Novel multilevel techniques for convergence acceleration in the solution of systems of equations arising from RBF-FD meshless discretizations

Abstract

The present paper develops two new techniques, namely additive correction multicloud (ACMC) and smoothed restriction multicloud (SRMC), for the efficient solution of systems of equations arising from Radial Basis Function-generated Finite Difference (RBF-FD) meshless discretizations of partial differential equations (PDEs). RBF-FD meshless methods employ arbitrary distributed nodes, without the need to generate a mesh, for the numerical solution of PDEs. The proposed techniques are specifically designed for the RBF-FD data structure and employ simple restriction and interpolation strategies in order to obtain a hierarchy of coarse-level node distributions and the corresponding correction equations.Both techniques are kept as simple as possible in terms of code implementation, which is an important feature of meshless methods. The techniques are verified on 2D and 3D Poisson equations, defined on non-trivial domains , showing very high benefits in terms of both time consumption and work to convergence when comparing the present techniques to the most common solver approaches. These benefits make the RBF-FD approach competitive with standard grid-based approaches when the number of nodes is very high, allowing large size problems to be tackled by the RBF-FD method.