A double-hybrid finite element formulation for Stokes flows using a divergence-free approximation space

This paper presents a study of Stokes flows using a fully-hybrid finite element formulation. Incompressibility appears in Stokes differential equations as an additional constraint that enforces the velocity field to be divergence-free. A straightforward option to discretise those kind of problems would be to use a space of vector functions that includes the divergence-free constraint. When using standard De Rham compatible $H(/text{div},\Omega) - L^2(\Omega)$ spaces a scalar $H^1(\Omega)$ space is used as a Lagrange multiplier to enforce the divergence-free condition weakly. In recent developments, the research group at LabMeC developed $H(\text{div})$ approximation spaces derived from $H(\text{Curl})$ spaces whose divergence is constant. This work demonstrates that this pair, that also satisfies the De Rham sequence, can be used to approximate the velocity and pressure fields, respectively. This approach greatly reduces the number of global degrees of freedom. In previous work, $H(\text{div})$ spaces, which automatically guarantee continuity of the normal components, were combined with a Lagrange multiplier space to enforce the tangent velocity weakly. The Lagrange multiplier is associated with a shear stress applied between elements. In this contribution, a second hybridization is applied to the shear stress approximation, leading to a Lagrange multiplier associated with the tangent velocity. Condensing internal degrees of freedom leads to an element stiffness matrix that is positive definite with a single pressure Lagrange multiplier enforcing the incompressibility. Examples are presented to test and verify the developed numerical methodology.