A Structure-Preserving Stagewise Rescaling Algorithm for a Two-Dimensional Nonlocal MEMS Equation in an Asymptotically Constant-Feedback Regime

TL;DR

This paper introduces a stagewise rescaling algorithm for a 2D nonlocal MEMS equation, capturing finite-time touchdown phenomena with energy-based analysis and reproducible numerical tests.

Contribution

It develops a novel stagewise rescaling method with energy dissipation analysis for nonlocal MEMS equations in a specific feedback regime.

Findings

The algorithm accurately detects touchdown events.

Energy decay is verified within each stage.

Numerical diagnostics identify conditions for finite feedback.

Abstract

Nonlocal MEMS equations exhibit finite-time quenching, or touchdown, which is difficult to capture numerically. We study a stagewise rescaling algorithm for a two-dimensional nonlocal MEMS equation in an asymptotically constant-feedback touchdown regime. The equation is not exactly invariant under the $A^{3/2}$--$A^3$ scaling used here; the scaling is justified when the reciprocal-integral feedback $K(t)=1+\int_\Omega(1-u)^{-1}dx$ remains bounded and converges to a finite positive limit, as in the single-point touchdown profiles of Duong--Zaag. In this regime the leading-order core dynamics reduce to a local MEMS equation with an asymptotically constant coefficient. Using a fixed-stage scaling of the deficit variable, we obtain a gradient flow for a rescaled energy at frozen amplitude and prove an exact energy dissipation identity within each stage. We introduce a minimizing-movement stage solver and derive a discrete energy inequality. Since strict monotonicity need not hold across stage transitions, we separate the switch and outer-update defects and prove an exact defect balance. Under a uniform switch-defect estimate, this yields quantitative almost monotonicity and a defect-aware criterion for nonexistence of a global admissible continuation. The numerical section is organized around reproducible two-dimensional reference computations: a full-domain stagewise run showing trigger detection, fixed-stage energy decay, and geometric accumulation of physical time, and a direct fixed-domain energy check. These tests are not used as proof of the bounded-window criterion; instead, they report finite-feedback diagnostics and identify the ideal-transfer switch-energy diagnostics required for a posteriori verification.