Choice of spaces for mixed formulation for Poisson Equation Or Darcy equation

Question

Consider the mixed formulation of the Poisson/Darcy system for a region $\Omega$:

$\alpha \mathbf{v} + \nabla p = f \\ \mathrm{div}[\mathbf{v}] = 0 $

with the boundary conditions

$\mathbf{v}\cdot \mathbf{n} = v_n$ on $\Gamma_v$,

$p = p_0$ on $\Gamma_p$

For the weak form: $a(\mathbf{v}, \mathbf{w}) := \int_\Omega \mathbf{v} \cdot \mathbf{w}$,

while $b_1(\mathbf{w},p) := \int_\Omega p\;\mathrm{div}[\mathbf{w}]$ or

$b_2(\mathbf{w},p) := \int_\Omega \nabla p\cdot\mathbf{w}$.

Depending on the choice for $b_1$ or $b_2$, the solution spaces can be chosen as:

1) $\mathbf{v} \in H(div; \Omega)^n $ while $p \in L^2(\Omega)$ for the choice $b_1$. One can choose for this the BDM elements for the velocity and piecewise constant pressures for each element and get a stable solution. In this formulation, the condition on velocity is essential while the boundary condition for pressure is natural.

2) $\mathbf{v} \in L^2(\Omega)^n, p \in H^1_0(\Omega)$ for the choice $b_2$. In this case, the velocity can be approximated by the discontinous Raviart-Thomas elements of polynomial order $k$ and the pressure by Lagrangian elements of polynomial order $k+1$, to get a stable solution. In this formulation, the condition on velocity is natural while the boundary condition for pressure is essential.

Q1. Is one of 1), 2) to be preferred over the other choice? Will the solutions obtained differ ?

Q2. However, $H^1_{0}(\Omega)^n \subset H(div;\Omega)^n$. So suppose I choose $\mathbf{v} \in H^1_0(\Omega)^n, p \in L^2(\Omega)$ for the choice $b_1$ and use the standard Taylor-Hood element, enforcing velocity boundary condition as essential and the pressure boundary condition as natural. Is this a stable approximation ?

PS: I am unable to create the tags "Sobolev-spaces", "mixed-formulations". If you agree that they are apt for this question, could you please create them.

Answer 1

Assuming you choose stable pairings of elements, the main consideration is which variable you are more interested in. For example, if you choose the $H(div),L^2$ formulation, then you have the choice of piecewise constants for the pressure and a BDM element for the velocity. This yields only first order accuracy for the pressure in the $L^2$ norm -- in other words, very slow convergence. You would do this if your primary concern is not the pressure, but the velocity, for example because you want to compute the velocity field as the input for an advection equation transporting along some contaminant.

On the other hand, if you choose the $L^2,H^1$ formulation, then you need to use at least a continuous $P_1$ element for the pressure, and you get second order convergence. If you care about the pressure, then that would be an option. Of course, you could also choose a higher order mixed element and the first formulation in that case to get second order convergence in the pressure. If you do care about the pressure, which formulation to use is then a question of trading off the relative sizes of the velocity spaces.

Answer 2

Q1)

In terms of finite elements approximation the main question is about stability (aka invertibility of the discrete system). For the primal formulation (elliptic systems) usually any subspace $V_h \subset H_0^1$ yields a stable approximation because the bilinear form is coercive. However the condition that the $a$ form be coercive is not satised for most mixed methods. In fact elements that are not chosen with respect to stability will usually prove to be unstable. But $a$ can be made coercive on certain subspaces of $H(div)^n$. For a good reference I recommend http://www.ima.umn.edu/~arnold//papers/mixed.pdf.

Q2) if you choose an unstable approximation such as $V_h\subset H_0^1 \subset H(div)^n$ your discrete system is not invertible.

All the best