Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The scheme is shown to be convergent to both nonlocal diffusion and its corresponding local limit as nonlocal interaction vanishes. The analysis is carried out on a special family of rectilinear Cartesian grids for linear RK method with designed kernel support. The key idea for the stability of the RK collocation scheme is to compare the collocation scheme with the standard Galerkin scheme which is stable. In addition, there is a large computational cost for assembling the stiffness matrix of the nonlocal problem because high order Gaussian quadrature is usually needed to evaluate the integral. We thus provide a remedy to the problem by introducing a quasi-discrete nonlocal diffusion operator for which no numerical quadrature is further needed after applying the RK collocation scheme. The quasi-discrete nonlocal diffusion operator combined with RK collocation is shown to be convergent to the correct local diffusion problem by taking the limits of nonlocal interaction and spatial resolution simultaneously. The theoretical results are then validated with numerical experiments. We additionally illustrate a connection between the proposed technique and an existing optimization based approach based on generalized moving least squares (GMLS).

中文翻译：

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非局部扩散的渐近兼容再生核搭配和无网格集成

再现核 (RK) 近似是一种无网格方法，可从散布数据集构造形状函数。我们为具有 Dirichlet 边界条件的非局部扩散模型提出了一种渐近兼容 (AC) RK 搭配方法。当非局部相互作用消失时，该方案被证明收敛于非局部扩散及其相应的局部极限。该分析是在具有设计内核支持的线性 RK 方法的特殊直线笛卡尔网格族上进行的。RK搭配方案稳定性的关键思想是将搭配方案与稳定的标准Galerkin方案进行比较。此外，组装非局部问题的刚度矩阵需要大量的计算成本，因为通常需要高阶高斯求积来评估积分。因此，我们通过引入准离散非局部扩散算子来解决该问题，在应用 RK 搭配方案后，该算子不再需要数值求积。通过同时考虑非局部相互作用和空间分辨率的限制，表明准离散非局部扩散算子与 RK 搭配相结合，可以收敛到正确的局部扩散问题。然后通过数值实验验证理论结果。我们还说明了所提出的技术与基于广义移动最小二乘法 (GMLS) 的现有优化方法之间的联系。通过同时考虑非局部相互作用和空间分辨率的限制，表明准离散非局部扩散算子与 RK 搭配相结合，可以收敛到正确的局部扩散问题。然后通过数值实验验证理论结果。我们还说明了所提出的技术与基于广义移动最小二乘法 (GMLS) 的现有优化方法之间的联系。通过同时考虑非局部相互作用和空间分辨率的限制，表明准离散非局部扩散算子与 RK 搭配相结合，可以收敛到正确的局部扩散问题。然后通过数值实验验证理论结果。我们还说明了所提出的技术与基于广义移动最小二乘法 (GMLS) 的现有优化方法之间的联系。