MEMS with fringing field: curvature-dependent electrostatic field and numerical techniques for recovering the membrane profile

As known, a 1$D$ membrane MEMS semi-linear elliptic model with fringing field can be written as egin{equation}otag u''=-rac{lambda^2(1+delta |u'|^2)}{(1-u)^2};;; ext{in};;Omega subset mathbb{R}, ;;u=0;; ext{on};;partial Omega, end{equation} oindent where $lambda^2$ and $delta$ are positive parameters and $u$ is the deflection of the membrane. Since the electric field $ mathbf{E} $ on the membrane is locally orthogonal to the straight tangent line to the membrane, $|mathbf{E}|$ can be considered locally proportional to the curvature $K$ of the membrane, so that a well-known model with fringing field in which $|mathbf{E}|^2propto lambda^2/(1-u)^2$ has been here considered. In this paper, starting from this model, we present a new algebraic condition of uniqueness for the solution of this model depending on the electromechanical properties of the material constituting the membrane, which weighs more than the condition of existence known in literature. Furthermore, shooting-Dekker-Brent, Keller-Box-Scheme, and III/IV Stage Lobatto IIIA formulas were exploited and their performances compared to recover $ u $ under convergence conditions in the presence/absence of ghost solutions. Finally, a criterion that is able to choose the material constituting the membrane starting from the applied electric voltage $ V $ and vice versa, in conditions of convergence, and in the presence and absence of ghost solutions, is presented.