强非线性偏微分方程的数值分析

IF 16.3 1区数学 Q1 MATHEMATICS

Acta Numerica Pub Date : 2016-10-25 DOI:10.1017/S0962492917000071

M. Neilan, A. Salgado, Wujun Zhang

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引用次数: 66

摘要

本文综述了强非线性偏微分方程数值方法的构造和分析，重点介绍了凸和非凸全非线性方程及其收敛性。我们首先描述了这一领域的一个基本结果，即当离散化参数趋于零时，稳定、一致和单调方案收敛。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格式的方法，并且，此外，讨论了一些相当新颖的工具，这些工具为在此框架内推导收敛速率铺平了道路。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Numerical analysis of strongly nonlinear PDEs *

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.

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来源期刊

Acta Numerica MATHEMATICS-

CiteScore

26.00

自引率

0.70%

发文量

期刊介绍： Acta Numerica stands as the preeminent mathematics journal, ranking highest in both Impact Factor and MCQ metrics. This annual journal features a collection of review articles that showcase survey papers authored by prominent researchers in numerical analysis, scientific computing, and computational mathematics. These papers deliver comprehensive overviews of recent advances, offering state-of-the-art techniques and analyses. Encompassing the entirety of numerical analysis, the articles are crafted in an accessible style, catering to researchers at all levels and serving as valuable teaching aids for advanced instruction. The broad subject areas covered include computational methods in linear algebra, optimization, ordinary and partial differential equations, approximation theory, stochastic analysis, nonlinear dynamical systems, as well as the application of computational techniques in science and engineering. Acta Numerica also delves into the mathematical theory underpinning numerical methods, making it a versatile and authoritative resource in the field of mathematics.