Algorithms and Numerical Analysis for Partial Differential and Integral Equations

摘要针对偏微分方程和积分方程的有限元数值分析中存在的几个问题进行了研究。A.二阶双曲型问题的有限元方法的最大范数估计：与椭圆型和抛物型问题的情况相比，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（何时不）超收敛符合边界。C.各向异性扩散问题的有限元方法：在奇异摄动情况下，所知甚少。D.局部积分-微分方程的有限元方法：主要问题是在具有积分型记忆项的问题中，其内存和工作量巨大。据估计，在所有涉及科学或工程模型近似值的计算机运行中，有三分之二是无用的：不是因为它们一定是错的，而是因为你不知道它们是对的，因此不能相信它们。当你正在构建一个新模型，并希望找到它的预测来测试现实时，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得来的。目前的建议旨在研究如何选择广泛使用的模拟方法的行为与各种怪癖的数学问题。因此，它将为使用数值方法的人们提供指导，并帮助他们判断一系列特定的计算机运行是否可靠，是否代表科学模型。

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.