A least-squares Galerkin approach to gradient and Hessian recovery for nondivergence-form elliptic equations

TL;DR

This paper introduces a least-squares Galerkin method for elliptic equations in nondivergence form, enabling gradient and Hessian recovery without inf-sup stabilization, and demonstrates convergence through numerical experiments.

Contribution

It presents a novel least-squares approach that uses standard finite elements for nondivergence elliptic equations, avoiding stabilization and providing convergence guarantees.

Findings

Method achieves convergence with standard finite elements.

Numerical experiments confirm theoretical results.

Approach works with uniform and adaptive meshes.

Abstract

We propose a least-squares method involving the recovery of the gradient and possibly the Hessian for elliptic equation in nondivergence form. As our approach is based on the Lax--Milgram theorem with the curl-free constraint built into the target (or cost) functional, the discrete spaces require no inf-sup stabilization. We show that standard conforming finite elements can be used yielding apriori and aposteriori convergnece results. We illustrate our findings with numerical experiments with uniform or adaptive mesh refinement.