FINITE ELEMENT EXPANSION FOR VARIABLE COEFFICIENT ELLIPTIC PROBLEMS

The Ritz projections of the elliptic problem(?)(a_(ij)(?)_ju) c_0u=f in Ω,u=0 on (?)Ω,and the quasilinear eiliptic problem(?)(a_i(x,Du)) a_0(x,Du)=0 in Ω,u=0 on (?)Ωwith linear finite elements admit an asymptotic error expansion,respectively,for certain classes of“uniform” meshes.This provides the theoretical justification for the use of Richardson extrapolationfor increasing the accuracy of the scheme from O(h~2) to O(h~4 |ln h|).     

FINITE ELEMENT ERROR EXPANSIONS FOR THE EIGENVALUE APPROXIMATION TO THE MULTIGROUP DIFFUSION EQUATIONS

Asymptotic expansions for the finite element approximation to the eigenvalues of the mult-igroup diffusion equations on Ω(?)R~n(n=2,3)of reactor theory are given,firstly for a piecewiseuniform triangulation,then for nonuniform quadrilateral meshes,and finally for nonuniformhexahedral meshes.The effect of certain classes of numerical integrations is studied.As applicationsof the expansions,several extrapolation formulas and a posteriori error estimates are obtained.     

QUADRATURE AND EXTRAPOLATION FOR THE VARIABLE COEFFICIENT ELLIPTIC EIGENVALUE PROBLEM

For the variable coefficient elliptic eigenvalue problem on a smooth domain or aconvex polygonal domain,a numerical quadrature scheme over triangles is used for computingthe coefficient of the resulting linear finite element system.The effect of numerical integrationis studied.The corresponding discrete eigenvalue with linear finite elements is shown to admitasymptotic error expansions for certain classes of“uniform”meshes.Hence,the Richardsonextrapolation increases the accuracy of the scheme from second to fourth order.     

多维问题有限元逼近的误差展式及其应用 ——Ⅰ.特征值问题

对于多维区域Ω~R~X(N＞2),采用适当的单元剖分。给出Ω上特征值问题的有限元逼近的渐近误差展式,从而从理论上说明通过Richardson外推。可以将计算精度从二阶提高到四阶。     

有限元导数的渐近展式和外推

对区域施行某类一致剖分,关于线性元给出椭圆Ritz投影的微商的渐近展式,应用Richardson外推,证明可以提高线性有限元微商的计算精度。证明基于有限元技巧。     

二阶Hamilton系统在固定能量面上的周期解

一、引言 本文讨论下述二阶Hamilton系统在固定能面上的周期解的存在性问题:其中V∈C~2(Ω,R~1)是给定的位势函数,Ω(?)R~n是一开集,V′=grad V,h是一实数。 近来有许多文章研究所谓奇异Hamilton系统的固定能量周期解(参阅文献[1,2])。其实,系统(H1,2)有无周期解,似应只与V在给定的能量面上的性态有关,而与它在Ω上是否有奇性无重大关系。本文试图部分地回答这一猜测。令