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I am currently working on finite element simulations about electrokinetics. My solver (getdp) accepts directly galerkin-type weak formulation of equations. I am thus trying to write my equations in adequate formulation.

Electrokinetics, in my work, is about studying the static electric charges without any magnetic effects. The charges are volumetric-type (no particles like ions in a liquid). The following equations should be enough to describe the problem : $$ \vec{\nabla} \times \vec{E}=\vec{0} $$ $$ \vec{\nabla} \cdot \vec{E}=\frac{\rho}{\epsilon}$$ $$ \vec{\nabla} \cdot \vec{J}+\frac{\partial \rho}{\partial t}=0$$ $$ \vec{J}=\sigma \vec{E}$$ Where $\vec{E}$ is the electric field, $\vec{J}$ the current density, $\rho$ the charge density. Assuming that: $$\vec{E}=-\vec{\nabla}V$$ We got: $$\sigma \Delta V + \epsilon \frac{\partial}{\partial t} \Delta V = 0$$

The weak formulation would then be ($a$ being the test function): $$\int{\sigma \vec{\nabla} V \cdot \vec{\nabla} a + \int{\epsilon \frac{\partial}{\partial t} \vec{\nabla} V \cdot \vec{\nabla} a}}=0$$ Or: $$(\sigma \vec{\nabla} V, \vec{\nabla} a)+\frac{\partial}{\partial t}(\epsilon \vec{\nabla} V , \vec{\nabla} a)=0$$

However, I cannot get any good results... Is my formulation naive ? Do you see some errors ?

As a reference, the correct weak formulation for electrostatics is: $$(\epsilon \vec{\nabla} V , \vec{\nabla} a)=0$$

Thanks in advance,

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