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# 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 $\lambda$. 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.

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

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

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

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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