An Adaptive Method of Fundamental Solutions for Solving the Laplace Equation
Mathematics and Natural Sciences
In this paper, we propose a residual-type adaptive method of fundamental solutions (AMFS) for solving the two-dimensional Laplace equation. An error estimator is defined only on the boundary of the domain. Initial distributions of source points and collocation points are determined by using approaches proposed in Chen et al. (2006). The adding, removing, and stopping strategies are designed so that the required accuracy can be satisfied within finite steps. Numerical experiments reveal that AMFS improves the accuracy of the MFS approximation obtained from uniformly distributed sources and collocation points, which makes the MFS more practical for non-harmonic and non-smooth boundary conditions. Moreover, it is shown that the error estimator becomes equidistributed after an adaptive iteration. A detailed comparison between AMFS and MFS using uniformly distributed points is also presented for each numerical example.
Computers & Mathematics With Applications
(2019). An Adaptive Method of Fundamental Solutions for Solving the Laplace Equation. Computers & Mathematics With Applications, 77(7), 1828-1840.
Available at: https://aquila.usm.edu/fac_pubs/15996