Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations

Philip L. Lederer*, Christian Merdon

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

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Abstract

This paper aims to improve guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e., for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the velocity errors of divergence-free primal and perfectly equilibrated dual mixed methods for the velocity stress. The first main result of the paper is a framework with relaxed constraints on the primal and dual method. This enables to use a recently developed mass conserving mixed stress discretisation for the design of equilibrated fluxes and to obtain pressure-independent guaranteed upper bounds for any pressure-robust (not necessarily divergence-free) primal discretisation. The second main result is a provably efficient local design of the equilibrated fluxes with comparably low numerical costs. Numerical examples verify the theoretical findings and show that efficiency indices of our novel guaranteed upper bounds are close to one.

Original languageEnglish
Pages (from-to)267-294
Number of pages28
JournalJournal of Numerical Mathematics
Volume30
Issue number4
DOIs
Publication statusPublished - 1 Dec 2022
Externally publishedYes

Keywords

  • A posteriori error estimators
  • Adaptive mesh refinement
  • Equilibrated fluxes
  • Incompressible Stokes equations
  • Mixed finite elements
  • Pressure-robustness

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