Numerical analysis of strongly nonlinear PDEs

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.

An optimal switching approach toward cost‐effective control of a stand‐alone photovoltaic panel system under stochastic environment

“…Because of their nonlinearity and the degeneration of coefficients, HJB equations do not always have smooth solutions that are defined in a classical sense, while they admit solutions with lower regularity called viscosity solutions . Due to their rich mathematical properties, HJB equations and viscosity solutions have been analyzed from both analytical and numerical viewpoints . To our knowledge, PV systems have not been mathematically analyzed from the viewpoint of stochastic control and viscosity solutions, although they are potential mathematical tools for efficiently analyzing the systems.…”

“…35 Due to their rich mathematical properties, HJB equations and viscosity solutions have been analyzed from both analytical [36][37][38] and numerical viewpoints. [39][40][41] To our knowledge, PV systems have not been mathematically analyzed from the viewpoint of stochastic control and viscosity solutions, although they are potential mathematical tools for efficiently analyzing the systems. Addressing this issue would open new doors for a novel framework for modeling, analysis, and control of PV systems.…”

The Monge–Ampère equation

“…The following result is from [73], see also [86,Proposition 6.13]. It is remarkable that the convexity assumption on the solution is not enforced in (2.76), it is rather a consequence of the formulation.…”

“…For this reason, the theory regarding fully nonlinear operators can guide us to develop a notion of solution (viscosity solution) that is weaker than classical. We refer the reader to [55,Chapter 17], [26], [31] and [86,Section 2] for additional details.…”

Improved Accuracy of Monotone Finite Difference Schemes on Point Clouds and Regular Grids

“…However, there is a large gap between the rate of convergence, and the accuracy, which is more consistent with computational results. For a recent review, see [NSZ17].…”