Stabilisation of high aspect ratio mixed finite elements for incompressible flow

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Ainsworth, Mark and Barrenechea, Gabriel R. and Wachtel, Andreas (2015) Stabilisation of high aspect ratio mixed finite elements for incompressible flow. SIAM Journal on Numerical Analysis, 53 (2). 1107–1120. ISSN 0036-1429 (https://doi.org/10.1137/140972755)

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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.

ORCID iDs

Ainsworth, Mark, Barrenechea, Gabriel R. ORCID logoORCID: https://orcid.org/0000-0003-4490-678X and Wachtel, Andreas;
  • Item type:Article
    ID code:55338
    Dates:
    Date
    Event
    2015
    Published
    16 April 2015
    Published Online
    17 February 2015
    Accepted
    Subjects:Science > Mathematics
    Department:Faculty of Science > Mathematics and Statistics
    Depositing user: Pure Administrator
    Date deposited:15 Jan 2016 10:10
    Last modified:11 Nov 2024 11:18
    URI:https://strathprints.strath.ac.uk/id/eprint/55338
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