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

Hawraa Alsayed; Hussein Fakih; Alain Miranville; Ali Wehbe

doi:10.3934/era.2019.26.005

Article Contents

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

This volume Previous Article On higher-order anisotropic perturbed Caginalp phase field systems Next Article

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: March 2019

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Abstract

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.

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Mathematics Subject Classification: Primary: 65N06, 65N12, 35K55.

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Figure 1. $ \lambda = 10,\; f(x,y) = \sqrt{x^2 +y^2},\; \tau = 0.01,\; M = 29 $

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Figure 2. $ \lambda = 10,\; f(x,y) = \sqrt{x^2 +y^2},\; \tau = 20,\; M = 29 $

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Figure 3. $\lambda = 11.5, \; f(x, y) = \sqrt{x^2 +y^2}, \; \tau = 0.001, \; M = 35$, 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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