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1. Helmholtz方程HDG离散的网格一致稳定性与误差估计
Mesh-Uniform Stability and Error Estimates for HDG Discretizations of the Helmholtz Equation
作者: Yukun Yue
类别: math.NA
发布日期: 2026-07-05
arXiv ID: 2607.04507v1
摘要翻译: 我们重新审视Cui和Zhang针对带有一阶吸收边界条件的Helmholtz方程的可杂交间断Galerkin方法。他们的分析在没有施加耦合$h$和$\kappa$的网格约束的情况下证明了稳定性，但显式稳定性界仍然包含网格尺寸的负幂次。我们证明，这种依赖于网格的爆破并非HDG离散所固有的。在与参考文献分析相同的几何和网格框架下，对于固定的多项式次数以及上下一致有界的稳定化参数，我们建立了一个广义稳定性估计，其常数与网格尺寸无关。对于每一个指定的紧区间$0<\kappa_0\le\kappa\le\kappa_1<\infty$，稳定性常数可以关于$\kappa$局部一致地选取。证明用基于HDG离散分布梯度和Crouzeix–Raviart提升机制的紧性论证取代了Rellich恒等式和逆估计论证。作为推论，我们获得了适定性、对最小$L^2$数据的收敛性以及基于投影的误差估计，其中先前理论中出现的网格尺寸的负幂次被消除。数值实验证实了所预测的网格一致行为，并显示出与先前理论所暗示的依赖网格的增长之间的明显对比。
Comment: 无
链接: http://arxiv.org/abs/2607.04507v1
1. 研究对象和背景
亥姆霍兹方程是时谐波传播问题的基本模型，其典型形式为 $-\\Delta u - \\kappa^2 u = f$，$\\kappa>0$ 为波数。该方程出现在声波、电磁散射、地震成像等众多应用中。从偏微分方程角度看，亥姆霍兹方程对应的变分形式包含负质量项 $-\\kappa^2(u,v)$，因此在 $H^1(\\Omega)$ 上不是强制椭圆型的。这使得有限元逼近面临本质困难：离散系统稳定性的波数依赖、数值色散造成的污染效应，以及预渐近区间的非标准收敛行为，都严重制约了实用计算的精度与效率。此外，非光滑区域上角点奇性的高频渐近进一步影响可达到的收敛阶。因此，构造对波数稳健、对网格尺寸不敏感的离散格式及其稳定性分析，是亥姆霍兹方程数值解的核心课题。
本文关注混合间断伽辽金（HDG）方法。HDG 方法通过引入单元局部未知量和骨架迹未知量，将全局自由度限制在网格骨架上，从而继承了间断伽辽金方法的局部守恒、高并行性等优势，同时显著缩减了系统规模。这类方法在诸多领域已有成功应用，例如对流占优扩散、斯托克斯方程、线弹性、不可压 Navier–Stokes 和 Cahn–Hilliard 方程等。在亥姆霍兹问题中，已有若干混合或可杂交格式的研究，包括针对异质介质的多尺度混合方法以及基于特征变量的 HDG 方法。特别地，Cui 和 Zhang [11] 对带有一阶吸收边界条件 $\\partial_n u + i\\kappa u = g$ 的亥姆霍兹方程提出了 HDG 格式，并在二维和三维情形下证明了：对任意 $h>0$ 和 $\\kappa>0$，该方法无需满足 $h\\kappa^2$ 充分小这一较强的网格约束即可稳定并适定。这一成果较之前仅适用于狄利克雷问题的 HDG 分析[17] 有本质推进。
Cui–Zhang 的稳定性分析虽然避开了 $h\\kappa^2$ 约束，但其明确给出的稳定性常数 $C_{\\mathrm{CZ}}(h,\\kappa,\\tau)$ 仍包含网格尺寸 $h$ 的负幂次项，如 $\\tau_{\\min}^{-1}h^{-3} + \\tau_{\\max}h^{-1}$。具体地，对于由最大单元直径 $h$、分片常数稳定参数 $\\tau$（边界上为正）构成的 HDG 空间，[11] 定理 3.1 中的广义稳定因子为
$$ C_{\\mathrm{CZ}}(h,\\kappa,\\tau) = \\big( \\kappa^2\\tau_{\\min}^{-1} + \\kappa + \\tau_{\\min}^{-1}h^{-3} + \\tau_{\\max}h^{-1} \\big)^2 + 1. $$当 $h\\to 0$ 时，该上界趋于无穷，特别是常见选择 $\\tau=\\kappa$ 时，括号内主导项为 $\\kappa^{-1}h^{-3}$，导致稳定常数像 $h^{-6}$ 一样增长。这并非单纯的美观问题：同一稳定因子直接进入投影误差估计，从而最终 $L^2$ 误差界也带有 $h$ 的负幂次，理论上暗示误差可能在网格加密时恶化，这与实际计算经验相悖。该网格奇异的根源在于 [11] 的证明技巧：其核心工具是单元上的 Rellich 恒等式，结合乘子型离散检验函数以及逆估计来处理切向跳跃项，逆估计不可避免地带入了 $h$ 的负幂次。
本文试图填补的空白是：证明 Cui–Zhang HDG 格式的稳定常数可以摆脱网格尺寸 $h$ 的负幂次依赖，即稳定性对网格一致。在保持与 [11] 相同的几何假设（区域对于某内点 $x_\\Omega$ 是严格星形的）和网格假设（形状正则的单纯形网格族）下，本文构造了新的证明框架，完全避免了逆估计，转而利用 HDG 离散分布梯度和 Crouzeix–Raviart 提升的紧性论证。最终建立的稳定性估计中，常数 $C_{\\mathrm{st}}$ 与 $h$ 无关，且对于任意给定的紧波数区间 $0<\\kappa_0\\le\\kappa\\le\\kappa_1<\\infty$ 可局部一致选取。由此，适定性、对最低 $L^2$ 正则性数据的收敛性以及投影误差估计中 $h$ 负幂项均被清除。数值实验验证了理论预测的网格一致行为，并与先前包含网格奇异的稳定性曲线形成鲜明对比。
2. 主要定理或主要结果
本文在固定多项式次数 $p$ 和稳定参数 $\\tau$ 满足一致上下界（对波数局部一致）的假设下，建立了如下主要结果：
广义辅助 HDG 系统的网格一致稳定性（定理 4.3）。对于引入辅助右端项 $\\mathbf{Q}\\in \\mathbf{V}_h$ 的广义系统（2.6），存在与 $h$ 无关的常数 $C_{\\mathrm{st}}$，使得对任意 $h>0$ 及任意解 $(\\mathbf{q}_h, u_h, \\widehat{u}_h)\\in \\mathbf{V}_h\\times W_h\\times M_h$，成立
$$ \\|\\mathbf{q}_h\\|_{\\mathcal{T}_h}^2 + \\kappa^2\\|u_h\\|_{\\mathcal{T}_h}^2 + \\kappa\\|\\widehat{u}_h\\|_{\\partial\\Omega}^2 + \\|\\tau^{1/2}(u_h-\\widehat{u}_h)\\|_{\\partial\\mathcal{T}_h}^2 \\le C_{\\mathrm{st}}\\big( \\|f\\|_{\\mathcal{T}_h}^2 + \\|g\\|_{\\partial\\Omega}^2 + \\|\\mathbf{Q}\\|_{\\mathcal{T}_h}^2 \\big). $$常数 $C_{\\mathrm{st}}$ 可对 $\\kappa$ 在任意紧区间 $[\\kappa_0,\\kappa_1]$ 上一致选取，只要 $\\tau|_{F}$ 在该区间上满足 $0<\\tau_0\\le\\tau|_F\\le\\tau_1<\\infty$。该估计去除了 [11] 中出现的 $h^{-3}$ 和 $h^{-1}$ 项，实现了真正意义上的网格一致稳定。
小网格情形的紧性稳定性（定理 4.2）。首先证明存在仅依赖于域、多项式次数、形状正则常数、波数区间和稳定参数界的 $h_0>0$ 和 $C_{\\mathrm{sm}}>0$，使得对所有 $0
适定性与收敛性（推论 4.4 及 §5.1）。网格一致稳定性直接推出齐次系统只有零解，从而原 HDG 系统（$\\mathbf{Q}=0$）对任何右端项均唯一可解。此外，利用紧性工具可将稳定性用于极限过程：若数据 $f\\in L^2(\\Omega), g\\in L^2(\\partial\\Omega)$，则可证明 HDG 解收敛到连续问题的弱解。
投影误差估计中 $h$ 负幂的消除（§5.2）。将网格一致稳定性估计应用于 HDG 投影误差方程（此时 $\\mathbf{Q} = \\mathbf{\\Pi q} - \\mathbf{q}$ 为局部投影残差），得到形如
$$ \\|\\mathbf{\\Pi q} - \\mathbf{q}_h\\|_{\\mathcal{T}_h} + \\kappa\\|\\Pi u - u_h\\|_{\\mathcal{T}_h} \\le C_{\\mathrm{st}}\\big( \\|\\mathbf{q} - \\mathbf{\\Pi q}\\|_{\\mathcal{T}_h} + \\|u - \\Pi u\\|_{\\mathcal{T}_h} + \\text{类似边界项} \\big) $$的误差界，右端不再含有 $h$ 的负幂次，从而最优收敛阶得以恢复。数值实验证实了预期的收敛速率，并显示实测稳定比常数的网格一致行为，与 [11] 中形式界给出的 $h^{-3}$ 增长趋势形成鲜明对比。
3. 研究方法、关键技术和核心工具
本文的核心突破在于用紧性论证替代了以往导致网格奇异的逆估计–Rellich 恒等式路径。具体的技术创新与证明框架如下：
HDG 离散分布梯度与广义系统表示。为隔离分析，引入一辅助广义 HDG 系统 (2.6)，其中第一方程包含一个额外的向量右端项 $\\mathbf{Q}\\in \\mathbf{V}_h$。定义 HDG 离散分布梯度算子 $\\mathbf{G}_h: W_h\\times M_h \\to \\mathbf{V}_h$，通过
$$ (\\mathbf{G}_h(v_h,\\widehat{v}_h), \\mathbf{r}_h)_{\\mathcal{T}_h} = -(v_h, \\nabla\\!\\cdot\\!\\mathbf{r}_h)_{\\mathcal{T}_h} + \\langle\\widehat{v}_h, \\mathbf{r}_h\\!\\cdot\\!\\mathbf{n}\\rangle_{\\partial\\mathcal{T}_h} \\qquad \\forall \\mathbf{r}_h\\in\\mathbf{V}_h $$或等价地通过单元分部积分定义。该算子将单元未知量 $u_h$、骨架迹 $\\widehat{u}_h$ 与通量 $\\mathbf{q}_h$ 通过关系式 $\\mathbf{q}_h + \\mathbf{G}_h(u_h,\\widehat{u}_h) = \\mathbf{P}_V\\mathbf{Q}$ 联系起来。这一构造是紧性分析的关键：$\\mathbf{G}_h$ 的行为类似于弱梯度，且与连续分布梯度满足一致性 $\\mathbf{G}_h(P_W v, P_M v) = \\mathbf{P}_V\\nabla v$。
能量恒等式。对广义系统可导出两个能量关系（引理 4.1），实部和虚部分别为
$$ \\|\\mathbf{q}_h\\|_{\\mathcal{T}_h}^2 - \\kappa^2\\|u_h\\|_{\\mathcal{T}_h}^2 = \\Re\\big[(f,u_h) + (\\mathbf{q}_h,\\mathbf{Q}) + \\langle g,\\widehat{u}_h\\rangle_{\\partial\\Omega}\\big], $$$$ \\|\\tau^{1/2}(u_h-\\widehat{u}_h)\\|_{\\partial\\mathcal{T}_h}^2 + \\kappa\\|\\widehat{u}_h\\|_{\\partial\\Omega}^2 = \\Im\\big[(f,u_h) + (\\mathbf{q}_h,\\mathbf{Q}) + \\langle g,\\widehat{u}_h\\rangle_{\\partial\\Omega}\\big]. $$这两个恒等式将解的范数直接与数据线性项控制，是接下来稳定性估计的出发点。
HDG 紧性工具。本文的核心技术成分是定理 3.2 给出的 HDG 离散紧性。考虑一族满足
$$ \\|w_h\\|_{L^2(\\Omega)} + \\|\\mathbf{G}_h(w_h,\\widehat{w}_h)\\|_{L^2(\\Omega)} + \\|w_h - \\widehat{w}_h\\|_{L^2(\\partial\\mathcal{T}_h)} \\le C $$的序列 $(w_h,\\widehat{w}_h)\\in W_h\\times M_h$，当 $h\\to 0$ 时必存在子列及 $w\\in H^1(\\Omega)$，使得
$$ w_h \\to w\\;\\text{于}\\;L^2(\\Omega),\\quad \\mathbf{G}_h(w_h,\\widehat{w}_h) \\rightharpoonup \\nabla w\\;\\text{于}\\;L^2(\\Omega)^d,\\quad \\widehat{w}_h|_{\\partial\\Omega} \\rightharpoonup w|_{\\partial\\Omega}\\;\\text{于}\\;L^2(\\partial\\Omega). $$该紧性结论依赖于 Crouzeix–Raviart 提升技巧：首先对骨架迹 $\\widehat{w}_h$ 取面平均得到分片面常数函数 $\\overline{\\widehat{w}}_h$，再将其提升为 Crouzeix–Raviart 非协调元函数 $\\mathcal{L}_h\\overline{\\widehat{w}}_h$。利用已知的 CR 空间估计可得
$$ \\|w_h - \\mathcal{L}_h\\overline{\\widehat{w}}_h\\|_{L^2(\\Omega)} \\le C\\big( h\\|\\mathbf{G}_h(w_h,\\widehat{w}_h)\\|_{L^2(\\Omega)} + h^{1/2}\\|w_h-\\widehat{w}_h\\|_{L^2(\\partial\\mathcal{T}_h)} \\big), $$从而 $w_h$ 与 CR 提升函数之间的差距随 $h$ 趋向于零。CR 空间本身的紧性（其离散范数对梯度一致有界蕴含强收敛子列）与迹紧性即可导出上述结论。这一工具完全避开了逆估计，为后面稳定性证明中取极限提供了途径。
小网格稳定性通过反证法与紧性证明。定理 4.2 的证明采用标准紧性反证法。若结论不成立，则存在网格尺寸 $h_n\\to 0$ 的序列、波数 $\\kappa_n\\to \\kappa_*$、一致有界的稳定参数、以及趋于零的数据 $(f_n,g_n,\\mathbf{Q}_n)$，使得相应的解满足归一化条件
$$ \\|\\mathbf{q}_n\\|_{\\mathcal{T}_{h_n}}^2 + \\kappa_n^2\\|u_n\\|^2 + \\kappa_n\\|\\widehat{u}_n\\|_{\\partial\\Omega}^2 + \\|\\tau_n^{1/2}(u_n-\\widehat{u}_n)\\|_{\\partial\\mathcal{T}_{h_n}}^2 = 1, $$但右端数据范数之和趋于零。利用能量恒等式可逐步推出 $\\|\\mathbf{q}_n\\|^2 - \\kappa_n^2\\|u_n\\|^2 \\to 0$，$\\|u_n-\\widehat{u}_n\\|_{\\partial\\mathcal{T}_{h_n}}\\to 0$，且 $\\kappa_n^2\\|u_n\\|^2 \\to 1/2$，$\\|\\mathbf{q}_n\\|^2 \\to 1/2$。进而由 $\\mathbf{q}_n = \\mathbf{P}_{V_n}\\mathbf{Q}_n - \\mathbf{G}_{h_n}(u_n,\\widehat{u}_n)$ 知 $\\mathbf{G}_{h_n}(u_n,\\widehat{u}_n)$ 在 $L^2$ 中有界，满足紧性定理的条件。于是可抽出子列使 $(u_n,\\widehat{u}_n)$ 收敛到某个 $u\\in H^1(\\Omega)$，且 $\\mathbf{G}_{h_n}(u_n,\\widehat{u}_n)\\rightharpoonup \\nabla u$，$\\mathbf{q}_n\\rightharpoonup \\mathbf{q}$ 满足 $\\mathbf{q} + \\nabla u = 0$。利用离散方程的配点形式（引理 2.1）及嵌入映射 $\\iota_h v = (P_W v, P_M v)$ 的逼近性质，取极限可得 $u$ 满足
$$ -\\Delta u - \\kappa_*^2 u = 0,\\quad u|_{\\partial\\Omega}=0,\\quad \\partial_n u = 0, $$由星形区域上齐次 Cauchy 数据的唯一延拓结论（附录 B）得 $u\\equiv 0$。但强 $L^2$ 收敛 $u_n\\to u$ 与 $\\kappa_n\\|u_n\\| \\to 1/\\sqrt{2}$ 矛盾，从而完成证明。该过程完全未使用逆估计，所得稳定性常数不显含 $h$ 的负幂。
全局稳定性结合粗网格情形。对小网格 $0
4. 与之前工作的比较
本文与最相关的已有工作——Cui 和 Zhang [11]（以下简称 CZ）——进行系统比较，从假设、证明技术、所得稳定性界及误差估计等维度阐述推进与差异。
稳定性常数对网格的依赖。CZ 证明了 HDG 方法在无 $h\\kappa^2$ 网格约束下的稳定性和适定性，但其明确给出的广义稳定因子 $C_{\\mathrm{CZ}}(h,\\kappa,\\tau)$（见 (1.1)）包含 $h^{-3}$ 和 $h^{-1}$ 项。这意味着当网格加密时，稳定常数的理论上界无界增长，即使数值上稳定性可能仍然很好。本文的直接贡献是证明了稳定常数可与 $h$ 无关，即整个 HDG 解算子在适当的范数下具有关于 $h$ 一致的界。这一结论不仅消除了理论上的奇性，也使得随后导出的误差估计不再受 $h$ 负幂困扰。
证明技术的根本差异。CZ 的证明基于每个单元上的 Rellich 恒等式，运用一种乘子型离散检验函数，再结合逆估计处理切向跳跃项。逆估计步骤不可避免地产生了 $h^{-1}$ 等因子，并最终累积为 $h^{-3}$。而本文完全舍弃了这一路线，代之以紧性方法。核心是构造 HDG 离散分布梯度 $\\mathbf{G}_h$ 并利用 Crouzeix–Raviart 提升技巧建立离散紧性（定理 3.2），该紧性工具使得小网格情形可通过反证法直接得出稳定性，无需任何逆估计。证明框架更接近于连续问题的紧性分析，将 HDG 格式看作对梯度的相容逼近，从而在 $h\\to 0$ 时自动保证稳定性。这一技术途径可能会对其它非强制问题的 HDG 稳定性分析有所启发。
假设条件与适用范围。本文保持了与 CZ 完全相同的几何条件（区域对于某个内点严格星形，满足 $(x-x_\\Omega)\\cdot \\mathbf{n}_\\Omega \\ge c_\\Omega>0$）和网格条件（形状正则的单纯形剖分族）。对 HDG 多项式的次数 $p$ 固定，稳定参数 $\\tau$ 要求分片面常数正且有一致上下界，对波数局部一致（即对任一紧波数区间 $[\\kappa_0,\\kappa_1]$，存在与 $h,\\kappa$ 无关的界 $0<\\tau_0\\le\\tau|_F\\le\\tau_1$）。CZ 的明确估计中稳定参数以 $\\tau_{\\min}^{-1}$ 和 $\\tau_{\\max}$ 形式出现，虽也有界假设，但其常数更显式地依赖于这些界。本文放宽了对 $\\tau$ 与 $\\kappa$ 精确比例的要求，只须它们随波数保持有界且远离零即可，从而覆盖了 $\\tau = c_\\tau\\kappa$ 等常见取法。
波数依赖的形式。CZ 的稳定常数可显式写出对 $\\kappa$ 的多项式依赖，如 $\\kappa^2\\tau_{\\min}^{-1}+\\kappa$，但含有 $h$ 负幂。本文在波数上仅保证了局部一致界：对任何紧区间 $[\\kappa_0,\\kappa_1]$，可选取统一的 $C_{\\mathrm{st}}$。然而，若 $\\kappa$ 趋于零或无穷，常数可能恶化，文中未给出这种情形的定量分析。一个未完全澄清的问题是：当 $\\kappa\\to 0$ 或 $\\kappa\\to\\infty$ 时，本文的常数如何增长？由于证明是紧性论证，不能直接读出其对波数的显式依赖，这可能是未来值得探索的方向。
误差估计的提升。CZ 的投影误差估计最终形式为
$$\\|\\mathbf{q}-\\mathbf{q}_h\\| + \\kappa\\|u-u_h\\| \\lesssim C_{\\mathrm{CZ}}(h,\\kappa,\\tau)\\, \\big(\\text{投影误差}\\big),$$其中稳定性因子带有 $h$ 负幂。这就意味着即使投影误差给出最优逼近阶，整体误差界仍可能随网格加密而反常增大。本文将该因子替换为与 $h$ 无关的 $C_{\\mathrm{st}}$，从而恢复了标准意义下的最优误差估计：当精确解具有足够正则性时，误差以 $h^{p+1}$ 等阶次衰减，不再有来自稳定因子的阻碍。数值实验明确显示了这一差异：实际计算的稳定比常数随 $h$ 减小保持有界，与 CZ 预测的 $h^{-3}$ 型增长完全不同。
对数据正则性的要求与收敛性。CZ 的分析中，为获得误差估计通常需要解具有一定的 $H^2$ 等正则性，以保证投影误差的阶。本文借助网格一致稳定性还证明了另一个副产品：只要右端数据 $f\\in L^2(\\Omega), g\\in L^2(\\partial\\Omega)$，无需更高正则性，HDG 解即收敛到连续问题的弱解（见 §5.1）。这是因为网格一致有界性结合离散紧性可直接取极限，而不依赖正则性假设。CZ 文中未明确讨论这一最低正则性收敛结果。
综上，本文在保持 CZ 已有几何和网格框架的前提下，通过引入紧性论证彻底消除了稳定常数中的网格奇异项，从而获得了真正网格一致的稳定性与误差估计，是一次实质性的理论完善。
2. 边多尺度有限元方法
Edge Multiscale Finite Element Methods
作者: Shubin Fu, Guanglian Li
类别: math.NA
发布日期: 2026-07-04
arXiv ID: 2607.03710v1
摘要翻译: 本文旨在综述近期发展的边多尺度有限元方法，用于求解含非均匀系数或高度振荡解的偏微分方程。以含非均匀系数的椭圆型方程为例，我们阐述了该方法的核心思想。在考虑多尺度基函数离散误差的情况下，我们对方法进行了分析。文中提供了大量数值实验来验证方法的性能。
Comment: 无
链接: http://arxiv.org/abs/2607.03710v1
1. 研究对象和背景
具有多尺度系数的偏微分方程广泛出现于应用科学与工程中，例如油藏模拟中的多尺度渗透率椭圆方程。这类问题中，系数在不同空间尺度上剧烈变化，导致解呈现高振荡或多尺度特征。标准的数值方法（如基于局部多项式基的有限元法）需要在最细尺度上进行离散化，才能以足够精度捕捉解的微观行为，这使得计算量在许多实际情形中变得不可承受。因此，过去数十年间发展出了均质化理论与多尺度数值方法。本文系统综述了边缘多尺度有限元方法（Edge Multiscale Finite Element Methods, EMsFEM）的近期进展。该方法由[17,10]首次提出，并已被成功应用于多类异质系数偏微分方程。其核心思想是在重叠子区域上局部求解，并通过单位分解函数将局部表示组合为全局分解，从而将全局误差估计归结为局部误差分析。本文以具有异质系数的椭圆方程为例，给出了方法的离散层次构造，建立了系统的误差分析，并通过数值实验验证了方法性能。模型问题为
$$ \begin{cases} A u = f & \text{在 } \Omega \text{ 内},\\ B_j u = g_j & \text{在 } \Gamma = \partial \Omega \text{ 上}, \ j=0,\dots,m-1, \end{cases} $$其中 $A: H^1(\Omega) \to H^{-1}(\Omega)$ 是形如
$$ A u = \sum_{|p|,|q|\le 1} (-1)^{|p|} \frac{\partial^{p}}{\partial x^{p}} \left( a_{p,q}(x) \frac{\partial^{q} u}{\partial x^{q}} \right) $$的二阶偏微分算子，边界算子 $B$ 的阶为 1。系数 $a_{p,q}\in L^\infty(\Omega)$ 可能包含多个尺度，从而在解中传播多尺度行为。$f\in L^2(\Omega)$，$g\in H^{-1/2}(\Gamma)$。这一模型涵盖了均匀椭圆但粗糙系数的算子、高波数 Helmholtz 算子、对流占优扩散方程以及不定 Maxwell 方程等情形。EMsFEM 并非仅限于稳态线性算子，它已被推广到半线性抛物问题等。问题的弱形式为：求 $u\in H^1(\Omega)$ 满足
$$ a(u,v) = (f,v)_\Omega + \langle g, v \rangle_\Gamma \quad \forall v\in H^1(\Omega), $$其中双线性形式 $a(\cdot,\cdot): H^1(\Omega)\times H^1(\Omega)\to\mathbb{R}$ 定义为
$$ a(v,w) = \sum_{|p|,|q|\le 1} \int_\Omega a_{p,q}(x) \frac{\partial^{q} v}{\partial x^{q}} \frac{\partial^{p} w}{\partial x^{p}} \,dx. $$假设 $a(\cdot,\cdot)$ 强制定有界，即存在常数 $0<\alpha\le\beta<\infty$，使得对所有 $v,w\in H^1(\Omega)$ 有 $a(v,v)\ge \alpha \|v\|_{H^1(\Omega)}^2$ 和 $a(v,w)\le \beta \|v\|_{H^1(\Omega)}\|w\|_{H^1(\Omega)}$，则 Lax–Milgram 定理保证解存在唯一。然而，多尺度系数要求细网格 $h$ 远小于宏观尺度，导致有限元空间 $V_h$ 维数巨大。为降低计算代价，多尺度方法构造一个远小于 $V_h$ 的近似空间，使其仍能以可控精度逼近真解。本文综述的 EMsFEM 正是基于子域边缘上的层次基构造局部多尺度空间，再通过单位分解拼装为全局空间。该方法的关键在于：局部空间仅依赖于边界上的低维自由度（按参数 $\ell$ 控制），且构造过程与右端项 $f$ 无关，因此在处理多右端项问题（如不确定性量化）时特别高效。然而，已有的 EMsFEM 分析多基于连续水平或对基函数做代价较高的全局修正，本文则给出了直接考虑离散（即有限元）基函数误差的严格收敛性分析，填补了该理论空白。
2. 主要定理或主要结果
论文的主要结果是定理 3.1 中给出的误差估计。记 $u_h\in V_h$ 为细网格上的标准有限元解（见 (3.2)），$u_{ms,\ell}\in V_{ms,\ell}$ 为边缘多尺度方法解（见 (3.12)），网格参数 $H$ 为粗网格尺寸，$h$ 为细网格尺寸，$\delta$ 为子域重叠宽度，$\bar d$ 为子域直径的最大值，且子域直径满足 $d_i \le H+2\delta$。令 $\ell$ 为边界层次基的级数（每个粗边上的节点数为 $n_i 2^\ell$）。在以下关键假设下：
- 对任意满足局部离散调和方程 $a_i(v_{h,i}, w_h)=0$ (对所有 $w_h\in V^0_{h,i}$) 的 $v_{h,i}\in V_{h,i}$，有 $L^2$ 先验估计 $\|v_{h,i}\|_{L^2(\Omega_i)} \lesssim d_i^{1/2} \|v_{h,i}|_{\partial\Omega_i}\|_{L^2(\partial\Omega_i)}$ (3.19)，以及 Caccioppoli 型估计 $\|\chi_i \nabla v_{h,i}\|_{L^2(\Omega_i)} \lesssim \delta_i^{-1} \|v_{h,i}\|_{L^2(\Omega_i)}$ (3.20)，此处 $\chi_i$ 为满足 (3.3) 的局部单位分解函数， 则以下误差界成立： $$ \|u_h - u_{ms,\ell}\|_{H^1(\Omega)} \lesssim \bar d \left(1 + \frac{\bar d}{\delta} + \frac{2^{-\ell/2}}{\delta}\right) \|f\|_{L^2(\Omega)}. $$ 该估计表明，当重叠宽度 $\delta$ 固定时，通过提高边界离散层次 $\ell$（即增加每边的基函数数目）可使误差以 $2^{-\ell/2}$ 衰减；增加重叠 $\delta$ 可改善常数，但当 $\delta$ 过大时 $\bar d$ 也随之增大，因此存在最优重叠选取。上述假设已在 Helmholtz 算子和不定对流-扩散问题中得到验证。数值实验（第 4 节）针对 Darcy 问题、对流-扩散问题和 Helmholtz 问题验证了理论结果，并重点考察了重叠宽度对精度的影响，显示出与理论一致的收敛趋势。
3. 研究方法、关键技术和核心工具
误差分析的核心障碍在于：多尺度空间 $V_{ms,\ell}$ 的局部基由离散调和延拓构成，其逼近性质取决于局部椭圆算子的正则性以及离散化误差。作者采用了如下创新框架：
1. 局部解的分裂与投影构造 首先将细网格解 $u_h$ 在每个子域上拆分为两部分：$u_h|_{\Omega_i} = u^0_{h,i} + u^\partial_{h,i}$，其中 $u^0_{h,i}\in V^0_{h,i}$ 满足带右端项 $f$ 的局部问题 (3.15)，而 $u^\partial_{h,i}$ 是调和部分（满足齐次方程但具有非齐次 Dirichlet 边界 $u_h|_{\partial\Omega_i}$）。对 $u^0_{h,i}$ 利用泡函数 $b_{h,i}$（由 (3.10) 定义）构造其逼近 $\hat u^0_{h,i} = \bar f_i b_{h,i}$，并利用引理 3.1（能量估计与 Friedrichs 不等式）得出误差 $e^0_{h,i}$ 的 $H^1$ 界 $\lesssim d_i^2 \|f\|_{L^2(\Omega_i)}$。对调和部分 $u^\partial_{h,i}$，构造其投影 $\hat u^\partial_{h,i} = P_{i,\ell}(u_h|_{\partial\Omega_i})$，其中 $P_{i,\ell}: V_{h,i}(\partial\Omega_i) \to V^{i;\partial}_{ms,\ell}$ 由 $A^{-1}_{h,i}(I_{i,\ell} v)$ 定义，$I_{i,\ell}$ 是边界上的 $L^2$ 正交投影。这样得到全局投影 $P_\ell$ (3.14)，并利用单位分解性质 (3.5) 将全局误差表示为局部误差之和。
2. 关键估计与假设 估计边界部分误差 $e^\partial_{h,i} = u^\partial_{h,i} - \hat u^\partial_{h,i}$ 时，核心是结合两个不等式：利用离散调和函数的 $L^2$ 先验估计 (3.19) 将内部范数转为边界范数，再利用边界 $L^2$ 投影 $I_{i,\ell}$ 的逼近性质：
$$ \|u_h - I_{i,\ell} u_h\|_{L^2(\partial\Omega_i)} \lesssim d_i^{1/2} 2^{-\ell/2} \|u_h\|_{H^1(\Omega_i)}. $$该逼近性质来源于层次基的构造（类似小波，网格尺寸约为 $d_i 2^{-\ell}$）。然后应用 Caccioppoli 估计 (3.20) 控制 $\|\chi_i \nabla e^\partial_{h,i}\|_{L^2(\Omega_i)}$。对于 $e^0_{h,i}$ 部分，同样需要 Caccioppoli 型估计处理单位分解的梯度项，并利用 $\nabla \chi_i$ 的界 $\lesssim \delta_i^{-1}$。最终通过有限交叠常数 $\Lambda$ 和加权延拓算子的稳定性，得到全局 $H^1$ 误差界。
3. 离散水平分析的意义 与以往 EMsFEM 工作相比，本文直接考虑细网格 $V_h$ 上的离散基函数误差，避免了连续问题中可能需要的高正则性假设。引理 3.1 与估计 (3.16)–(3.18) 均是在离散框架下推导的，只依赖细网格的正则性。假设 3.1 虽然需要在具体算子下验证，但已被作者在先前工作中对多种不定问题建立，为分析提供了统一框架。此外，证明中通过单位分解将全局投影 $P_\ell$ 构造为局部投影的加权和，并利用 (3.5) 实现精确的恢复性质，这一处理既保持了 Galerkin 正交性，又使得局部误差可以直接估计，从而将多尺度方法的全局收敛性完整归约为局部边值问题和边界逼近性质。
4. 与之前工作的比较
本文作为一篇综述，其贡献主要体现在 EM-sFEM 框架下离散误差分析的严格化与系统化。与已有工作的比较可以从以下几方面展开：
1. 离散误差分析 vs. 连续水平分析 早期 EMsFEM 的工作（如[10,11,12]）多是在连续水平上构造局部多尺度基函数（即利用算子真解的广义调和延拓），再通过有限元离散求解。这类分析往往假设真解具有足够的正则性以保证局部基函数的逼近阶。本文则直接在离散水平（细网格 $V_h$）定义局部基为离散调和延拓 (3.9) 和泡函数 (3.10)，允许系数可能只有 $L^\infty$ 正则性，且不要求解的光滑性高于 $H^1$。引理 3.1 和估计 (3.21) 展示了即使在一般系数下，用离散泡函数逼近局部强制部分仍能达到最优误差。此离散框架的误差分析是本文的主要推进之一。
2. 对假设的系统整合 以往针对不同算子的 EM-sFEM 分析需要根据具体物理背景逐个验证关键估计（如对 Helmholtz 方程的稳定性估计）。本文将两类本质估计抽象为统一的假设 3.1（即 $L^2$ 先验和 Caccioppoli 型估计），并在该假设下给出一般性的误差界。作者指出，这些假设已在 Helmholtz、对流-扩散等算子的离散设置下得到验证[13,14]，从而表明所给框架的通用性。同时，这也是首次在 EMsFEM 文献中显式以假设形式固定这些条件，便于后续应用只需检验这两个不等式即可获得收敛性。
3. 重叠大小与层次参数的影响 已有工作 [10,15] 通常仅给出误差对网格尺寸 $H$ 的依赖，而对重叠宽度 $\delta$ 的影响分析不足。本文定理 3.1 明确展示了三项误差：$\bar d$ 项、$\bar d^2/\delta$ 项和 $\bar d 2^{-\ell/2}/\delta$ 项。第一项来自泡函数逼近，第二项来自单位分解梯度与内部误差的耦合，第三项来自边界逼近。此表达式定量说明了增大重叠可以降低 $\delta^{-1}$ 的放大效应，但同时通过 $\bar d$ 与 $H+2\delta$ 的关系引入了平衡。数值实验部分专门考察了 $\delta/h$ 从 1 到 16 的误差变化，证实了中等重叠（如 $\delta/h=2$）即可获得精度和效率的较好折衷。与之前的数值实验相比，这里的参数研究更为系统。
4. 适用范围与代价 本文的方法构造适用于任何满足强制有界性的二阶椭圆型问题（允许不定但要求假设 3.1 成立）。与标准有限元相比，多尺度空间维数由局部边界自由度数 $\sum_i n_i 2^\ell$ 决定，当 $\ell$ 较小时远小于细网格维数。但代价在于需要求解每个子域上的多个局部问题（每个边界基函数对应一个离散调和延拓）以构造基。本文未与其他多尺度方法如异质多尺度方法（HMM）或广义多尺度有限元（GMsFEM）进行量化比较，但指出 EMsFEM 的特点在于基函数与右端项无关，适合多右端项情景。此外，对于高波数 Helmholtz 问题，文中仅展示了最精细层次（$\ell=2$）的结果，提示该方法在处理弥散误差方面仍有待改进。
综上，本文的贡献在于为 EMsFEM 提供了一个清晰且可操作的离散误差分析模板，明确了重叠和边界逼近阶的影响，并以统一假设的形式降低了新应用的理论验证门槛。
了解领域论文（摘要翻译）
1. 压力鲁棒的Stokes方程$\boldsymbol{H}(\mathrm{div})$-协调间断Galerkin方法的$hp$-后验误差估计
Pressure-robust $hp$-a posteriori error estimates of $\boldsymbol{H}(\mathrm{div})$-conforming discontinuous Galerkin methods for the Stokes equations
作者: Zhaonan Dong, Zuodong Wang, Lina Zhao
类别: math.NA
发布日期: 2026-07-06
arXiv ID: 2607.04787v1
链接: http://arxiv.org/abs/2607.04787v1
摘要翻译
中文标题: 压力鲁棒的Stokes方程$\boldsymbol{H}(\mathrm{div})$-协调间断Galerkin方法的$hp$-后验误差估计
摘要翻译: 我们设计并分析了一个压力鲁棒、基于残差的$hp$-后验误差估计子，该估计子适用于二维和三维多面体Lipschitz区域上Stokes问题的$\boldsymbol{H}(\mathrm{div})$-协调间断Galerkin（dG）方法。该估计子给出了能量范数意义下速度误差的上界和局部下界，两者均对黏性具有鲁棒性且与压力无关。我们的分析依赖于将误差分解为协调部分和非协调部分。非协调误差通过单位分解框架并结合顶点补丁上的局部Helmholtz分解得到控制。协调误差则利用[14]中的广义Bogovskiĭ算子，在二维和三维情形下进行分析，得到了分别对应于不同插值算子的两个与压力无关的基于残差的估计子。在第一种方法中，协调误差的上界由五个误差指示子以及一个数据振荡项组成。其中四个指示子具有$p$-最优的尺度依赖，而剩余的一个在尺度上差一个$p^{1/2}$因子。在第二种方法中，上界仅包含两个残差指示子以及数据振荡项，代价是损失了一个$p$的阶。此外，受四阶偏微分方程相关技术的启发，使用$H^2$-气泡函数建立了压力鲁棒的局部下界。二维和三维的数值结果证实了所提出估计子的可靠性、有效性和压力鲁棒性。
2. 流形上椭圆型方程的基于物理信息神经网络的区域分解方法
Domain decomposition methods with Physics-informed neural networks for elliptic equations on manifolds
作者: Yufang Jiang, Lizhen Qin, Feng Wang
类别: math.NA, math.DG
发布日期: 2026-07-05
arXiv ID: 2607.04285v1
链接: http://arxiv.org/abs/2607.04285v1
摘要翻译
中文标题: 流形上椭圆型方程的基于物理信息神经网络的区域分解方法
摘要翻译: 我们针对紧致黎曼流形上的椭圆型方程，基于物理信息神经网络(PINNs)，提出了两种数值区域分解方法(DDMs)。我们的方法结合了流形的区域分解技术以及神经网络在高维情形下的优势。所提方法通过在不同流形上的数值实验得到了验证，这些流形带有或不带边界，维度从$5$到$10$。
3. 基于梯度流的欧拉非线性弹性与扩散离散化
Discretisation of Eulerian nonlinear elasticity and diffusion using gradient flows
作者: Andrea Zafferi, Dirk Peschka
类别: math.NA, math.AP
发布日期: 2026-07-04
arXiv ID: 2607.03829v1
链接: http://arxiv.org/abs/2607.03829v1
摘要翻译
中文标题: 基于梯度流的欧拉非线性弹性与扩散离散化
摘要翻译: 本文中，我们针对在拉格朗日和欧拉框架下均具有扩散输运的粘性多孔弹性材料，提出一种基于能量的通用建模方法。通过采用欧拉构型中的参考映射概念，我们得到了改进的弱形式。我们提出并实现了一种新颖的保结构离散化策略，利用混合有限元方法。通过对拉格朗日格式和欧拉格式的比较分析，本文展示了所提方法在空间和时间上的数值收敛性，从而证明了该方法的稳健性和可用性。此外，在欧拉多相流的背景下，特别是拟静态欧拉-欧拉类型，我们的研究证明了多孔粘弹性介质中存在孤立流体波。这种基于能量的方法为更深入地理解耦合多孔弹性、流动和扩散的热力学建模及相应离散化格式奠定了基础。
4. 面向重力测量区域反问题的水平集物理信息神经网络
Level-set physics-informed neural networks for domain inverse problems of gravimetry
作者: Jingnan Yao, Wenbin Li, Jianliang Qian
类别: math.NA
发布日期: 2026-07-04
arXiv ID: 2607.03772v1
链接: http://arxiv.org/abs/2607.03772v1
摘要翻译
中文标题: 面向重力测量区域反问题的水平集物理信息神经网络
摘要翻译: 我们针对重力测量的区域反问题提出水平集物理信息神经网络（PINNs）。该区域反问题为不适定的重力反演问题建立了一个适定性类，我们在PINNs框架下求解该类问题。由于界面不连续的性质，直接通过神经网络表示区域反问题是有问题的。我们考虑了一种水平集形式，其中神经网络表示一个连续的水平集函数，其零水平集刻画尖锐界面。为了克服训练过程中尖锐界面导致的梯度爆炸和消失的挑战，我们提出了一种界面感知的反向传播策略。通过重新定义与界面演化相关的导数，该策略使得更广的支持区域能够驱动演化过程。我们给出了详细分析以论证该策略的有效性。此外，我们引入了一种简单的过程，用于在界面附近自适应加密配点。通过研究解空间和神经网络的逼近性质，我们考察了网络架构的选择。最后，大量的二维和三维数值算例验证了所提方法的有效性。
不相关论文（基本信息）
1. 格点QCD中基于探测和多层蒙特卡洛的方差缩减
Variance reduction with probing and Multilevel Monte Carlo in Lattice QCD
作者: Andreas Frommer, Jose Jimenez-Merchan, Bruno Lang, Mario Papace, Gustavo Ramirez-Hidalgo
类别: hep-lat, math.NA
发布日期: 2026-07-06
arXiv ID: 2607.05157v1
链接: http://arxiv.org/abs/2607.05157v1
摘要: Trace estimation is central in many lattice QCD computations, but the accuracy of the standard, stochastic Hutchinson method improves only with the square root of the sample size, making precise results expensive. We investigate two complementary variance reduction strategies. First, multigrid multilevel Monte Carlo uses a multigrid hierarchy to construct an unbiased multilevel estimator via recursive coarse grid corrections available from the multigrid hierarchy of the solver. Second, stochastic probing uses distance-$d$ graph colorings; we propose a torus based coloring that requires substantially fewer colors than hierarchical probing at the same distance. We test these approaches on two representative problems: the connected pseudoscalar correlator and disconnected fermion loops. For the connected pseudoscalar two-point function, the multilevel decomposition yields a variance reduction of up to $\mathcal{O}(10^5)$ at large time separations and translates into a clear cost reduction at fixed accuracy, thus confirming earlier results of arXiv:2412.06347. For the disconnected loops, in contrast, the multilevel decomposition provides only moderate gains, whereas probing combined with dilution delivers a substantial cost reduction that improves as the number of probing vectors is increased. Overall, the results highlight a pronounced complementarity: deflation schemes are most effective for observables dominated by long distance propagation, while probing is most effective for localized quantities.
2. 正压欧拉方程组 MAC 格式的收敛性分析
Convergence analysis of a MAC scheme for the barotropic Euler system
作者: Bangwei She, Congying Wang, Jin Zhao
类别: math.NA
发布日期: 2026-07-06
arXiv ID: 2607.05105v1
链接: http://arxiv.org/abs/2607.05105v1
摘要: We study a Marker-and-Cell (MAC) scheme for the barotropic Euler system. First, we apply the recently developed Lax-type convergence theorem to show the convergence of the MAC scheme to i) a dissipative weak solution unconditionally and ii) a strong solution as long as it exists. Second, We derive relative energy error estimates up to the lifespan of a strong solution, without assuming uniform boundedness of the numerical sequence. Additionally, assuming the boundedness of the numerical solutions, we obtain the optimal relative energy rate of 1, corresponding to a convergence rate of 1/2 for the numerical solutions. Finally, we corroborate our theoretical results by numerical experiments.
3. 一种用于不可分离复合$\ell_0$-$\ell_2$正则化的精确乘子增广拉格朗日方法
An augmented Lagrangian method with exact multipliers for non-separable composite $\ell_0$-$\ell_2$ regularization
作者: Huan Ren, Guiyun Xiao
类别: math.OC, math.NA
发布日期: 2026-07-06
arXiv ID: 2607.05073v1
链接: http://arxiv.org/abs/2607.05073v1
摘要: This paper studies a non-separable composite $\ell_0$-$\ell_2$ regularization model that simultaneously enforces sparsity and smoothness for inverse problems. The $\ell_0$ norm induces inherent nonconvexity and nonsmoothness, while linear transformations further introduce nonseparability, making the problem computationally challenging to solve. The existing inexact augmented Lagrangian method suffers from high computational complexity and unstable convergence. To overcome these difficulties, we develop two novel augmented Lagrangian algorithms with exact multipliers, designed respectively for the full row-rank case and the general matrix case, where all subproblems are globally optimized via closed-form solutions. Furthermore, we prove linear convergence of the proposed method when the transformation matrix is full row rank. In the general setting, all accumulation points of the generated sequence are KKT points for the original problem. Numerical experiments on synthetic data, trend filtering, and image smoothing demonstrate the superior efficiency and accuracy of the proposed methods over the existing method, confirming our theoretical analysis.
4. 关于2-、3-和4-单形上波动方程有限元质量与刚度矩阵的求积规则
Quadrature rules for the mass and stiffness matrices of finite elements for the wave equation on the 2-, 3- and 4-simplex
作者: W. A. Mulder
类别: math.NA
发布日期: 2026-07-06
arXiv ID: 2607.04995v1
链接: http://arxiv.org/abs/2607.04995v1
摘要: Mass lumping enables explicit time stepping for finite-element discretisation of the wave equation if the resulting quadrature weights are positive and accuracy is preserved. New rules for elements of degree two and three on the 4-simplex are presented. Numerical quadrature for the stiffness matrix can be more efficient than exact evaluation if it requires fewer nodes. For the latter, new rules in two and four space dimensions for the lower-degree elements on the simplex were found, as well as some additional results for three dimensions.
5. 压缩Anderson加速与压缩GMRES的等价性结果
An Equivalence result for sketched Anderson Acceleration and sketched GMRES
作者: Alberto Bucci, Fabio Durastante
类别: math.NA
发布日期: 2026-07-06
arXiv ID: 2607.04904v1
链接: http://arxiv.org/abs/2607.04904v1
摘要: In this paper we present an equivalence result between a randomized version of Anderson Acceleration and of randomized GMRES for linear problems. Namely, we extend the classical result of Walker and Ni (2011) to the case in which the least-squares problem in Anderson Acceleration is solved in a sketched space defined by a random projection. This equivalence suggests potential avenues for further research in the design and analysis of randomized acceleration methods.
6. 走向完全动态全能树：基于小波的矩守恒各向异性压缩
Towards Fully Dynamic Omnitrees: Moment-Conserving Anisotropic Compression With Wavelets
作者: Theresa Pollinger, Masado Ishii, Jens Domke
类别: cs.DS, cs.CG, cs.GR, math.NA
发布日期: 2026-07-06
arXiv ID: 2607.04881v1
链接: http://arxiv.org/abs/2607.04881v1
摘要: Recently, omnitrees were introduced as a flexible space partitioning tree that improves upon the benefits of both octrees and k-d trees: Omnitrees’ efficient encoding of anisotropic refinements holds particular interest for applications with anisotropic features and high dimensionality. These include, but are not limited to, computer graphics, databases, machine learning, and physics simulations. The present paper defines new operations on the omnitree encoding that extend its capabilities from the existing refinement to also include coarsening and therefore fully adaptive compression. It demonstrates natural integration of omnitrees with wavelets, which conserves moments of the stored function by design. For omnitrees, the wavelet coefficients can be interpreted as local refinement priorities, which can be used to guide the adaptation process. We derive algorithms for coarsening and downsplit that are guided by wavelet coefficients, and show their application to a large dataset of 3D shapes, as well as the continuous-valued density field of a cloud. The comparison to OpenVDB, a widely-used data structure for sparse volumetric data in computer graphics, enables a demonstration of the practical benefits of omnitrees even for moderately anisotropic three-dimensional data. Compared to OpenVDB, objects can be stored using up to 28x less space, and asymptotically show savings that exceed theoretical expectations. Using lossy compression, the cloud dataset can be compressed by $\approx5\times$ compared to OpenVDB, with negligible loss of visual quality. This demonstrates the potential of omnitrees for efficient storage and processing, and motivates further research into their applications in various domains.
7. 具有非平凡相容性条件的线性薛定谔方程的任意高阶分裂方法
Arbitrary high order splitting methods for linear Schr{ö}dinger equations with non-trivial compatibility conditions
作者: Joackim Bernier, Ramona Häberli, Gilles Vilmart
类别: math.NA
发布日期: 2026-07-06
arXiv ID: 2607.04835v1
链接: http://arxiv.org/abs/2607.04835v1
摘要: Splitting methods are a natural choice for the numerical time integration of partial differential equations, and arbitrary high order splitting schemes exist for Schr{ö}dinger equations with periodic boundary conditions. However, in the presence of non-periodic boundary conditions, we show that they suffer in general from an order reduction, even for smooth initial conditions. The reason for such order reduction phenomena are so-called compatibility conditions, which are not preserved by classical splitting schemes. In this paper, we introduce a family of modified splitting methods for one-dimensional linear Schr{ö}dinger equations with homogeneous Dirichlet boundary conditions, which achieve an arbitrary high order, and do not suffer from any order reduction. This is illustrated with a fourth order splitting scheme considering initial conditions with various regularity properties.
8. 用于边坡稳定性评估的流-固耦合模型：一种多边形免稳定化离散方法
Hydro-mechanical Model for Slope Stability Assessment: A polygonal stabilization-free discretization
作者: Stefano Berrone, Francesca Marcon, Gioana Teora
类别: math.NA
发布日期: 2026-07-06
arXiv ID: 2607.04778v1
链接: http://arxiv.org/abs/2607.04778v1
摘要: Rainfall-induced landslides are governed by the interaction between subsurface water flow and soil mechanics, requiring robust numerical methods for the simulation of variably saturated porous media. In this work, we consider a semi-coupled hydro-mechanical model based on Richards’ equation and linear elasticity and propose a numerical framework based on a stabilization-free Virtual Element Method for its spatial discretization. The proposed approach naturally accommodates general polygonal meshes while avoiding problem-dependent stabilization terms, whose design may become challenging when heterogeneous and strongly non-linear coefficients are involved. The approach is combined with a mass-lumping technique to improve stability in the treatment of the storage term and with Nitsche’s method to weakly impose seepage-face and infiltration boundary conditions, allowing for the automatic switching between Neumann and Dirichlet conditions. Time integration is performed using the backward Euler scheme, while non-linearities are handled through a Picard iteration. Numerical experiments demonstrate the stability and robustness of the proposed methodology and show its effectiveness in simulating rainfall infiltration and evaluating slope stability through the Local Factor of Safety.
9. 亚纯延拓中可见极点的行列式特征与辐角原理验证
Determinant Characteristics and Argument-Principle Certification for Visible Poles in Meromorphic Continuation
作者: Xiaomei Yang, Zhiliang Deng
类别: math.NA
发布日期: 2026-07-06
arXiv ID: 2607.04568v1
链接: http://arxiv.org/abs/2607.04568v1
摘要: We study outward meromorphic continuation from circular boundary data in the unit disk. The unknown function is holomorphic in the unit disk and admits a meromorphic continuation to a larger disk, where finitely many exterior simple poles are superposed on an unknown holomorphic background. The positive Fourier coefficients of the boundary trace are Taylor coefficients at the origin, and exterior poles generate a finite exponential-sum component in these coefficients. We introduce shifted determinant characteristics and prove that, in the pure finite-pole model, the determinant for the correct order factors exactly into a nonzero constant times the polynomial whose zeros are the reciprocals of the exterior poles. The same zero set is obtained for noiseless equispaced discrete Fourier coefficients; sampling changes only the amplitudes through an aliasing factor. For data containing a holomorphic background, discretization effects, and noise, roots of a single empirical determinant are only candidate reciprocal poles. We therefore propose a root-propose and contour-certify procedure: determinant roots generate candidate regions, while local argument-principle counts, contour moments, empirical margins, and persistence over determinant orders and shifts certify visible poles. A Rouché-type perturbation analysis gives sufficient conditions for stable local zero counts and explains how residues, pole separation, distance to the target annulus boundary, shifts, and noise affect visibility. Numerical experiments verify the pure-pole identity, demonstrate certification under background and noise, and show that high noise, weak residues, boundary-near poles, and close poles naturally lead to partial recovery of contour-certified visible poles.
10. 定点运算下硬件安全神经网络的李雅普诺夫引导训练
Lyapunov-Guided Training for Hardware-Safe Neural Networks Under Fixed-Point Arithmetic
作者: Anis Hamadouche, Amir Hussain
类别: cs.LG, cs.AI, eess.SP, math.NA
发布日期: 2026-07-05
arXiv ID: 2607.04531v1
链接: http://arxiv.org/abs/2607.04531v1
摘要: Low-precision neural networks are attractive for resource-constrained hardware, but fixed-point arithmetic introduces failure modes that are often hidden by idealised quantisation models. In particular, two’s-complement overflow wrapping can corrupt hidden activations by changing both their magnitude and sign, leading to unstable numerical error propagation and severe accuracy degradation. This paper proposes a Lyapunov-stabilised quantisation framework for low-precision neural networks operating under hardware-style wrapping arithmetic. The hidden-state energy is monitored through a layerwise Lyapunov function, and a monotone projection is applied to enforce bounded and non-increasing state evolution across depth. The method is evaluated on MNIST using a compact patch-based transformer under post-training quantisation and quantisation-aware training with fixed-point bit-widths from 4 to 16 bits. Monte Carlo results show that unconstrained wrapped quantisation-aware training collapses to near-chance accuracy across 6-16 bits, with activation overflow rates exceeding 11%. In contrast, the proposed monotone Lyapunov projection suppresses activation overflow to below 0.012% and restores stable low-precision learning, achieving 86.55% accuracy at 12 bits. These results demonstrate that Lyapunov-based state control can act as a hardware-aware stabilisation mechanism for reliable fixed-point neural inference and training.
11. 紧支撑RBF的平滑与偏心平滑
Smoothing by, and eccentric smoothing of, compactly supported RBFs
作者: Thomas Hangelbroek, Christian Rieger
类别: math.NA
发布日期: 2026-07-05
arXiv ID: 2607.04512v1
链接: http://arxiv.org/abs/2607.04512v1
摘要: We consider compactly supported RBFs having algebraically decaying Fourier transforms. Here we focus especially on generalized Wendland RBFs and their modification by making them smoother away from zero, a process we call eccentric smoothing. Specifically, we consider mapping properties of the integral operators (sometimes known as covariance operator) for these novel type RBFs. Moreover, we show that eccentric smoothing makes wavelet-inspired compression technique for the kernel matrix feasible.
12. 谱伽辽金方法的保证特征值下界及其在薛定谔算子中的应用
Guaranteed Lower Eigenvalue Bounds for Spectral Galerkin Methods with Application to Schrödinger Operators
作者: Xuefeng Liu
类别: math.NA
发布日期: 2026-07-05
arXiv ID: 2607.04247v1
链接: http://arxiv.org/abs/2607.04247v1
摘要: Spectral Galerkin methods are renowned for high-precision eigenvalue approximation, yet a rigorous lower bound obtained directly from a spectral discretisation has remained unavailable: the classical Kato and Weinstein–Temple enclosures do apply, but require a~priori information on a neighbouring eigenvalue. This paper resolves the issue by extending the author’s projection-based framework for guaranteed lower eigenvalue bounds – so far realised only through finite element methods – to conforming spectral Galerkin methods. For trial spaces of exact eigenfunctions the required projection constant is the closed-form optimal value $C_N=λ_{M+1}^{-1/2}$, the inverse square root of the first omitted eigenvalue. For $-Δ+V$ with $0\le V\in L^\infty$, a \emph{projection-gap estimate} yields an explicit constant for the standard Galerkin matrix (exact at $V=0$), and a composite discretisation removes the $||V||_{L^\infty}$-dependence for large potentials. With Neumann domain truncation these give certified two-sided bounds on $R^d$; for two benchmark potentials on $R^2$ the spectral enclosures match or surpass certified finite element ones at two orders of magnitude fewer degrees of freedom. The same auxiliary-projector mechanism extends to singular potentials with an unbounded $L^\infty$ norm – in particular to attractive Coulomb singularities in three dimensions, via a localised Hardy inequality – which we develop in a companion paper.
13. 扩散与流匹配采样器的渐近保持后验分析
Asymptotic-Preserving A Posteriori Analysis of Diffusion and Flow-Matching Samplers
作者: Shiheng Zhang
类别: cs.LG, math.NA
发布日期: 2026-07-05
arXiv ID: 2607.04113v1
链接: http://arxiv.org/abs/2607.04113v1
摘要: Diffusion and flow-matching samplers integrate a learned probability-flow ODE from a large noise scale down to a small terminal floor $σ_{\min}$, at which the score is stiff and the flow develops a boundary layer. We treat $σ_{\min}$ as a singular-perturbation parameter and determine which fixed-step samplers are asymptotic-preserving (AP), that is, stable and uniformly accurate as $σ_{\min}\to0$, casting the criteria as an a posteriori audit: residual functionals with $σ_{\min}$-uniform coefficients, computable on a pretrained checkpoint without ground-truth scores or exact trajectories. On the terminal layer, Euler in the $σ$-clock, the deterministic DDIM update, is the unique layer-exact discretization up to affine reparameterization, with rectified flow its flow-matching counterpart; the $λ$-clock is stable only for steps $h\le h_\star=1+W(1/e)$, and the uniform-$σ^2$ heat clock stalls a $σ_{\min}$-independent distance from the data. On two solvable models (rank-deficient Gaussian, symmetric two-point mixture), deterministic samplers remain first-order uniformly accurate with no $\log(1/σ_{\min})$ factor, even across a symmetric posterior-switching interface whose distributional budget is a universal constant; the logarithm is charged entirely to the Itô term of stochastic samplers, whose path-KL scales as $Λ^2/N$ against the ODE’s $O(Λ^2/N^2)$ budget, with $Λ=\log(σ_{\max}/σ_{\min})$. On the EDM CIFAR-10 checkpoint, spectra measured once predict held-out residual budgets across step count, schedule, and noise level against pre-specified gates with no per-configuration refitting, and calibrate the Itô coefficient at $M_1=1.00\pm0.01$. The clock decides stability; the noise, not the geometry, charges the logarithm.
14. 再论DeTurck方法在曲线缩短流中的应用及其最优误差分析
A revisit to DeTurck method on curve shortening flow with optimal error analysis
作者: Beiping Duan
类别: math.NA
发布日期: 2026-07-05
arXiv ID: 2607.04105v1
链接: http://arxiv.org/abs/2607.04105v1
摘要: The curve shortening–DeTurck flow introduced by Elliott and Fritz (IMA J. Numer. Anal. 37(2): 543–603, 2017) is a reparameterization of the curve shortening flow via the DeTurck trick. For corresponding discrete schemes, although the $H^1$-error estimate has been established, the optimal $L^2$-error estimate remains open due to technical difficulties. In this paper, we prove optimal $L^2$-error estimates for linearized Euler and Crank–Nicolson discretizations combined with finite elements of degree $k\ge 1$ in space. Moreover, we provide an extrinsic approach to derive the curve shortening–DeTurck flow. Numerical experiments are presented to illustrate the mesh distribution properties and to verify the convergence rates in both space and time.
15. 多孔介质中两种不混溶流体的准不可压缩Cahn-Hilliard-Darcy模型
A quasi-incompressible Cahn-Hilliard-Darcy model for two immiscible fluids in porous media
作者: Daozhi Han, Sayantan Sarkar
类别: math.AP, math.NA
发布日期: 2026-07-05
arXiv ID: 2607.04060v1
链接: http://arxiv.org/abs/2607.04060v1
摘要: We derive a quasi-incompressible Cahn-Hilliard-Darcy phase-field model (qCHD) with the logarithmic Flory-Huggins free energy density function for two-phase flows in porous media. The model satisfies an energy-dissipation law. In the formal sharp interface limit, the qCHD model gives rise to the classical Muskat’s problem. By exploiting estimates of the pressure from Darcy’s equations, we establish global existence of weak solutions in both 2D and 3D to the qCHD model.
16. 具有各向异性散射的输运问题的二阶矩方法
Second-Moment Method for Transport Problems with Anisotropic Scattering
作者: India J. Allan, Dmitriy Y. Anistratov
类别: math.NA, physics.comp-ph
发布日期: 2026-07-04
arXiv ID: 2607.03967v1
链接: http://arxiv.org/abs/2607.03967v1
摘要: This paper presents a new nonlinear two-level acceleration method for solving the particle transport equation with anisotropic scattering. The method is formulated with the projection operator approach. The low-order equations are defined for the angular moments using projection operators and closures of the second-moment method. A nonlinear prolongation operator is applied to the scattering term to derive the high-order transport equation. The nonlinear system of high-order and low-order equations is equivalent to the original transport equation. The equations are approximated in space by the lumped linear-discontinuous Galerkin method. Numerical results are presented to demonstrate the performance of the proposed numerical method.
17. 求解稳态Boltzmann方程的加速迭代方法
Accelerated iterative method for solving the steady-state Boltzmann equation
作者: Pei Zhang, Zhenning Cai, Yanli Wang
类别: math.NA, physics.comp-ph
发布日期: 2026-07-04
arXiv ID: 2607.03910v1
链接: http://arxiv.org/abs/2607.03910v1
摘要: The efficient simulation of steady-state rarefied gas flows remains a significant computational challenge due to the high dimensionality of the collision integral and the severe numerical stiffness in the near-continuum regime. In this work, we propose a modified Newton method equipped with a macroscopic synthetic system (Newton-MS) for the steady-state Boltzmann equation with the quadratic collision operator. In Newton-MS, the modified Newton iteration is utilized as the outer nonlinear solver, while each Newton correction equation is solved by an inner source iteration, where the linearized collision operator is utilized to approximate the quadratic collision model, and it is reduced into a linear iteration. Moreover, a macroscopic synthetic system based on Chapman-Enskog closure is derived to accelerate the convergence of the linear inner iteration in the continuum limit. Besides, the fully discrete macroscopic synthetic system is deduced under the framework of the discontinuous Galerkin method to reduce computational cost compared to directly discretizing the continuous macroscopic synthetic system. Several numerical examples, including the 1D Fourier, Couette flow problem, and the 2D cavity flow and thermal-driven cavity flow, are studied to validate the high efficiency of Newton-MS.
18. 一类完全非线性椭圆型偏微分方程的虚拟元方法
Virtual element methods for a class of fully nonlinear elliptic PDEs
作者: Guillaume Bonnet, Andrea Cangiani, Andreas Dedner, Ricardo H. Nochetto
类别: math.NA
发布日期: 2026-07-04
arXiv ID: 2607.03850v1
链接: http://arxiv.org/abs/2607.03850v1
摘要: We study virtual element discretizations of a well-known variational formulation in $H^2$ of Hamilton-Jacobi-Bellman and Isaacs equations with Cordes coefficients. We show that the use of $H^2$ conforming virtual element spaces leads to a relatively simple analysis, bypassing the need for discrete Miranda-Talenti estimates that typically arises when using nonconforming schemes. We investigate how the polynomial degree of projection operators, especially for lower order terms, affects both the error analysis and robustness of the proposed schemes. We also show the possibility of imposing weakly the Dirichlet boundary condition, which simplifies the implementation of the method in some virtual element codes. Our results are complemented by numerical experiments in which we compare the convergence of different variants of the scheme for some test problems.
19. 趋化驱动的肿瘤—免疫空间模式形成与稳定性的计算肿瘤学研究
Computational Oncology of Chemotaxis-Driven Tumour–Immune Spatial Patterning and Stability
作者: Zonghao Liu, Jiguang Yu, Lei Su, Louis Shuo Wang, Yang Du, Jingfeng Liu
类别: math.AP, math.NA
发布日期: 2026-07-04
arXiv ID: 2607.03813v1
链接: http://arxiv.org/abs/2607.03813v1
摘要: Spatial tumour–immune heterogeneity is a key feature of solid-tumour progression, immune infiltration, and immune exclusion. We develop a computational oncology model in which tumour cells, immune effector cells, and a chemokine signal interact through a reaction–diffusion–chemotaxis system on a bounded tissue domain with no-flux boundaries. Chemokine is produced by tumour cells and tumour–immune contact, recruits immune cells, and guides chemotactic migration. After nondimensionalization, we establish positivity, a tumour-density bound, and immune/chemokine mass estimates. We identify the tumour-free equilibrium, derive the immune-control threshold $σ_0>δ$, and reduce coexistence to a scalar equation. Linear stability analysis about coexistence yields a mode-wise dispersion relation in which chemotaxis appears as a wavenumber amplified coupling, producing finite-wavelength instability above a critical sensitivity. A conservative finite-volume scheme with upwind chemotactic flux verifies the thresholds, dominant unstable modes, sensitivity maps, positivity, convergence, and residual consistency.
20. 弹性波逆介质散射问题的直接采样方法
Direct sampling methods for inverse medium scattering problems of elastic waves
作者: Lu Zhao, Yiling Li, Zhiyong Cheng
类别: math.NA
发布日期: 2026-07-04
arXiv ID: 2607.03745v1
链接: http://arxiv.org/abs/2607.03745v1
摘要: This paper concerns the inverse elastic scattering problem of determining an unknown penetrable obstacle from far-field data. Using the Helmholtz decomposition, the coupled boundary value problem is reformulated as a coupled scalar Helmholtz system. We prove the uniqueness of the associated coupled Helmholtz boundary value problem and of the corresponding boundary integral equation system, which is then discretized by a Nyström method for efficient numerical computation. Leveraging the relation between the compressional and shear far-field patterns of the Navier system and those of the coupled Helmholtz system, we employ three indicators to reconstruct the location and shape of the obstacle. Furthermore, we analyze the decay properties of these indicators and establish corresponding stability estimates. Numerical experiments are presented to illustrate the effectiveness and robustness of the proposed method, even for limited-aperture data.
21. PIEFS：物理信息特征函数特征与可学习缩放
PIEFS: Physics-Informed Eigenfunction Features with Learnable Scaling
作者: Varvara Nazarenkko, Timur Lidzhiev, Alexander Tarakanov
类别: cs.LG, math.NA, math.ST
发布日期: 2026-07-04
arXiv ID: 2607.03692v1
链接: http://arxiv.org/abs/2607.03692v1
摘要: Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised neural representation-learning framework with a spectral inductive bias, based on a modified Dirichlet energy. In PIEFS, scalar coordinate maps are trained under empirical Gram orthogonality, a supervised linear readout, and a Dirichlet penalty in which the input gradient is transformed by a learnable metric $A(x)=Λ(x)U(x)$. The diagonal factor $Λ(x)$ controls anisotropic scaling, while the orthogonal factor $U(x)$ is parameterized by a structured product of Givens rotations. This construction yields task-adaptive Dirichlet-regularized coordinates rather than eigenfunctions of a fixed supervision-independent operator. Experiments on synthetic, tabular, and image-based benchmarks study the effect of identity, diagonal, and rotation-scaling metrics, and compare the resulting coordinates with classical baselines and NeuralEF. The results support PIEFS as a compact supervised spectral representation method and identify optimization stability, validation on explicit operator eigenproblems, and richer metric parameterizations as the main directions for future work.