Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems

This paper is dedicated to study the fully discretized semi implicit and implicit schemes of a 2D parabolic semi linear problem modeling MEMS devices. Starting with the analysis of the semi-implicit scheme, we proved the existence of the discrete solution which converges under certain conditions on the voltage \begin{document}$ \lambda $\end{document} . On the other hand, we consider a fully implicit scheme, we proved the existence of the discrete solution, which also converges to the stationary solution under certain conditions on the voltage \begin{document}$ \lambda $\end{document} and on the time step. Finally, we did some numerical simulations which show the behavior of the solution.