Application of cardinal radial basis interpolation operators to numerical solution of the Poisson equation

Abstract

We consider the application of a new scattered data approximation scheme to numerically solving the Dirichlet problem for the Poisson equation. This collocation method, which is mesh-free and substantially independent on the space dimension, makes use of interpolation operators with cardinal radial basis and differs from the well-known discretization approach introduced by E. J. Kansa in 1990 and then extensively developed, based on Hardy’s multiquadrics or others radial basis functions. In our method the discretization matrix, whose dimension equals the number of internal points in the domain, is symmetric and strictly diagonally dominant, so that the discrete problem is well-posed and also well-conditioned, since the matrix condition number is small. Numerical experiments show that the performance of our method is comparable in many cases with that of Kansa’s method; moreover, the former works well even if the number of collocation points is large.

Keywords

• elliptic partial differential equations
• approximation scheme
• scattered data
• cardinal radial basis