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重，该方法既不需要网格划分，也不需要进行矩计算。权重在预定义的散乱节点上确定，作为通过配点法或无网格有限差分法离散化适当边值问题所产生的稀疏欠定线性系统的最小范数解。该方法易于实现，且独立于域的表示方式，因为它仅需输入所有求积节点的位置以及属于边界的每个节点处的外法线方向。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）上的鲁棒性和高精度。