Minimal residual discretization of a class of fully nonlinear elliptic PDE

TL;DR

This paper presents finite element methods for fully nonlinear elliptic PDEs using a minimal residual approach based on the Alexandroff--Bakelman--Pucci estimate, with proven convergence and adaptive mesh refinement in multiple dimensions.

Contribution

It introduces a novel minimal residual finite element framework for nonlinear elliptic PDEs with convergence proofs and adaptive algorithms.

Findings

Convergence of $C^1$ conforming and discontinuous Galerkin methods in $L^ abla$ norm.

Effective adaptive mesh refinement driven by residuals.

Numerical validation in 2D and 3D showing method robustness.

Abstract

This work introduces finite element methods for a class of elliptic fully nonlinear partial differential equations. They are based on a minimal residual principle that builds upon the Alexandrov--Bakelman--Pucci estimate. Under rather general structural assumptions on the operator, convergence of $C^1$ conforming and discontinuous Galerkin methods is proven in the $L^\infty$ norm. Numerical experiments on the performance of adaptive mesh refinement driven by local information of the residual in two and three space dimensions are provided.