A Cartesian-grid integrated-RBF Galerkin technique

This paper describes a high-order Galerkin technique, which is based on indirect/integrated radial-basis-function networks (IRBFNs) and Cartesian grids, for the discretisation of elliptic problems in two dimensions. The field variable is approximated by high-order IRBFNs that can work on uniform grids without suffering from Runge’s phenomenon. Unlike conventional Galerkin techniques, derivative boundary values are incorporated into the approximations and their imposition is conducted in an exact manner. The Galerkin formulation is then applied to force IRBFNs to satisfy the governing equation. The present technique is verified numerically through the solution natural convection in a square slot - a benchmark problem in CFD. Highly accurate solutions are obtained using relatively coarse grids, which show the effectiveness of using RBFs as trial functions in the Galerkin formulation