Bounds for eigenvalues and condition numbers in the p-version of the finite element method
Bounds for eigenvalues and condition numbers in the p-version of the finite element method
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DOI：
10.1090/s0025-5718-98-00983-1
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发表时间：
1998-10
期刊：
影响因子：
--
通讯作者：
Ning Hu;Xianwen Guo;I. Katz
中科院分区：
文献类型：
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作者：
Ning Hu;Xianwen Guo;I. Katz
在本文中，我们提出了一种理论，用于限制由p版本有限元分析产生的刚度矩阵的最小特征值、最大特征值和条件数。边界的特征值和条件数，这是有效的刚度矩阵的基础上，可以使用在p-版本的一组一般的基函数。对于一组分层基函数满足通常的局部支持条件，已普遍使用的p-版本，明确的界限推导出的最小特征值，最大特征值，和条件数的刚度矩阵。我们证明了刚度矩阵的条件数的增长像p4（d-1），其中d是维数。我们的研究结果反驳了奥尔森和道格拉斯的猜想，其中作者断言，无论选择的基础，条件数的增长像p 4d或更快。数值结果也验证了我们的理论界是正确的。
In this paper, we present a theory for bounding the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices arising from the p-version of finite element analysis. Bounds are derived for the eigenvalues and the condition numbers, which are valid for stiffness matrices based on a set of general basis functions that can be used in the p-version. For a set of hierarchical basis functions satisfying the usual local support condition that has been popularly used in the p-version, explicit bounds are derived for the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices. We prove that the condition numbers of the stiffness matrices grow like p 4(d-1) , where d is the number of dimensions. Our results disprove a conjecture of Olsen and Douglas in which the authors assert that regardless of the choice of basis, the condition numbers grow like p 4d or faster. Numerical results are also presented which verify that our theoretical bounds are correct.