Morley Finite Element Method for the von K\'{a}rm\'{a}n Obstacle Problem

TL;DR

This paper develops and analyzes a Morley finite element method for solving the von Kármán obstacle problem, establishing well-posedness, regularity, and optimal error estimates with numerical validation.

Contribution

It introduces a nonconforming Morley finite element approach for the von Kármán obstacle problem, providing new error estimates and numerical evidence for convergence under small data assumptions.

Findings

Well-posedness of the obstacle problem established

Optimal convergence rates demonstrated for small data

Numerical results confirm theoretical error estimates

Abstract

This paper focusses on the von K\'{a}rm\'{a}n equations for the moderately large deformation of a very thin plate with the convex obstacle constraint leading to a coupled system of semilinear fourth-order obstacle problem and motivates its nonconforming Morley finite element approximation. The first part establishes the well-posedness of the von K\'{a}rm\'{a}n obstacle problem and also discusses the uniqueness of the solution under an a priori and an a posteriori smallness condition on the data. The second part of the article discusses the regularity result of Frehse from 1971 and combines it with the regularity of the solution on a polygonal domain. The third part of the article shows an a priori error estimate for optimal convergence rates for the Morley finite element approximation to the von K\'{a}rm\'{a}n obstacle problem for small data. The article concludes with numerical results that illustrates the requirement of smallness assumption on the data for optimal convergence rate.