Convergence Analysis of Virtual Element Method for Nonlinear Nonlocal Dynamic Plate Equation

TL;DR

This paper develops a high-order virtual element method for a nonlinear nonlocal dynamic plate equation, proving its well-posedness and optimal convergence, supported by numerical experiments.

Contribution

It introduces a $C^1$ conforming virtual element method of arbitrary order for the nonlinear nonlocal plate equation, with theoretical analysis and numerical validation.

Findings

Proved well-posedness of the fully discrete scheme.

Derived optimal order of convergence in space and time.

Numerical experiments confirm the effectiveness of the method.

Abstract

In this article, we have considered a nonlinear nonlocal time dependent fourth order equation demonstrating the deformation of a thin and narrow rectangular plate. We propose $C^1$ conforming virtual element method (VEM) of arbitrary order, $k\ge2$, to approximate the model problem numerically. We employ VEM to discretize the space variable and fully implicit scheme for temporal variable. Well-posedness of the fully discrete scheme is proved under certain conditions on the physical parameters, and we derive optimal order of convergence in both space and time variable. Finally, numerical experiments are presented to illustrate the behaviour of the proposed numerical scheme.