An Extension of the Morley Element on General Polytopal Partitions Using Weak Galerkin Methods

TL;DR

This paper extends the Morley finite element method for the biharmonic equation from triangles to general polytopal meshes using weak Galerkin techniques, enabling broader applicability and maintaining computational efficiency.

Contribution

It introduces a novel extension of the Morley element to polytopal partitions via weak Galerkin methods, preserving degrees of freedom and expanding its use.

Findings

Error estimates in energy and L2 norms are established.

Numerical experiments confirm theoretical accuracy.

The method effectively handles general polytopal meshes.

Abstract

This paper introduces an extension of the well-known Morley element for the biharmonic equation, extending its application from triangular elements to general polytopal elements using the weak Galerkin finite element methods. By leveraging the Schur complement of the weak Galerkin method, this extension not only preserves the same degrees of freedom as the Morley element on triangular elements but also expands its applicability to general polytopal elements. The numerical scheme is devised by locally constructing weak tangential derivatives and weak second-order partial derivatives. Error estimates for the numerical approximation are established in both the energy norm and the $L^2$ norm. A series of numerical experiments are conducted to validate the theoretical developments.