We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and non-convex 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.

中文翻译：

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强非线性偏微分方程的数值分析

我们回顾了强非线性 PDE 数值方法的构建和分析，重点是凸和非凸完全非线性方程以及对粘度解的收敛性。我们首先描述该领域的一个基本结果，该结果表明当离散化参数趋于零时，稳定、一致和单调的方案会收敛。我们回顾了构建满足这些标准的有限差分、有限元和半拉格朗日方案的方法，此外，还讨论了一些相当新颖的工具，它们为在该框架内推导收敛速度铺平了道路。