反向不等式估计与符号计算
Inverse Inequality Estimates with Symbolic Computation
AI总结：
本文通过符号计算方法改进了有限元方法中广义特征值问题的最大特征值上界估计，提出改进的holonomic ansatz方法并推导出新的紧上界和下界。
摘要：
在数值方法求解偏微分方程（如有限元方法）的收敛性分析中，会遇到某些广义特征值问题，其最大特征值需要尽可能精确地估计。我们应用符号计算方法到正方形元素的情况，将之前已知的上界（见Schwab, 1998）提高了8倍。更具体地说，我们尝试使用holonomic ansatz方法来评估对应的行列式，这是一种由Zeilberger在2007年提出的强大工具。然而，这种方法在当前问题上并未成功。作为解决方案，我们提出了一种适用于更广泛行列式类别的holonomic ansatz变种，包括我们正在处理的行列式。我们得到了行列式的显式闭式表达式，其特殊形式使我们能够推导出新的、紧的上界和下界，以及其渐进行为。
英文摘要：
In the convergence analysis of numerical methods for solving partial differential equations (such as finite element methods) one arrives at certain generalized eigenvalue problems, whose maximal eigenvalues need to be estimated as accurately as possible. We apply symbolic computation methods to the situation of square elements and are able to improve the previously known upper bound, given in "p- and hp-finite element methods" (Schwab, 1998), by a factor of 8. More precisely, we try to evaluate the corresponding determinant using the holonomic ansatz, which is a powerful tool for dealing with determinants, proposed by Zeilberger in 2007. However, it turns out that this method does not succeed on the problem at hand. As a solution we present a variation of the original holonomic ansatz that is applicable to a larger class of determinants, including the one we are dealing with here. We obtain an explicit closed form for the determinant, whose special form enables us to derive new and tight upper resp. lower bounds on the maximal eigenvalue, as well as its asymptotic behaviour.
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