# 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
##### References:
 [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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##### References:
 [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.
$\lambda = 10,\; f(x,y) = \sqrt{x^2 +y^2},\; \tau = 0.01,\; M = 29$
$\lambda = 10,\; f(x,y) = \sqrt{x^2 +y^2},\; \tau = 20,\; M = 29$
$\lambda = 11.5, \; f(x, y) = \sqrt{x^2 +y^2}, \; \tau = 0.001, \; M = 35$, 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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