SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1465-1491, August 2024.
Abstract. A robust substructuring type preconditioner is developed for high order approximation of problem for which the element matrix takes the form [math] where [math] and [math] are the mass and stiffness matrices, respectively. A standard preconditioner for the pure stiffness matrix results in a condition number bounded by [math] where [math] blows up as [math]. It is shown that the best uniform bound in [math] that one can hope for is [math]. More precisely, we show that the upper envelope of the bound [math] is [math]. What, then, can be done to obtain a preconditioner that is robust for all [math]? The solution turns out to be a relatively minor modification of the basic substructuring algorithm of [I. Babuška et al., SIAM J. Numer. Anal., 28 (1991), pp. 624–661]: one can simply augment the preconditioner with a suitable Jacobi smoothener over the coarse grid degrees of freedom. This is shown to result in a condition number bounded by [math] where the constant is independent of [math]. Numerical results are given which shows that the simple expedient of augmentation with nodal smoothening reduces the condition number by a factor of up to two orders of magnitude.

中文翻译：

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三角形高阶有限元的均匀子结构预处理器以及节点基函数的影响


《SIAM 数值分析杂志》，第 62 卷，第 4 期，第 1465-1491 页，2024 年 8 月。

抽象的。为问题的高阶近似开发了鲁棒子结构类型预处理器，其中元素矩阵采用 [math] 形式，其中 [math] 和 [math] 分别是质量矩阵和刚度矩阵。纯刚度矩阵的标准预处理器会产生由 [math] 界定的条件数，其中 [math] 会膨胀为 [math]。结果表明，人们可以期望的[数学]中的最佳统一界限是[数学]。更准确地说，我们证明了边界 [math] 的上包络线是 [math]。那么，如何才能获得对所有[数学]都稳健的预条件子呢？事实证明，该解决方案是对[I.的基本子结构算法进行相对较小的修改。 Babuška 等人，SIAM J. Numer。 Anal., 28 (1991), pp. 624–661]：可以简单地在粗网格自由度上使用合适的雅可比平滑器来增强预处理器。这表明会产生一个以 [math] 为界的条件数，其中常数独立于 [math]。给出的数值结果表明，节点平滑增强的简单方法可将条件数降低多达两个数量级。