# PDE weak form and FEM approximation

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.

## 1 Answer

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