Hessian Recovery for Finite Element Methods

TL;DR

This paper introduces a Hessian recovery method for finite element analysis that achieves superconvergence and polynomial preservation on various mesh types, validated by numerical experiments.

Contribution

It presents a novel Hessian recovery strategy that preserves polynomial degrees and achieves superconvergence for arbitrary order finite elements on different meshes.

Findings

Superconvergence at rate O(h^k) on mildly structured meshes

Polynomial preservation of degree k+1 and k+2 on specific meshes

Symmetric Hessian matrix when using symmetric sampling points

Abstract

In this article, we propose and analyze an effective Hessian recovery strategy for the Lagrangian finite element of arbitrary order $k$. We prove that the proposed Hessian recovery preserves polynomials of degree $k+1$ on general unstructured meshes and superconverges at rate $O(h^k)$ on mildly structured meshes. In addition, the method preserves polynomials of degree $k+2$ on translation invariant meshes and produces a symmetric Hessian matrix when the sampling points for recovery are selected with symmetry. Numerical examples are presented to support our theoretical results.