Estimation of interpolation error constants for the P0 and P1 triangular finite elements
Estimation of interpolation error constants for the P0 and P1 triangular finite elements
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DOI：
10.1016/j.cma.2006.10.029
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发表时间：
2007-08
影响因子：
7.2
通讯作者：
F. Kikuchi;Xuefeng Liu
中科院分区：
文献类型：
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作者：
F. Kikuchi;Xuefeng Liu
我们给出了分段常数插值函数和分段线性三角形插值函数的误差常数的一些基本结果。对于分段线性，我们主要分析符合情况，但也给出了非符合情况的一些结果。我们获得了此类误差常数与三角形几何参数的依赖关系。特别是，我们明确确定了 Babuška-Aziz 常数，该常数在线性三角形有限元的插值误差估计中起着至关重要的作用。确定方程为超越方程λ+tanλ=0，这样就可以以期望的精度和可验证性进行数值求解。这种高精度的常数近似值以及其他常数的估计可广泛用于有限元解的自适应计算和数值验证中的先验和后验误差估计。
We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but some results are also given for the non-conforming case. We obtain explicit relations for the dependence of such error constants on the geometric parameters of triangles. In particular, we explicitly determine the Babuška–Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The equation for determination is the transcendental equation λ+tanλ=0, so that the solution can be numerically obtained with desired accuracy and verification. Such highly accurate approximate values for the constant as well as estimates for other constants can be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of finite element solutions.