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 {u sub t /sub (x,t) = u sub xx /sub (x,t) + λf(x)(1 - u(x,t)) sup -p /sup , 1 lt; x lt; 1, t gt; 0, u(-l,t) = 0, u(l,t) = 0, t gt;0, u(x,0) = u sub 0 /sub (x) ≥0, -1 ≤ x ≤ 1, where p gt; 1, λ gt; 0 and f(x) € C¹ )[-1,1]), symmetric and nondecreasing on the interval (-1,0), 0 lt; f(x) ≤ 1, f(-1) = 0, f(l) = 0 and 1 = 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.