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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).

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  • $\begingroup$ I've wondered about this too. Is it better to enforce the incompressibility constraint as $u_{i,i} = 0$ or $\epsilon_{ii} = 0$? $\endgroup$ – Daniel Shapero Apr 25 '16 at 17:29
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You can for instance have a look at the different HDG papers by Cockburn et. al. They use a similar form to formulate their hybridizable approaches. An overview is given in this review paper: http://www.sciencedirect.com/science/article/pii/S0045793013004568

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There is also some work on this in the literature on least-squares finite element methods, in which everything is reformulated as a first-order system; see for example this paper by Bochev and Gunzburger. They use the velocity/pressure/vorticity formulation rather than velocity/pressure/strain rate.

Since everything is positive-definite with LSFE, there's no LBB condition to satisfy (!) and you can use P1 elements for everything, instead of e.g. Taylor-Hood or MINI elements or what have you. I think the downside is that the divergence-free condition is not satisfied exactly with LSFE.

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