An Interpolation Error Estimate on Anisotropic Meshes in Rn and Optimal Metrics for Mesh Refinement
An Interpolation Error Estimate on Anisotropic Meshes in Rn and Optimal Metrics for Mesh Refinement
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DOI：
10.1137/060667992
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发表时间：
2007-10
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影响因子：
--
通讯作者：
Weiming Cao
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文献类型：
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作者：
Weiming Cao
本文推广了[W. Cao，Math. Comp.，$n$维的函数。我们测量的各向异性行为的高阶导数张量的“最大”（在某种意义上）的椭圆/椭球中包含的水平曲线/曲面的方向导数的多项式。在给定插值函数的各向异性测度的情况下，我们得到了拟一致网格上的分段多项式插值在给定度量下的误差估计。利用矩阵特征值的惯性性质[R. C. Thompson，J. Math. Anal.应用程序、58（1977），pp. 572-577]和保持器的不等式，我们可以确定最佳的网格度量导致最小的误差界在各种规范。此外，我们发展了一种降维方法来近似地找到各向异性测度。我们给出了两个数值示例，用于在使用本文开发的最佳网格度量生成的各种各向异性网格上进行线性和二次插值。数值结果表明，在相应误差范数最优的网格上，插值误差最小。
In this paper, we extend the work in [W. Cao, Math. Comp., to appear] to functions of $n$ dimensions. We measure the anisotropic behavior of higher-order derivative tensors by the “largest” (in certain sense) ellipse/ellipsoid contained in the level curve/surface of the polynomial for directional derivatives. Given the anisotropic measure for the interpolated functions, we derive an error estimate for piecewise polynomial interpolations on meshes that are quasi-uniform under a given metric. By using the inertia properties for matrix eigenvalues [R. C. Thompson, J. Math. Anal. Appl., 58 (1977), pp. 572-577] and Holder's inequality, we can identify the optimal mesh metrics leading to the smallest error bound in various norms. Furthermore, we develop a dimensional reduction method to find the anisotropic measure approximately. We present two numerical examples for linear and quadratic interpolation on various anisotropic meshes generated with the optimal mesh metrics developed in this paper. Numerical results show that the smallest interpolation error is attained exactly on meshes optimal for the corresponding error norm as predicted.