# American Institute of Mathematical Sciences

2019, 26: 54-71. doi: 10.3934/era.2019.26.005

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

 1 Lebanese University, Khawarizmi Laboratory for Mathematics and Applications, Hadath, Mont Liban, Beirut, Lebanon 2 Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France 3 Lebanese International University, Department of Mathematics and Physics, Lebanon 4 Xiamen University, School of Mathematical Sciences, Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing, Xiamen, Fujian, China

Received  May 2019 Revised  July 2019 Published  July 2019

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 $\lambda$. 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 $\lambda$ and on the time step. Finally, we did some numerical simulations which show the behavior of the solution.

Citation: Hawraa Alsayed, Hussein Fakih, Alain Miranville, Ali Wehbe. Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems. Electronic Research Announcements, 2019, 26: 54-71. doi: 10.3934/era.2019.26.005
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##### References:
$\lambda = 10,\; f(x,y) = \sqrt{x^2 +y^2},\; \tau = 0.01,\; M = 29$
$\lambda = 10,\; f(x,y) = \sqrt{x^2 +y^2},\; \tau = 20,\; M = 29$
$\lambda = 11.5, \; f(x, y) = \sqrt{x^2 +y^2}, \; \tau = 0.001, \; M = 35$, in 3(a) touchdown is observed at $t = 682\tau$ however in 3(b) it is observed at $t = 1.3339\tau$
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