p-refined RBF-FD solution of a Poisson problem

TL;DR

This paper investigates p-refined RBF-FD methods with polyharmonic splines for solving Poisson problems, analyzing convergence and performance improvements from spatially variable approximation orders.

Contribution

It introduces and analyzes p-refined RBF-FD methods with monomials, demonstrating their convergence properties and computational performance for Poisson problems.

Findings

Higher convergence rates with p-refinement.

Improved numerical performance on Poisson problems.

Analysis of convergence orders and execution times.

Abstract

Local meshless methods obtain higher convergence rates when RBF approximations are augmented with monomials up to a given order. If the order of the approximation method is spatially variable, the numerical solution is said to be p-refined. In this work, we employ RBF-FD approximation method with polyharmonic splines augmented with monomials and study the numerical properties of p-refined solutions, such as convergence orders and execution time. To fully exploit the refinement advantages, the numerical performance is studied on a Poisson problem with a strong source within the domain.