Boundary methods for mixed boundary problems of Laplace׳s equation in elliptic domains with elliptic holes

The particular solutions (PS) and fundamental solutions (FS) in polar coordinates can be found in many textbooks, but with much less coverage in elliptic coordinates (Chen et al., 2010 [5], Chen et al., 2012 [6], Morse and Feshbach, 1953 [20], Li et al., 2015 [18]). Since the elliptic domains with elliptic holes may be found in some engineering problems, the PS and the FS expansions in elliptic coordinates are essential for numerical computations. For Dirichlet problems of Laplace׳s equation in elliptic domains, the null field method (NFM), the interior field method (IFM) and the collocation Trefftz method (CTM) are reported in [18]. There seems to exist few reports for mixed problems, where the Dirichlet and Neumann conditions are assigned on the exterior and the interior boundaries, simultaneously. This paper is devoted to such mixed problems by the NFM and the IFM, and the explicit algebraic equations are derived for elliptic domains. Besides, other effective particular solutions (PS) are sought, and the collocation Trefftz method (CTM) [16] is employed. The CTM may be used for Robin problems in elliptic domains. The effective algorithms for the mixed problems of Laplace׳s equation on elliptic domains are the main goal of this paper. The techniques of the mixed techniques in this paper can be applied to Dirichlet problems, the dual techniques are called in Chen and Hong (1999 [4]), Hong and Chen (1988 [8]), and Portela et al. (1992 [21]). A preliminary study for the dual techniques is one goal of this paper.