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.

翻译：我们提出一种方法，能够同时生成Lipschitz域及其边界上积分的高阶求积权值，该方法既不需要网格剖分，也不涉及矩量计算。权值在预先定义的散乱节点上确定，通过配置法或无网格有限差分法离散合适的边值问题，得到一个稀疏欠定线性系统，权值即为该系统的最小范数解。该方法实现简便且与域的表征方式无关，仅需输入所有求积节点的位置以及边界上各节点处外法向的方向。数值实验证明了该方法在二维和三维中多个光滑及分段光滑域（包括一些具有凹角和凹边的域）上的鲁棒性与高精度。