Continuous Piecewise Linear Finite Elements for the Kirchhoff-Love Plate Equation

TL;DR

This paper introduces a family of continuous piecewise linear finite elements for the Kirchhoff-Love plate equation, employing quadratic reconstructions to enable discontinuous Galerkin methods and demonstrating their convergence properties.

Contribution

The paper presents a novel approach using quadratic reconstructions from linear elements for plate problems, including new error estimates and analysis of convergence.

Findings

Morley reconstruction is equivalent to Basic Plate Triangle

Fully quadratic reconstruction achieves optimal convergence

Morley reconstruction does not converge on unstructured meshes

Abstract

A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff-Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the Basic Plate Triangle. Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and $L^2$ norm. Numerical results indicate that the Morley reconstruction/Basic Plate Triangle does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.