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

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...

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Bibliographic Details
Published in:Journal of scientific computing 2023-06, Vol.95 (3), p.91, Article 91
Main Authors: Gopalakrishnan, J., Kogler, L., Lederer, P. L., Schöberl, J.
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Computational Mathematics and Numerical Analysis
Coupling
Divergence
Fluid flow
Galerkin method
Incompressible flow
Incompressible fluids
Lagrange multiplier
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Methods
Robustness (mathematics)
Stability analysis
Tensors
Theoretical
Variables
Velocity
Vorticity
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Summary: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.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-023-02203-8