Meshless moment-free quadrature formulas arising from numerical differentiation

Abstract

We suggest a method for simultaneously generating high order quadrature weights for integrals over Lipschitz domains and their boundaries that requires neither meshing nor moment computation. The weights are determined on pre-defined scattered nodes as a minimum norm solution of a sparse underdetermined linear system arising from a discretization of a suitable boundary value problem by either collocation or meshless finite differences. The method is easy to implement independently of the domain's representation, since it only requires as inputs the position of all quadrature nodes and the direction of outward-pointing normals at each node belonging to the boundary. Numerical experiments demonstrate the robustness and high accuracy of the method on a number of smooth and piecewise smooth domains in 2D and 3D, including some with reentrant corners and edges. Comparison with quadrature schemes provided by the state-of-the-art open source packages Gmsh and MFEM shows that the new method is competitive in terms of accuracy for a given number of nodes.

References in corpus (7)

• QMC designs: optimal order Quasi Monte Carlo Integration schemes on the sphere
• On generation of node distributions for meshless PDE discretizations
• A Meshfree Generalized Finite Difference Method for Surface PDEs
• Isogeometric Analysis on V-reps: first results
• Stable high-order cubature formulas for experimental data
• Fast variable density node generation on parametric surfaces with application to mesh-free methods
• Error Bounds for a Least Squares Meshless Finite Difference Method on Closed Manifolds