一类非线性抛物型方程广义差分法的变分原理及H^1模误差估计

VARIATIONAL PRINCIPLES AND ERROR ESTIMATES OF GENERALIZED DIFFERENCE METHODS FOR A CLASS OF TWO-DIMENSIONAL PARABOLIC EQUATION

和有限元方法类似,广义差分法属于基于变分原理的差分格式,是解偏微分方程的一种有效的数值方法.因此,寻求对应于定解问题的广义差分法的变分原理是很重要的,本文第一部分内容即属此.本文还给出了用此方法解一类非线性抛物型方程的H^1模误差估计.

In this paper, we consider the following problems:(?u/?t)+Bu=f(x,y,t,u,?u),t∈(0,T],u(x, y, 0) = u_0(x, y), (x. y)∈Q,u|?Q=0 or(ω·n +σu)|_?Q = g(s, t),where Bu = -?·ω, ω=(ω_1, ω_2)~T, ω_i= α_(j1)(x, y, t)(?u/?x)+α_(j2)(x, y, t)(?u/?y). The variational principles and the H^1 error estimates of generalized differencemethods are given and the approximation orders obtained are optimal.