Stabilisation of high aspect ratio mixed finite elements for incompressible flow

Mark Ainsworth, Gabriel R. Barrenechea, Andreas Wachtel

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Abstract

Anisotropically refined mixed finite elements are beneficial for the resolution of local features such as boundary layers. Unfortunately, the stability of the resulting scheme is highly sensitive to the aspect ratio of the elements. Previous analysis revealed that the degeneration arises from a relatively small number of spurious (piecewise constant) pressure modes. The present article is concerned with resolving the problem of how to suppress the spurious pressure modes in order to restore stability yet at the same time not incur any deterioration in the approximation properties of the reduced pressure space. Two results are presented. The first gives the minimal constraints on the pressure space needed to restore stability with respect to aspect ratio and shows that the approximation properties of the constrained pressure space and the unconstrained pressure space are essentially identical. Alternatively, one can impose the constraint weakly through the use of a stabilized finite element scheme. A second result shows that the stabilized finite element scheme is robust with respect to the aspect ratio of the elements and produces an approximation that satisfies an error bound of the same type to the mixed finite element scheme using the constrained space.
Original languageEnglish
Pages (from-to)1107–1120
Number of pages14
JournalSIAM Journal on Numerical Analysis
Volume53
Issue number2
Early online date16 Apr 2015
DOIs
Publication statusPublished - 2015

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Keywords

  • anisotropic meshes
  • stabilization
  • spurious pressure mode

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