A finite element method for fully nonlinear elliptic problems

TL;DR

This paper introduces a continuous finite element method tailored for fully nonlinear elliptic equations, leveraging a discretisation that directly addresses the strong form of linear PDEs, and demonstrates its effectiveness through numerical benchmarks.

Contribution

The paper develops two novel finite element methodologies for solving second order fully nonlinear PDEs using a discretisation that works directly on the PDE's strong form.

Findings

The method converges for test problems including Monge-Ampère and Pucci's equations.

The approach produces a finite element Hessian as a byproduct of the solution process.

Numerical results validate the convergence and effectiveness of the proposed schemes.

Abstract

We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretisation method is that a recovered (finite element) Hessian is a biproduct of the solution process. We build on the linear basis and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems including the Monge-Amp\`ere equation and Pucci's equation.