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

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(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.