NCKU Math Colloquium / RCTS Seminar DATE 2015-12-10　16:10-17:00 PLACE 數學館3174教室 SPEAKER 李明恭 教授（中華休閒系） TITLE The Null Field and the Interior Field Methods for Dirichlet and Mixed boundary problems of Laplace's Equations in Elliptic Domains with Elliptic Holes ABSTRACT Abstract： Recently, the null field method (NFM) is proposed by J.T. Chen with his groups (for simplicity, we only outline one paper of Chen in elliptic domain [1]). In NFM, the fundamental solutions (FS) with the field nodes Q outside of the solution domains are used in the Green formulas [2,3]. First, the FS is expanded by the infinite series. When the Fourier approximations of the boundary conditions on the boundaries are chosen, the explicit algebraic equations are derived, and the semi-analytic solutions can be found. In [2,3], the solution domain is circular domain with circular holes, but since elliptic domains also happen in many engineering problems, so in this talk, the NFM is developed for elliptic domains with elliptic holes [4,5]. Another issue is the implementation problem of NFM, the interior field method (IFM) is described, which is equivalent to the NFM when the field nodes are located exactly on the domain boundary. Moreover, the collocation Trefftz method (CTM) is also implemented by using the particular solutions in elliptic coordinates. The CTM is the simplest algorithm, has no risk of degenerate scales, and can be applied to non-elliptic domains. Numerical experiments are carried out for elliptic domains with one elliptic hole by the IFM, the NFM and the CTM. This talk also covers some mixed problems by the NFM and the IFM. By the inefficiency of NFM in some mixed problems, the CTM may be used for Robin problems [5] in elliptic domains. In summary, for Laplace's equation in elliptic domains, a comparative study of algorithms, errors, stability and numerical results is explored in this talk for three boundary methods: the NFM, the IFM and the CTM. The effective algorithm and easy of programming for the Dirichlet and Mixed problems of Laplace's equation on elliptic domains are the main goal of this talk.