An efficient RBF-FD method using polyharmonic splines alongside polynomials for the numerical solution of two-dimensional PDEs held on irregular domains and subject to Dirichlet and Robin boundary conditions

In the present paper, the relatively new method of Radial Basis Function-Generated Finite Difference (RBF-FD) is used to solve a class of Partial Differential Equations (PDEs) with Dirichlet and Robin boundary conditions. For this approximation, Polyharmonic Splines (PHS) are used alongside Polynomials. This combination has many benefits. On the other hand, Polyharmonic Splines have no shape parameter and therefore relieve us of the hassle of calculating the optimal shape parameter. As the first problem, a two-dimensional Poisson equation with the Dirichlet boundary condition is investigated in various domains. Then, an elliptic PDE with the Robin boundary condition is solved by the proposed method. The results of numerical studies indicate the excellent efficiency, accuracy and high speed of the method, while for these studies, very fluctuating and special test functions have been used.