The Hellan-Herrmann-Johnson method with curved elements

TL;DR

This paper analyzes the finite element approximation of the Kirchhoff plate equation on curved domains using the HHJ method, proving optimal convergence and examining effects of curved elements and geometrical approximation.

Contribution

It introduces a parametric curved HHJ space for optimal convergence on curved boundaries and discusses the method's robustness against the Babuška paradox.

Findings

Optimal convergence on domains with curved boundaries.

Degradation of convergence with insufficient polynomial degree curved triangles.

The HHJ method avoids the Babuška paradox on polygonal approximations.

Abstract

We study the finite element approximation of the Kirchhoff plate equation on domains with curved boundaries using the Hellan-Herrmann-Johnson (HHJ) method. We prove optimal convergence on domains with piecewise $C^{k+1}$ boundary for $k \geq 1$ when using a parametric (curved) HHJ space. Computational results are given that demonstrate optimal convergence and how convergence degrades when curved triangles of insufficient polynomial degree are used. Moreover, we show that the lowest order HHJ method on a polygonal approximation of the disk does not succumb to the classic Babu\v{s}ka paradox, highlighting the geometrically non-conforming aspect of the HHJ method.