Divergence-Conforming Velocity and Vorticity Approximations for Incompressible Fluids Obtained with Minimal Facet Coupling

J. Gopalakrishnan, L. Kogler, P. L. Lederer*, J. Schöberl

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

2 Citations (Scopus)
17 Downloads (Pure)

Abstract

We introduce two new lowest order methods, a mixed method, and a hybrid discontinuous Galerkin method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi–Douglas–Marini space for approximating the velocity and the lowest order Raviart–Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.

Original languageEnglish
Article number91
Number of pages25
JournalJournal of scientific computing
Volume95
Issue number3
Early online date11 May 2023
DOIs
Publication statusPublished - Jun 2023

Keywords

  • UT-Hybrid-D
  • Hybrid discontinuous Galerkin methods
  • Incompressible Stokes equations
  • Mixed finite elements
  • Pressure-robustness
  • Discrete Korn inequality

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