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首页> 外文期刊>Engineering analysis with boundary elements >The modified localized method of approximated particular solutions for solving elliptic equations with mixed boundary conditions on scattered data
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The modified localized method of approximated particular solutions for solving elliptic equations with mixed boundary conditions on scattered data

机译：散乱数据混合边界条件椭圆方程的近似特殊解的改进局部化方法

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摘要

The localized method of approximated particular solutions (LMAPS) was first introduced in 2011 [31]. The method is then modified by employing integrated polyharmonic splines with polynomial basis. However, the current reported results on LMAPS still limited to the evenly distributed data points and Dirichlet boundary conditions. On the other hand, the traditional point-wise moving least square method is improved by piece-wise moving least squares (PMLS) in [19] for scattered data approximation. The paper proved that the PMLS is is an optimal design for data approximation. In this paper, the modified LMAPS is further improved by involving the Hermite-type PMTS to construct shape functions. The improved LMAPS is called piece-wise smoothed LMAPS (PS-LMAPS). Together with the original LMAPS, PS-LMAPS is used to solve elliptic partial differential equations with Dirichlet and Neumann mixed boundary conditions on the scattered data points. Performance of PS-LMAPS in comparison with LMAPS is tested on two Poisson equation with mixed boundary conditions using evenly-spaced nodes and scattered nodes, modified Helmholtz equations and a non-smooth problem. Particularly, PS-LMAPS can avoid some of the ill-conditioning issues of the system as shown in Example 3. The computational complexity ratio and relative error ratio indicate that the PS-LMAPS is much more efficient than that original LMAPS. The conclusion is supported by theoretical analysis of computational complexity and numerical experiments on the error analysis.
机译：近似特定解的局部化方法（LMAPS）于2011年首次提出[31]。然后通过采用具有多项式的积分多调和样条对方法进行修改。但是，当前报告的关于LMAPS的结果仍然限于均匀分布的数据点和Dirichlet边界条件。另一方面，传统的逐点移动最小二乘法在[19]中通过分段移动最小二乘（PMLS）进行了改进，用于离散数据逼近。本文证明了PMLS是用于数据逼近的最佳设计。在本文中，通过使用Hermite型PMTS构造形状函数，对改进的LMAPS进行了进一步改进。改进的LMAPS称为分段平滑LMAPS（PS-LMAPS）。与原始LMAPS一起，PS-LMAPS用于在分散的数据点上用Dirichlet和Neumann混合边界条件求解椭圆型偏微分方程。在带有混合边界条件的两个Poisson方程上使用均匀间隔的节点和分散的节点，修改后的Helmholtz方程和一个非光滑问题，对PS-LMAPS与LMAPS的性能进行了测试。特别是，PS-LMAPS可以避免如示例3所示的系统的某些不适情况。计算复杂度比和相对误差比表明PS-LMAPS比原始LMAPS效率更高。结论得到了计算复杂度的理论分析和误差分析的数值实验的支持。

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  • 来源
    《Engineering analysis with boundary elements》 |2019年第3期|164-174|共11页
  • 作者

    Li Wen; Yao Guangming; Niu Jing;

  • 作者单位

    Taiyuan Univ Technol, Coll Big Data Sci, Taiyuan 030024, Peoples R China|Clarkson Univ, Dept Math, 8 Clarkson Ave, Potsdam, NY 13699 USA;

    Clarkson Univ, Dept Math, 8 Clarkson Ave, Potsdam, NY 13699 USA;

    Harbin Normal Univ, Sch Math & Sci, Harbin 150025, Heilongjiang, Peoples R China;

  • 收录信息 美国《科学引文索引》(SCI);美国《工程索引》(EI);
  • 原文格式 PDF
  • 正文语种 eng
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