An efficient implementation of the radial basis integral equation method

In this paper, we propose an efficient implementation of the radial basis integral equation method (RBIEM) that does not involve discretization of the circular subdomains. By avoiding discretization on the boundaries of the subdomains, a major source of error in the numerical scheme can be eliminated. The proposed implementation is tested on the Helmholtz equation with higher gradients in the exact solution. Three different radial basis functions are investigated, namely the augmented thin plate spline, r3 and r4log(r). The latter two functions are augmented with the second order global polynomial. Numerical results show that the new implementation of the RBIEM produces more accurate results and is more robust in handling problems with highly variable solutions. By avoiding the boundary discretization, the tasks of keeping track of the boundary elements and the boundary nodes are not needed, which can be a daunting task especially in three-dimensional problems with complicated geometries. The proposed implementation of the RBIEM is promising and the feasibility of the approach in three-dimensional problems is currently being investigated.