  We propose a finite element method for the numerical simulation of
electroconvection of thin liquid crystals. The liquid is located in between two
concentric circular electrodes which are either assumed to be of infinite
height or slim. Each configuration results in a different nonlocal
electro-magnetic model defined on a two dimensional bounded domain. The
numerical method consists in approximating the surface charge density, the
liquid velocity and pressure, and the electric potential in the two dimensional
liquid region. Finite elements for the space discretization coupled with
standard time stepping methods are put forward. Unlike for the infinite
electrodes configuration, our numerical simulations indicate that slim
electrodes are favorable for electroconvection to occur and are able to sustain
the phenomena over long period of time. Furthermore, we provide a numerical
study on the influence of the three main parameters of the system: the Rayleigh
number, the Prandtl number and the electrodes aspect ratio.
