Finite Difference Method and Finite Element Method on Manifolds, and Their Applications

(1)黎曼曲面上的有限元逼近方法。本文的方法符合黎曼曲面的抽象定义，不仅对黎曼曲面的情形，而且对平面区域的情形，都将提供一种新的数值计算方法和很高的实用性。本文方法的一个特点是对黎曼曲面的每个临界点采用参数圆盘上的有限元逼近，从而得到高精度的逼近。(2)用有限单元法确定四边形的模量。本文建立了一种方法，利用这种方法可以得到复平面上四边形的模的一个较好的近似。我们的方法的一个特点是，在边界上每个临界点的邻域上，采用与（1）相同的方法。

(1) A method of finite element approximations on a Riemann surface. Our method matches the abstruct definition of a Riemann surface,and also will offer a new technique and high utility in numerical calculation not only for the case of Riemann surfaces but also for the case of plane domains. It is a peculiarity of our method that by means of adopting a finite element approximation on a parametric disk of each critical point of a Riemann surface, approximations of high accuracy is obtained.(2) Determination of the modulus of quadrilaterals by finite element methods. We establish a method by which a fairly good approximation of the modulus of quadrilaterals on the complex plane is obtained. It is a peculiarity of our method that on a neighborhood of each critical point on the boundary, the same method as (1) is adopted.