The MFS and MAFS for solving Laplace and biharmonic equations

The method of fundamental solutions (MFS) has been known as an effective boundary meshless method for solving homogeneous differential equations with smooth boundary conditions and boundary shapes. Despite many attractive features of the MFS, the determination of the source location and the boundaries with sharp corners still pose a certain degree of challenges. In this paper, we revisit another powerful boundary method, the method of approximate fundamental solutions (MAFS), which approximates the fundamental solution using trigonometric functions. In the MAFS, the fundamental solutions for various governed equations can be easily constructed. The placement of the source points is also simple. In this paper, we will apply the MAFS for solving the Laplace equation with non-harmonic boundary conditions and the biharmonic equation with non-biharmonic boundary conditions with highly irregular or non-smooth domains. We will compare the performance of the MAFS and the MFS in these types of problems.