Steady Stokes

The Stokes equations describes the flows of incompressible viscous fluid when the importance of inertia is low with respect to viscous forces

\[\begin{split} \left\{ \begin{array}{ll} \nu\Delta \mathbf{u}-\nabla p = 0 & \mbox{in }\Omega\\ \nabla\cdot \mathbf{u} = 0 & \mbox{in }\Omega\\ \mathbf{u}=\mathbf{u}_D & \mbox{on }\Gamma_D\\ \nu\frac{\partial \mathbf{u}}{\partial \mathbf{n}}-p\mathbf{n}=g\mathbf{n} & \mbox{on }\Gamma_N \end{array} \right. \end{split}\]

This problem has a saddle point structure, making its numerical solution non-trivial.

Before entering into the details of the Galerkin problem let us derive the weak formulation. Let \(\mathcal{V}\subset[\mathcal{H}^1]^d,\; \mathcal{V}_0[\subset\mathcal{H}^1]^d\) the velocity trial and test spaces defined as

\[ \mathcal{V} = \left\{\mathbf{v}\in[\mathcal{H}^1(\Omega)]^d:\;\left. \mathbf{v}\right|_{\Gamma_D} = \mathbf{u}_D\right\}\qquad \mathcal{V}_0 = \left\{\mathbf{v}\in[\mathcal{H}^1(\Omega)]^d:\;\left. \mathbf{v}\right|_{\Gamma_D} = \mathbf{0}\right\} \]

and let \(\mathcal{Q}= L^2(\Omega)\) the pressure trial and test space. The momentum equation can be multiplied by the test function \(\mathbf{v}\in\mathcal{V}_0\) and the integration by parts is applied

\[ -\int_\Omega \nu\nabla \mathbf{u}\cdot \nabla\mathbf{v}\,d\Omega +\int_\Omega p \nabla\cdot \mathbf{v}\,d\Omega + \int_{\partial\Omega} \left(\nu\frac{\partial \mathbf{u}}{\partial \mathbf{n}}-p\mathbf{n}\right)\cdot \mathbf{v}\,d\sigma=0 \]

whereas the continuity equation is multiplied by \(q\in\mathcal{Q}\)

\[ \int_\Omega q \nabla\cdot \mathbf{u}\,d\Omega=0 \]

Imposing the boundary conditions, the weak formulation reads: find \((\mathbf{u}, p)\in\mathcal{V}\times\mathcal{Q}\) s.t.

(1)\[\begin{equation} \int_\Omega \nu\nabla \mathbf{u}\cdot \nabla\mathbf{v}\,d\Omega -\int_\Omega p \nabla\cdot \mathbf{v}\,d\Omega - \int_\Omega q \nabla\cdot \mathbf{u}\,d\Omega = \int_{\Gamma_N} g\mathbf{n}\cdot \mathbf{v}\,d\sigma \qquad \forall (\mathbf{v}, q)\in\mathcal{V}_0\times\mathcal{Q} \end{equation}\]

Derivation of the linear system

When the finite dimensional spaces are introduced an important remark should be made. The Galerkin problem has a stable solution \((\mathbf{u}_h,p_h)\) if the finite dimensional spaces are inf-sup compatible. In fact, there exists a connection between the finite dimensional functional space of velocity and pressure referred to as the Taylor-Hood compatible spaces. In order to have a stable solution [Qua16], the most common couple is given by parabolic FE \(P2\) for the velocity and linear finite element for pressure \(P1\).

Let us consider the finite dimensional representation of the spaces (using Taylor-Hood elements), the correspondent linear system results in

(2)\[\begin{equation} \left[ \begin{array}{cc} A & B^T \\ B & 0 \end{array} \right]\cdot \left[ \begin{array}{c} \mathbf{U} \\ \mathbf{P} \end{array} \right] = \left[ \begin{array}{c} \mathbf{F} \\ \mathbf{0} \end{array} \right] \end{equation}\]