Numerical Solution and Errors Analysis of Iterative Method for a Nonlinear Plate Bending Problem

Abstract

This paper uses the HCT finite element method and mesh adaptation technology to solve the nonlinear plate bending problem and conducts error analysis on the iterative method, including a priori and a posteriori error estimates. Our investigation exploits Hermite finite elements such as BELL and HSIEH-CLOUGH-TOCHER (HCT) triangles for conforming finite element discretization. Then, the existence and uniqueness of the approximation solution are proven by using a variant of the Brezzi-Rappaz-Raviart theorem. We solve the approximation problem through a fixed-point strategy and an iterative algorithm, and study the convergence of the iterative algorithm, and provide the convergence conditions. An optimal a priori error estimation has been established. We construct a posteriori error indicators by distinguishing between discretization and linearization errors and prove their reliability and optimality. A numerical test is carried out and the results obtained confirm those established theoreticall.