Asymptotically exact a posteriori error estimates for the BDM finite element approximation of mixed Laplace eigenvalue problems

Philip L. Lederer*

*Corresponding author for this work

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Abstract

We derive optimal and asymptotically exact a posteriori error estimates for the approximation of the eigenfunction of the Laplace eigenvalue problem. To do so, we combine two results from the literature. First, we use the hypercircle techniques developed for mixed eigenvalue approximations with Raviart-Thomas finite elements. In addition, we use the post-processings introduced for the eigenvalue and eigenfunction based on mixed approximations with the Brezzi-Douglas-Marini finite element. To combine these approaches, we define a novel additional local post-processing for the fluxes that appropriately modifies the divergence without compromising the approximation properties. Consequently, the new flux can be used to derive optimal and asymptotically exact upper bounds for the eigenfunction, and optimal upper bounds for the corresponding eigenvalue. Numerical examples validate the theory and motivate the use of an adaptive mesh refinement.

Original languageEnglish
Article number34
Number of pages21
JournalBIT Numerical Mathematics
Volume63
Issue number2
DOIs
Publication statusPublished - Jun 2023

Keywords

  • UT-Hybrid-D
  • Brezzi-Douglas-Marini finite element
  • Mixed Laplace eigenvalue problem
  • Prager-Synge
  • A posteriori error analysis

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