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1. 具有线性滑移界面的弹性问题的杂交高阶方法
A Hybrid High-Order Method for the Elasticity Problem with Linear Slip Interface
作者: Erik Burman, Peiqi Huang
类别: math.NA
发布日期: 2026-09-17
arXiv ID: 2609.20444v1
摘要翻译: 我们设计并分析一种用于具有线性滑移界面的线弹性问题的杂交高阶（HHO）方法，即具有弹簧型界面条件，其中界面两侧位移的跳跃通过柔度张量 $\bK=α\bI+(β-α)\udl{n}\otimes\udl{n}$ 与面力成正比。网格与界面贴合，离散未知量在网格面上为次数 $k\ge 1$ 的多项式，在网格单元内为次数 $k+1$ 的多项式，并且允许一般多面体单元。该方法的两个创新点在于局部对称应变重构，它融入了界面条件，以及界面稳定化，它按照 Hansbo 和 Hansbo \cite{HH04} 的思想由正则化界面刚度 $\bS_h$ 构造。因此，单一格式覆盖整个柔度范围，从完全粘结界面 $α=β=0$，此时界面稳定化起 Nitsche 型罚项作用，到由 $α,β\to+\infty$ 得到的无面力界面，并且没有未知量附加于跳跃。我们证明离散双线性形式是强制的，并且误差在能量范数下以 $h^{k+1}$ 阶收敛，在 $L^2$ 范数下以 $h^{k+2}$ 阶收敛，其中的常数独立于柔度参数和 Lamé 系数 $λ$，因此该方法也是无锁定的。提供了各种数值算例和比较以验证理论结果。
Comment: 无
链接: http://arxiv.org/abs/2609.20444v1
1. 研究对象和背景
本文研究带线性滑移界面的线性弹性问题. 设 $\Omega\subset\mathbb{R}^d$ , $d=2,3$ , 是连通凸多边形或多面体区域, 由内部光滑界面 $\Gamma=\partial\Omega_1\cap\partial\Omega_2$ 分成两个子区域 $\Omega_1$ 与 $\Omega_2$ . 位移未知量为 $u:\Omega\to\mathbb{R}^d$ . 材料是分段各向同性线弹性体, 本构关系与平衡方程为
$$\sigma(u)=2\mu\epsilon(u)+\lambda\nabla\cdot u\,I,\quad -\nabla\cdot\sigma(u)=f\quad \text{in }\Omega_1\cup\Omega_2,$$其中 $\epsilon(u)=\frac12(\nabla u+\nabla u^T)$ , $\mu,\lambda$ 是分段常数的 Lamé 系数, 满足 $0
其中 $[v]=v_1|_\Gamma-v_2|_\Gamma$ 表示跳跃, $n$ 是 $\Omega_1$ 的外法向. 柔度张量取各向同性形式
$$K=\alpha I+(\beta-\alpha)n\otimes n,$$其中 $\alpha\ge0$ 与 $\beta\ge0$ 分别是切向与法向柔度. 该模型覆盖两个极端情形: 当 $\alpha=\beta=0$ 时 $K=0$ , 界面完美粘接, 位移连续, 界面条件退化为约束; 当 $\alpha,\beta\to+\infty$ 时两个子区域解耦, 界面趋于无牵引边界条件. 该问题在复合材料, 地质介质, 胶接或焊接组件, 以及薄粘接层或损伤区建模中自然出现. 直接解析薄层通常代价很高, 因此采用牵引连续而位移跳跃与牵引成比例的界面模型是常见的降维建模方式.
其弱形式可写为: 寻找 $u\in V$ 使得
$$a(u,v)=l(v)\quad \forall v\in V,$$其中
$$a(u,v)=(\sigma(u),\epsilon(v))_{\Omega_1\cup\Omega_2}+\langle S[u],[v]\rangle_\Gamma,\quad l(v)=(f,v)_\Omega.$$这里 $S$ 是界面刚度, 根据 $\alpha,\beta$ 是否为零分段定义: 当 $\alpha>0,\beta>0$ 时 $S=K^{-1}$ ; 当 $\alpha>0,\beta=0$ 时 $S=\alpha^{-1}(I-n\otimes n)$ ; 当 $\alpha=0,\beta>0$ 时 $S=\beta^{-1}n\otimes n$ ; 当 $\alpha=\beta=0$ 时 $S=0$ . 文中指出该变分问题与原始强形式等价, 并引用 Hansbo 和 Hansbo 的已有工作.
数值逼近该问题有三个核心困难. 第一, 精确解跨界面不连续, 因此网格要么拟合界面, 要么离散空间需要能表示单元内部的跳跃. 第二, 柔度参数 $\alpha,\beta$ 在实际问题中可能跨越多个数量级, 若离散格式在完美粘接极限或高柔度极限下损失稳定性或精度, 则实用价值有限. 完美粘接极限对应一个必须弱强加的约束, 例如 Nitsche 型罚方法, 其罚参数必须与网格尺寸正确匹配; 高柔度极限则要求准确表示物理界面刚度. 第三, 当某个材料接近不可压时, 第二个 Lamé 系数 $\lambda$ 变得很大, 离散格式必须锁定自由, 即误差估计中的常数不能随 $\lambda\to+\infty$ 恶化.
已有文献分别处理了这些困难. Hansbo 和 Hansbo 提出了非拟合有限元方法, 在界面切割单元中加倍自由度, 采用 Nitsche 型格式并引入正则化界面刚度 $S_h=(h/\delta+K)^{-1}$ , 对分段仿射近似证明了关于 $\alpha,\beta$ 一致的最优收敛性. 基于 Nitsche 方法的锁定自由间断 Galerkin 方法也被用于近不可压弹性. 混合高阶方法由扩散问题与线性弹性问题发展而来, 具有任意阶, 多面体网格, 局部重构与稳定, 单元未知量可局部消去等优点, 并且已有锁定自由的线性弹性 HHO 格式. Cardenas 和 Solano 发展了线性弹性的高阶非拟合 HDG 方法并建立了最优误差估计; 另有一些 HDG 与弱 Galerkin 工作处理线性弹性界面问题, 但通常假设位移跨界面连续. Carstensen 和 Tran 引入了线性化 Green 应变张量的重构算子, 以避免经典 HHO 分解为偏量与球量部分. 然而, 据作者所言, 在本文之前尚无任意阶方法能够处理完整界面柔度谱并得到严格分析. 本文正是为填补这一空白, 设计并分析任意阶 $k\ge1$ 的 HHO 方法, 网格拟合界面 $\Gamma$ , 并证明误差常数独立于柔度参数与 $\lambda$ .
2. 主要定理或主要结果
本文的主要结论是: 所设计的 HHO 离散格式适定, 离散双线性形式具有强制性, 并且在合适的正则性假设下达到能量范数 $h^{k+1}$ 与 $L^2$ 范数 $h^{k+2}$ 的最优收敛阶, 且误差估计中的常数不依赖于柔度参数 $\alpha,\beta$ 和 Lamé 系数 $\lambda$ . 该结论由摘要与引言明确给出, 并分别对应正文中的命题 5.1 与命题 5.2. 由于提供的 PDF 文本只到第 10 页的能量范数定义处, 命题 5.1 与命题 5.2 的完整定理陈述, 包括正则性假设, 对偶问题设定以及常数依赖关系的逐项说明, 未在提供内容中展开; 因此下面只陈述可以由摘要, 引言以及已给技术准备严格确认的内容.
第一, 离散问题为: 寻找 $\hat u_h\in\hat V_{h0}$ 使得
$$a_h(\hat u_h,\hat w_h)=l(\hat w_h)\quad \forall \hat w_h\in\hat V_{h0},$$其中
$$a_h(\hat u_h,\hat w_h)=\sum_{T\in\mathcal{T}_h}\big(a_T(\hat u_T,\hat w_T)+s_T(\hat u_T,\hat w_T)\big).$$局部双线性形式 $a_T$ 由重构对称应变 $E$ 与重构散度 $D=\operatorname{tr}E$ 组成, 并包含界面修正项
$$a_T(\hat v_T,\hat w_T)=2\mu(E(\hat v_T),E(\hat w_T))_T+\lambda(D(\hat v_T),D(\hat w_T))_T+\frac12\langle (I-KS_h)KG(\hat v_T)n,G(\hat w_T)n\rangle_{\partial T\cap\Gamma},$$其中 $G(\hat v_T)=(2\mu E+\lambda I\operatorname{tr}E)(\hat v_T)$ . 稳定项 $s_T$ 由非界面部分与界面部分组成, 界面部分采用正则化界面刚度 $S_h$ 加权
$$s_{T\Gamma}(\hat v_T,\hat w_T)=2\langle S_h(v_{\partial T\cap\Gamma}-\Pi_{\partial T\cap\Gamma}v_T),w_{\partial T\cap\Gamma}-\Pi_{\partial T\cap\Gamma}w_T\rangle_{\partial T\cap\Gamma}.$$非界面部分采用通常的 HHO 罚, 权重为 $h_T^{-1}(2\mu I+\lambda n\otimes n)$ .
第二, 正则化界面刚度定义为
$$S_h|_T=Q\tilde L^{-1}Q^T,\quad \tilde L=L+P,\quad P=\operatorname{diag}\!\left(\frac{h_T}{\mu+\lambda},\frac{h_T}{\mu},\frac{h_T}{\mu}\right),$$其中 $K=Q L Q^T$ , $L=\operatorname{diag}(\beta,\alpha,\alpha)$ . 该定义保证方法在端点处自动退化到正确极限: 当 $\alpha=\beta=0$ 时 $I-KS_h=I$ , 且
$$S_h=\frac{\mu+\lambda}{h_T}n\otimes n+\frac{\mu}{h_T}\sum_{i=1,2}t_i\otimes t_i,$$因此界面稳定项表现为 Nitsche 型罚, 界面面被当作普通面处理; 当 $\alpha,\beta\gg h_T/\mu$ 时 $S_h\simeq K^{-1}$ , 界面稳定项表示物理界面刚度, 而 $I-KS_h\simeq0$ , 因此重构与局部双线性形式中的界面修正项消失. 在极限 $\alpha,\beta\to+\infty$ 下界面未知量变得不活跃, 离散问题解耦为两个子问题, 每个子问题在 $\Gamma$ 上带无牵引边界条件, 这与连续问题的极限行为一致.
第三, 关于界面未知量的关键结构性结论是: 界面面属于 $\mathcal{F}_h$ , 因此界面上的未知量是单值的, 它们逼近位移的平均值 $\{u\}$ , 而跳跃 $[u]$ 不是未知量, 可由两个相邻单元未知量的迹之差后验恢复. 通过消去界面未知量可以验证, 当两个相邻单元 $T_1,T_2$ 的 $h_T$ 和 $S_h$ 相同时, 最小化两个界面稳定项给出界面未知量 $v_F=\frac12\Pi_F(v_{T_1}+v_{T_2})$ , 且其值为 $\langle S_h\Pi_F[v],\Pi_F[v]\rangle_F$ . 稳定项 (3.6) 中的因子 $2$ 正是为了使该消去过程重现弱形式中的柔度项, 只是把 $S$ 替换为正则化 $S_h$ .
第四, 在技术准备层面, 文中证明了若干支撑主结果的稳定性与逼近性质. 引理 4.1 给出 $S_h$ 的谱范数控制:
$$n^T S_h n\le\frac{\mu+\lambda}{h},\quad t_i^T S_h t_i\le\frac{\mu}{h},\quad |S_{h,t}|\le\frac{\mu}{h},$$以及
$$|I-KS_h|\le1,\quad |S_hK|\le1,\quad |(I-KS_h)K|\le\frac{h}{\mu}.$$引理 4.3 给出单元多项式与其面投影之间距离的控制: 对任意 $v_T\in P^{k+1}(T)^d$ ,
$$h_T^{-1/2}\|v_T-\Pi_{\partial T}v_T\|_{\partial T}\le C_{dtr}C_P C_K\|\epsilon(v_T)\|_T,$$其中不出现 $\lambda$ . 引理 4.4 给出局部稳定性的关键不等式: 存在 $0<\alpha^\flat<+\infty$ 使得
$$\alpha^\flat\|\hat v_T\|_E\le\left(2\mu\|E(\hat v_T)\|_T^2+\|[(I-KS_h)K]^{1/2}G(\hat v_T)n\|_{\partial T\cap\Gamma}^2+\frac{2\mu}{h_T}\|v_{\partial T\setminus\Gamma}-\Pi_{\partial T\setminus\Gamma}v_T\|_{\partial T\setminus\Gamma}^2+s_{T\Gamma}(\hat v_T,\hat v_T)\right)^{1/2}.$$该估计表明精确对称梯度 $\epsilon(v_T)$ 的能量范数贡献可由重构对称梯度与稳定项控制, 且相关常数不依赖于 $\lambda$ 与柔度参数.
第五, 在正则性足够时, 摘要明确指出误差收敛为能量范数 $h^{k+1}$ 与 $L^2$ 范数 $h^{k+2}$ , 并且常数独立于柔度参数与 $\lambda$ , 因而方法同时是柔度鲁棒与锁定自由的. 数值实验部分验证了理论结果, 包括八个数量级的柔度变化, 退化情形 $\alpha=0$ 或 $\beta=0$ , 以及准不可压极限. 但数值实验的具体网格, 右端项, 误差表与收敛率的逐项数据不在提供的截断文本中, 无法在此确认.
3. 研究方法、关键技术和核心工具
本文的技术核心不是简单地把标准 HHO 方法套用到弹性界面问题, 而是重新设计局部对称应变重构与界面稳定项, 使一个格式同时覆盖完美粘接, 有限柔度与无牵引三个极限, 并且误差常数不依赖于柔度参数与 $\lambda$ . 主要障碍可以分成四类: 界面条件如何在重构中一致地进入; 界面跳跃没有独立未知量时如何恢复弱形式中的柔度项; 如何避免完美粘接极限失去强制性与高柔度极限失去物理刚度; 以及如何避免近不可压极限下的锁定.
第一项关键工具是正则化界面刚度 $S_h$ . 作者先利用 $K=Q L Q^T$ , 其中 $Q=[n|t_1|t_2]$ , $L=\operatorname{diag}(\beta,\alpha,\alpha)$ , 然后扰动特征值为
$$\tilde L=L+P,\quad P=\operatorname{diag}\!\left(\frac{h_T}{\mu+\lambda},\frac{h_T}{\mu},\frac{h_T}{\mu}\right),\quad S_h=Q\tilde L^{-1}Q^T.$$这样 $S_h$ 在法向与切向分别具有尺度 $(\beta+h_T/(\mu+\lambda))^{-1}$ 与 $(\alpha+h_T/\mu)^{-1}$ . 该扰动同时实现两个目的: 当物理柔度为零时, $S_h$ 自动给出量级为 $(\mu+\lambda)/h_T$ 与 $\mu/h_T$ 的 Nitsche 罚; 当物理柔度很大时, $S_h$ 自动趋于 $K^{-1}$ . 引理 4.1 中的恒等式
$$I-KS_h=QPQ^T S_h$$以及范数界
$$|(I-KS_h)K|\le\frac{h}{\mu}$$是后续稳定性证明的关键, 它们把界面修正项的大小控制在网格尺度与 $\mu$ 上, 而不是让 $\lambda$ 或 $\alpha,\beta$ 失控.
第二项关键工具是局部对称应变重构. 对每个 $T\in\mathcal{T}_h$ , 重构 $E(\hat v_T)\in P^k(T)^{d\times d}$ 由变分方程定义: 对任意 $\tau\in P^k(T)^{d\times d}$ ,
$$(E(\hat v_T),\tau)_T+\frac12\langle (I-KS_h)K(2\mu E+\lambda I\operatorname{tr}E)(\hat v_T)n,\tau n\rangle_{\partial T\cap\Gamma}=(\epsilon(v_T),\tau)_T+\langle v_{\partial T\setminus\Gamma}-v_T,\tau n\rangle_{\partial T\setminus\Gamma}+\langle (I-KS_h)(v_{\partial T\cap\Gamma}-v_T),\tau n\rangle_{\partial T\cap\Gamma}.$$与标准 HHO 不同的是, 界面条件通过因子 $I-KS_h$ 被嵌入重构方程. 在完美粘接极限 $K=0$ 时该因子为 $I$ , 界面项变成通常的 Nitsche 型界面项; 在高柔度极限该因子趋于零, 界面项自动退出, 重构退化为各子区域上的标准重构. 作者还通过取 $\tau=qI$ 得到离散散度重构 $D=\operatorname{tr}E$ , 满足类似方程. 重构的稳定性由引理 4.4 证明, 其中核心是测试 $\tau=2\mu\epsilon(v_T)$ 并把界面项移到右边, 再用离散迹不等式, $|(I-KS_h)K|\le h_T/\mu$ 以及因子分解 $|I-KS_h|\le(h_T/\mu)^{1/2}|S_h^{1/2}|$ 控制各项. 第二步则用引理 4.3 控制 $v_T$ 与其 $L^2$ 投影之间的面差异, 该引理通过刚性运动投影, Poincaré 不等式与第二 Korn 不等式证明, 且不涉及 $\lambda$ .
第三项关键工具是界面稳定项. 稳定项被设计为
$$s_T(\hat v_T,\hat w_T)=s_{T\setminus\Gamma}(\hat v_T,\hat w_T)+s_{T\Gamma}(\hat v_T,\hat w_T),$$其中非界面部分采用标准 HHO 罚
$$s_{T\setminus\Gamma}=\langle h_T^{-1}(2\mu I+\lambda n\otimes n)(v_{\partial T\setminus\Gamma}-\Pi_{\partial T\setminus\Gamma}v_T),w_{\partial T\setminus\Gamma}-\Pi_{\partial T\setminus\Gamma}w_T\rangle_{\partial T\setminus\Gamma},$$而界面部分则用量
$$s_{T\Gamma}=2\langle S_h(v_{\partial T\cap\Gamma}-\Pi_{\partial T\cap\Gamma}v_T),w_{\partial T\cap\Gamma}-\Pi_{\partial T\cap\Gamma}w_T\rangle_{\partial T\cap\Gamma}.$$界面部分的权重不是简单的 $h_T^{-1}$ , 而是由 $S_h$ 给出. 这一点非常关键: 在完美粘接极限, $S_h$ 量级为 $\mu/h_T$ 或 $(\mu+\lambda)/h_T$ , 因此界面稳定项起到 Nitsche 罚的作用; 在高柔度极限, $S_h\simeq K^{-1}$ , 界面稳定项直接表示物理界面柔度. 此外, 由于界面未知量是单值的, 跳跃不是未知量, 稳定项中的界面项在消去界面未知量后正好产生 $\langle S_h\Pi_F[v],\Pi_F[v]\rangle_F$ , 其中的因子 $2$ 是经过消去计算设计的, 以便重现弱形式中的 $\langle S[u],[v]\rangle_\Gamma$ 项.
第四项关键工具是面向锁定自由的插值策略. 在误差分析中, 作者定义局部插值算子 $\hat I_T(v)=(\Pi_T(v),\Pi_{\partial T\setminus\Gamma}(v),\Pi_{\partial T\cap\Gamma}(v))$ . 但为了避免近不可压极限下出现 $\lambda$ 因子, 在定义 $D\hat I_T(v)$ 时, 对单元内部使用保持法向迹并交换散度的 BDM 型 $H(\operatorname{div})$ 插值. 文中 Remark 4.1 明确解释: $L^2$ 正交投影使重构 $E$ 具有最优一致性, 但它不保持法向迹, 若直接用于散度部分, 则 $D(\hat I_T v)-\pi_k(\nabla\cdot v)$ 的界会带 $\lambda$ , 破坏准不可压极限下的鲁棒性. BDM 插值满足 $(v-\Pi_T v)\cdot n=0$ 且 $\nabla\cdot\Pi_T v=\pi_k(\nabla\cdot v)$ , 因此涉及法向分量的插值误差项消失. 这一双投影技巧是获得锁定自由估计的关键, 作者指出它沿用了线性弹性 HHO 中的同类装置.
总体证明框架可以概括为: 先建立网格上的迹不等式, 反不等式, Poincaré 不等式与第二 Korn 不等式; 再证明 $S_h$ 的谱性质; 然后证明局部重构的稳定性与插值逼近; 由此得到离散双线性形式的强制性与一致性误差; 最后在能量范数下组合得到 $O(h^{k+1})$ , 并通过 $L^2$ 对偶论证得到 $O(h^{k+2})$ . 提供的文本详细展示了强制性与局部稳定性所需的核心工具, 但没有给出命题 5.1 与 5.2 的完整证明, 因此对偶论证的具体形式, 正则性假设以及一致性误差的逐项组合在此无法进一步确认.
4. 与之前工作的比较
本文与已有工作的比较可以从界面处理方法, 多项式阶数, 柔度参数鲁棒性, 锁定自由性, 网格与自由度结构等几个维度展开.
首先, 与 Hansbo 和 Hansbo 的非拟合有限元方法相比, 本文明确借用了其正则化界面刚度思想. Hansbo 和 Hansbo 引入
$$S_h=\left(\frac{h}{\delta}+K\right)^{-1}$$以统一处理弹簧型界面, 并在界面切割单元中加倍自由度, 采用 Nitsche 型格式, 对分段仿射近似证明关于 $\alpha,\beta$ 一致的最优收敛. 本文则把该思想与 HHO 的网格相关权重结合, 使用
$$S_h|_T=Q\operatorname{diag}\!\left(\beta+\frac{h_T}{\mu+\lambda},\alpha+\frac{h_T}{\mu},\alpha+\frac{h_T}{\mu}\right)^{-1}Q^T.$$这一正则化不仅保留 $\alpha=0$ 或 $\beta=0$ 的退化情形, 还与 HHO 的局部重构和稳定项一致. 相对于 Hansbo 和 Hansbo 的分段仿射结果, 本文把方法推进到任意阶 $k\ge1$ , 单元未知量为 $P^{k+1}$ , 面未知量为 $P^k$ , 并允许一般多面体单元. 代价是本文要求网格拟合界面 $\Gamma$ , 而 Hansbo 和 Hansbo 的非拟合方法不需要网格与界面贴合; 但本文避免了切割单元中加倍自由度, 并且界面未知量是单值的, 不附加跳跃未知量.
第二, 与锁定自由的间断 Galerkin 与 Nitsche 型弹性方法相比, 本文的 HHO 格式也达到了误差常数不依赖于 $\lambda$ 的性质. 已有基于 Nitsche 的锁定自由 DG 方法主要处理近不可压弹性, 本文则同时处理近不可压与任意柔度界面. 其锁定自由性依赖两个设计: 一是重构与稳定中的权重使用 $2\mu I+\lambda n\otimes n$ 与 $S_h$ 的尺度使得 $\lambda$ 只出现在可控的法向罚中; 二是在误差分析中采用 BDM 型插值处理散度, 避免 $\lambda$ 进入一致性误差. 文中指出这种双投影装置与线性弹性 HHO 中的已有做法相同, 因此本文的推进在于把它与界面滑移条件耦合起来.
第三, 与标准 HHO 方法相比, 标准 HHO 最初用于扩散问题, 随后用于线性弹性, 具有任意阶, 多面体网格, 局部重构与稳定, 单元未知量可消去等优点. 本文保留了这些结构, 但界面问题的核心困难在于位移跨界面不连续. 标准 HHO 处理界面时通常假设位移连续或界面条件较弱. 本文的创新在于: 局部对称应变重构中加入 $I-KS_h$ 因子, 使界面条件直接进入重构; 界面稳定项用 $S_h$ 加权, 使一个格式覆盖从完美粘接到无牵引的全柔度范围; 同时不引入跳跃未知量, 而由界面单值未知量近似平均位移, 跳跃后验恢复. 文中 Remark 3.1 说明, 消去界面未知量后正好重现 $\langle S_h\Pi_F[v],\Pi_F[v]\rangle_F$ , 这与弱形式中的柔度项匹配. 该结构在已有 HHO 文献中没有被严格分析过, 特别是任意阶且关于柔度参数一致的最优误差估计.
第四, 与 Cardenas 和 Solano 的高阶非拟合 HDG 方法相比, 两者都追求高阶与最优误差. 但本文是 HHO, 采用拟合网格, 局部重构与稳定, 而 Cardenas 和 Solano 是 HDG 非拟合方法. 文中并未显式展开两者在界面条件, 柔度极限与锁定自由常数依赖方面的逐项比较; 从提供内容只能确认本文声称是首个任意阶且能处理完整柔度谱并给出严格分析的方法. 与 HDG 和弱 Galerkin 的线性弹性界面工作相比, 文中明确指出这些工作通常假设位移跨界面连续, 因而不能直接处理本文的 $[u]=-K\sigma(u)n$ 跳跃条件. 这是本文的主要推进之一: 允许位移不连续, 并且一个格式覆盖 $\alpha,\beta$ 从零到无穷的整个范围.
第五, 与 Carstensen 和 Tran 的重构方法相比, 他们引入线性化 Green 应变张量的重构以避免经典 HHO 分解为偏量与球量部分. 本文则采用局部对称应变重构, 并把界面条件嵌入重构方程. 两者都关注 HHO 的重构设计, 但目标不同: Carstensen 和 Tran 主要处理重构的构造与锁定自由性, 本文处理带滑移界面的弹性问题, 并需要同时控制柔度参数与 $\lambda$ 的依赖. 文中没有显式比较两种重构在界面问题上的优劣, 从提供内容无法确认更细致的差别.
第六, 在适用范围与局限方面, 本文的模型限于线性弹性, 各向同性界面柔度 $K=\alpha I+(\beta-\alpha)n\otimes n$ , 界面 $\Gamma$ 光滑且网格拟合界面. 柔度参数 $\alpha,\beta\ge0$ , 且 $K$ 在界面上取常数矩阵; Lamé 系数在子区域上分段常数, 满足 $0
最后, 文中明确指出的空白是: 在本文之前没有任意阶方法能够处理完整界面柔度谱并得到严格分析. 本文填补该空白, 但并未声称在所有方面优于非拟合方法. 从提供内容中无法确认本文是否也适用于非拟合网格, 是否处理非线性接触或损伤演化, 以及 $L^2$ 估计是否在更弱正则性下成立. 这些比较维度需要完整论文的后续定理与数值部分才能判断.
了解领域论文（摘要翻译）
1. 超越 PINNs：面向神经与混合 PDE 求解器的统一 Gauss–Newton 与 Petrov–Galerkin 框架
Beyond PINNs: A Unified Gauss–Newton and Petrov–Galerkin Framework for Neural and Hybrid PDE Solvers
作者: Nilo Schwencke, Roland Maier
类别: math.NA, cs.LG
发布日期: 2026-09-17
arXiv ID: 2609.20641v1
链接: http://arxiv.org/abs/2609.20641v1
摘要翻译
中文标题: 超越 PINNs：面向神经与混合 PDE 求解器的统一 Gauss–Newton 与 Petrov–Galerkin 框架
摘要翻译: 物理信息神经网络与有限元方法为偏微分方程的数值逼近提供了两种不同的范式：前者通常通过最小化逐点强残差来训练，而后者则自然地由弱变分形式以及离散后得到的有限维系统构建。在这项工作中，我们引入一个共同框架，它基于用有限族线性测量对泛函 Gauss–Newton 问题进行离散化。我们证明，通过适当的对偶配对，这些线性测量可以由检验函数表示。所得的 Gauss–Newton 系统于是恰好是线性化泛函问题的 Petrov–Galerkin 离散化。这一视角将逐点配点和自然梯度构造恢复为特例，同时使检验函数的选择成为一个显式的算法设计选择。我们将该框架具体应用于椭圆问题；在那里，它自然导向弱残差形式以及作用于互补逼近空间的混合有限元–神经网络构造。数值实验支持所提出的框架，并证明了弱 Gauss–Newton 形式以及混合有限元–神经网络逼近的有效性。
2. Bhatnagar–Gross–Krook 模型的自适应稀疏网格间断 Galerkin 逼近
Adaptive Sparse-grid Discontinuous Galerkin Approximations the Bhatnagar–Gross–Krook Model
作者: Stefan Schnake, Miroslav Stoyanov, Eirik Endeve, Cory Hauck
类别: math.NA
发布日期: 2026-09-17
arXiv ID: 2609.20395v1
链接: http://arxiv.org/abs/2609.20395v1
摘要翻译
中文标题: Bhatnagar–Gross–Krook 模型的自适应稀疏网格间断 Galerkin 逼近
摘要翻译: 本文研究 Bhatnagar–Gross–Krook (BGK) 模型的自适应稀疏网格间断 Galerkin (DG) 离散化，该模型是定义在四维和六维相空间中的动理学方程。维数灾难使得标准 DG 方法对于 BGK 模型变得不可行，这促使人们采用随时间适应解的压缩表示。使用自适应稀疏网格 DG 方法，我们通过将自适应自由度与全网格 DG 方法进行比较，并通过评估在流体和稀薄两种区域中得到的动理学量和流体力学会量，来量化精度和压缩效果。测试算例包括弛豫问题、多维 Sod 激波管，以及先前低秩 BGK 研究中使用的剪切/膨胀流动。为了在不违反守恒性的前提下构建高效的 Maxwell 分布求值——这是结构保持 BGK 模拟的核心障碍——我们引入一种混合插值策略，该策略利用速度可分性来恢复正确的离散碰撞不变量，并证明所得离散碰撞算子在自适应稀疏网格上的守恒性。我们的结果表明，自适应稀疏网格策略能够以尖锐梯度恢复准确且物理相关的解，并且该方法将活跃自由度减少数倍至数个数量级，其中最大的减少出现在六维算例中。所有计算均使用开源 ASGarD 自适应稀疏网格 DG 库完成。
3. 随机哈密顿系统动力学 I - Wong-Zakai 变分积分器
Dynamics of Stochastic Hamiltonian systems I - Wong-Zakai Variational integrators
作者: Jacky Cresson, Rouba SAfI
类别: math.DS, math.NA
发布日期: 2026-09-17
arXiv ID: 2609.19949v1
链接: http://arxiv.org/abs/2609.19949v1
摘要翻译
中文标题: 随机哈密顿系统动力学 I - Wong-Zakai 变分积分器
摘要翻译: 我们构造变分积分器，利用随机哈密顿系统的 Wong-Zakai 逼近来研究其动力学。该方法可用于解释并证明 Wang 等人（L. Wang, J. Hong, R. Scherer, F. Bai. Dynamics and variational Integrators of stochastic Hamiltonian systems. International Journal of Numerical Analysis and Modeling, 6(4), 2009）先前关于随机变分积分器的工作的合理性。文中给出了数值算例。
4. 一种带气泡富集的虚拟区域格式
A Fictitious Domain Formulation with Bubble Enrichment
作者: Stefano Berrone, Lorenzo Neva, Stefano Scialò, Fabio Vicini
类别: math.NA
发布日期: 2026-09-17
arXiv ID: 2609.19836v1
链接: http://arxiv.org/abs/2609.19836v1
摘要翻译
中文标题: 一种带气泡富集的虚拟区域格式
摘要翻译: 虚拟区域方法通过将复杂的物理区域嵌入到一个更简单、非匹配的背景网格中，为绕开复杂的网格生成过程提供了一种强有力的替代方案。在这种网格上施加边界条件的各种策略中，一种可能是通过 Lagrange 乘子弱施加边界条件。已有遵循该策略的方法通常要求区域和边界分别使用两套独立的均匀网格，且其网格尺寸满足预先给定的比例关系。本文通过直接由背景三角剖分的迹构造边界网格，放宽了这一要求。为在这一更弱的网格假设下恢复一致离散 inf-sup 条件，我们用气泡函数富集解的离散空间。我们的主要结果是构造一个合适的算子，使我们能够证明该一致离散 inf-sup 条件，从而建立所得格式的稳定性。作为结果，我们推导出最优先验误差估计，并提供一个数值实验来验证理论结果。
5. 一种用于应变梯度弹性的内罚非协调有限元方法
An interior penalty nonconforming finite element method for strain gradient elasticity
作者: Xuehai Huang, Zheqian Tang
类别: math.NA
发布日期: 2026-09-17
arXiv ID: 2609.19567v1
链接: http://arxiv.org/abs/2609.19567v1
摘要翻译
中文标题: 一种用于应变梯度弹性的内罚非协调有限元方法
摘要翻译: 本文针对任意维数下的应变梯度弹性（SGE）模型，提出了一种内罚非协调有限元方法，采用由向量值二次多项式并用无散度面泡函数富集构造的 $H^1$-非协调位移元。严格分析建立了关于 Lamé 系数 $λ$ 和尺寸参数 $ι$ 均最优且稳健的误差估计。其底层的散度交换结构进一步导出了二维和三维的非协调有限元 Stokes 复形。数值实验支持了所预测的收敛行为和参数稳健性，而单轴拉伸下的圆孔基准算例说明了该方法对多连通区域的适用性。
6. 基于几何自适应聚类树的电场积分方程宽频带定向 $\mathcal{H}^2$-矩阵压缩
Wideband Directional $\mathcal{H}^2$-Matrix Compression for the Electric Field Integral Equation with Geometry-Adaptive Cluster Trees
作者: Joshua M. Tetzner, Simon B. Adrian
类别: math.NA
发布日期: 2026-09-16
arXiv ID: 2609.19431v1
链接: http://arxiv.org/abs/2609.19431v1
摘要翻译
中文标题: 基于几何自适应聚类树的电场积分方程宽频带定向 $\mathcal{H}^2$-矩阵压缩
摘要翻译: 我们提出一种用于电场积分方程的高效宽频带定向 $\mathcal{H}^2$-矩阵构造，与现有构造相比，它不仅支持盒树，还支持几何自适应聚类树。为适应几何自适应聚类树，我们根据每个聚类的电尺寸（而非盒树的层数）确定方向数，使用 Spherical-Fibonacci 点构造方向，并通过角度最近邻映射在聚类与其子聚类的方向集之间建立层级。我们使用不完全自适应交叉近似构造嵌套定向表示，为此引入一种稳健化的树模拟主元策略，以防止由某些几何和网格产生的块结构矩阵过早收敛。数值结果表明，所提方法达到预期精度，所需存储不多于基于八叉树的构造，并且当几何或离散化与八叉树聚类匹配不佳时所需存储显著更少，同时对高频问题表现出预期的 $\mathcal{O}(N\log N)$ 复杂度增长。
7. 奇异偏微分方程的深度学习：一种加权神经网络方法
Deep Learning for Singular PDEs: A Weighted Neural Network Approach
作者: Badr Oulgiht
类别: math.NA
发布日期: 2026-09-16
arXiv ID: 2609.19335v1
链接: http://arxiv.org/abs/2609.19335v1
摘要翻译
中文标题: 奇异偏微分方程的深度学习：一种加权神经网络方法
摘要翻译: 我们提出一个深度学习框架，用于数值逼近带有齐次 Dirichlet 边界条件和正性约束的弱奇异半线性椭圆方程。这些问题具有挑战性，因为当正解在边界附近趋于零时，非线性源项会变得无界。 所提出的方法结合了三个组成部分。第一，通过硬约束表述将 Dirichlet 边界条件直接嵌入到神经网络表示中。第二，Softplus 变换在计算区域内部保持正性，从而确保奇异非线性项在训练期间保持良定义。第三，奇异性感知的加权残差损失强调预测解变小且奇异行为更显著的区域。 该方法在弱奇异情形下通过数值参考解和受控验证测试进行评估。与标准残差表述的比较用于评估近似精度和收敛行为。数值实验说明了将硬边界约束、正性保持和奇异性感知残差加权相结合用于弱奇异椭圆问题的潜力。
不相关论文（基本信息）
1. 多维双曲平衡律的灵敏度演算及其数值实现
Sensitivity Calculus and its Numerical Implementation for Multi-D Hyperbolic Balance Laws
作者: Olivia Dreßen, Michael Herty, Adrian Kolb, Siegfried Müller
类别: math.NA, math.OC
发布日期: 2026-09-17
arXiv ID: 2609.20770v1
链接: http://arxiv.org/abs/2609.20770v1
摘要: We investigate the sensitivity of solutions to multi-dimensional scalar balance laws with respect to perturbations of the initial data analytically and numerically. Reliable first-order sensitivity information is essential in gradient-based optimization and inverse problems constrained by hyperbolic balance laws. In hyperbolic problems, such perturbations affect both the smooth components of the solution and the locations of shocks. Consequently, classical difference quotients of the solution operator generally fail to converge in $L^1$, even in one space dimension. Although generalized tangent-vector techniques have been developed for one-dimensional problems, extending these ideas to multiple space dimensions is considerably more challenging because it requires a geometric description of hypersurfaces of discontinuity. We represent perturbations of hypersurfaces of discontinuity by normal displacements and account for the induced variation of the normal direction. This representation yields evolution equations for the components of a generalized tangent vector. Based on this calculus, we develop a numerical method for computing first-order variations with respect to the initial data. Numerical experiments in two space dimensions confirm the expected first-order accuracy of the resulting approximation.
2. 超越对数凹性的 ULMC 方法 Wasserstein 收敛的统一框架：旧与新
A Unified Framework for Wasserstein Convergence of ULMC Methods beyond Log-Concavity: Old and New
作者: Wanjie Lyu, Xiaojie Wang, Bin Yang
类别: math.NA, math.ST
发布日期: 2026-09-17
arXiv ID: 2609.20713v1
链接: http://arxiv.org/abs/2609.20713v1
摘要: As a fundamental task across computational statistics, scientific computing and machine learning, sampling from high-dimensional probability distributions has received increasing attention in recent years. Numerous sampling algorithms have been proposed, among which underdamped Langevin Monte Carlo (ULMC) methods based on underdamped Langevin dynamics (ULD) have emerged as a class of efficient ones. In this work, we introduce a universal" predictor-corrector formulation that bridges Euler-type, UBU-type and randomized schemes through different choices of method parameters. Notably, the universal" integrator induces two novel classes of low-cost integrators, termed low-cost randomized integrators (LC-RIs) and low-cost UBU integrators (LC-UBUIs), as well as their exponential-free variants based on polynomial and rational approximations. The resulting new UBU-type and randomized schemes require only one gradient evaluation and two Gaussians per iteration, considerably reducing the number of gradient evaluations or Gaussians per iteration required by existing counterparts. Further, a general framework of long-time error analysis is developed for general discretization schemes in a probability metric. Under certain smoothness and non-log-concavity conditions, we rely on the unified framework to establish non-asymptotic $\mathcal{W}_1$-error bounds of both old and new schemes, revealing convergence rates of order $\mathcal{O}(d^{\frac{1}{2}}h)$ for Euler-type schemes, order $\mathcal{O}(d h^2)$ for UBU-type ones and order $\mathcal{O}(d^{\frac{1}{2}}h^{\frac{3}{2}})$ for randomized ones. In the strongly convex setting, the same non-asymptotic error bounds can be recovered in $\mathcal{W}_2$-distance. Numerical experiments corroborate the theoretical findings.
3. 基于条件行列式点过程的增量列子集选择
Incremental Column Subset Selection via Conditional Determinantal Point Processes
作者: Laura Grigori, Zhipeng Xue
类别: math.NA
发布日期: 2026-09-17
arXiv ID: 2609.20556v1
链接: http://arxiv.org/abs/2609.20556v1
摘要: Column subset selection aims to seek a small set of representative columns that accurately approximates a given matrix. We study this problem from the perspective of determinantal point processes (DPPs) and their relation to several existing algorithms. To support incremental column selection, we introduce conditional DPPs, which can be applied to any column selection strategy. In particular, we combine it with adaptive randomized pivoting(ARP), and develop a multi-stage ARP (MSARP) algorithm. We establish theoretical guarantees on the expected Frobenius-norm approximation errors for these methods. In addition, we propose a fixed-precision variant of ARP that adaptively determines the number of selected columns. Numerical experiments on column subset selection and Nyström approximation show that MSARP achieves accuracy comparable to standard ARP while enabling incremental selection.
4. 求解伴随 Volterra 积分微分方程的最优阶分数阶向后配置法
An optimal order fractional backward collocation method for adjoint Volterra integro-differential equations
作者: Mahmoud A. Zaky
类别: math.NA
发布日期: 2026-09-17
arXiv ID: 2609.20356v1
链接: http://arxiv.org/abs/2609.20356v1
摘要: This paper develops and analyzes an optimal-order fractional backward collocation method for adjoint Volterra integro-differential equations with weakly singular kernels. The backward Volterra structure, together with the weakly singular kernel, induces fractional-power singularities at the terminal endpoint, thereby reducing the classical regularity of the exact solution and causing order deterioration in standard polynomial collocation methods. We first establish a regularity result that characterizes the terminal singular behavior of the solution, showing that the solution is continuously differentiable, whereas its second derivative may exhibit a weak singularity at the terminal endpoint. Motivated by this regularity structure, we introduce a terminally graded mesh and construct a fractional backward collocation scheme whose local approximation space is adapted to the endpoint singularity. Rigorous convergence and superconvergence estimates are derived for both the solution and its derivative. With suitable choices of the fractional parameter and the mesh-grading exponent, the proposed method attains the optimal convergence orders dictated by the local approximation degree. Numerical experiments confirm the theoretical predictions and demonstrate the accuracy and effectiveness of the method for adjoint weakly singular Volterra integro-differential equations.
5. 有界区域中时间非局部扩散的蒙特卡洛近似
Monte Carlo Approximations of Time-Nonlocal Diffusions in Bounded Domains
作者: Ivan Biočić, Daniel E. Cedeño-Girón, Aleksandar Mijatović, Bruno Toaldo
类别: math.PR, math.NA
发布日期: 2026-09-17
arXiv ID: 2609.20128v1
链接: http://arxiv.org/abs/2609.20128v1
摘要: We develop and analyze a Monte Carlo method for sampling killed anomalous diffusions obtained by time-changing Brownian motion with drift by the inverse of a subordinator. The method targets probabilistic representations of time-nonlocal, including time-fractional, Cauchy–Dirichlet problems on bounded domains. Since inverse subordinators can be sampled exactly in broad classes, while Brownian exit times are generally unavailable in arbitrary domains, we approximate the killed Brownian component by an Euler scheme with discrete boundary detection. We prove a square-root weak error bound with explicit dependence on the Laplace exponent of the subordinator, and derive mean-square and central limit results for the resulting Monte Carlo estimator. A numerical example in the disk and one in a high-dimensional anisotropic shell illustrate the theoretical rates, computation time, and the mesh-free character of the method.
6. 通过 EIFEM 加速结构优化：一种基于 ROM 的预条件子
Accelerating structural optimization via EIFEM: a ROM-based preconditioner
作者: Raul Rubio, Alex Ferrer, Joaquin Alberto Hernandez, Pablo Antolin
类别: math.NA
发布日期: 2026-09-17
arXiv ID: 2609.20053v1
链接: http://arxiv.org/abs/2609.20053v1
摘要: Topology optimization (TopOpt) is a standard tool for structural conceptual design, providing optimal structures with great design freedom. However, it has a high computational costs due to the repeated evaluation of high-fidelity finite element models and frequently produces complex geometries that require extensive interpretation for manufacturability. As an alternative, this work presents a parameterized optimization framework based on Reduced Order Modeling (ROM) and preconditioning techniques. We employ the Empirical Interscale Finite Element Method (EIFEM) coupled with the Discrete Empirical Interpolation Method (DEIM) to construct localized, parameter-dependent reduced operators. From an optimization perspective, this approach can be interpreted as a preconditioning strategy in which inexact gradients are used to accelerate convergence. Furthermore, parameterizing the design space in terms of explicit geometric features, such as inclusion radii or lattice widths, significantly improves the manu facturability of the resulting optimized designs. We assess the performance of the method in terms of computational time, design topology and structural performance with TopOpt as baseline for three differ ent unit cell geometries and three different benchmarks. While the parametric ROM framework requires an initial offline training phase, our comparative analysis demonstrates that it drastically accelerates the online optimization loop and that it can produce even stiffer design in some of our experiments.
7. 多目标相遇传播子：Dirichlet-to-Neumann 谱形式
Encounter Propagator for Multiple Targets: A Dirichlet-to-Neumann Spectral Formalism
作者: Denis S. Grebenkov
类别: cond-mat.stat-mech, math-ph, math.NA, math.SP
发布日期: 2026-09-17
arXiv ID: 2609.19995v1
链接: http://arxiv.org/abs/2609.19995v1
摘要: We develop an encounter-based formulation of restricted diffusion in a bounded domain with multiple targets, resolving separately the boundary local time accumulated on each target. Unlike the single-target problem, the target projection operators do not generally commute with the governing Dirichlet-to-Neumann (DtN) operator, so that its eigenbasis does not diagonalize the local-time dependence. We overcome this difficulty by decomposing the DtN operator into target-restricted blocks. Multi-dimensional Laplace inversion then yields a convergent switching expansion, whose successive terms describe alternating diffusive transfers between the targets. We also derive an equivalent operator-valued renewal equation. For two targets, diagonalizing the target-restricted blocks provides an explicit spectral representation in terms of two DtN block spectra and an inter-target coupling matrix. When the coupling preserves spectral modes, the switching series can be resummed exactly in terms of modified Bessel functions. In general, as the off-diagonal DtN block is smoothing for separated targets, one can resort to low-rank finite-dimensional approximations. We examine an effective two-mode reduction for small, well-separated targets and identify a regime in which repeated inter-target transfers are progressively suppressed. Probabilistic interpretations and implications for diffusion-controlled reactions are discussed.
8. 通过边界验证六面体
Validating Hexahedra through their Boundaries
作者: Paul Zhang
类别: math.NA
发布日期: 2026-09-17
arXiv ID: 2609.19926v1
链接: http://arxiv.org/abs/2609.19926v1
摘要: Trilinear hexahedral mesh elements are used in finite element analysis to simulate volumetric phenomena. Realism of the simulation mandates that mesh elements through the trilinear map maintain positive volume, or mathematically, that the trilinear map maintains positive Jacobian determinant (jacdet). While jacdet positivity generally needs to be verified in the full element volume, we prove Knupp’s conjecture which posits that for trilinear hexahedra, jacdet positivity on the boundary of the element is sufficient to guarantee positivity in the entire element. We further prove that for any valid hexahedron, its globally minimal jacdet must reside on the boundary. Lastly, we prove that the globally minimal jacdet on the boundary of a hexahedron, valid or not, must be achieved within a finite set of candidate points that can be determined by quartic root finding. Combining these results, we can algorithmically validate a hexahedron through evaluation of its jacdet on a finite set of boundary points.
9. 基于熵的局部特征分解
Entropy-Based Local Characteristic Decomposition
作者: Shaoshuai Chu, Michael Herty, Alexander Kurganov
类别: math.NA
发布日期: 2026-09-17
arXiv ID: 2609.19838v1
链接: http://arxiv.org/abs/2609.19838v1
摘要: Local characteristic decomposition (LCD) is widely used in high-order numerical methods for hyperbolic systems of conservation laws to reduce spurious oscillations appearing in the computed solutions. The LCD implementation requires a representative average interface state, typically obtained using arithmetic or Roe-type averages of the nearly grid values. We propose an entropy-based LCD (ELCD), in which the states in the two cells adjacent to an interface are considered together with several nearby states in phase space. Among these candidates, we select the state that locally minimizes the entropy and use it as an average interface state for linearizing the flux Jacobian. We incorporate the ELCD into several second-order finite-volume and fifth-order finite-difference schemes. Numerical experiments for the two-dimensional Euler equations of gas dynamics show that the schemes, which utilize the ELCD procedure generally resolve complex wave structures more sharply than their counterparts, which use the arithmetic averages.
10. 用于基于 Raviart-Thomas 有限元的高效湍流流动模拟的混合精度 GPU 算法
Mixed-precision GPU algorithms for efficient turbulent flow simulations with Raviart-Thomas finite elements
作者: Ivan Prusak, Enes Mustafa Soydan, Ivan Pribec, Martin Kronbichler
类别: physics.flu-dyn, cs.MS, math.NA
发布日期: 2026-09-17
arXiv ID: 2609.19828v1
链接: http://arxiv.org/abs/2609.19828v1
摘要: We propose GPU algorithms for high-fidelity simulation of incompressible turbulent flows. Discretization in space is performed with H(div)-conforming high-order Raviart-Thomas finite elements for the velocity and an $L^2$-conforming discontinuous Galerkin approximation for the pressure. In time, a consistent splitting scheme based on higher-order BDF time stepping is used, with convection treated explicitly. In this scheme, a pressure Poisson equation and a symmetric reaction-diffusion-type equation for the velocity need to be solved in each time step. We develop a solution framework with fast matrix-free operator evaluation for all ingredients, combined with multigrid solvers for the Poisson problem, and propose a robust mixed-precision algorithmic framework. A key to mixed-precision efficiency is a least-squares projection to generate accurate initial guesses for the iterative linear solvers, enabling us to work with relative residual tolerances of $10^{-3}$. In this regime, running the solvers entirely in single precision leads to almost no change in overall iteration counts and maintains the crucial turbulence statistics, while showing up to $1.7\times$ speedup over pure double-precision simulations.
11. 曼哈顿格上镜像模型中的边界奇偶性
Boundary parity in the mirror model on the Manhattan lattice
作者: Jian Gu, Qin Hao
类别: math-ph, math.NA
发布日期: 2026-09-16
arXiv ID: 2609.19498v1
链接: http://arxiv.org/abs/2609.19498v1
摘要: We study a finite-domain event for the mirror model on the Manhattan lattice: every boundary-to-boundary trajectory crosses a marked connector an even number of times, while internal cycles are unrestricted. Our main theorem gives a finite-scale confinement criterion: if this event’s probability reaches a universal threshold in one even two-square rectangle, then almost surely every trajectory in the plane is periodic. We construct an exactly normalized three-color tensor representation and prove deterministic bounds for approximate contractions using the signed partial-permutation structure of directed continuations. These tools turn numerical evaluations into rigorous finite-volume statements. At mirror density (p=2/5), we prove that the boundary-parity probability is nonmonotone in the rectangle width, with a decrease followed by a certified increase. These values remain below the sufficient confinement threshold and do not resolve planar localization at this density. Further experiments track how the measured finite-size minimum varies with density, compare matching-based and entrywise error bounds, and exhibit endpoint-sensitive and cooperative responses to local mirror changes.
12. 结合观测 nudging 的用于二叠纪盆地甲烷排放监测的站点尺度、多分辨率 WRF 风场建模
Site-scale, Multi-resolution WRF Wind Modeling with Observational Nudging for Methane Emission Monitoring in the Permian Basin
作者: Ehsan Kharazmi, Erin Menezes, George Karniadakis, Anantha Sundaram, Arash Fathi
类别: math-ph, math.NA
发布日期: 2026-09-16
arXiv ID: 2609.19442v1
链接: http://arxiv.org/abs/2609.19442v1
摘要: Accurate methane source localization and emission-rate estimation at oil and gas facilities require wind fields that resolve site-scale spatial and temporal variability, which cannot be characterized by a single anemometer or coarse operational weather products. We develop a multiscale Weather Research and Forecasting (WRF) framework for a Permian Basin facility that dynamically downscales hourly, 3 km High-Resolution Rapid Refresh (HRRR) fields through four nested domains with grid spacings of 3 km, 1 km, 200 m, and 40 m. The two outer domains use planetary-boundary-layer parameterization, whereas the two inner domains operate in large-eddy-simulation mode. One-minute wind observations from two on-site anemometers are assimilated through wind-only observational nudging in the innermost domain, and the effects of nesting feedback and turbulence-closure choices are examined. Simulations for winter and summer 2025 periods are evaluated using near-surface wind-speed time series, power spectra, spatial fields, and statistical metrics. The baseline simulation reproduces the broad evolution of observed wind events but drifts during weak-wind periods. Observational nudging with one- and two-way nesting reduces the absolute bias of wind-speed by %65 at one sensor and %87 at the other, while reducing the corresponding root-mean-square errors by %33 and %44. Nudging also increases high-frequency energy and resolved spatial gradients, although improvements in turbulence statistics are not uniform. These results demonstrate a practical physics-based pathway for generating high-resolution wind fields for methane-plume modeling while emphasizing the need for independent spatial observations to validate accuracy away from assimilated sensors.
13. Stiefel 注意力：当 Transformer 投影矩阵的几何主导优化器选择时——以及何时并不主导
Stiefel Attention: When the Geometry of Transformer Projection Matrices Dominates Optimizer Choice—and When It Does Not
作者: Rubén Darío Guerrero
类别: cs.LG, math.NA
发布日期: 2026-09-16
arXiv ID: 2609.19363v1
链接: http://arxiv.org/abs/2609.19363v1
摘要: The query and key projections $\WQ,\WK$ in attention are almost always trained by Euclidean optimizers with no constraint on their geometry. We constrain them to the Stiefel manifold and optimize them there with a Riemannian Adam that carries one scalar second moment per frame, caps its step by a trust region, and retracts polarly. Four propositions prove this update is steepest descent in the embedded metric, independent of gradient scale, well conditioned, and exactly $\mathrm{O}(d)$-equivariant, each certified numerically in \texttt{float64}. A fifth supplies the mechanism: weight decay has \emph{identically zero} Riemannian gradient on $\St(d,r)$, since $W = W I_r$ lies in the normal space, so the learned attention geometry survives the collapse cycles that decay drives through the rest of the model. On modular arithmetic grokking, a single run holds $97.0\%$ validation accuracy at epoch 20,000 against the baseline’s $61.1\%$—an unstable endpoint we report as evidence for the mechanism rather than as an effect size. On CIFAR-10 patches the same rule gains $\mathbf{+8.98}$,pp over 12 paired starts ($t{=}60.6$, $12/12$), and the gap widens with data rather than eroding. The step rule earns this: a fixed-step Riemannian update is degree one in the gradient, so it moves $24$–$40\times$ less per step than an identically shaped AdamW matrix—its frames barely leave their initialization, and freezing them outright costs only $0.28$,pp. An ablation credits the whole gain to making the step scale free, and nothing measurable to the projector or to equivariance. A negative result sharpens the account: gauge removal cannot motivate the method, because a direction along which the loss is invariant carries no gradient at all.