PDE system with a transient Poisson equation and a transportation (convection) equation

Question

Dear all,

I am solving a system with a Poisson equation for pressure field and a passive scalar transportation equation for a component fraction. The equations are as
$${{\partial p} \over {\partial t} } = c\nabla \cdot \left ( \bf K \cdot \nabla p \right ) $$,
$$ \bf u = -\bf K\cdot \nabla p $$,
$${\partial (\rho v) \over \partial t } + \nabla \cdot (\rho v \bf u) = 0 $$,
where $\rho$ is a function of $p$, such as $\rho = \rho_0 [1+c_p(p-p_0)]$. Here the unknowns $p$ and $v$ are scalars. $\bf K$ is a tensor with only diagonal elements $K_x$, $K_y$, and $K_z$.

From the FEniCS tutorial, I learned how to solve the transient Poisson equation. My questions are ,
1. how to handle the anisotropic coefficient tensor $\bf K$?
2. how to solve the convection equation for $v$?

Is there any example for this kind of problem? Your suggestion and experience are highly appreciated.

Regards,

Yong Chang

Answer 1

1, you could create a tensor easily by using 'as_tensor', like:

K = as_tensor([\
    [1.0, 0.0, 0.0],\
    [0.0, 2.0, 0.0],\
    [0.0, 0.0, 3.0],\
    ])

2, The third equation is a convection problem, which the standard finite element method is not good at. I suggest you using DG method or SUPG method.

Comments

Is there any possibility to solve the convection problem with fenics, especially with the DG or the SUPG method?