Algorithms and Numerical Analysis for Partial Differential and Integral Equations

本文提出了偏微积分方程组有限元数值分析中的几个问题。A.二阶双曲问题有限元方法的最大模估计：与椭圆和抛物型问题相比，双曲型问题的理论是非常不完整的。B.有限元方法中的超收敛：主要问题是确定何时(以及何时不)超收敛达到边界。各向异性扩散问题的有限元方法：在奇异摄动的情况下，我们所知的很少。D.偏积分-微分方程解的有限元方法：在具有积分型记忆项的问题中，主要问题在于巨大的存储和工作需求。据估计，在所有涉及科学或工程模型近似的计算机运行中，有三分之二是无用的：不是因为它们必然是错误的，而是因为你不知道它们是正确的，因此不能信任它们。当你正在制定一个新的模型，并想要找到它的预测来与现实进行测试时，这尤其令人恼火。你不知道一个特殊的预测是模型固有的，还是通过错误的计算机模拟悄悄产生的。本提案的目的是调查一些广泛使用的模拟方法在具有各种怪癖的数学问题上的表现。因此，它将为使用数值方法的人提供指导，并帮助他们判断特定的计算机运行序列在表示科学模型时是否可靠。

Lars Wahlbin Abstract It is proposed to work on several problems in the numerical analysis of finite element methods for partial differential and integral equations. A. Maximum-norm estimates for finite element methods in second order hyperbolic problems: In contrast to the situation for elliptic and parabolic problems, the theory in the case of hyperbolic problems is very incomplete. B. Superconvergence in finite element methods: The main problem is to ascertain when (and when not) superconvergence holds up to boundaries. C. Finite element methods in problems with anisotropic diffusion: In the singularly perturbed case, very little is known. D. Finite element methods in partial integro-differential equations: The main problem lies with enormous memory and work requirements in problems with an integral-type memory term. It has been estimated that 2/3 of all computer runs involving approximation of scientific or engineering models are useless: not because they are necessarily wrong but because you do not know that they are right and so cannot trust them. This is particularly annoying when you are formulating a new model and want to find its predictions to test against reality. You do not know whether a peculiar prediction is inherent in the model or crept in via a faulty computer simulation of it. The present proposal is aimed at investigating how a selection of widely used simulation methods behave on mathematical problems with various quirks. It will thus give guidelines to people using numerical methods and help them in judging whether a particular series of computer runs is reliable or not in representing a scientific model.