A D-N Alternating Method For Exterior Anisotropic Problems With Mixed Boundary Value Conditions

The elliptic boundary value problem in unbounded domains have a widely practical background,such as the electromagnetic wave radiation and diffraction.elastic mechanics,fracture mechanics,fluid mechanics etc.Many of these issues can be naturalized as the mathematical model of the partial differential equations in the unbounded domain.So,numerical solutions of unbounded region problems have important significance.Among them,the non-overlapping regional decomposition algorithm based on natural boundary reduction(Dirichlet-Neumann alternating method)is a popular method in recent years.In this paper,we study a D-N alternating method for exterior anisotropic problems with mixed boundary value conditions.The following is an introduction of this article:Chapter 1 firstly introduces the application background of the elliptic differential equation in the unbounded region,and some common methods of solving the problem and their advantages and disadvantages.Secondly,it introduces the previous research results.Finally,the mathematical theory of the D-N alternating method is introduced.Chapter 2 considers the exterior anisotropic problems with mixed boundary value conditions.Firstly,the original equation is transformed into harmonic equation by coordinate transformation.Secondly,an elliptical artificial boundary is introduced to enclose the original boundary,and divide the unbounded region into two non-overlapping sub-regions:the unbounded sub-region and the bounded sub-region.Then,the D-N alternating method of the problem described in this paper is constructed.Finally,the variational form,convergence and relaxation factor of the D-N alternating method are discussed.Chapter 3 discusses the D-N alternating method in the discrete case.The geometrical convergence of the D-N alternating method of the discrete case is proved based on the principle of curved finite element and natural boundary element,keeping the coordinated finite element.Chapter 4 makes several numerical experiments to obtain the factors of the convergence of the D-N alternating method,verify the correctness of the theoretical analysis and the feasibility of the method.