A Partition of Unity Method for Divergence-free or Curl-free Radial Basis Function Approximation

TL;DR

This paper introduces a partition of unity approach for efficiently constructing divergence-free or curl-free vector field approximations using radial basis functions, ensuring physical constraints are preserved in applications like fluid dynamics and electromagnetism.

Contribution

It develops a local approximation method combining div/curl-free RBFs within a partition of unity framework, reducing computational cost while maintaining divergence-free or curl-free properties.

Findings

The method effectively approximates divergence-free and curl-free fields in 2D and on surfaces.

It provides error estimates demonstrating accuracy of the local approximants.

The approach is applicable to scalar potentials and higher-dimensional vector fields.

Abstract

Divergence-free (div-free) and curl-free vector fields are pervasive in many areas of science and engineering, from fluid dynamics to electromagnetism. A common problem that arises in applications is that of constructing smooth approximants to these vector fields and/or their potentials based only on discrete samples. Additionally, it is often necessary that the vector approximants preserve the div-free or curl-free properties of the field to maintain certain physical constraints. Div/curl-free radial basis functions (RBFs) are a particularly good choice for this application as they are meshfree and analytically satisfy the div-free or curl-free property. However, this method can be computationally expensive due to its global nature. In this paper, we develop a technique for bypassing this issue that combines div/curl-free RBFs in a partition of unity framework, where one solves for local approximants over subsets of the global samples and then blends them together to form a div-free or curl-free global approximant. The method is applicable to div/curl-free vector fields in $\R^2$ and tangential fields on two-dimensional surfaces, such as the sphere, and the curl-free method can be generalized to vector fields in $\R^d$. The method also produces an approximant for the scalar potential of the underlying sampled field. We present error estimates and demonstrate the effectiveness of the method on several test problems.