Error Bounds for a Least Squares Meshless Finite Difference Method on Closed Manifolds

Abstract

We present an error bound for a least squares version of the kernel based meshless finite difference method for elliptic differential equations on smooth compact manifolds of arbitrary dimension without boundary. In particular, we obtain sufficient conditions for the convergence of this method. Numerical examples are provided for the equation $-{\Delta }_{\mathcal{M}}u+u=f$ on the 2- and 3-spheres, where ${\Delta }_{\mathcal{M}}$ is the Laplace-Beltrami operator.

References in corpus (8)

• A Meshfree Generalized Finite Difference Method for Surface PDEs
• An RBF-FD closest point method for solving PDEs on surfaces
• Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant
• The Overlapped Radial Basis Function-Finite Difference (RBF-FD) Method: A Generalization of RBF-FD
• Minimal positive stencils in meshfree finite difference methods for the Poisson equation
• Improved Stencil Selection for Meshless Finite Difference Methods in 3D
• Approximation with Conditionally Positive Definite Kernels on Deficient Sets
• Selection of Sparse Sets of Influence for Meshless Finite Difference Methods

Cited by in corpus (3)

• An unfitted RBF-FD method in a least-squares setting for elliptic PDEs on complex geometries
• Meshless moment-free quadrature formulas arising from numerical differentiation
• Overlap Splines and Meshless Finite Difference Methods