Comparison of meshfree and mesh-based methods for boundary value problems in physics

Abstract

Meshfree methods using basis expansion are common in computational physics, but their efficiency and accuracy depend on the quality of basis sets. We describe the application of radial basis functions (RBF), a class of basis functions attracting considerable interest and showing promising results. In the RBF approach, the solutions are interpolated over a chosen set of RBFs scattered over the domain. The value of a given RBF depends on the distance to the center, in contrast to the usual approach where the basis functions depend on the position. The advantage of the RBF method is that it can be efficient, simple to implement, and easily adaptable to irregular domains or higher dimensions. We study the application of this meshfree method to boundary value problems, including electrostatics and quantum systems. We show that the method is robust and flexible, and can be a compelling alternative to mesh-based methods.