惯性流形应用于科学计算

Applications of Inertial Manifolds to Scientific Computation

在科学计算中多级方法是我们熟知的,且在研究中经常与多格或区域分解法有关.此外,还可以利用惯性流形作为发展问题的物理意义更为密切相关的一种不同类型的多级方法,主要用于非线性耗散方程. 起源于利用惯性流形的新的多级方法,目前已经引入的有增加未知量法(Incremental Unknown Meth-od)和非线性伽辽金法;这些方法适用于对演化进行长时间积分.本文的目的是介绍这些方法及用这些算法进行数值分析所得到的一些新的结果.

Mnltilevel methods are well known in scientific computation and much studied in relation with multigrid or domain decomposition methods. The utilization of inertial manigolds in scientific computation opens a way for the development of a different type of multilevel methods more closely related to the physics of problems, with an emphasis on nonlinear dissipative evolution equations. The new multilevel methods stemming from the utilization of inertial manifolds which have been introduced so far include the incremental unknown quantity method and the noulinear unknown quantity method; these methods are adapted to the large time integration of evolution equations. Our aim in this article is to present these methods and the new results concerning the numerical analysis of these algorithms.