2D Stokes problem#

Keywords: PDE:Stokes Method:FEM FE:CrouzeixRaviart FE:TaylorHood Condition:Dirichlet Solver:Direct

We consider the problem of a flow entrained in a cavity, say $Ω =] 0, 1 [\times] 0, 1 [$ :

\begin{array}{r} {\begin{cases} - ν △ u + \nabla p = f & in Ω \\ div u = 0 & in Ω \\ u = g & on Σ = {0} \times] 0, 1 [ \\ u = 0 & on Γ = \partial Ω ∖ Σ \end{cases} \end{array}

with $f \in L^{2} (Ω)$ and $g = (0, v_{0}), v_{0} \in H_{00}^{\frac{1}{2}} (Σ)$ . Note that $g | n = 0$ .

This problem has the following variational formulation: find $(u, p) \in H^{1} (Ω)^{2} \times L_{0}^{2} (Ω)$ such that for all $(v, p) \in H_{0}^{1} (Ω)^{2} \times L_{0}^{2} (Ω)$ :

\begin{array}{r} {\begin{cases} ν \int_{Ω} \nabla u : \nabla v - \int_{Ω} p div v - \int_{Ω} div u q = \int_{Ω} f \cdot v \\ u = g on Σ and u = 0 on Γ \end{cases} \end{array}

The following XLiFE++ implementation of this problem solves the 2D Stokes problem using different couples of approximation for the velocity $v$ and the pressure $p$ : (P2, P0), (P2, P1), (CR, P0); CR corresponds to a P1 Crouzeix-Raviart finite element approximation leading to a non-conforming approximation of $H^{1}$ . Note that the (P1, P0) approximation is not working because of dof locking.

Using Paraview with the Glyph filter that represents 2D vectors with arrows, we get the following pictures:

Note that to approximate the space $L_{0}^{2} (Ω)$ , an essential condition of the zero mean type of pressure p has been used. Such a condition is quite costly because it interacts with all the pressure dofs. If we remove it, the matrix becomes singular (the pressure being defined to within a constant) and direct solvers fail. But iterative solvers do converge to the right velocity, and pressure to the right constant, determined by the initial guess of the iterative solver !

Note also that, when using Crouzeix-Raviart element, in order to visualize the velocity, it is required to project the CR solution in a standard Lagrange finite element space (here $P_{1}$ ).