A Collocation Finite Element Method For Parabolic Problems In Irregular Domains

The collocation method is a numerical method which has developed for several decades. It is a method which search for the approximate solutions of the operator functions by satisfying pure interpolation condition. The collocation method has many advantages , for example, it needn't calculate numerical integral; it can form the appproximate equation easily; its compute is simple and convenient, and it has high-order accuracy. Hence, the collocation is widely used for solving elliptie equations , prarbolic equations and hyperbolic equations. But most study was ristricted in regular domains.In this paper, a collocation finite element method for solving certain initial boundary value problems in irregular domains. The method combines the simplicity of orthogonal collocation with the versatility of deformable finite elements. Bicubic Hermite elements with four degrees-of-freedom per node are used. Analysis of this paper show that by taking advantage of the boundary conditions, a minimum number of collocation points can be used.This article is divided into two chapters.Chapter 1 introduces a potentially powerful numerical method for solving certain boundary value problems which is developed in [1]. The method combines the simplicity of orthogonal collocation with the versatility of deformable finite elements. Bicubic Hermite elements with four degrees-of-freedom per node are used. A sub- parametric transformation permits the precise positioning of the collocation points for maximum accuracy as well as a unique representation of irregular boundaries.This chapter is devided into three sections. The first section is introduction, which introduces the general situation of the collocation method simply. The second section gives the introduction of the arithmetic. This method sets up a collocation finite method for potential problems in irregular domains: and gives the compare with the Galerkin finite method in the computational efficiency . The third section concludes the method simply.Chapter 2 applies the method to a parabolic equation, and gives the corresponding collocation method of certain heat exchanging problem in irregular domains.This chapter is devided into four sections. The first section is the introductions which introduces the development of the collocation method for parabolic problems simply, and it also gives the main problems of this article as well. The second section gives a heat exchanging problem in sector-cirque-shaped domain:Using the method introduced in the Chapter 1 , we gain the collocation method . The third section extends the method of this article, and indicates that the method has very important value; The fourth section gives the shortage of this paper.