发表机构
McGill University; National University of Mongolia; Mongolian Academy of Sciences（麦吉尔大学; 蒙古国立大学; 蒙古科学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明一致inf-sup稳定的嵌套Petrov-Galerkin方法满足松弛拟正交性，并构造反例说明完全拟正交性需额外层次信息。
AI 中文摘要
我们给出一个初等的、无坐标的证明，证明在Hilbert空间上一致inf-sup稳定的嵌套Petrov-Galerkin方法满足松弛的一般拟正交性。更精确地说，在任意$N$个连续层次的窗口上，累积的Galerkin增量平方和被窗口开始时的误差平方乘以$N^\sigma$所界定，其中$\sigma<1$，且$\sigma$和常数前因子都仅显式地依赖于Galerkin投影的一致界。该证明利用了相容投影链中连续块之间的Hilbert空间角，结合二进分解和对偶性。它避免了矩阵表示、小波基和LU分解。对于对称不定问题，我们将论证与Galerkin细节空间的正负谱分解联系起来，并得到符号分解方法的有效有限窗口版本。我们还构造了一个固定的自伴对合和一个固定的嵌套、一致inf-sup稳定的Galerkin序列，对于该序列，完全的一般拟正交性不成立。同样的构造产生，对于每个$0<\alpha<1$，具有有限支撑的目标，其全尾比率至少以$N^\alpha$的速度增长。因此，一致inf-sup稳定性保证了次线性的有限窗口拟正交性，而完全拟正交性通常需要额外的层次信息。
英文摘要
We give an elementary, coordinate-free proof that uniformly inf-sup stable nested Petrov-Galerkin methods on Hilbert spaces satisfy relaxed general quasi-orthogonality. More precisely, the accumulated squared Galerkin increments over any window of $N$ consecutive levels are bounded by the squared error at the beginning of the window times $N^σ$, where $σ<1$, and both $σ$ and the constant prefactor depend explicitly only on a uniform bound for the Galerkin projections. The proof uses the Hilbert space angle between consecutive blocks of a uniformly bounded compatible projection chain, together with a dyadic decomposition and duality. It avoids matrix representations, wavelet bases, and LU-factorization. For symmetric indefinite problems, we relate the argument to the positive and negative spectral splittings of the Galerkin detail spaces and obtain a valid finite-window version of the sign-decomposition approach. We also construct a fixed self-adjoint involution and a fixed nested, uniformly inf-sup stable Galerkin sequence for which full general quasi-orthogonality fails. The same construction yields, for every $0<α<1$, finite-support targets whose full-tail ratios grow at least like $N^α$. Thus uniform inf-sup stability guarantees sublinear finite-window quasi-orthogonality, whereas full quasi-orthogonality requires additional hierarchical information in general.
Comments18 pages
