My question is about Finite Element Method.

I want to know how to implement "gradient zero" conditions to advection-diffusion equations in conservative form like,

$\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{\Gamma} = 0$

where, $\mathbf{\Gamma} = \mathbf{u}\rho - k\nabla \rho$

$\mathbf{u}$ and $k$ is not constant variables.

The boundary integral part of weak-form can be written as follows,

$\int_{\Gamma^N} \rho^* \mathbf{\Gamma} \cdot \mathbf{n} d\Gamma^N$

where, $\mathbf{n}$ is normal vector and $\rho^*$ is weighed function.

But it seems that this integral is specialized to "Mixed boundary condition".

So my question is how to modify this integral to that of "Gradient zero conditon".

• The term "gradient zero condition" is not commonly used. What do you mean by it? – Wolfgang Bangerth Dec 26 '13 at 14:47
• Thank you for your speedy response. I mean, $\frac{\partial \rho}{\partial \mathbf{n}} = 0$ as "gradient zero condition". – Phoenix Kyoma Dec 26 '13 at 16:48

$$\int_{\Gamma^N} \rho^* \mathbf{\Gamma} \cdot \mathbf{n} d\Gamma^N = 0$$
If you were to apply the condition $\frac{\partial n}{\partial x} = 0$ at the boundary this would be equivalent to preventing diffusion through the boundary, but it would be an open boundary condition for the advection component of the flux. This may or may not be what you want.