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I need to derive the weak form for the following PDE system:

$\rho_{f}\frac{\partial \mathbf{u}}{\partial t}+\nabla P=0$ in $\Omega_{s}$

$\nabla\cdot \mathbf{u}=0$ in $\Omega_{s}$

where $P=P(x,y,t):\mathbb{R^+}\times \mathbb{R^2}\rightarrow \mathbb{R}$ , $\mathbf{u}=\mathbf{u}(x,y,t):\mathbb{R^+}\times \mathbb{R^2}\rightarrow \mathbb{R^2}$ are both unknowns.

The boundary conditions are the following:

$P=\bar{P}$ on $\Gamma^{1}_{f}\cup \Gamma^{2}_{f}$

$\mathbf{u}\cdot \mathbf{n}=0$ on $\Gamma^{3}_{f}$

$\mathbf{u}\cdot \mathbf{n}=w$ on $\Gamma$

An illustration of the computation domain can be seen here $\rightarrow$1.

I try to multiply by the test function and integrating, but then I don't see how I can go from there, because I can't seem to determine both $P$ and $\mathbf{u}$ with only these two equations after approximating with linear finite elements.

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1 Answer 1

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You have the case of a mixed finite element problem. This a special case of time-dependent Stoke's equation. You need to use a pair of finite elements that will solve these equations. I suggest reading Mixed FEM for Stokes equation to get an idea of how to do the problem. NOTE: You have $\partial u/\partial t$, i.e., you have to apply a finite difference to approximate the time derivative

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