Divergence-conforming velocity and vorticity approximations for incompressible fluids obtained with minimal facet coupling

TL;DR

This paper presents two new minimal-coupling, divergence-conforming finite element methods for incompressible fluid flow that ensure exactly divergence-free velocities and pressure robustness, with optimal error estimates and practical stability.

Contribution

Introduction of two novel lowest order methods, a mixed and a hybrid DG, using minimal facet coupling for divergence-conforming velocity and vorticity approximation.

Findings

Both methods produce exactly divergence-free velocity solutions.

Numerical results confirm optimal error estimates and stability.

Comparison of condition numbers demonstrates practical advantages.

Abstract

We introduce two new lowest order methods, a mixed method, and a hybrid Discontinuous Galerkin (HDG) 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.