RBF Multiscale Collocation for Second Order Elliptic Boundary Value Problems
RBF Multiscale Collocation for Second Order Elliptic Boundary Value Problems
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DOI：
10.1137/120898383
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发表时间：
2013-08
期刊：
影响因子：
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通讯作者：
P. Farrell;H. Wendland
中科院分区：
文献类型：
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作者：
P. Farrell;H. Wendland
本文讨论了求解有界域上椭圆型偏微分方程组的多尺度径向基函数配置法。近似解是以多层方式构造的，每一层都使用越来越精细的网格上规模较小的紧支的径向基函数。在每个层面上，都采用了标准的对称搭配。给出了一个收敛理论，该理论建立在紧支径向基函数多尺度逼近的最新理论进展的基础上。我们能够证明收敛在能级数上是线性的。我们还讨论了出现的系统的条件数和简单的对角预条件子的影响，现在严格地证明了以前的数值观测。
In this paper, we discuss multiscale radial basis function collocation methods for solving elliptic partial differential equations on bounded domains. The approximate solution is constructed in a multilevel fashion, each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh. On each level, standard symmetric collocation is employed. A convergence theory is given, which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions. We are able to show that the convergence is linear in the number of levels. We also discuss the condition numbers of the arising systems and the effect of simple, diagonal preconditioners, now proving rigorously previous numerical observations.