From Electrostatics to Almost Optimal Nodal Sets for Polynomial Interpolation in a Simplex
From Electrostatics to Almost Optimal Nodal Sets for Polynomial Interpolation in a Simplex
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DOI：
10.1137/s003614299630587x
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发表时间：
1998-04
影响因子：
2.9
通讯作者：
J. Hesthaven
中科院分区：
文献类型：
--
作者：
J. Hesthaven
利用雅可比-高斯求积点的静电解释来获得适合于逼近单形上定义的光滑函数的插值点。此外，几个新的估计，广泛的数值研究的基础上，近似沿着线使用Jacobi-高斯-Lobatto求积点作为节点集。静电类比扩展到二维的情况下，重点是节点集内的三角形的两个非常好的矩阵的节点集。通过计算Lebesgue常数来评估矩阵，并且它们共享这样的属性，即沿着单纯形的边缘的节点沿着是Chebyshev多项式和Legendre多项式的Gauss-Lobatto求积点。这使得所得到的节点集特别适合与传统的谱方法集成，并提供了一个新的节点的基础上的H-P有限元方法。
The electrostatic interpretation of the Jacobi--Gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex. Moreover, several new estimates, based on extensive numerical studies, for approximation along the line using Jacobi--Gauss--Lobatto quadrature points as the nodal sets are presented. The electrostatic analogy is extended to the two-dimensional case, with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented. The matrices are evaluated by computing the Lebesgue constants and they share the property that the nodes along the edges of the simplex are the Gauss--Lobatto quadrature points of the Chebyshev and Legendre polynomials, respectively. This makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h-p finite element methods.