The Coupling Method With The Natural Boundary Reduction On An Ellipse For The Poisson Equation

In practical applications, many scientific and engineering computing problems, such as the radiation and diffraction of electromagnetic waves, the fluid flow around obstacles, etc., often come down to boundary value problems of partial differential equations in unbounded domains. How to find the numerical solution of such problems has always been the focus of attention. The easiest way is to ignore the unbounded region, but this will cause the accuracy too low or the computational cost too high. In recent years, a lot of more effective methods have been created, such as infinite element method, spectral method, perfectly matched layer method, boundary element method, domain decomposition method, coupling method and so on. Because the natural boundary reduction maintained entirely some basic characteristics of the original elliptic boundary value problems, and natural boundary element method and the finite element method based on the same variational principle. Therefore, the coupling between natural boundary element method and classical finite element is very natural and direct, and can be included into finite element analysis system simply. Moreover, this method not only can overcome the limitations of the region, but also can make the classical finite element method applicable on unbounded domains and the crack regions It has more advantages than other types of coupling method.It is usually to make a circular or spherical artificial boundary in the coupling method for solving unbounded problems before. But for exterior problem with a long bar as the inside boundary, it is obviously not the best choice. It will lead to a large number of calculations, not even with satisfactory results. However using an ellipse or ellipsoid boundary can greatly reduce the computational domain, and reduce the calculation amount to get better resultsThis paper investigates the coupling method of the finite element and the natural boundary element using an elliptic artificial boundary for solving the Poisson equation and exterior anisotropic problems, and obtains a new error estimate that depends on the mesh size, the location of the elliptic artificial boundary, the number of terms after truncating from the infinite series in the integral. Numerical examples are presented to demonstrate the effectiveness and the properties of this method.