Stokes FEM derivation

For modelling the equations above, we will employ the finite element method (FEM). To use this method, we transform the equations into their variational form.

\[\begin{split} \int_{\Omega_{CSF}} -\mu \Delta \mathbf{u}\cdot \mathbf{v}~\mathrm{d}x + \int_{\Omega_{CSF}} \nabla p \cdot \mathbf{v}~\mathrm{d}x = 0\\ \int_{\Omega_{CSF}} \nabla\cdot \mathbf{u}q~\mathrm{d} x = 0 \end{split}\]

Integrating the first equation by parts, and choosing \(\mathbf{u}\in H_{div}(\Omega)\), we get

\[ \int_{\Omega_{CSF}} \mu \nabla \mathbf{u} \cdot \nabla \mathbf{v} - \nabla \cdot \mathbf{v} p ~\mathrm{d}x + \int_{\partial\Omega_{CSF}} \left(-\mu \nabla \mathbf{u} \cdot \mathbf{n} + p\mathbf{n}\right) \cdot \mathbf{v} ~\mathrm{d}s=0. \]

We observe that we can split the boundary integral into its normal and tangential component \(\mathbf{A}_t = \mathbf{A} - (\mathbf{A}\cdot \mathbf{n}) \mathbf{n}\)

\[ \int_{\partial\Omega_{CSF}} \left(-\mu \nabla \mathbf{u} \cdot \mathbf{n} + p\mathbf{n}\right) \cdot \mathbf{v} ~\mathrm{d}s= \int_{\partial\Omega_{CSF}} \left(-\mu \nabla \mathbf{u} \cdot \mathbf{n} + p\mathbf{n}\right)\cdot \mathbf{n} \mathbf{n} \cdot\mathbf{v} + \left(-\mu \nabla \mathbf{u} \cdot \mathbf{n} + p\mathbf{n}\right)_t \cdot \mathbf{v} ~\mathrm{d}x. \]

As we enforce \(\mathbf{u}\cdot n=\mathbf{0}\) on \(\Gamma_{AM-L}\cup\Gamma_{pia}\cup\Gamma_{SSAS}\) and \(\mathbf{u}\cdot n =\frac{1}{\vert \Gamma \vert}u_{in}\) through our choice of function space, these integrals dissapear on these boundaries.

On \(\Gamma_{AM-u}\) we use the boundary condition relation to rewrite the equation above as

\[ \int_{\partial\Omega_{CSF}} \left(-\mu \nabla \mathbf{u} \cdot \mathbf{n} + p\mathbf{n}\right) \cdot \mathbf{v} ~\mathrm{d}s= \int_{\Gamma_{AM-u}} R_0 \mathbf{u}\cdot \mathbf{n} \mathbf{n} \cdot\mathbf{v}~\mathrm{d}s + \int_{\partial \Omega_{CSF}} \left(-\mu \nabla \mathbf{u} \cdot \mathbf{n} + p\mathbf{n}\right)_t \cdot \mathbf{v} ~\mathrm{d}x. \]

It is clear that \((p\mathbf{n})_t=0\). As \(\mathbf{u}\cdot \mathbf{t} =0\) across the whole boundary, we have that \((\nabla \mathbf{u})_t=0\), and we are left with only the first integral.

This leaves us with the variational form

\[\begin{split} \int_{\Omega_{CSF}} \mu \nabla \mathbf{u} \cdot \nabla \mathbf{v} - \nabla \cdot \mathbf{v} p ~\mathrm{d}x + \int_{\Gamma_{AM-u}} R_0 \mathbf{u}\cdot \mathbf{n} \mathbf{n} \cdot\mathbf{v} ~\mathrm{d}s=0\\ \int_{\Omega_{CSF}} \nabla\cdot \mathbf{u}q~\mathrm{d} x = 0 \end{split}\]

Discretized problem

When the domain is discretized into tetrahedra, and a discrete \(H_{div}\) space is chosen, one cannot guarantee the tangential continuity of \(\mathbf{u}\) across element surfaces \(\mathcal{F}\). To enforce this, we add a standard interior penalty method of the form

\[ \int_{\mathcal{F}} -\mu (\{\nabla \mathbf{u}\}\cdot\mathbf{n}_F) \cdot \left[\mathbf{v}_t\right] -\mu (\{\nabla \mathbf{v}\}\cdot \mathbf{n}_F) \cdot \left[\mathbf{u}_t\right] + \frac{\sigma \mu}{h_F}\left[\mathbf{u}_t\right]~\mathrm{d}s \]