Adaptive Morley FEM for the von K\'{a}rm\'{a}n equations with optimal convergence rates

TL;DR

This paper establishes that the adaptive Morley finite element method achieves optimal convergence rates for the von Kármán equations by leveraging an axiomatic framework and novel error estimates.

Contribution

It introduces a comprehensive analysis of the adaptive Morley FEM for von Kármán equations, including stability, reliability, and quasiorthogonality, with novel error estimates for nonlinearity.

Findings

Optimal convergence rates achieved with adaptive Morley FEM

Error estimator based on residuals is effective

Novel piecewise H^1 error estimate enhances analysis

Abstract

The adaptive nonconforming Morley finite element method (FEM) approximates a regular solution to the von K\'{a}rm\'{a}n equations with optimal convergence rates for sufficiently fine triangulations and small bulk parameter in the D\"orfler marking. This follows from the general axiomatic framework with the key arguments of stability, reduction, discrete reliability, and quasiorthogonality of an explicit residual-based error estimator. Particular attention is on the nonlinearity and the piecewise Sobolev embeddings required in the resulting trilinear form in the weak formulation of the nonconforming discretisation. The discrete reliability follows with a conforming companion for the discrete Morley functions from the medius analysis. The quasiorthogonality also relies on a novel piecewise $H^1$ a~priori error estimate and a careful analysis of the nonlinearity.