On solving the direct/inverse cauchy problems of laplace equation in a multiply connected domain, using the generalized multiple-source-point boundary-collocation Trefftz method & characteristic lengths

In this paper, a multiple-source-point boundary-collocation Trefftz method, with characteristic lengths being introduced in the basis functions, is proposed to solve the direct, as well as inverse Cauchy problems of the Laplace equation for a multiply connected domain. When a multiply connected domain with genus p (p>1) is considered, the conventional Trefftz method (T-Trefftz method) will fail since it allows only one source point, but the representation of solution using only one source point is impossible. We propose to relax this constraint by allowing many source points in the formulation. To set up a complete set of basis functions, we use the addition theorem of Bird and Steele (1992), to discuss how to correctly set up linearly-independent basis functions for each source point. In addition, we clearly explain the reason why using only one source point will fail, from a theoretical point of view, along with a numerical example. Several direct problems and inverse Cauchy problems are solved to check the validity of the proposed method. It is found that the present method can deal with both direct and inverse problems successfully. For inverse problems, the present method does not need to use any regularization technique, or the truncated singular value decomposition at all, since the use of a characteristic length can significantly reduce the ill-posed behavior. Here, the proposed method can be viewed as a general Trefftz method, since the conventional Trefftz method (T-Trefftz method) and the method of fundamental solutions (F-Trefftz method) can be considered as special cases of the presently proposed method.