High Order Morley Elements for Biharmonic Equations on Polytopal Partitions

TL;DR

This paper extends the Morley finite element method for biharmonic equations to polytopal partitions using weak Galerkin techniques, achieving high accuracy with minimal degrees of freedom.

Contribution

It introduces a novel higher-order Morley element extension on polytopal meshes via weak Galerkin methods, enhancing flexibility and efficiency in biharmonic problem approximation.

Findings

Achieves optimal error estimates in discrete $H^2$ and $L^2$ norms.

Demonstrates the method's effectiveness through numerical experiments.

Provides a stable and accurate scheme for biharmonic equations on complex meshes.

Abstract

This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite element methods to accommodate higher degrees of polynomials and the flexibility of general polytopal elements. By utilizing the Schur complement of the weak Galerkin method, the extension allows for fewest local degrees of freedom while maintaining sufficient accuracy and stability for the numerical solutions. The numerical scheme incorporates locally constructed weak tangential derivatives and weak second order partial derivatives, resulting in an accurate approximation of the biharmonic equation. Optimal order error estimates in both a discrete $H^2$ norm and the usual $L^2$ norm are established to assess the accuracy of the numerical approximation. Additionally, numerical results are presented to validate the developed theory and demonstrate the effectiveness of the proposed extension.