9