Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion
Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion
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DOI：
10.1137/19m1277801
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发表时间：
2019-07
期刊：
影响因子：
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通讯作者：
Y. Leng;Xiaochuan Tian;Nathaniel Trask;J. Foster
中科院分区：
文献类型：
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作者：
Y. Leng;Xiaochuan Tian;Nathaniel Trask;J. Foster
再生核（RK）近似是无网格方法，从分散的数据集构造形状函数。对Dirichlet边界条件下的非局部扩散模型，提出了一种渐近相容的RK配置方法.当非局部相互作用消失时，该格式收敛于非局部扩散及其相应的局部极限。在一类特殊的直角坐标网格上对线性RK方法进行了分析。RK配置格式稳定性的关键思想是将其与稳定的标准Galerkin格式进行比较。另外，由于通常需要高阶高斯求积来计算非局部问题的刚度矩阵，因此计算量很大。因此，我们提供了一个补救的问题，通过引入一个准离散的非局部扩散算子，没有数值求积后，进一步应用RK配置方案。通过同时考虑非局部相互作用的极限和空间分辨率，证明了拟离散的非局部扩散算子与RK配置相结合是收敛于正确的局部扩散问题的.最后通过数值实验对理论结果进行了验证。我们还说明了所提出的技术和现有的基于广义移动最小二乘（GMLS）的优化方法之间的连接。
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).