广义差分法及其应用

综合报道了关于广义差分法的研究。简要概述了作者及国内同行近十几年来在广义差分法方面的主要成果，包括了椭圆、双曲和抛物方程广义差分格式的构造，误差的Ｓｏｂｏｌｅｖ模估计和超收敛性，以及广义差分法在电磁场、计算流体等领域的应用。

A Survey on the Generalized Defference Method and Its Application

In 1978, the author established a class of so-called generalized difference methods(abbrev.as GDM)on the basis of the Petrov-Galerkin form,in which we took the finite element spaces as triedspaces and the span of the common terms of the local Taylor expansions on the dual subdivisions asthe test space. In the past teens of years, the author and his colleagues didextensive and deep re-searchs on the theory and application of GDM,including constructing first order or higher order dif-ference schemes on elliptic,parabolic and hyperbolic equations, establishing the optimal Sobolev normestimates of errors,and applying GDM to underground fluid, electromagnetic field and other fields.Theoretical researches and realistic computations show that GDM not only keeps the computationalsimplicity of difference methods, bue also enjoys the accuracy of finite element methods. This paper isa summary of the results which we have obtained.In section 2 we describe GDM for elliptic equa-tions. In section 3 we state some basical results due to the author and etc, In sections 4 and 5, we givethe upwind GDM for the first order hyperbolic system and convection-dominated diffusion equationsrespectively. Finally, in section 6, a general outline of the appliation to GDM is provided.