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

TL;DR

This work develops precise anisotropic interpolation and consistency error estimates for the Morley finite element method applied to fourth-order elliptic equations on semi-regular meshes, bypassing the traditional shape-regularity requirement. By exploiting the relationship between Raviart–Thomas and Morley spaces, it derives robust L2-projection and nonconforming interpolation bounds that accommodate anisotropy through mesh-geometric parameters. The authors apply these results to a modified Morley method for fourth-order problems and extend the framework to stream function formulations of Stokes flow, achieving near-optimal convergence under convex-domain regularity assumptions. The findings support the effectiveness of modified Morley-type methods on anisotropic meshes for boundary-layer and related problems, with careful attention to 3D extensions and stability considerations.

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.

Figures (1)

• Figure 1: New parameters $\widetilde{\mathscr{H}}_i$, $i=1,2,3$