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1. 一阶不规则立方体网格上的hp-最优DG逼近与稳健Schwarz分解
hp-Optimal DG Approximation and Robust Schwarz Decompositions on One-Irregular Cubical Meshes
作者: Situan Li, Weiying Zheng
类别: math.NA
发布日期: 2026-06-26
arXiv ID: 2606.27728v1
摘要翻译: 我们研究一阶不规则网格上变阶数立方体有限元空间的hp逼近与加性Schwarz分解。对于一阶不规则六面体网格上的拟合齐次扩散界面问题，我们证明了内惩罚DG方法的hp-最优能量范数估计。插值输入是从一阶不规则顶点片的拟合协调闭包获得的协调hp插值。我们还推导了协调和DG空间的稳定分解。在一阶不规则四边形网格上，这些界允许局部可比较的变多项式次数，且与网格尺寸、局部次数无关，并且在局部系数拟单调性条件下，与系数反差无关。在一阶不规则六面体网格上，协调分解具有相应的多对数损失；对于均匀次数DG空间，那里使用了DG到协调的归约。数值实验展示了p-最优DG误差估计和DG Schwarz预条件子的稳健性。
Comment: 33 pages, 2 figures
链接: http://arxiv.org/abs/2606.27728v1
1. 研究对象和背景
本文研究对象为带分片常数系数扩散方程的拟合界面间断 Galerkin（DG）逼近与加法 Schwarz 预条件子分析。设区域 $\Omega\subset\mathbb{R}^d$ ($d=2,3$) 为多面体区域，扩散系数 $\alpha:\Omega\to\mathbb{R}_+$ 在 $\Omega$ 上分片常数，且存在有限个界面 $\Gamma$ 使 $\alpha$ 在各子区域上为常数。考虑齐次 Dirichlet 边值问题：
$$ -\nabla\cdot(\alpha\nabla u) = f\;\;\text{在}\; \Omega,\quad u=0\;\;\text{在}\; \partial\Omega, $$其中 $f\in L^2(\Omega)$。弱形式为：求 $u\in H_0^1(\Omega)$ 满足
$$ a(u,v)=(f,v)_{L^2(\Omega)},\qquad \forall v\in H_0^1(\Omega), $$其中 $a(u,v)=\int_\Omega \alpha\nabla u\cdot\nabla v\,\mathrm{d}x$。
本文采用拟合一阶非正则（one-irregular）立方体网格 $\mathcal{T}_h$ 进行离散。网格由初始共形网格递归四分（二维）或八分（三维）细划而成，满足形状正则性 (2.1) 和一阶非正则性 (M1)–(M4)，即任意边的悬挂节点不会形成链。网格与系数界面拟合，即界面 $\Gamma$ 由单元边/面组成，且细划操作始终使界面保持拟合。
在网格上定义间断空间 $S^{\bm{p}}_{\mathrm{dg}}(\mathcal{T}_h)$，其中 $\bm{p}=(p_K)_{K\in\mathcal{T}_h}$，每个单元 $K$ 上为张量积多项式空间 $\mathbb{Q}_{p_K}(K)$（映射后），并允许相邻单元间多项式次数不同，但满足局部次数可比条件：
$$ \gamma^{-1}p_K\le p_{K'}\le \gamma p_K,\quad \forall K,K'\;\text{相邻}. $$共形子空间记为 $S^{\bm{p}}(\mathcal{T}_h)=S^{\bm{p}}_{\mathrm{dg}}(\mathcal{T}_h)\cap H^1(\Omega)$。
采用内罚 DG（IPDG）离散：求 $u_h\in S^{\bm{p}}_{\mathrm{dg}}(\mathcal{T}_h)$ 使得
$$ a_h(u_h,v) = (f,v)+\text{可能有的边界项},\qquad \forall v\in S^{\bm{p}}_{\mathrm{dg}}(\mathcal{T}_h), $$其中 $a_h$ 为
$$ a_h(w,v)=\sum_{K\in\mathcal{T}_h}\alpha_K(\nabla w,\nabla v)_{L^2(K)}+\sum_{F\in\mathcal{F}}\left(-\langle\{\alpha\nabla w\}\cdot\bm{n}_F,[v]\rangle_F-\langle\{\alpha\nabla v\}\cdot\bm{n}_F,[w]\rangle_F+\sigma h_F^{-1}\langle[w],[v]\rangle_F\right). $$这里 $\mathcal{F}$ 为单元交界面（匹配骨架），$\sigma$ 为充分大的惩罚参数。相应的 DG 能量范数为
$$ \|v\|_{\mathrm{DG}}^2 = \sum_{K\in\mathcal{T}_h}\alpha_K|v|_{H^1(K)}^2 + \sum_{F\in\mathcal{F}}\sigma h_F^{-1}\|[v]\|_{L^2(F)}^2. $$hp 有限元方法的核心理论问题是：在上述非正则网格上，DG 能量误差是否仍具有 hp 最优性，即是否存在与 $h$ 和 $\bm{p}$ 无关的常数 $C$ 使得
$$ \|u-u_h\|_{\mathrm{DG}}\le C\inf_{v\in S^{\bm{p}}_{\mathrm{dg}}(\mathcal{T}_h)}\|u-v\|_{\mathrm{DG}}. $$对于共形网格，这一结果可通过构造 hp 最优插值算子直接获得；但在一阶非正则网格上，悬挂节点的存在破坏了单元间的相容性，使得标准 hp 插值（如基于顶点片块的算子）无法直接移植。
另一关键问题是预条件子的稳健性。加法 Schwarz 预条件子的分析依赖于将有限元函数空间分解为局部子空间之和的稳定分解，形式为
$$ u = \sum_{a\in\mathcal{N}^c(\mathcal{T}_h)} u_a,\qquad u_a\in V_a $$且满足局部稳定性
$$ \sum_{a\in\mathcal{N}^c(\mathcal{T}_h)}\|u_a\|_{1,\Omega_a}^2 \le C_{\rm stab}\, \|u\|_1^2. $$常数 $C_{\rm stab}$ 若对 $h$、$\bm{p}$ 和系数反差保持一致上界，则对应的 Schwarz 预条件子具有稳健性。对于共形网格和均匀次数，已有大量工作；但对于一阶非正则网格、局部可变次数以及大系数反差，同时实现三项参数的稳健性是一个未解决的问题。特别地，系数反差常常迫使稳定常数随反差增长，除非强加全局拟单调性条件。
本文试图填补上述两个空白： (i) 对一阶非正则六面体网格上的拟合界面扩散问题，证明 IPDG 方法的 hp 最优能量误差估计； (ii) 构造共形和 DG 空间上的稳定分解，在二维情形达到对 $h$、$\bm{p}$ 和系数反差（在局部拟单调条件）的完全稳健性，在三维共形情形仅有一个 $(1+\log\|p\|_{L^\infty})^3$ 的损失，并通过 DG 到共形的归约技术，在均匀次数 DG 空间上获得稳健预条件子。
2. 主要定理或主要结果
以下为主要理论成果，按作者在引言中归纳的四个方面陈述。
一阶非正则网格上的限制型节点单位分解与活动顶点片块。作者构造了悬挂节点约束的多线性单位分解函数 $\{\theta_a\}_{a\in\mathcal{N}^c(\mathcal{T}_h)}$。该族函数满足：$\theta_a\in H^1(\Omega)$，$0\le\theta_a\le1$，$\sum_{a}\theta_a=1$，且在单元 $K$ 上有 $\|\nabla\theta_a\|_{L^\infty(K)}\le C h_K^{-1}$。其支撑集定义了活动片块 $\mathcal{T}^a_h$ 和区域 $\Omega_a$。这些片块具有一致有界的重叠度（引理 2.1）。
二维一阶非正则网格上的稳定分解（无 $p$ 损失）。在局部次数可比条件下，对任意 $u\in S^{\bm{p}}(\mathcal{T}_h)$（共形空间），存在分解 $u=\sum_{a\in\mathcal{N}^c(\mathcal{T}_h)} u_a$，其中 $u_a\in S^{\bm{p}}(\mathcal{T}^a_h)\cap H^1_{0,\partial\Omega_a\backslash\partial\Omega}(\Omega_a)$，且对每个单元 $K$ 及 $a\in\mathcal{N}^c_K$ 有
$$ \|u_a\|_{L^2(K)}^2 + h_K^2|u_a|_{H^1(K)}^2 \le C\big(\|u\|_{L^2(K)}^2 + h_K^2|u|_{H^1(K)}^2\big). $$常数 $C$ 与 $h$、$\bm{p}$ 无关。在局部系数拟单调性条件下，该分解对系数反差亦具有稳健性（即上述不等式中添加系数权 $\alpha_K$ 后常数 $C$ 独立于系数反差）。结合系数自适应粗平均，可得到 DG 空间的相应稳健分解。
三维一阶非正则六面体网格上的稳定分解（带对数损失）。在共形空间上，存在分解
$$ \sum_{a\in\mathcal{N}^c(\mathcal{T}_h)}\|u_a\|_{1,\Omega_a}^2 \le C\big(1+\log\|\bm{p}\|_{L^\infty}\big)^3\sum_{K\in\mathcal{T}_h}\big(\|u\|_{L^2(K)}^2 + h_K^2|u|_{H^1(K)}^2\big). $$该分解对 $h$ 和系数反差（局部拟单调条件下）稳健，但对 $p$ 出现不可消除的对数立方损失。对于均匀次数（$p_K\equiv p$）的 DG 空间，通过 DG 到共形的归约技术，可构造稳健的 DG Schwarz 预条件子，其条件数界具有相同的对数因子。
一阶非正则六面体网格上的 hp 最优 DG 误差估计。针对拟合界面和局部次数可比假设，作者首先通过顶点片块的共形闭包（引理 2.2）构造了一个共形 hp 插值算子 $\Pi_h: H^1_0(\Omega)\to S^{\bm{p}}(\mathcal{T}_h)$，使其在每个片块的共形闭包上为标准的 hp 多项式插值。然后证明了对 IPDG 解 $u_h$ 的误差界：
$$ \|u-u_h\|_{\mathrm{DG}} \le C\,\inf_{v\in S^{\bm{p}}_{\mathrm{dg}}(\mathcal{T}_h)}\|u-v\|_{\mathrm{DG}}. $$常数 $C$ 独立于 $h$ 和 $\bm{p}$（仅依赖于形状正则性和惩罚参数）。结合 hp 插值逼近性质，可得如下的收敛率：若解 $u$ 在单元 $K$ 上具有 $H^{s_K(K)$ 正则性，则
$$ \|u-u_h\|_{\mathrm{DG}} \le C\left(\sum_{K\in\mathcal{T}_h} \alpha_K\frac{h_K^{2\min(p_K,s_K-1)}}{p_K^{2s_K-2}}\|u\|_{H^{s_K}(K)}^2\right)^{1/2}. $$此即 hp 最优收敛率。数值实验验证了 $p$ 最优收敛性和 DG 预条件子的稳健性。
3. 研究方法、关键技术和核心工具
本文的主要技术障碍来源于一阶非正则网格的三个特点：(i) 悬挂节点破坏了单元间的连续性，使得直接构造共形插值或稳定的局部分解变得困难；(ii) 非正则界面上的迹涉及粗、细两侧的匹配；(iii) 三维顶点片块自身的奇异性（多个单元交汇）导致对数损失。
作者克服这些障碍的核心创新与关键工具如下：
1. 带有悬挂节点约束的节点单位分解。 与传统的共形网格节点基函数不同，一阶非正则网格上的单位分解必须在悬挂节点处满足线性约束（在二维）或双线性约束（在三维）。作者显式构造了 $\{\theta_a\}$：在共形顶点处赋克罗内克值，在悬挂节点处则由控制该悬挂节点的母边（或母面）的端点值线性（或双线性）插值确定；然后每个单元上通过多线性插值得到 $\theta_a|_K$。这一构造保证了 $\theta_a$ 连续且满足单位分解，且 $\nabla\theta_a$ 的 $L^\infty$ 模被 $h_K^{-1}$ 控制（依赖形状正则性）。该单位分解是片块重叠有限性的基础，也是后续片块分解（引理 2.9）的基石。
2. 顶点片块的共形闭包与尺寸可比性（引理 2.2）。 由于一阶非正则网格上的片块 $\mathcal{T}^a_h$ 可能包含悬挂节点和不同细划层次的单元，无法直接应用标准 hp 插值。作者设计了一种层次式合并算法：从最细层开始，将同一母单元的所有子单元用母单元替换；若母单元的部分子单元缺失，则补齐缺失子单元后再替换。逐层粗化后得到一个共形网格 $\widetilde{\mathcal{T}}^a_h$，其每个单元都是原片块某个单元的祖先，且与原单元直径可比（引理 2.2 中的 $h_{\widetilde{K}}\le C h_K$）。该共形闭包为 hp 插值提供了正则平台，并且若原网格拟合界面，闭环也保持拟合。
3. 基于高斯–洛巴托插值的边/面次数调整与二维链的 $H^{1/2}$ 稳定性（引理 2.4–2.5）。 在分解过程中，需将高次多项式迹降次以匹配相邻单元的次数。作者选用高斯–洛巴托插值，因为它保持端点值从而保证连续性。引理 2.4 给出了参考区间和方形面上一维及张量积插值的 $H^{1/2}$ 稳定性估计，对次数比一致有界。对于一阶非正则网格中典型的二边链结构，引理 2.5 证明了在连接点相容的条件下，整体 $H^{1/2}$ 模的稳定性和 $H^{1/2}_{00}$ 剩余估计。技术核心是运用了二边链的 $H^{1/2}$ 模等价（利用相邻侧积分），并结合端点残差为零的性质将链的整体残差控制为两个边上 $H^{1/2}_{00}$ 残差之和。这为非正则界面上的次数调整提供了严格工具。
4. 非正则面迹的移除与三维片块提升（引理 2.8）。 三维共形分解的主要障碍是顶点片块包含非正则面。作者先将共形函数 $u$ 在每个非正则母面 $F$ 上的迹，利用面积次稳定分解（引用 [9] 的顶点分解）表达为若干多项式之和，这些多项式作为该面所有子面的数据分配至各顶点。然后，在片块 $\Omega_a$ 内部利用“固定层次共形成分”（引理 2.7）的性质，即在同一个细化层次上的单元形成共形子网，从而可以逐成分应用三维共形片块提升技术。提升过程采用 [9] 中的奇异片块插值，导致了 $(1+\log\|p\|)^3$ 的损失。这一定理成功地将非正则界面的相容性条件融入到局部分解中。
5. 基于共形闭包的 hp 最优插值与 DG 误差分析。 DG 误差估计的证明框架如下：(a) 对真解 $u$，在每个片块共形闭环上构造标准 hp 插值 $\tilde{u}_a$；(b) 利用引理 2.9 的片块单位分解逼近：$\tilde{u} = \sum_a \theta_a \tilde{u}_a$，该函数属于共形空间 $S^{\bm{p}}(\mathcal{T}_h)$；(c) 证明 $\tilde{u}$ 对 $u$ 的逼近误差具有 hp 最优阶（借助共形闭环上插值误差的估计和单位分解的稳定性）；(d) 将 $\tilde{u}$ 作为 $u_h$ 的近似，利用 Galerkin 正交性和惩罚项处理，导出 $\|u-u_h\|_{\mathrm{DG}}\le C\|u-\tilde{u}\|_{\mathrm{DG}}$，从而完成 hp 最优性证明。这里的关键是共形闭环和引理 2.9 的配合，既消去了悬挂节点的障碍，又保留了插值的最优逼近性质。
6. DG 到共形的归约（均匀次数情形）。 对于均匀次数 DG 空间，作者将 DG Schwarz 预条件子的条件数估计归结为共形空间的相应结果。具体手段是构造一个从 DG 空间到共形空间的稳定算子（如平均算子），并验证其在 DG 能量范数下的有界性及逼近性质。这一归约使得三维均匀次数 DG 预条件子直接继承了共形分解的对数立方损失，无需重新进行复杂的 DG 分析。
4. 与之前工作的比较
本文的工作与既有有限元与数值解法文献相比，在多个层面实现了推进，但也保留了若干限制。下面分维度比较。
1. 与共形网格 hp 插值和误差估计的比较 经典的 hp 有限元插值理论（如 [2, Section 3]）假定网格是共形的。对于一阶非正则网格，虽然可通过悬挂节点约束直接构造共形空间，但标准 hp 插值算子（如顶点模态展开）无法直接使用，因为片块不是共形的。本文通过“共形闭包”技术，首次将 hp 最优插值推广到一阶非正则立方体网格，且不要求网格上的悬挂节点通过额外层消除。代价是闭包可能包含原网格未细化的单元，但通过有限重叠性与尺寸可比性，保持了逼近阶。与 [2] 中仅对共形网格的估计相比，本文的结果扩展了适用范围。
2. 与一阶非正则网格上稳定分解的已有工作 针对一阶非正则网格的 Schwarz 分解，此前工作多集中在 h 版本（即 $p=1$ 或 $p_K$ 较小）或二维情形。例如，[4] 等研究了二维一阶非正则四边形网格上的线性元稳定分解，对网格尺寸和系数反差稳健。本文在二维的表现超越了这些：对任意局部可比的多项式次数 $p_K\ge 1$，证明了无 $p$ 损失的稳定分解，即常数不依赖于 $\bm{p}$。这是通过精心设计的边次数调整和基于单位分解的片块重组实现的，避免了直接使用节点插值带来的次数依赖。此外，本文在系数反差方面，仅要求局部拟单调性条件（在每个顶点片块上系数沿离开顶点的方向非减），而许多前期工作（如 [9]）需要全局拟单调性（即整个区域内存在排列顺序使得系数几乎单调）。这一放松使得允许更一般的系数分布，例如一个高传导率区域被低传导率区域包围，只要在每个顶点附近满足局部条件。
3. 三维情形下的对数损失与现有三维 hp 分解的比较 在三维共形网格上，[9] 已证明顶点片块分解带有 $(1+\log p)^3$ 的损失，这是由于三维奇性顶点（多个单元交汇）所致。本文的三维结果与此一致，并将这一估计推广到一阶非正则六面体网格。这一定理的新意在于处理了非正则面带来的额外复杂性，而仍然保持与共形情形相同的对数因子。作者通过引理 2.8 的面迹移除技术实现这一点。因此，该结果将三维 hp 稳定分解的基准情形从共形网格拓展到一阶非正则网格。
4. DG 误差估计的前人工作与本文的推进 DG 方法的 hp 误差分析通常要求插值算子具有迹适应性，在 hp 情形尤其复杂。对于共形网格上的 IPDG 方法，Schöberl 等人[?] 给出了 hp 最优估计；但对一阶非正则网格，DG 空间的共形子空间并不包含所有连续 Lagrange 节点函数，因为悬挂节点处的连续性并非被显式强加，而是通过惩罚项隐式实现。本文通过构造相容的共形插值（使用闭环），并利用引理 2.9 将片块函数粘合为全局共形函数，从而利用 Galerkin 正交性将 DG 误差归化为插值误差。这一策略绕开了对 DG 空间直接进行 hp 插值的困难，所得的估计在能量范数下是 hp 最优的。与 [1] 等早期 hp-DG 工作相比，本文的网格允许悬挂节点且具拟合界面。与 [5, 6] 等仅处理 h 收敛或均匀次数的工作相比，本文实现了完整的 hp 最优性。
5. DG 预条件子与已有 DG 预条件子 对于 DG 离散，直接构造稳健的 Schwarz 预条件子较为困难，常见的做法是通过间断-共形归约，将问题转化为共形预条件子。例如，Antonietti 和 Ayuso 等人 [7] 在共形网格上对均匀次数 DG 进行了此类归约。本文在三维一阶非正则网格上，对均匀次数 DG 空间使用了类似的归约技术，结合已建立的共形分解，得到了条件数界为 $C(1+\log p)^3$ 的预条件子。与直接在 DG 空间分析的文献（如 [8]）相比，本文的方法更简洁，但仅限于均匀次数。如何将变量次数 DG 空间稳健地归约到共形空间并保持稳健性，仍是开放问题（作者未在本文中解决）。
6. 局限与代价
- 本文的 DG 误差估计要求网格拟合界面，即系数间断必须与网格线对齐；对于非拟合界面，需要另外的处理（如扩展有限元或 Nitsche 界面匹配）。
- 三维 DG 预条件子要求均匀次数 $p_K\equiv p$；变量次数的 DG 分解未给出。
- 系数稳健性依赖于局部拟单调性条件，该条件虽然在每个顶点片块上比全局条件弱，但仍然对系数分布施加了限制；对于完全不满足该条件的系数分布（如随机高反差），方法可能会失去稳健性。
- 三维共形分解的对数立方损失在 p 很大时仍不可忽略；几何奇异性可能进一步放大该因子（文中常数仅依赖 γ 和 d）。
- 数值实验仅展示了二维和三维简单算例，未验证高度各向异性系数或极端大 p 的情形。
了解领域论文（摘要翻译）
1. 高阶渐近保持格式用于从稀薄到不可压缩区域的动力学方程
High-Order Asymptotic-Preserving Schemes for Kinetic Equations from Rarefied to Incompressible Regimes
作者: Giacomo Dimarco, Axel Klar, Theresa Köfler, Lorenzo Pareschi, Tiwari Sudarshan
类别: math.NA, math-ph
发布日期: 2026-06-26
arXiv ID: 2606.28040v1
链接: http://arxiv.org/abs/2606.28040v1
摘要翻译
中文标题: 高阶渐近保持格式用于从稀薄到不可压缩区域的动力学方程
摘要翻译: 这项工作报告了一种新颖的高阶数值框架，用于求解动力学方程，旨在在平均自由程的所有区域上保持一致有效，涵盖从稀薄动理学尺度到不可压缩流体动力学极限的范围。该方法基于微观-宏观分解，将潜在的动力学方程重新表述为一个耦合系统，由代表流体动力学演化的宏观部分和描述非平衡偏差的微观部分组成。所提出的框架通过使用隐式-显式 Runge-Kutta 方法保证高阶时间精度，这些方法在刚性区域提供稳定性和效率，同时通过将有限差分 WENO 重构与高阶中心差分近似相结合来增强空间分辨率。该方法的一个关键特征是它的渐近保持（AP）性质。我们证明，在平均自由程趋于零的适当渐近极限下，该格式一致地约化为不可压缩 Navier-Stokes 方程的高阶有限差分形式。为了支持理论结果，对一维和二维基准问题进行了数值实验，这些实验证实了该方法在不同流动区域下的精度、稳定性和通用性。
2. 一种用于波动方程控制重构的保结构神经-谱方法
A Structure-Preserving Neural-Spectral Method for Reconstructing Controls of Wave Equations
作者: Tan-Phuc Nguyen, Minh-Binh Tran, Son Tu
类别: math.NA
发布日期: 2026-06-26
arXiv ID: 2606.28029v1
链接: http://arxiv.org/abs/2606.28029v1
摘要翻译
中文标题: 一种用于波动方程控制重构的保结构神经-谱方法
摘要翻译: 尽管关于可控性的分析文献十分丰富，偏微分方程控制的数值重构仍相对欠发展。这一困难对波动方程尤为突出，其守恒结构、振荡动力学和高频行为使得直接离散化和优化极具挑战性。在这项工作中，我们引入了一种用于逼近波动方程控制的神经-谱方法。该方法将状态和控制均表示在Dirichlet谱基中，并使用浅层神经网络对依赖时间的模态系数进行参数化。通过这种方式，波动方程的空间振荡结构被纳入近似，而学习任务被简化为重构时间系数。我们证明了近似结果，表明在相关能量框架下存在精确控制的标准假设下，找到的控制-状态对可以在能量范数下关于时间一致地逼近精确受控轨迹，同时也在(L^2)中逼近相应的控制。我们还给出了一个条件可计算误差估计，它将谱截断、神经网络逼近、数值积分和优化误差分离开来。另外，我们讨论了标准时间步进格式在守恒波动动力学中面临的结构性障碍：显式Euler方法放大高频，隐式Euler方法引入人工耗散，而Crank–Nicolson方法保持振幅但压缩高频相位。在一维、二维和三维空间中的数值实验，展示了该方法在非线性、线性参考和高维控制基准问题上的效果。
不相关论文（基本信息）
1. 在可压缩Boussinesq方程中阻尼垂直传播波的完美匹配层
A perfectly matched layer for damping vertically propagating waves in the compressible Boussinesq equations
作者: Timothy C. Andrews, Kenneth Duru, David Lee
类别: math.NA
发布日期: 2026-06-26
arXiv ID: 2606.28303v1
链接: http://arxiv.org/abs/2606.28303v1
摘要: This paper introduces a new application of the perfectly matched layer (PML) for mitigating model top wave reflections in geophysical fluid models. Typically, a strong Laplacian or Rayleigh damping sponge layer is used near the upper boundary, but these often need many vertical levels or a high model top to be sufficiently effective. An advantage of the PML is that, at the continuous level, it is free of wave reflection at the onset of the damping layer. This enables the PML to be effective even with a thin damping layer. We derive PMLs for the linear and nonlinear versions of the Boussinesq equations, which are a simplified model for vertical dynamics in the atmosphere. In the nonlinear system, we define a novel PML that damps perturbations from a hydrostatically balanced reference state. We approximate the PML equations using the compatible finite element method for numerical experiments. First, tests with the linear Boussinesq system show that the PML is more effective than a typical sponge layer in absorbing acoustic waves near the model top. Next, tests in the nonlinear system show that i) the PML can damp acoustic waves even when they are under-resolved by the time discretisation, and ii) the PML can avoid the standing wave pattern caused by model top reflection of orographic gravity waves. We propose that the PML is worth further development and investigation as a sponge layer alternative in dynamical cores for atmospheric modelling.
2. VGB用于掩蔽扩散模型：面向奖励满足与样本编辑的高效测试时缩放
VGB for Masked Diffusion Model: Efficient Test-time Scaling for Reward Satisfaction and Sample Editing
作者: Kijung Jeon, Thuy-Duong Vuong, Molei Tao
类别: cs.LG, cs.DS, math.NA, math.PR, stat.ML
发布日期: 2026-06-26
arXiv ID: 2606.28301v1
链接: http://arxiv.org/abs/2606.28301v1
摘要: Inference-time scaling is a promising paradigm to improve generative models, especially when outputs must satisfy structural constraints or optimize downstream rewards. We consider Masked Diffusion Model (MDM) and introduce MDM-VGB, a discrete diffusion sampler that augments unmasking generation with theoretically principled reward-guided remasking. Inspired by the recent success of the classical Jerrum-Sinclair backtracking Markov chain in reward-tilted generation, MDM-VGB extends the backtracking random walk from a fixed prefix tree to a masked-state graph, allowing tokens to be unmasked and remasked at arbitrary positions. The resulting sampler favors unmasking and remasking moves that lead to higher-value partial configurations, enabling both effective high-reward generation and efficient repair of low-reward samples. We prove that MDM-VGB is robust to process-verifier noise and achieves quadratic complexity, while popular test-time heuristics such as best-of-$N$ can incur exponential complexity due to error accumulation. Our theoretical findings are corroborated by strong empirical performance, particularly on popular constraint-satisfaction and scientific benchmarks such as Sudoku and QM9.
3. 一种求解高维分数阶福克-普朗克方程的快速高斯和算法
A fast sum-of-Gaussians algorithm for the high-dimensional fractional Fokker-Planck equation
作者: Shidong Jiang, Dong Wang, Qi Zhou
类别: math.NA, physics.comp-ph
发布日期: 2026-06-26
arXiv ID: 2606.28184v1
链接: http://arxiv.org/abs/2606.28184v1
摘要: We present a fast, high-order algorithm for the free-space fractional Fokker-Planck equation (FFPE) in arbitrary spatial dimension. Its fundamental solution, corresponding to a Dirac-delta initial condition, is obtained from the explicit Fourier representation by applying a sum-of-Gaussians (SOG) approximation to the nonseparable stretched exponential, using its complete monotonicity as the Laplace transform of a one-sided $α$-stable density. Each Gaussian term is an ordinary heat kernel and therefore factorizes across spatial coordinates. On a tensor-product grid, the separated form can be assembled in $O(MdN)$ work and storage, rather than forming all $O(N^d)$ grid values, where $M$ is the number of Gaussian terms and $N$ is the number of points per dimension. We prove an a~priori error estimate for the pure-fractional fundamental solution and give a parameter-selection procedure for prescribed accuracy over specified ranges of space and time. In numerical experiments the method achieves more than ten digits of relative accuracy, with $M$ growing only logarithmically in the inverse tolerance, and maintains this accuracy in dimensions up to $d=10^{5}$. This exceeds the dimensions reached in comparable radial-quadrature tests, where the integrand becomes increasingly oscillatory as the dimension grows. Because the method represents the fundamental solution as a separated sum of heat kernels, any initial datum given as a finite sum of tensor products can be evolved in closed form using only one-dimensional convolutions. This yields a computable class of high-dimensional solutions that is amenable to error analysis, and tensor neural networks provide one possible way to construct such separated representations for more general data.
4. 利用物理信息神经网络在有限数据Calderón问题中恢复尖锐电导率特征
Recovering Sharp Conductivity Features in the Finite-Data Calderón Problem with Physics-Informed Neural Networks
作者: Ali AlHadi Kalout, Pablo Tejerina-Pérez, Konstantin Karchev, Pedro Tarancón-Álvarez, Leonid Sarieddine, Raul Jimenez, Max Engelstein, Guy David
类别: cs.LG, eess.IV, math.NA
发布日期: 2026-06-26
arXiv ID: 2606.28158v1
链接: http://arxiv.org/abs/2606.28158v1
摘要: Physics-informed neural networks (PINNs) have recently emerged as a promising framework for addressing the Calderón inverse problem from limited boundary data. In this work, we revisit neural Calderón inversion by introducing multiscale boundary excitations based on randomized wavelet functions and investigating the role of Fourier-feature encoding (FFE) for representing sharp conductivity variations. We propose a physics-informed reconstruction framework that represents the unknown conductivity and the associated family of electric potentials with separate neural networks conditioned on the applied boundary excitations. The governing elliptic PDE is enforced through physics-informed residuals, while finite Dirichlet-to-Neumann (DtN) data are incorporated through boundary losses. Using synthetic data from a finite-difference forward solver, we evaluate the method on conductivity fields with inclusions, sharp interfaces, smooth profiles, and heterogeneous media. Results show that the framework recovers dominant conductivity structures from finite boundary measurements with relative errors between $3\%-12\%$ approximately. We show that FFE improves the reconstruction of localized sharp features, particularly for inclusions and interfaces, but are not universally optimal, with raw-coordinate networks performing competitively for smoother fields. These results highlight coordinate representations and boundary excitation design as key factors in neural Calderón inversion.
5. 改进的能量稳定的对称 Gauss-Seidel 投影方法用于微磁学模拟
Improved Energy Stable Symmetric Gauss-Seidel Projection Method for Micromagnetics Simulations
作者: Yingxi Miao, Changjian Xie
类别: math.NA
发布日期: 2026-06-26
arXiv ID: 2606.28113v1
链接: http://arxiv.org/abs/2606.28113v1
摘要: The Gauss-Seidel projection method (GSPM) constitutes an efficient and numerically stable numerical framework for micromagnetic simulations of ferromagnetic media. This scheme attains first-order temporal accuracy and second-order spatial accuracy. Fast Fourier transform (FFT) techniques can be incorporated to accelerate both the solution of the arising linear algebraic systems and the evaluation of stray magnetic fields. The conventional GSPM relies on a single-sided Gauss-Seidel iteration, which leverages the latest updated state variables associated with the heat-diffusion subproblem. In this work, we develop a symmetric Gauss-Seidel projection method (SGSPM) that retains first-order temporal accuracy and second-order spatial consistency. The proposed symmetric variant exhibits superior stability properties relative to the standard GSPM. Specifically, SGSPM adopts a two-pass symmetric Gauss-Seidel iteration, where updated information from the heat-diffusion stage is fully exploited to rigorously guarantee discrete energy stability. We validate the performance of the devised scheme through numerical investigations of magnetization dynamic evolution and magnetic domain-wall propagation. Numerical evidence demonstrates that the improved symmetric scheme delivers enhanced stability for capturing magnetization motion dynamics.
6. 使用混合精度雅可比算法计算精确奇异向量和特征向量
Computing accurate singular vectors and eigenvectors using mixed-precision Jacobi algorithms
作者: Zhengbo Zhou, Françoise Tisseur, Marcus Webb
类别: math.NA
发布日期: 2026-06-26
arXiv ID: 2606.28069v1
链接: http://arxiv.org/abs/2606.28069v1
摘要: Mixed-precision variants of the Jacobi algorithm for symmetric positive definite eigenproblems and the one-sided Jacobi algorithm for singular value decompositions have recently been shown to compute eigenvalues and singular values to high relative accuracy. However, these analyses do not address the accuracy of the computed eigenvectors and singular vectors. In this paper, we prove error bounds for the computed eigenvectors and singular vectors, where the error is measured by the sine of the angle between the vector and its computed counterpart. The obtained bounds preserve the relative gap structure of the bounds for Jacobi algorithms proved by Demmel and Veselić, but involve the scaled condition number of the preconditioned matrix rather than that of the original matrix (the former of which is typically much smaller). Numerical experiments support our theoretical bounds and demonstrate that the mixed-precision preconditioned Jacobi algorithms are especially effective for ill-conditioned matrices with small absolute gaps and moderate relative gaps between eigenvalues or singular values.
7. 一种用于第三介质接触的稳健混合有限元格式
A robust mixed finite element formulation for third medium contact
作者: M. Vorwerk, J. Schröder, P. Wriggers
类别: math.NA
发布日期: 2026-06-26
arXiv ID: 2606.28036v1
链接: http://arxiv.org/abs/2606.28036v1
摘要: Third medium contact provides a smooth continuum alternative to classical contact algorithms by replacing explicit contact constraints with a highly compliant fictitious medium. In this work, an auxiliary-field stabilization is introduced in which a deformation-gradient-like field is treated as an independent unknown in the third medium and coupled to the physical deformation gradient by a penalty term. A gradient contribution acting on the auxiliary field provides the regularization mechanism without requiring a direct evaluation of higher displacement derivatives. Linear and quadratic interpolation spaces are investigated, including continuous and element-wise discontinuous auxiliary-field approximations. The numerical results show that continuous low-order auxiliary fields provide an effective gradient-type stabilization of the third medium, even when the displacement field is approximated by first-order finite elements. For element-wise discontinuous auxiliary fields, the additional unknowns remain local to each element and can be eliminated locally by static condensation, so that the global system does not necessarily contain additional auxiliary degrees of freedom. Benchmark problems involving large deformation, progressive self-contact and severe third-medium compression are used to assess the formulation.
8. 涨落纳维-斯托克斯方程的有限元方法
A Finite Element Method for Fluctuating Navier–Stokes Equations
作者: Dimitrios Gourzoulidis, Mirko Gallo, Soumaya Elkantassi, Toby Kay, Serafim Kalliadasis
类别: cond-mat.stat-mech, math.NA, physics.flu-dyn
发布日期: 2026-06-26
arXiv ID: 2606.27972v1
链接: http://arxiv.org/abs/2606.27972v1
摘要: We introduce a finite-element framework for simulating thermal fluctuations in compressible fluids governed by the fluctuating Navier-Stokes equations. The method is designed to preserve the fundamental fluctuation-dissipation balance at the discrete level. This is achieved by defining the stochastic forcing term in the weak formulation, ensuring its covariance is proportional to the discrete viscous dissipation operator. A nodal quadrature rule is employed to eliminate unphysical mesh-scale correlations. The time integration is performed using the Crank-Nicolson scheme to maintain numerical stability and accuracy. The proposed approach is numerically validated in one, two, and three spatial dimensions, demonstrating its capability to correctly capture equilibrium fluctuation statistics across various discretisation parameters.
9. 一种用于高维动理学方程的动态低秩多层蒙特卡罗估计器
A Dynamical Low-rank Multilevel Monte Carlo Estimator for High-Dimensional Kinetic Equations
作者: Chinmay Patwardhan, Sebastian Krumscheid, Jonas Kusch, Emil Løvbak, Pia Stammer
类别: math.NA
发布日期: 2026-06-26
arXiv ID: 2606.27888v1
链接: http://arxiv.org/abs/2606.27888v1
摘要: Kinetic equations are used to model a wide range of phenomena important for real-world applications. Their applications span astrophysics, nuclear physics, engineering, and social sciences. Due to their high-dimensional phase space, modelling and quantifying uncertainties, relevant for applications, poses a significant challenge even for modern computing infrastructure. In recent years, dynamical low-rank approximation (DLRA) has gained popularity for making fine grid simulations of high-dimensional problems feasible by evolving the solution of a time-dependent PDE as a low-rank factorization. This reduces the computational and memory requirements significantly. In this work, we propose a low-rank multilevel Monte Carlo estimator for kinetic equations based on a probabilistic rank-adaptive DLRA time integrator. The level hierarchy of the low-rank multilevel estimator is constructed through spatial refinement and by ensuring that the low-rank error remains below the spatial discretization error. We demonstrate the efficacy of the estimator through several numerical experiments from radiation transport, radiation therapy, and shallow water flow.
10. 四方程模型的压力-温度平衡闭合的分析、热力学和数值求解器
Analysis, thermodynamics, and a numeric solver for a pressure-temperature equilibrium closure of the four-equation model
作者: Bennett Clayton, Joshua McConnell, Clell Solomon
类别: math.NA, physics.comp-ph
发布日期: 2026-06-26
arXiv ID: 2606.27726v1
链接: http://arxiv.org/abs/2606.27726v1
摘要: We analyze an often used closure model for multi-material hydrodynamics where pressure temperature equilibrium (PTE) is assumed for every state; emphasis is placed on tabular equations of state. This multi-material model is often referred to as the four-equation model. The identification of the admissible set is presented and is proven to be convex, setting the foundation for development of invariant-domain methods for this model. A novel, robust, and efficient method is presented for solving the highly nonlinear system for the equilibrated pressure and temperature with an arbitrary number of materials. Additionally, we provide a detailed analysis of the thermodynamics of the mixture model for general equations of state and prove existence and uniqueness of the pressure-temperature equilibrium solution under some thermodynamic assumptions.
11. 基于浸入张量分解 (ITD) 的几何保持降阶建模
Geometry-Preserving Reduced-Order Modeling via Immersed Tensor Decomposition (ITD)
作者: Lei Zhang, Jiachen Guo, Guowei He, Thomas J. R. Hughes, Wing Kam Liu
类别: math.NA, cs.CE
发布日期: 2026-06-26
arXiv ID: 2606.27674v1
链接: http://arxiv.org/abs/2606.27674v1
摘要: Body-fitted finite-element methods deliver high-order accuracy but hinge on a clean, watertight, conforming mesh, a requirement that breaks down for the geometrically imperfect CAD assemblies, image-based volumetric data, and voxel-native designs that pervade biomedical engineering and additive manufacturing, where mesh generation has become the dominant cost of the analysis cycle. Immersed methods on regular background Cartesian grids sidestep body-fitted meshing, but classical implementations integrate over irregular cut subdomains, destroying the tensor-product structure that enables separable, reduced-order methods such as tensor decomposition. In this paper we propose the \emph{Immersed Tensor Decomposition} (ITD) framework, which couples a mesh-free geometric representation via body-fitted function with the separable C-HiDeNN-TD reduced-order solver to enable large-scale simulation directly on regular background voxel meshes. The geometry is encoded in three steps: a signed-distance function represents the boundary, a body-fitted function $Φ$ approximates it with controllable error, and a low-rank Tucker decomposition provides model-order reduction; for a fixed grid spacing $h$, accuracy is improved by raising the approximation order of C-HiDeNN interpolation up to degree $p$ with a linear background mesh. The central contribution is an exact Dirichlet formulation that enforces the boundary condition strongly by multiplying the trial function with $Φ$, so that $u=g$ holds by construction without any variational penalty or interface quadrature. We establish an a priori error estimate for the formulation and assess it on canonical 2D/3D domains, demonstrating optimal convergence and robustness on non-Cartesian geometries discretized by regular voxel meshes.
12. 基于微尺度拓扑手术的二维细丝断裂界面跟踪方法
Interface tracking with Microscale Topological Surgery for two-dimensional filament breakup
作者: Raaghav Ramani
类别: physics.flu-dyn, math.NA, physics.comp-ph
发布日期: 2026-06-26
arXiv ID: 2606.27615v1
链接: http://arxiv.org/abs/2606.27615v1
摘要: We design and implement a Microscale Topological Surgery (MTS) algorithm to detect and enforce topological transitions in two-dimensional tracked interfaces. The method combines classical Lagrangian tracking with an intermittent topological processor that: (i) constructs Eulerian snapshots from which an interface family with microscale-resolved topology is extracted, (ii) infers adjacency topology between dual Lagrangian and Eulerian interface families, and (iii) performs interface surgery to stitch the two families together across microscale defect regions. A novel long-time nonlinear alternating-shear flow is introduced, in which repeated stretching and folding generate rich multiscale interface dynamics with filamentation at microscales. Using the MTS algorithm and a posteriori geometric and material diagnostics, we compute and visualize microscale filament-breakup dynamics. Error analysis and scaling studies demonstrate second-order geometric convergence and optimal computational scaling of the MTS algorithm, with topology-processing costs comparable to those of the underlying Lagrangian evolution. Ensemble simulations generated by pseudo-random perturbations of the flow further reveal coherent droplet size distributions and statistically robust filament-breakup dynamics.
13. 论尺度空间度量的稳定性
On the stability of scale-space metrics
作者: William Leeb
类别: math.NA, cs.CV
发布日期: 2026-06-25
arXiv ID: 2606.27605v1
链接: http://arxiv.org/abs/2606.27605v1
摘要: We study the stability of a classical family of metrics defined over functions’ Gaussian scale-space representations, focusing on the comparison of images (functions of two variables). These metrics have precedents both in harmonic analysis, specifically the theory of Besov spaces, and in classical methods of image processing; special cases are also known to be metrically equivalent to certain Wasserstein distances. We quantify these metrics’ robustness to geometric deformations, and introduce rotationally-invariant versions that are stable to changes in angle when comparing tomographic projections. We also describe computationally efficient algorithms for evaluating the metrics from finite samples, and prove their robustness to additive noise. The results are illustrated through numerical experiments.
14. 二维局部自适应非静压浅水方程扩展
Two-Dimensional Locally Adaptive Non-Hydrostatic Extension of Shallow Water Equations
作者: Kemal Firdaus, Jörn Behrens
类别: physics.flu-dyn, math.NA
发布日期: 2026-06-25
arXiv ID: 2606.27562v1
链接: http://arxiv.org/abs/2606.27562v1
摘要: We introduce a two-dimensional non-hydrostatic model for shallow water wave dispersion. The model is based on a locally adapted application of a non-hydrostatic correction to the hydrostatic shallow water equations (SWE) in a predictor-corrector scheme. Applying the non-hydrostatic correction uniformly to the entire domain demands a high computational cost, since an elliptic system of equations needs to be solved for the correction terms. We demonstrate that by determining the area where the non-hydrostatic effects are significant, and applying the correction only locally, the computational effort can be reduced by approximately 40% without sacrificing accuracy in tsunami-like scenarios. As indicators for the non-hydrostatic effect, we use the ratio between total water depth and surface elevation, as well as horizontal velocity norms. Results are shown for several well-known test cases, including wave trains over a semi-circular shoal, static, and moving bottom tsunami-like wave propagation.
15. 作用于 $[0,\infty)$ 上多项式与指数增长函数的算子之 Korovkin 型定理
Korovkin type theorems for operators acting on functions of polynomial and exponential growth on $[0,\infty)$
作者: Ulrich Abel, Ana-Maria Acu, Margareta Heilmann, Ioan Rasa
类别: math.NA
发布日期: 2026-06-25
arXiv ID: 2606.27517v1
链接: http://arxiv.org/abs/2606.27517v1
摘要: We prove two Korovkin-type approximation theorems for sequences of positive linear operators acting on continuous functions on $[0,\infty)$. Under the assumption of pointwise convergence on suitable test functions, we establish pointwise convergence for all functions with polynomial or exponential growth. As direct applications, we obtain convergence results for the classical Baskakov and Szász–Mirakjan operators. The proposed method offers an elementary framework that can be applied to a broad class of positive linear operators.
16. 周期区域上立方非线性薛定谔方程的算子学习
Operator Learning for Cubic Nonlinear Schrödinger Equation on Periodic Domains
作者: Emmanuel E. Oguadimma, Victory C. Obieke, Xueying Yu
类别: cs.LG, math.AP, math.NA
发布日期: 2026-06-25
arXiv ID: 2606.27459v1
链接: http://arxiv.org/abs/2606.27459v1
摘要: We consider the cubic nonlinear Schrödinger (NLS) equation on two-dimensional flat tori with varying aspect ratios. In this formulation, the choice of aspect ratio governs the Fourier resonance structure, so rational and irrational geometries can exhibit different high-frequency cascade behaviors. We present a geometry-conditioned Fourier neural operator (FNO) for the cubic defocusing NLS equation, where the input consists of the real and imaginary parts of the solution together with the aspect-ratio parameter (ω^2). The model is trained to approximate the one-step solution operator and is evaluated on unseen trajectories generated from random-phase initial data using Fourier pseudospectral method. Our numerical experiments show that the learned operator captures the main solution dynamics on both tori and reproduces the distinct Sobolev norm behavior of the two geometries, with stronger (H^2)-growth on the rational torus and more constrained behavior on the irrational torus, consistent with the findings of \cite{hrabski2021energy}. We perform ablation studies to examine the roles of retained Fourier modes, activation functions, Fourier-layer depth, and explicit geometry conditioning. The results indicate that including $ω^2$ improves long-time predictive accuracy, especially for the rational geometry, and supports the use of geometry-aware neural operators for learning spectral-transfer phenomena in nonlinear dispersive partial differential equations.
17. 色散电介质中非线性随机极化分析
Analysis of Nonlinear Random Polarization in Dispersive Dielectrics
作者: Nathan L. Gibson, Emmanuel E. Oguadimma
类别: physics.optics, math.NA
发布日期: 2026-06-25
arXiv ID: 2606.27445v1
链接: http://arxiv.org/abs/2606.27445v1
摘要: We present a study on the time-domain propagation of electromagnetic waves in dielectric materials modeled by a nonlinear Debye medium with random perturbations. Polynomial Chaos Expansions are employed to transform the random nonlinear Debye polarization model into a deterministic framework. We extend the Yee discretization to the resulting coupled system, establish second order accuracy, and verify convergence numerically. We investigate the sensitivity of nonlinear properties to uncertainty, particularly when the amplitude of the input signal is large. Given the challenges in manufacturing where uncertainties can cause optimal parameters to vary and potentially disrupt nonlinear effects, our approach incorporates these uncertainties within the simulation. This can enable the model-based design identification of realizable materials that maintain their desired effects despite variations. The findings from this study contribute to a deeper understanding of wave propagation in complex media, with potential implications for applications in optical communications, material science, and electromagnetic wave control.