# Darcy flow finite elements

The Darcy equations for porous media flow are given by:

$\frac{\mu}{\kappa}\mathbf{u} - \nabla p = \mathbf{0}$

$\nabla\cdot\mathbf{u} = 0$

where $\kappa$ is the permeability and can in general be spatially dependent.

This can be put into a weak formulation that looks like:

$a(\mathbf{u},\mathbf{v}) + b(p,\mathbf{v}) = -\int_\Gamma p\mathbf{v}\cdot\mathbf{n}d\Gamma$

$b(q,\mathbf{u}) = 0$

where:

$a(\mathbf{u},\mathbf{v}) = \int_\Omega \frac{\mu}{\kappa} \mathbf{u}\cdot\mathbf{v}d\Omega$

$b(p, \mathbf{v}) = -\int_\Omega p\nabla\cdot\mathbf{v}d\Omega$

Now I'd like to have no through boundary conditions on part of the domain $\Gamma$, say $\Gamma_1$. Then I can say that $\mathbf{u}$ and $\mathbf{v}$ belong to the space:

$\hat{\mathbf{H}} = \{\mathbf{v}\in\mathbf{H}(\text{div}) : \mathbf{v}\cdot\mathbf{n} = 0 \text{ on } \Gamma_1\}$

I'd like to prescribe $p$ on the remaining part of the boundary $\Gamma_2$. Then I have $p$ belonging to the space: $L^2_a = \{p \in L^2 : p = a \text{ on } \Gamma_2\}$

and $q$ belonging to $L^2_0$. If there's anything wrong with what I've done so far please let me know.

My question now is, what basis for my discrete space should I choose? For Stokes flow the common choice is Taylor-Hood; I think that should work here too. The only possible complication I see is that now $\mathbf{u}$ has to be in $\mathbf{H}(\text{div})$ instead of $\mathbf{H}^1$.

I think that $a(\mathbf{u}, \mathbf{v})$ is still bounded and coercive (for smooth $\kappa$), and the Taylor-Hood element guarantees that $b(p,\mathbf{v})$ is also bounded and coercive. In literature however I'm finding it difficult to find many references to support this choice. Is there a better choice for Darcy flow?

Edit: If I expect my solution $\mathbf{u}$ to be in $\mathbf{H}^1$, due to a smooth domain and smooth $\kappa$, would it be acceptable to use the Taylor-Hood element pair? If so, would this choice of spaces be better if I am interested in the viscous stresses, which depend on $\nabla\mathbf{u}$ (and might not exist if $\mathbf{u}\in\mathbf{H}(\text{div})$)?

• What makes you think that the solution will be in $H^1$? From the bilinear form, you can only expect $H(\mathrm{div})$, which is strictly larger than $H^1$. Jul 29 '15 at 9:46
• Yeah you're right, I got my spaces backwards there, I'll edit the question. Jul 29 '15 at 9:54

I think this is more the mixed finite element method for Poisson's equation with Dirichlet boundary data. In that case, a(.,.) is not coercive over the whole velocity space, but rather only coercive over the kernel of b(p,.), i.e. coercive over the space of all divergence free velocities (I believe).

This shows up in the approximation spaces as you need the div(velocity space) = pressure space. Taylor-Hood elements do not have this property and you need to use the Raviart-Thomas elements.

Prescribing discontinuous pressure boundary conditions shouldn't be a problem as long as they are in $L^2$. However, just becareful with your notation, I once used the notation $L^{2}_{0}$ and was repremanded that $L^{2}$ functions dont enough regularity to have a trace! :)

Two good references are:

http://www.amazon.com/Mathematical-Element-Methods-Applied-Mathematics/dp/0387759336

as well as Arnold's notes:

http://www.ima.umn.edu/~arnold/8445-8446.14-15/notes.pdf

• Thanks for your answer. You're right about this being like the mixed Poisson equation. There's a few examples out there using with Raviart-Thomas elements or Brezzi-Douglas-Marini elements for $\mathbf{u}$, along with discontinuous or Lagrange elements for $p$. I still don't exactly understand why though, let alone which combination would be prefered. The notes you posted won't download for me for some reason. Jul 19 '15 at 13:39