非线性抛物型初边值问题的差分-边界有限元耦合方法及其误差估计

A COUPLING METHOD OF DIFFERENCE AND BOUNDARY FINITE ELEMENT AND ITS ERROR ESTIMATES FOR NONLINEAR PARABOLIC INITIAL BOUNDARY VALUE PROBLEMS

一、引言边界元方法以其对于无界区域问题的独特有效性及其它一些性质,在工程技术和计算数学领域得到越来越广泛的重视、应用和研究.对于椭圆型边值问题,边界元方法的应用和理论研究已是硕果累累,对于发展型的初边值问题,近十年来,其理论研究在某些方面已取得了突破性进展,但仍有许多方面处于空白.发展型方程的边界元方法基本上分为三种类型:第一种类型是利用发展型方程的基本解导出发展型的边界积分方程;第二种类型是通过可逆积分变换将发展方程转化为椭圆型方程;第三种类型是对于时间变量采用差分离散化,将发展型方程转化成一组椭圆型方程.对于第一种类型方法的应用和理论研究已日臻完善.但对于第三种类型方法的理论分析尚属空白.本文研究第三种类型方法的应用及其误差分析,给出了数值计算格式和近似解的先验误差估计.

In this paper,a new coupling numerical method of difference and boundary finite ele-ment is presented to solve the initial boundary problem of nonlinear parabolic partial differen-tial equations defined on a bounded or an unbounded domain.In this method,a differencemethod is first used to discretige the time derivative to change the parabolic equation into afamily of semi-discrete elliptic difference partial differential equations;then its boundary in-tegral equation and its boundary variational formula and finally numerical approximate schemesbased on the boundary finite element method are derived.The prior error estimates of the ap-proximate solutoin in L^2-norm,H^1-norm and L~∞-norm are obtained.