Morley finite element analysis for fourth-order elliptic equations under a semi-regular mesh condition

TL;DR

This paper develops an anisotropic error estimate for the Morley finite element method applied to fourth-order elliptic equations, removing the need for shape-regular meshes and demonstrating optimal convergence.

Contribution

It introduces a new proof of term consistency enabling anisotropic error estimates without shape-regularity assumptions.

Findings

Optimal convergence rates achieved

Anisotropic meshes effectively used

Enhanced understanding of Morley FEM error behavior

Abstract

In this study, we present a precise anisotropic interpolation error estimate for the Morley finite element method (FEM) and apply it to fourth-order elliptic equations. We do not impose the shape-regularity mesh condition in the analysis. Anisotropic meshes can be used for this purpose. The main contributions of this study include providing a new proof of the term consistency. This enables us to obtain an anisotropic consistency error estimate. The core idea of the proof involves using the relationship between the Raviart--Thomas and Morley finite-element spaces. Our results indicate optimal convergence rates and imply that the modified Morley FEM may be effective for errors.