The method of transformed angular basis function for solving the Laplace equation

In this paper, we propose a new approach to improve the method of angular basis function (MABF) proposed by Young et al. (2015) for the Laplace equation in two-dimensional settings. Instead of the fundamental solution ln r used in the traditional Method of Fundamental Solution (MFS), MABF employs a different basis function θ and produces good approximate solutions on the domains with acute, narrow regions and exterior problems (Young et al., 2015). However, the definition of θ inevitably incurs a singularity situation for many different types of domains. Therefore, the selection of source points of MABF is not as convenient as the traditional MFS. To avoid the singularity situation in implementing, we introduce a transformation so that the transformed angular basis function does not exhibit this type of singularity for commonly used distributions of source points. As a result, source points for the method of transformed angular basis function (MTABF) can then be chosen in a similar way to traditional MFS. Numerical experiments demonstrate that the proposed approach significantly simplifies the selection of source points in MABF for different types of domains, which makes MABF more applicable. Numerical results of MTABF and MFS are presented for comparison purposes.