Domain Decomposition Methods Based on the Natural Boundary Reduction for the Anisotropic Elliptic Boundary Value Problem in the Exterior Domain
|School||North China University of|
|Keywords||unbounded domain anisotropic elliptic boundary value problem overlapping domain decomposition method non-overlapping domain decomposition method natural boundary reduction|
In the computation of science and engineering,such as the exploration of oil and gas,the design of spacecrafts and large structure engineering,and the weather forecast,along with the development of parallel computation,more and more attention is paid on the domain decomposition method.However,only applying the domain decomposition method is not enough for solving the elliptic boundary value problem in unbounded domain,because after joining the artificial boundary,there still is an infinite domain at least.This problem can be solved by applying boundary reduction.We usually use the coupling method of the finite element and boundary element to solve the problem in unbounded domain,or use an appropriate artificial boundary and the approximate boundary condition,and then apply finite element in bounded domain.In recent years,overlapping and non-overlapping domain decomposition methods based on natural boundary reduction in the unbounded domain were supposed,i.e.decompose the infinite domainΩinto a finite domainΩand a typical infinite domainΩ2,and solve the problem onΩ1 andΩ2 alternatively.A small scale problem inΩ1 is solved by the finite element method, and only simple computation on the typical boundary ofΩ2 is needed by natural boundary element method.This method can decrease the scale of computation and can also apply parallel computation.This method chose the circle or ball as artificial boundaries in the early time. But for the problem with long and thin strip boundary,it isn’t the best choice obviously to use a circle or ball as an artificial boundary;it will cause the computation too large to get a satisfactory result.According to the above situation,this paper adopts elliptic artificial boundary after coordinate transformation,and with natural boundary reduction for the foundation puts forward overlapping and non-overlapping domain decomposition methods for solving anisotropic elliptic problem. About overlapping domain decomposition method,the domain infinity outside the elongated boundary is resolved and the difficulty of the small coefficient is overcome by introducing elliptic artificial boundaries.Its geometric convergence is proved in the sense of‖·‖1 by using the projection theory.An optimal convergence rate formula dependent upon the overlapping degree of subdomains,the lowest frequency of the exact solution and the anisotropic coefficients is obtained by using Fourier analysis.The numerical example confirms the convergence theory and shows the practical application.A non-overlapping domain decomposition algorithm which is based on the elliptic arterial boundary for solving the two dimensional harmonic problems over unbounded domain is offered and iterative convergence of its dissected form is discussed.Numerical results show that this method is very efficient for exterior two-dimensional harmonic.