SHARP QUADRATURE ERROR BOUNDS FOR THE NEAREST-NEIGHBOR DISCRETIZATION OF THE REGULARIZED STOKESLET BOUNDARY INTEGRAL EQUATION
SHARP QUADRATURE ERROR BOUNDS FOR THE NEAREST-NEIGHBOR DISCRETIZATION OF THE REGULARIZED STOKESLET BOUNDARY INTEGRAL EQUATION
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DOI：
10.1137/18m1191816
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发表时间：
2019-01-01
影响因子：
3.1
通讯作者：
Smith, David J.
中科院分区：
文献类型：
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作者：
Gallagher, Meurig T.;Choudhuri, Debajyoti;Smith, David J.
正则斯托克小波法是求解生物流体力学问题中斯托克流方程的一种强有力的数值方法。该方法的最新变体包含了最近邻离散化，以提高精度和效率，同时保持原始无网格方法的易于实现。这种新方法包含三个数值误差来源：与使用正则化形式的边界积分方程（参数为epsilon）有关的正则化误差，以及与力和正交离散化(长度尺度为h（f)和h(q)）有关的两个离散化误差来源。需要解决的一个关键问题是正交误差；最初的工作并没有完全解释观测到的数值收敛现象。在本文中，我们为最近邻离散构造了尖锐的正交误差界，注意到核的单个评估的误差取决于这些离散集之间的最小距离（delta）。描述了两种情况下的正交误差界限：在h(q)上接近线性且对epsilon不敏感的不相交集（delta > 0），以及在h(q)上二次且与epsilon相反的包含集（delta = 0）。结合最近邻方法的矩阵系统的条件数，讨论了这些误差界的实际意义，分析表明，对于不相交集，条件数对不相交集不敏感，对于包含集，条件数与ε线性增长。一般情况下的误差界限（δ >= 0）与每种情况下的误差总和成正比。
The method of regularized stokeslets is a powerful numerical approach to solve the Stokes flow equations for problems in biological fluid mechanics. A recent variation of this method incorporates a nearest-neighbor discretization to improve accuracy and efficiency while maintaining the ease of implementation of the original meshless method. This new method contains three sources of numerical error: the regularization error associated with using the regularized form of the boundary integral equations (with parameter epsilon), and two sources of discretization error associated with the force and quadrature discretizations (with lengthscales h(f) and h(q)). A key issue to address is the quadrature error; initial work has not fully explained observed numerical convergence phenomena. In the present manuscript we construct sharp quadrature error bounds for the nearest-neighbor discretization, noting that the error for a single evaluation of the kernel depends on the smallest distance (delta) between these discretization sets. The quadrature error bounds are described for two cases: disjoint sets (delta > 0) that are close to linear in h(q) and insensitive to epsilon, and contained sets (delta = 0) that are quadratic in h(q) with inverse dependence on epsilon. The practical implications of these error bounds are discussed in reference to the condition number of the matrix system for the nearest-neighbor method, with the analysis revealing that the condition number is insensitive to epsilon for disjoint sets, and grows linearly with epsilon for contained sets. Error bounds for the general case (delta >= 0) are revealed to be proportional to the sum of the errors for each case.