Numerical analysis of strongly nonlinear PDEs

TL;DR

This paper reviews numerical methods for strongly nonlinear PDEs, focusing on convergence to viscosity solutions and discussing various schemes and new tools for convergence rates.

Contribution

It provides a comprehensive overview of constructing and analyzing stable, consistent, and monotone schemes for nonlinear PDEs, including novel tools for convergence rate estimation.

Findings

Stable, consistent, and monotone schemes converge as discretization tends to zero.

Construction methodologies for finite difference, finite element, and semi-Lagrangian schemes.

Introduction of novel tools for deriving convergence rates.

Abstract

We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and nonconvex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent, and monotone schemes converge as the discretization parameter tends to zero. We review methodologies to construct finite difference, finite element, and semi-Lagrangian schemes that satisfy these criteria, and, in addition, discuss some rather novel tools that have paved the way to derive rates of convergence within this framework.