Outside The Poisson Equation Problem Based On Elliptic Natural Boundary Domestication And Curved Edge Finite Element Coupling Method

In the computation of science and engineering, people often encounter on solving the problem of partial differential equations on unbounded domain, great difficulties are often meet for unbounded the regions, in recent decades, more and more mathematicians and engineers pay attention to this problem. In the study of solution, people produced many effective methods, the artificial boundary method is one of the effective and important methods. It converts the exterior problem to an equivalent problem with the appropriate artificial boundary condition in a bounded domain. Natural boundary element method is an effective method in all of the artificial boundary methods, it not only has the advantages of general boundary element method, and the symmetric positive definiteness of the stiffness matrix to maintain the original problem, and cyclicity or block cyclicity, so it makes the calculation greatly reduced, thus it also has more merits on the calculation. Because the natural boundary element method and finite element method are based on the same variational principle, the coupling between them is very natural and direct, and the total stiffness matrix is exactly that the natural boundary element method stiffness matrix adding the finite element method stiffness matrix, it is simply than the other coupling method. On the entirety artificial boundary condition, in practical calculation for a point value depends on the whole value of a function or its entire derivative boundary value, so needs large calculation and storage space, but the local boundary condition at a point value depends only on the function or its derivative at a point value, so the local artificial boundary condition has a certain superiority.When we use the coupling method for solving differential equations boundary value problems, we often select the circle or sphere as artificial boundary, but if the external area is a strip type domain, it will leads to amount of redundant computation. If we select an ellipse elliptic or ellipsoid as artificial boundary, it will greatly reduce the calculation. In addition, in the bounded region using finite element method, in generally, the mesh that people adoption is triangular or quadrilateral element, which is often difficult to achieve high accuracy. In order to overcome this shortcoming, we can adopt the curved finite element instead of the straight edge finite element mesh.Therefore, in this paper we take the exterior problem of the Poisson equation for example, firstly based on the natural boundary reduction, we discuss the coupling method with the elliptic artificial boundary, then use the curved finite element instead of the triangle element, give the error estimate of the numerical solution, especially when the original boundary is elliptical itself, give the error estimate with the approximate boundary condition. In the end of this article we will also discuss the use of local artificial boundary conditions to slove the exterior problem of the Poisson equation the exterior Poisson equation problem, not only give the error estimate of the numerical solution, but also give the relationship between the error estimate and the mesh size of a partition of the finite domain, the location of the artificial boundary, the approximation of artificial boundary condition。...