# Stokes Equation in "two-fold saddle point" form?

Are there papers that deal with the (nondimensionalized) Stokes equation for incompressible fluid flow in a "doubly mixed" form like the following?

\begin{align*} 0&=\underline{\epsilon} + \frac{1}{2} (\vec{\nabla} \vec{u}+ (\vec{\nabla} \vec{u})^{T})\\ \vec{f}&= \vec{\nabla} \cdot \underline{\epsilon} + \nabla p\\ 0&= \vec{\nabla} \cdot \vec{u} \end{align*}

Or using Einstein notation

\begin{align*} 0 &= \epsilon_{ij}+ \frac{1}{2} (u_{i,j}+u_{j,i})\\ f_i &= \epsilon_{ij,j}+p_{,i}\\ 0&= u_{i,i} \end{align*}

Here, much like the mixed form for linear elasticity we write the symmetric tensor $\underline{\epsilon}$ as the symmetric gradient of $\vec{u}$. I am looking for any references in the literature for finite element methods that solve the Stokes equation this way (as a system of three first order PDE's).

• I've wondered about this too. Is it better to enforce the incompressibility constraint as $u_{i,i} = 0$ or $\epsilon_{ii} = 0$? Apr 25, 2016 at 17:29