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.