Numerical Quenching Solutions of a Parabolic Equation Modeling Electrostatic Mems

In this paper, we study the semidiscrete approximation for the following initial-boundary value problem  ut(x, t) = uxx(x, t) + λf(x)(1− u(x, t))−p, −l < x < l, t > 0, u(−l, t) = 0, u(l, t) = 0, t > 0, u(x, 0) = u0(x) ≥ 0, −l ≤ x ≤ l, where p > 1, λ > 0 and f(x) ∈ C([−l, l]), symmetric and nondecreasing on the interval (−l, 0), 0 < f(x) ≤ 1, f(−l) = 0, f(l) = 0 and l = 1 2 . We find some conditions under which the solution of a semidiscrete form of above problem quenches in a finite time and estimate its semidiscrete quenching time. Moreover, we prove that the semidiscrete solution must quench near the maximum point of the function f(x), for λ sufficiently large. We also establish the convergence of the semidiscrete quenching time to the theoretical one when the mesh size tends to zero. Finally, we give some numerical experiments for a best illustration of our analysis.