在用于格心离散化的多重网格公理下，对称V循环可能发散
The symmetric V-cycle can diverge under the multigrid axioms for cell-centred discretisations
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中文总结 AI 辅助
研究格心离散化下多重网格方法中标准对称V循环的收敛性，通过两种构造表明其可能发散，即使满足相关公理假设，而W循环仍一致收缩，还给出了平滑计数阈值与相关常数的关系及具体离散化中的发散情况。
中文摘要 AI 辅助
多重网格方法应用于格心有限差分和有限体积离散化的公理收敛理论基于两个假设：不平衡伽辽金条件（G3），即$R_{\ell - 1}A_\ell P_{\ell - 1}=2A_{\ell - 1}$，其中$R_{\ell - 1}=\frac{1}{2}P_{\ell - 1}^T$，以及Bramble型的弱逼近性质$(A2)_\alpha$。在这些假设以及Richardson平滑下，对称W循环和可变V循环已知是一致收敛的，而标准对称V循环的一致收敛性仍未解决。我们通过两种构造给出否定答案。首先，对于每个平滑计数$m$，我们展示了满足（G3）、Richardson可容许性（$C_R = 1$）以及对于每个$\alpha\in(0,1]$的$(A2)_\alpha$（具有与层数无关的尖锐常数$C_{A2}^2 = 4m$）的任意深度的层次结构，其对称$V(m,m)$循环误差算子在三层时谱半径$\theta_m(1 + 2\theta_m)>1$，并随深度几何增长；该族表明任何能恢复一致V循环收敛的平滑计数阈值$m_0$必须至少以$C_{A2}$的二次方增长。其次，我们证明在完全标准的离散化中也会出现同样的失败：具有网格对齐系数跳跃$1:\kappa$和谐波（Samarskii）界面平均的一维扩散方程的格心有限体积层次结构恰好满足（G3）和具有与层数无关常数$C_{A2}=O(\kappa)$的$(A2)_{1/2}$，但对于每个$\kappa\geq3$，其具有任何可容许Richardson参数（包括最优参数）的对称$V(1,1)$循环在层数上几何发散。在两种构造中，W循环仍然是一致收缩的，所以这些假设区分了这两个循环。所有断言都通过数值验证。
英文摘要
The axiomatic convergence theory for multigrid methods applied to cell-centred finite-difference and finite-volume discretisations rests on two hypotheses: an imbalanced Galerkin condition (G3), which states that $R_{\ell-1}A_\ell P_{\ell-1}=2A_{\ell-1}$ with $R_{\ell-1}=\frac{1}{2}P_{\ell-1}^T$, and a weak approximation property $(A2)_α$ of Bramble type. Under these hypotheses, together with Richardson smoothing, the symmetric W-cycle and the variable V-cycle are known to be uniformly convergent, while the uniform convergence of the standard symmetric V-cycle has remained open. We answer this in the negative by two constructions. First, for every smoothing count $m$ we exhibit hierarchies of every depth satisfying (G3), Richardson admissibility with $C_R=1$, and $(A2)_α$ for every $α\in(0,1]$ with the sharp level-independent constant $C_{A2}^2=4m$, whose symmetric $V(m,m)$-cycle error operator has spectral radius $θ_m(1+2θ_m)>1$ already on three levels, where $θ_m=(1-\frac{1}{4m})^{2m}$, and growing geometrically with the depth; the family shows that any smoothing-count threshold $m_0$ that could restore uniform V-cycle convergence must grow at least quadratically in $C_{A2}$. Second, we prove that the same failure occurs in a completely standard discretisation: the cell-centred finite-volume hierarchy for a one-dimensional diffusion equation with a mesh-aligned coefficient jump $1:κ$ and harmonic (Samarskii) interface averaging satisfies (G3) exactly and $(A2)_{1/2}$ with a level-independent constant $C_{A2}=O(κ)$, yet for every $κ\ge3$ its symmetric $V(1,1)$-cycle with any admissible Richardson parameter, including the optimal one, diverges geometrically in the number of levels. In both constructions the W-cycle remains uniformly contractive, so the hypotheses separate the two cycles. All claims are verified numerically.
