Meshless moment-free quadrature formulas arising from numerical differentiation

TL;DR

This paper introduces a meshless, moment-free quadrature method that computes high-order weights for integrals over complex domains without meshing, using boundary value problem discretization and sparse linear systems.

Contribution

The proposed method generates high-order quadrature weights without meshing or moment calculations, applicable to complex 2D and 3D domains, and is competitive with existing schemes.

Findings

Demonstrates robustness and high accuracy on various smooth and piecewise smooth domains.

Effective in 2D and 3D, including domains with reentrant corners and edges.

Competitive with state-of-the-art open source quadrature packages.

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.