A least-squares Galerkin gradient recovery method for fully nonlinear elliptic equations

TL;DR

This paper introduces a least-squares Galerkin gradient recovery method for solving fully nonlinear elliptic equations, providing error bounds and adaptive strategies for improved numerical approximation.

Contribution

It develops a novel least-squares Galerkin approach for nonlinear elliptic equations, including error analysis and adaptive algorithms.

Findings

Effective approximation of nonlinear elliptic problems demonstrated.

Error bounds established for the proposed method.

Adaptive methods improve solution accuracy.

Abstract

We propose a least squares Galerkin based gradient recovery to approximate Dirichlet problems for strong solutions of linear elliptic problems in nondivergence form and corresponding apriori and aposteriori error bounds. This approach is used to tackle fully nonlinear elliptic problems, e.g., Monge-Amp\`ere, Hamilton-Jacobi-Bellman, using the smooth (vanilla) and the semismooth Newton linearization. We discuss numerical results, including adaptive methods based on the aposteriori error indicators.