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

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  • 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.

    Citation:

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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 $t = 1.3339\tau$

  • [1] E. L. Allgower and K. Georg, Continuation and path following, Acta Numerica, 2 (1993), 1–64. doi: 10.1017/s0962492900002336.
    [2] N. D. Brubaker and J. A. Pelesko, Non-linear effects on canonical MEMS models, European Journal of Applied Mathematics, 22 (2011), 455–470. doi: 10.1017/S0956792511000180.
    [3] L. Cherfils, A. Miranville, S. Peng and C. Xu, Analysis of discretized parabolic problems modelling Electrostatic Micro-Electromechanical systems, AIMS Journals, 2018.
    [4] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modelling Electrostatic MEMS, Courant Lecture Notes in Mathematics, 20. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010. doi: 10.1090/cln/020.
    [5] L. C. Evans, Partial Differential Equations , Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.
    [6] G. Flores, G. Mercado, J. A. Pelesko and N. Smyth, Analysis of the dynamics and touch down model of electrostatic MEMS, SIAM Jpournal on Applied Mathematics, 67 (2006/07), 434–446. doi: 10.1137/060648866.
    [7] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM Journal on Applied Mathematics, 38 (2006/07), 1423–1449. doi: 10.1137/050647803.
    [8] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices Ⅱ: Dynamic case, NoDEA Nonlinear Differential Equations and Applications, 15 (2008), 115–145. doi: 10.1007/s00030-007-6004-1.
    [9] Y. Guo, On the partial differential equations of electrostatic MEMS devices Ⅲ: Refined touchdown behavior, J. Differential Equations, 244 (2008), 2277–2309. doi: 10.1016/j.jde.2008.02.005.
    [10] Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM Journal on Applied Mathematics, 66 (2005), 309–338. doi: 10.1137/040613391.
    [11] S. H. Lui, Numerical Analysis of Partial Differential Equations, John Wiley & Sons, Inc., Hoboken, NJ, 2011. doi: 10.1002/9781118111130.
    [12] I. Stakgold, Green's Functions and Boundary Value Problems, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1998.
    [13] J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM Journal on Applied Mathematics, 62 (2002), 888–908. doi: 10.1137/S0036139900381079.
    [14] J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003.
    [15] J. A. Pelesko, D. H. Bernstein and J. McCuan, Symmetry and Symmetry Breaking in Electrostatic MEMS, Proceedings of MSM, (2003), 304–307.
    [16] J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control, Journal of Engineering Mathematics, 41 (2001), 345–366. doi: 10.1023/A:1012292311304.
    [17] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006.
    [18] Q. Wang, Quenching phenomenon for a parabolic MEMS equation, Chinese Annals of Mathematics, 39 (2018), 129–144. doi: 10.1007/s11401-018-1056-6.
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