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Inexpensive guaranteed and efficient upper bounds on the algebraic error in finite element discretizations

Abstract

We present new constructions of (approximate) H(div, Ω)-liftings of the algebraic residual leading to estimators of the algebraic error in h and p finite element discretizations of a model diffusion problem. The estimators provide guaranteed bounds without any uncomputable constants and they are globally efficient, similarly to some recent developments, but the cost of their construction is significantly reduced. We provide a set of numerical experiments to assess the performance of the new estimators.

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Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 647134 GATIPOR). The work of J. Papež was supported by the Grant Agency of the Czech Republic under grant no. 20-01074S in the framework of RVO 67985840.

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Correspondence to Jan Papež.

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Papež, J., Vohralík, M. Inexpensive guaranteed and efficient upper bounds on the algebraic error in finite element discretizations. Numer Algor (2021). https://doi.org/10.1007/s11075-021-01118-5

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Keywords

  • Finite element method
  • Iterative algebraic solver
  • Algebraic error
  • A posteriori error estimate
  • Guaranteed upper bound
  • Hierarchical splitting

Mathematics Subject Classification (2010)

  • 65N15
  • 65N30
  • 76M10
  • 65N22
  • 65F10