关于Lax-Milgram定理和Céa定理的一个注记

摘要

本文研究抽象变分问题(不必要求具有强制性)的Galerhin方法,利用泛函分析理论证明了:若变分问题的Galerkin逼近问题存在唯一解,那么它本身的解存在唯一且可由Galerhin逼近解无限逼近的充要条件是其Galerkin逼近格式具有某种稳定性.此结果是对Lax-Milgram定理和Céa定理的补充,可以应用于不必具有强制性的变分问题.%This paper investigates Galerkin methods for an abstract variational problem when its bilinear form has no eoereivity,and by the use of funetional analysis theory,proves such a result:the variational problem has a unique solution and the solution can be approximated by their Galerkin approximation solutions if and only if the Galerkin approximation problems sarisfy certain stability,where the Galerkin approximation problems are supposed to have unique solutions.This result may be viewed as replenishment for the Lax-Milgram theorem and the Céa theorem;it is applied to the variational problem with or without coercivity.