A stable algorithm for divergence-free radial basis functions in the flat limit

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

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 meshfr…

使用向量值有理逼近的平坦径向基函数稳定计算

Stable computations with flat radial basis functions using vector-valued rational approximations

本文提出基于向量值有理逼近的新算法RBF-RA，用于稳定计算平坦径向基函数，比Contour-Padé方法更准确，适用于各种光滑径向核和更广泛的任务。

A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations

A new projection method based on radial basis functions (RBFs) is presented for discretizing the incompressible unsteady Stokes equations in irregular geometries. The novelty of the method comes from the application of a new technique for computing the Leray-Helmholtz projection of a vector field using generalized interpolation with divergence-free and curl-free RBFs. Unlike traditional projection methods, this new method enables matching both tangential and normal components of divergence-free vector fields on the domain boundary. This allows incompressibility of the velocity field to be enfo…

一种基于径向基函数的方法用于计算霍普夫-霍奇分解

A Radial Basis Function Method for Computing Helmholtz-Hodge Decompositions

本文提出了一种基于矩阵值核的径向基函数方法，用于计算有界域上的两种向量分解，进而结合得到完整的霍普夫-霍奇分解，提供了误差估计和数值示例。

An analytically divergence-free collocation method for the incompressible Navier-Stokes equations on the rotating sphere

In this work, we develop a high-order collocation method using radial basis function (RBF) for the incompressible Navier-Stokes equation (NSE) on the rotating sphere. The method is based on solving the projection of the NSE on the space of divergence-free functions. For that, we use matrix valued kernel functions which allow an analytically divergence-free approximation of the velocity field. Using kernel functions which lead to rotation-free approximations, the pressure can be recovered by a simple kernel exchange in one of the occurring approximations, without solving an additional Poisson p…

An investigation of global radial basis function collocation methods applied to Helmholtz problems

Global radial basis function (RBF) collocation methods with inifinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential convergence properties for smooth solution functions. At the same time, the linear systems that arise are dense and severely ill-conditioned for large numbers of unknowns and small values of the shape parameter that determines how flat the basis functions are. We use Helmholtz equation as an application problem for the theoretical analysis and numerical experiments. We analyse and characterise the convergen…