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On Jamet’s Estimates for the Finite Element Method with Interpolation at Uniform Nodes of a Simplex
Siberian Advances in Mathematics Pub Date : 2018-03-08 , DOI: 10.3103/s1055134418010017
N. V. Baĭdakova

We suggest a new geometric characteristic of a simplex. This characteristic tends to zero together with the characteristic introduced by Jamet in 1976. Jamet’s characteristic was used in upper estimates for the error of approximation of the derivatives of a function on a simplex by the corresponding derivatives of the polynomial interpolating the values of the function at uniform nodes of the simplex. The use of our characteristic for controlling the form of an element of a triangulation allows us to perform a small finite number of operations. We present an example of a function with lower estimates for approximation of the uniform norms of the derivatives by the corresponding derivatives of the Lagrange interpolating polynomial of degree n. This example shows that, for a broad class of d-simplices, Jamet’s estimates cannot be improved on the set of functions under consideration. On the other hand, for d = 3 and n = 1, we present an example showing that, in general, Jamet’s estimates can be improved.

中文翻译:

关于单纯形均匀节点处插值的有限元方法的Jamet估计

我们建议单纯形的新几何特征。该特征与Jamet于1976年引入的特征趋于零。Jamet的特征用于通过多项式的相应导数对函数值进行插值的单纯式上函数导数的近似误差。单纯形的统一节点。使用我们的特征来控制三角剖分元素的形式使我们可以执行少量有限的操作。我们提供了一个函数示例,该函数具有较低的估计值,可以通过阶数为n的Lagrange插值多项式的相应导数来近似导数的统一范数。这个例子表明,对于广泛的d简而言之,Jamet的估计无法在所考虑的功能集合上得到改善。另一方面,对于d = 3和n = 1,我们给出一个示例,该示例通常显示Jamet的估计可以改进。
更新日期:2018-03-08
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