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1. 一种适用于Stokes界面问题的压力稳健非协调浸入有限元方法

A Pressure-Robust Nonconforming Immersed Finite Element Method for Stokes Interface Problems

作者: Haifeng Ji, Feng Wang

类别: math.NA

发布日期: 2026-08-07

arXiv ID: 2608.07235v1

摘要翻译: 已知通过对检验函数进行适当修正，可以在Stokes问题中实现压力稳健的混合方法。然而，对于非匹配网格上Stokes界面问题的浸入有限元逼近，由于速度和压力在其中一个界面条件中耦合，速度误差是否独立于压力仍不清楚。本文通过将间断压力新颖地分解为连续分量和依赖于速度的间断分量，给出了肯定回答。我们证明，浸入Crouzeix–Raviart/$P_0$元方法通过对右端项的检验函数进行$H(\operatorname{div})$-协调重构，实现了压力稳健性。对该方法的稳定性和最优误差估计的证明中，常数与界面相对于网格的位置无关。数值实验验证了理论结果。

Comment: 无

链接: http://arxiv.org/abs/2608.07235v1

1. 研究对象和背景

两相不可压缩流动广泛存在于多相流体动力学、生物流以及材料加工等科学和工程问题中. 当惯性效应可以忽略时, Stokes 界面问题能够作为合适的数学模型, 同时保留设计有效数值方法的核心挑战. 设 $\Omega\subset\mathbb{R}^2$ 为具有多边形边界 $\partial\Omega$ 的有界凸区域, $\Gamma$ 是浸没在 $\Omega$ 中的简单闭 $C^2$ 曲线, 将其分为两个子区域 $\Omega^+$ 和 $\Omega^-$. Stokes 界面问题为: 求速度场 $\mathbf{u}$ 和压力场 $p$, 满足

$$ \begin{aligned} -\nabla\cdot\boldsymbol{\sigma}(\mu,\mathbf{u},p) &= \mathbf{f} \quad &&\text{在 } \Omega^+\cup\Omega^-,\\ \nabla\cdot\mathbf{u} &= 0 \quad &&\text{在 } \Omega,\\ [\boldsymbol{\sigma}(\mu,\mathbf{u},p)\mathbf{n}]_\Gamma &= \mathbf{0} \quad &&\text{在 } \Gamma,\\ [\mathbf{u}]_\Gamma &= \mathbf{0} \quad &&\text{在 } \Gamma,\\ \mathbf{u} &= \mathbf{0} \quad &&\text{在 } \partial\Omega, \end{aligned} $$

其中应力张量 $\boldsymbol{\sigma}(\mu,\mathbf{u},p)=2\mu\boldsymbol{\epsilon}(\mathbf{u})-p\mathbf{I}$, 应变张量 $\boldsymbol{\epsilon}(\mathbf{u})=\frac12(\nabla\mathbf{u}+(\nabla\mathbf{u})^\top)$, $\mathbf{I}$ 是单位矩阵, $\mathbf{n}$ 是 $\Gamma$ 的单位法向量, 指向从 $\Omega^-$ 到 $\Omega^+$, $[v]_\Gamma$ 表示跨越 $\Gamma$ 的跳跃, 即 $[v]_\Gamma=v^+|_\Gamma-v^-|_\Gamma$ 且 $v^\pm=v|_{\Omega^\pm}$. 体积力 $\mathbf{f}\in L^2(\Omega)^2$, 黏性系数 $\mu$ 为分片常数且 $\mu|_{\Omega^\pm}=\mu^\pm>0$. 由不可压缩条件 (1.1b) 还可导出 $[\nabla\cdot\mathbf{u}]_\Gamma=0$ 在 $\Gamma$ 上.

与传统体拟合网格方法不同, 无拟合网格有限元方法不要求网格与界面对齐, 在模拟复杂或移动界面时具有高度的灵活性. 浸入有限元 (IFE) 方法是无拟合网格方法中的一类典型代表, 它通过修正界面单元上的形函数来强加界面跳跃条件, 而不改变单元的自由度. 其关键优势在于 IFE 空间与同一网格上的传统有限元空间同构, 这使得该方法易于在现有代码框架下实现.

将 IFE 方法应用于 Stokes 界面问题极具挑战性, 因为在界面条件中速度和压力相互耦合. 已有的若干研究开发了不同的 IFE 方法, 例如基于 Taylor-Hood 元的 IFE 方法 [10]、MINI 元的 IFE 方法 [23] 以及基于标准 Crouzeix-Raviart 元 (CR-P0) 的非协调 IFE 方法 [24]. 然而, 这些方法均不具备压力鲁棒性: 速度误差会依赖于压力, 当体积力被一梯度场修改时 (仅改变压力而不改变真实速度), 离散速度解会受到显著影响 [27]. 对于不含界面的标准 Stokes 问题, 已有多种途径可实现压力鲁棒格式, 如构造无散有限元空间 [38, 40] 或利用散度相容重构测试函数的技巧 [34]. 然而在无拟合网格下的界面问题中, 同时满足界面条件、无散约束和离散 inf-sup 稳定性似乎十分困难. 文献 [41] 提出了一种无散的 Petrov-Galerkin 浸入有限元方法 (PG-IFEM), 其速度空间通过最低阶 Raviart-Thomas 元空间进行增广, 但建立其 inf-sup 条件仍具挑战性.

本文基于 [24] 中引入的非协调 IFE 空间 (即对经典 CR-P0 元在界面单元上的修正), 首次在无拟合网格的 Stokes 界面问题中实现了压力鲁棒性. 核心思想是通过将压力分解为一个连续部分和一个由速度决定的不连续部分, 将耦合的界面条件重写为仅依赖于速度的形式. 在离散水平上, 这一分解自然导出将离散压力表示为分片常数部分 $\hat{p}_h$ 和依赖于离散速度的部分 $R_h(\mathbf{u}_h)$ 之和. 基于此, 本文的方案无需压力惩罚项即可获得 inf-sup 稳定性, 并可将离散问题简化为仅含速度的椭圆问题. 进一步, 对右端测试函数施加 H(div)-相容的最低阶 Raviart-Thomas 重构, 最终得到压力鲁棒的 CR-IFE 方法. 数值实验验证了理论结果.

2. 主要定理或主要结果

本文在连续问题解具有子区域正则性 $(\mathbf{u},p)\in\widehat{H}^2 H^1\cap (\mathbf{V}\times Q)$ 的假设下, 证明了所提压力鲁棒 CR-IFE 方法的如下主要结果:

离散 inf-sup 稳定性: 无须压力惩罚项, 离散速度和压力空间满足 inf-sup 条件, 即存在与 $h$ 和界面位置无关的常数 $\beta>0$, 使得
$$\sup_{\mathbf{v}_h\in X_h^{\rm IFE}} \frac{b_h(\mathbf{v}_h, q_h)}{\|\mathbf{v}_h\|_{1,h}} \ge \beta \|q_h\|_{L^2(\Omega)} \quad \forall q_h\in Y_h,$$
其中 $b_h$ 为带界面边积分项的离散散度形式, $\|\cdot\|_{1,h}$ 为适当的离散 $H^1$ 范数.
最优逼近性质: 对任一 $\mathbf{v}\in X$ (满足界面条件的函数空间), IFE 插值算子 $\pi_h^{\rm IFE}$ 满足 (推论 4.3)
$$\sum_{T\in\mathcal{T}_h} |\mathbf{v} - \pi_h^{\rm IFE}(\mathbf{v})|^2_{H^m(T)} \le C h^{4-2m} \|\mathbf{v}\|^2_{H^2(\cup\Omega^\pm)}, \quad m=0,1.$$
特别地, 该估计的右端仅依赖于速度的 $H^2$ 半范, 不出现压力范数.
压力鲁棒误差估计: 设 $(\mathbf{u},p)$ 为精确解, $(\mathbf{u}_h,p_h)$ 为方法 (3.10) 的离散解, $p_h=R_h(\mathbf{u}_h)+\hat{p}_h$. 存在与 $h$, $\mu$ 以及界面相对网格位置无关的常数 $C$, 使得
$$\|\mathbf{u} - \mathbf{u}_h\|_{1,h} + \|p - p_h\|_{L^2(\Omega)} \le C h \bigl( \|\mathbf{u}\|_{H^2(\cup\Omega^\pm)} + \|p\|_{H^1(\cup\Omega^\pm)} \bigr).$$
此外, 速度误差满足如下压力无关上界:
$$\|\mathbf{u} - \mathbf{u}_h\|_{1,h} \le C h \|\mathbf{u}\|_{H^2(\cup\Omega^\pm)}.$$
该估计表明, 即便压力梯度很大或包含高振荡分量, 只要真实速度光滑, 速度误差不受污染. 这即为压力鲁棒性.
带有表面张力的推广: 本文第 6 节将方法拓展至有表面张力的情况, 即界面条件 (1.1c) 非齐次时, 通过引入适当的界面力项并修改右端重构, 仍能保持最优收敛阶和压力鲁棒性.

数值实验 (第 7 节) 针对不同黏性比、不同界面位置以及带有大幅值压力梯度的算例, 验证了理论误差阶和压力鲁棒性.

3. 研究方法、关键技术和核心工具

本文取得压力鲁棒性的核心创新点与技术路线如下:

1. 连续与离散压力的分解. 传统的 Stokes 界面问题弱形式中, 速度和压力在界面条件 (1.1c) 中耦合: $[\boldsymbol{\sigma}(\mu,\mathbf{u},p)\mathbf{n}]_\Gamma=\mathbf{0}$. 作者引入一个线性算子 $R$, 对任意 $\mathbf{v}$ 构造满足 $[\boldsymbol{\sigma}(\mu,\mathbf{v},R(\mathbf{v}))\mathbf{n}]_\Gamma=\mathbf{0}$ 的函数 $R(\mathbf{v})$, 并将压力写作 $p=R(\mathbf{u})+\hat{p}$, 其中 $\hat{p}$ 连续跨越 $\Gamma$, 即 $[\hat{p}]_\Gamma=0$. 此分解将耦合条件转化为只依赖于速度的形式: $\mathbf{u}$ 自身连同 $R(\mathbf{u})$ 已自动满足界面应力跳跃条件, 剩下的 $\hat{p}$ 成为连续压力. 相应地, 解空间被分裂为 $\widehat{H}^2 H^1\cap(\mathbf{V}\times Q)=\{(\mathbf{v},R(\mathbf{v})):\mathbf{v}\in X\}\oplus(\{0\}\times Y)$, 其中 $X$ 和 $Y$ 分别为速度空间和连续压力空间.

在离散水平, 通过对 IFE 基函数结构的观察 (引理 3.2), 作者发现 IFE 空间中每个与速度自由度对应的基函数自动带有压力修正, 从而可定义一个逐单元的线性算子 $R_{h,T}$, 将离散速度 $\mathbf{u}_h$ 映射为一个分片常数的“速度诱导压力” $R_h(\mathbf{u}_h)$, 使得离散速度-压力对 $(\mathbf{u}_h,R_h(\mathbf{u}_h))$ 满足离散界面条件 (3.3a). 于是, 整个 IFE 空间具有直和分解:

$$\widetilde{\mathbf{V}\mathbf{Q}}_h^{\rm IFE}=\{(\mathbf{v}_h,R_h(\mathbf{v}_h)):\mathbf{v}_h\in X_h^{\rm IFE}\}\oplus(\{0\}\times Y_h),$$

其中 $X_h^{\rm IFE}$ 为修正的 CR 速度空间, $Y_h$ 为分片常数压力空间. 这一分解使得我们可以将离散解的压力表达为 $p_h=R_h(\mathbf{u}_h)+\hat{p}_h$, $\hat{p}_h\in Y_h$, 从而将原问题 (3.10) 分离为仅含速度的椭圆方程和连续压力所对应的散度自由约束.

2. 无惩罚项的 inf-sup 稳定性. 基于上述分解, 与速度耦合的压力部分 $R_h(\mathbf{u}_h)$ 已被自然地纳入速度空间中, 剩余的压力 $\hat{p}_h$ 是跨越界面连续的. 由于 $\hat{p}_h$ 没有跳跃, 离散散度形式 $b_h$ 中的界面边积分项 $\sum_{e\in\mathcal{E}_h^\Gamma}\int_e\{\hat{p}_h\}[\mathbf{v}_h]\cdot\mathbf{n}_e$ 自动消失 (因为 $\{\hat{p}_h\}$ 为常数, 而 $\mathbf{v}_h$ 的边平均值为零). 因此离散散度退化为标准的分片积分 $b_h(\mathbf{v}_h,\hat{p}_h)=-\sum_T\int_T \hat{p}_h\nabla\cdot\mathbf{v}_h$. 此时, 经典的 inf-sup 条件论证可直接从标准 CR 元继承 (因为 $X_h^{\rm IFE}$ 与 CR 空间 $X_h$ 同构, 且在非界面单元上完全相同). 这与文献 [24] 不同, 那里由于压力被全体离散, 必须引入压力惩罚项 $J_h(\cdot,\cdot)$ 来稳定 inf-sup 条件. 本文通过分解, 完全避免了这一惩罚项.

3. 右端测试函数的 H(div) 重构与压力鲁棒性. 压力鲁棒离散的关键在于消除右端项中压力对速度误差的影响. 对标准 Stokes 问题 (不含界面), 文献 [34] 的做法是: 将右端项中的测试函数 $\mathbf{v}_h$ 替换为其 H(div) 相容的最低阶 Raviart-Thomas 插值 $\pi_h^{\rm RT}\mathbf{v}_h$, 即 $l_h(\mathbf{v}_h)=\int_\Omega \mathbf{f}\cdot\pi_h^{\rm RT}\mathbf{v}_h$. 这样做后, 任何梯度力场 $\nabla\psi$ 在 $l_h$ 中将被自动消除, 因为 $\int_\Omega\nabla\psi\cdot\pi_h^{\rm RT}\mathbf{v}_h=-\int_\Omega\psi\,\nabla\cdot(\pi_h^{\rm RT}\mathbf{v}_h)$, 而离散无散性质保证了 $\nabla\cdot(\pi_h^{\rm RT}\mathbf{u}_h)=0$ (在某种意义下) 以及一致性. 本文将这一技巧推广到有界面的情况: 在界面背景下, 由于 $\mathbf{f}$ 也可能在界面处间断, 但 IFE 空间保证了 $\mathbf{u}_h$ 满足合适的界面条件, 同时压力分解使得离散无散约束 $b_h(\mathbf{u}_h,\hat{q}_h)=0$ 仍只涉及连续压力. 右端采用 $\pi_h^{\rm RT}$ 重构后, 误差方程分解为与压力和速度无关的两部分. 作者详细分析了由界面区域几何不一致产生的相容性误差 (即切割单元上的 $\mathcal{O}(h^2)$ 区域 $T^{\triangle}$), 这些误差通过 $\delta$-条带估计 (引理 3.1) 被控制为 $\mathcal{O}(h)$ 阶. 此外, 在界面附近, IFE 基函数与标准 CR 基函数的差异也需被考虑, 但压力分解保证了速度插值孤立地最优 (推论 4.3), 从而最终得到速度误差与压力无关的估计.

4. 交换图性质与插值分析. 文中定义了 IFE 插值算子 $\pi_h^{\rm IFE}$ 和散度重构插值算子 $\pi_h^{\rm RT}$, 并证明了重要性质: $\nabla\cdot(\pi_h^{\rm RT}\mathbf{v}_h)=\pi_h^0(\nabla\cdot\mathbf{v}_h)$ 对 $\mathbf{v}_h\in X_h^{\rm IFE}$ 成立; 更进一步, 这些算子与相应连续算子之间的交换图可保持. 结合 IFE 空间的逼近性质 (定理 4.1 和 4.2) 以及压力分解后得到的尖锐估计 (推论 4.3), 作者得以将整个误差分析置于标准框架内, 且常数与界面位置、黏性比无关.

5. 表面张力的扩展. 第 6 节中, 当界面应力跳跃非零 (即 $[\boldsymbol{\sigma}\mathbf{n}]_\Gamma = \kappa\mathbf{n}$) 时, 通过对右端力项添加适当的界面力贡献, 并相应调整压力分解, 上述分析可几乎逐字推广, 仍保持最优收敛和压力鲁棒性.

4. 与之前工作的比较

本文与已有相关工作的具体推进与区别主要体现在以下几个方面:

与文献 [24] 的非协调 IFE 方法比较. [24] 中提出的 CR-IFE 方法使用标准 CR-P0 配对, 在界面单元上修正形函数以满足界面条件, 并对离散压力施加了额外的惩罚项 $J_h(\cdot,\cdot)$ 以稳定 inf-sup 条件. 但该方法并不具备压力鲁棒性, 数值实验显示当体积力包含大梯度压力分量时, 速度误差会显著增大. 本文的方法在以下三方面与 [24] 不同: (i) 完全去除了压力惩罚项 $\eta_p J_h$, 因为新的压力分解使得 inf-sup 条件自然成立; (ii) 对右端项施加了 H(div)-相容的重构 $\pi_h^{\rm RT}$, 这使得速度解免受压力污染; (iii) 添加了对称/非对称内部惩罚项以处理非协调速度的界面连续性, 这与 [24] 的边项一致. 因此, 本文方法可视为 [24] 的实质性改进, 获得了压力鲁棒性而无需增加自由度.

与文献 [41] 的无散 Petrov-Galerkin IFE 方法比较. [41] 为获取压力鲁棒性, 采用了一种不同的思路: 将速度空间通过最低阶 RT 元增广来构造局部无散空间, 并采用 Petrov-Galerkin 格式. 但其 inf-sup 条件的严格证明直至最近 [17] 才在椭圆界面问题的框架下给出, 且依赖于系数对比度 (即 $\mu^+/\mu^-$) 的假定. 相比之下, 本文的方法只需对右端测试函数进行重构, 速度空间本身保持为修正的 CR 空间, 从而避免了增加额外自由度; 同时, 基于压力分解建立的 inf-sup 条件对任意正的 $\mu^\pm$ 都成立, 更具一般性.

与标准 Stokes 问题的压力鲁棒方法比较 (如 [34, 35, 7, 13]). 对于体拟合网格上的 Stokes 问题, 压力鲁棒性已有多重实现途径, 包括构造无散有限元 [38, 40] 或采用精确无散重构 [34], 以及针对光滑边界用 CutFEM 实现的无散方法 [35, 7, 13]. 然而, 这些方法大多需要额外自由度或弱强制边界条件. 本文首次将这些概念成功移植到无拟合网格下的 Stokes 界面问题中. 关键障碍在于界面条件中将速度和压力耦合在一起; 而传统无散方法假设速度与压力在界面上解耦. 本文通过压力的显式分解克服了这一障碍, 在不牺牲原始 IFE 方法简洁性 (自由度数不变) 的前提下实现了压力鲁棒性.

中文标题: 双曲守恒律的熵稳定间断伽辽金方法的误差估计

摘要翻译: 熵不等式是双曲守恒律适定性的基础，为从无穷多弱解中选取物理上可接受的解提供了关键准则。Chen 和 Shu [J. Comput. Phys. 345 (2017)] 提出了一个统一框架，通过特定的数值求积构造满足任意给定熵的熵不等式的高阶间断伽辽金方法；然而，他们附带的误差分析仅限于截断误差层面，在这些熵稳定格式的严格收敛理论中留下了关键空白。本文通过为双曲守恒律在一般非结构网格上的半离散熵稳定DG方法建立严格的先验误差估计来填补这一空白。分析适用于标量方程和方程组，并基于直接在节点层面进行的有限差分类型的相容性-稳定性论证。在多项式重构假设和$L^\infty$先验界下，我们在基于求积的范数下证明了$O(h^k)$误差估计，该范数等价于有限维重构空间上的断裂$L^2$范数。我们进一步将此框架推广到由Liu, Lu, and Shu [SIAM J. Sci. Comput. 46 (2024)]引入的熵稳定无振荡DG方法，证明了附加的阻尼项不会降低收敛阶。数值实验表明，观察到的收敛率可能超出理论预测最多半个阶。

不相关论文（基本信息）

1. 活性 Cahn-Hilliard 方程的数值分析与粗化动力学

Numerical analysis and coarsening dynamics of the Active Cahn-Hilliard equation

作者: Abramo Agosti, Andrea Giorgini

类别: math.NA, math.AP

发布日期: 2026-08-07

arXiv ID: 2608.07450v1

链接: http://arxiv.org/abs/2608.07450v1

摘要: We investigate the analysis and phase ordering dynamics of the active Cahn–Hilliard equation, providing novel results beyond the current state of the art concerning the well-posedness and the characterization of its coarsening dynamics. We consider both regular polynomial and singular logarithmic potentials. In particular, we exploit a new method based on heteroclinic trajectories in the phase plane to characterize static kink profiles and spherical droplet states, recovering the exact values of key quantities related to static phase-separated configurations; moreover, we develop a theory accounting for surface tension modifications driven by activity and local interface curvature, which explains the power-law shift $L(t)\sim t^{\frac{1}{z}}$ from $z=3$ to $z=4$ induced by activity for the characteristic domain length conjectured in the literature. This shows that there is a transitory effect before the attainment of a finite saturation length. We also design an efficient numerical scheme, based on finite elements, to approximate the model, proving its well-posedness and stability both for regular and singular potentials. In dimensions $d=2,3$ with singular potential, the convergence analysis of the finite element approximation proves the local-in-time existence and uniqueness of weak solutions satisfying the physical constraint $φ\in (-1, 1)$. In dimension $d=1$ with singular potential, we establish global-in-time well-posedness and regularity of weak solutions under a smallness condition on the activity parameter. Finally, we show numerical simulations for different test cases which prove that our numerical algorithm correctly reproduces the expected phase separation dynamics. Moreover, we show the results for coarsening dynamics at late times which present a power law shift from $z=3$ to $z=4$ prior to reaching late-time length saturation, which confirms our theoretical findings.

2. 通过求解多个GPU并行化重复系统加速共轭梯度法

Accelerating the Conjugate Gradient Method by Solving Multiple GPU-Parallelized Duplicate Systems

作者: Andrew Xu, Shaked Regev

类别: math.NA

发布日期: 2026-08-07

arXiv ID: 2608.07327v1

链接: http://arxiv.org/abs/2608.07327v1

摘要: We propose a method to accelerate the conjugate gradient method (CG) on parallel computing hardware by performing block conjugate gradient on one linear system. We aim to reduce the number of iterations until convergence by solving several copies of the system to explore multiple regions of the solution space at once. Although solving multiple systems requires more floating point operations than solving one, these computations are highly parallelizable, even without additional hardware resources. Our method does not replace preconditioning and can be used with any preconditioner. We developed linear regression models to predict iteration reduction and speedup as functions of the coefficient matrix’s size, its number of nonzeros, and the solve time with CG. Our experiments show that our method can reduce solve time by up to 6 times in dense systems and up to 5 times in sparse systems that require long solve times with CG relative to their size.

3. 非协调Helmholtz离散化的区域分解预条件子比较

Comparing domain decomposition preconditioners for non-conforming Helmholtz discretizations

作者: Moritz Gallauner, Emile Parolin, Paul Stocker, Igor Voulis

类别: math.NA

发布日期: 2026-08-07

arXiv ID: 2608.07326v1

链接: http://arxiv.org/abs/2608.07326v1

摘要: We compare additive and multiplicative domain decomposition preconditioners, without coarse correction, for three non-conforming polynomial discretizations of Helmholtz problems: discontinuous Galerkin, embedded Trefftz discontinuous Galerkin, and hybrid discontinuous Galerkin methods. The preconditioners are studied within stationary and Krylov iterative solvers. All three discretizations lead to complex-symmetric systems and local subproblems with inherited impedance-type boundary conditions. These properties enable the use of conjugate gradients and allow the local matrices to be obtained directly by restriction of the global matrix. Numerical experiments in two and three dimensions demonstrate promising performance for large-scale Helmholtz problems.

4. GPU加速的轨道传播与高保真太阳辐射压建模

GPU-Accelerated Orbit Propagation with High-Fidelity Solar Radiation Pressure Modeling

作者: Leandro Zardaín, Ariadna Farrés, Anna Puig, Kevin Muntal, Albert Mir

类别: math.DS, astro-ph.EP, astro-ph.IM, cs.GR, math.NA

发布日期: 2026-08-07

arXiv ID: 2608.07307v1

链接: http://arxiv.org/abs/2608.07307v1

摘要: Solar Radiation Pressure (SRP), the force exerted by photons emitted by the Sun, is one of the main non-gravitational perturbations affecting spacecraft trajectories, making its accurate modeling essential for high-fidelity orbit propagation. While physically-based ray-tracing models improve SRP accuracy, their computational cost becomes a limitation during numerical integration, where the SRP force must be evaluated repeatedly throughout the trajectory propagation. This paper investigates the integration of high-fidelity SRP models into orbit propagation through two complementary contributions: a Vulkan-based GPU implementation that accelerates direct SRP evaluation, and an extension of the SRP model to account for the dynamic orientation of solar panels. These contributions are evaluated independently through orbit propagation experiments, while a precomputed SRP interpolation strategy (SPAD) is included as an alternative approach for reducing computational cost through offline sampling and interpolation. Numerical validation shows that the Vulkan implementation preserves the accuracy of the original OpenGL-based method, showing relative differences below $5\times10^{-4}$ while achieving speed-ups of up to 9.4 for individual SRP computations and up to 15.2 for complete orbit propagation, particularly for geometrically complex spacecraft. The movable solar panel model shows that neglecting panel motion can produce significant long-term propagation errors, especially for spacecraft with large articulated solar panels, while introducing only a moderate computational overhead. Based on the experimental evaluation, this work concludes with practical guidelines for integrating high-fidelity SRP models into orbit propagation frameworks, identifying the scenarios under which interpolation-based methods are sufficient and when online high-fidelity SRP computation is justified.

5. 利用梯度信息增强的类稀疏网格代理模型

Sparse-grids-like surrogate models enhanced with gradient information

作者: Andrea Bressan, Sofia Imperatore, Francesca Locatelli, Lorenzo Tamellini

类别: math.NA

发布日期: 2026-08-07

arXiv ID: 2608.07210v1

链接: http://arxiv.org/abs/2608.07210v1

摘要: This work concerns surrogate modeling for quantities of interest (QoI) arising from parametric non-linear partial differential equations (PDEs). More specifically, we consider extending the sparse-grids surrogate modeling approach to incorporate derivatives of the QoI with respect to the PDE parameters. We discuss why this operation is not straightforward and propose a hybrid approach in which a sparse-grid scheme provides the collocation points in the parameter domain and a suitable polynomial space, but the surrogate model is built with a least-squares approach. We showcase our approach on several numerical tests, and we discuss in particular how its performance crucially depends on the relative cost and accuracy of evaluating the derivatives of the QoI compared to evaluating the QoI itself.

6. 用于三维流体-二维平板作用系统的舒尔补区域分解：分析与预条件

Schur complement domain decomposition for 3D fluid - 2D plate interaction system: analysis and preconditioning

作者: Lander Besabe, Hyesuk Lee

类别: math.NA

发布日期: 2026-08-07

arXiv ID: 2608.07200v1

链接: http://arxiv.org/abs/2608.07200v1

摘要: Fluid-structure interaction problems involving three-dimensional fluids coupled with thin elastic plates arise in many engineering applications and present significant computational challenges due to the strong coupling across the fluid-structure interface. In this work, we develop a Schur complement domain decomposition method for a 3D fluid-2D plate interaction problem in which the fluid is modeled by the unsteady Stokes equations and the structure by a reformulated Kirchhoff plate model. Starting from a mixed finite element discretization with Lagrange multipliers enforcing the interface conditions, we derive an interface Schur complement formulation that decouples the fluid and plate subproblems while preserving the strong interface coupling without requiring subiterations. We then analyze the conditioning of the resulting Schur complement system matrix and show how its condition number deteriorates under mesh refinement. Motivated by this analysis, we introduce an interface-based preconditioner and establish theoretical bounds showing that the proposed preconditioner significantly improves the conditioning of the resulting system. Numerical experiments verify the expected spatial and temporal convergence rates, demonstrate significant reductions in condition numbers and GMRES iteration counts for the proposed preconditioner compared with both the unpreconditioned system and a block Jacobi preconditioner, and illustrate the robustness of the proposed method for challenging added-mass regimes.

7. 渐近保持微观-宏观格式用于准中性和低马赫数区等离子体模拟的动力学升级

An Asymptotic-Preserving Micro–Macro Scheme for Plasma Simulations in Quasi-Neutral and Low-Mach-Number Regimes with Kinetic Upgrades

作者: Zeyu Liu, Fabrice Deluzet, Chang Yang

类别: math.NA

发布日期: 2026-08-07

arXiv ID: 2608.07197v1

链接: http://arxiv.org/abs/2608.07197v1

摘要: We propose an asymptotic-preserving micro–macro method bridging a kinetic description of electrons and a low-frequency reduced model in which the electrons are a massless, quasi-neutral fluid obeying the Boltzmann relation. Two features distinguish the construction. First, the fluid and low-Mach limits are coupled, so that the low-Mach stiffness is handled on a macroscopic system, where implicit treatment is affordable, rather than on the kinetic equations. Second, an auxiliary variable rescales the stiff force balance, turning the singular low-Mach limit into a regular limit of the augmented system, which is shown to remain non-degenerate uniformly in the Debye length as well. This matters at the discrete level: with an iterative linear solver, the stiffness induced by the small Mach number amplifies the solver residual, so that a scheme designed to be asymptotic-preserving in its time discretization alone loses that property once the full solution chain is taken into account. The proposed scheme retains it, with no tightening of the solver tolerance as the Mach number vanishes, and admits a post-processing variant that decouples the auxiliary variable and reduces the size of the linear system. Numerical experiments spanning distinct parameter regimes confirm the analysis: standard semi-implicit schemes lose low-Mach-number equilibrium under residual amplification, whereas the proposed schemes preserve it down to round-off.

8. 用于矩阵函数的Lanczos方法的最优近最优性界

Optimal near-optimality bounds for the Lanczos method for matrix functions

作者: Tyler Chen, David Persson

类别: math.NA

发布日期: 2026-08-07

arXiv ID: 2608.07160v1

链接: http://arxiv.org/abs/2608.07160v1

摘要: Let $A$ be Hermitian positive definite and let $f_m$ denote the Lanczos approximation to $f(A)b$. We prove that if $f(z)$ or $f(z) / z$ is Stieltjes, then the $A^α$-norm error of the Lanczos approximation is within a factor $\tfrac{1}{2}(κ(A)^{E/2} + κ(A)^{-E/2})$ of the the best possible Krylov Subspace Method, where $κ(A)$ is the condition number of $A$ and $E = \max\{α,1-α\}$. Our result strengthens and generalizes the upper bound of [Schweitzer; SIMAX, 46.3 (2025)]. Moreover, we prove that the constant $\tfrac{1}{2}(κ(A)^{E/2} + κ(A)^{-E/2})$ is optimal.

9. 近似域上的高阶虚拟元方法：Neumann边界条件

The High Order Virtual Element Method On Approximate Domains: Neumann Boundary Conditions

作者: Silvia Bertoluzza, Monica Montardini, Daniele Prada

类别: math.NA

发布日期: 2026-08-07

arXiv ID: 2608.07100v1

链接: http://arxiv.org/abs/2608.07100v1

摘要: In the framework of the virtual element method with shifted boundary type treatment of curved geometries, we propose a novel approach to the treatment of Neumann boundary conditions that does not require the approximate aligning of the physical and numerical normal directions and is therefore well suited to handle domain approximations obtained by agglomeration from underlying fine structured grids. For such an approach we give conditions under which we are able to prove optimal error estimates for arbitrary orders of the virtual element discretization.

10. 基于小批量最优传输的回流的极限点

Limit Points of Reflow with Minibatch Optimal Transport

作者: Antonin Chambolle, Johannes Hertrich

类别: math.PR, cs.LG, math.NA

发布日期: 2026-08-07

arXiv ID: 2608.07042v1

链接: http://arxiv.org/abs/2608.07042v1

摘要: Rectified flows, also called flow matching or stochastic interpolants, are generative models that learn a time-dependent vector field steering a probability curve between two probability distributions, usually referred to as latent and target distributions. Reflow accelerates inference by iteratively straightening the trajectories induced by this vector field. We study the asymptotic behavior of this iteration and characterize its limit points. First, we define weak rectified couplings which always exist. Next, when rectified flow updates are alternated with minibatch optimal transport steps of fixed batch size, we show that any limit is $N$-cyclically monotone, where $N$ is the batch size. Such $N$-cyclically monotone couplings enjoy favorable structural and stability properties such as rectifiability and straightness. Finally, restricting velocities to gradient fields and assuming additional support conditions, we prove that reflow limits coincide with the optimal transport map between the endpoint distributions.

11. 紧辛流形上的蒙特卡罗方法

Monte Carlo methods on compact symplectic manifolds

作者: Yuri A. Kordyukov

类别: math.DG, math-ph, math.NA, math.PR

发布日期: 2026-08-07

arXiv ID: 2608.07021v1

链接: http://arxiv.org/abs/2608.07021v1

摘要: We build an unbiased Monte Carlo estimator of the integral of any $C^1$ function on a prequantized compact symplectic manifold against a smooth Riemannian volume form, taking for quadrature nodes the determinantal point process associated with an appropriate spectral projection of the Bochner-Schrödinger operator. We show that the estimator satisfies a central limit theorem, and the decay rate of the mean squared error reaches the optimal worst-case rate investigated by Bakhvalov in Euclidean spaces. These results extend previous results of Lemoine and Bardenet on Monte Carlo methods on compact complex manifolds.

12. 线性和非线性散度型椭圆问题的扩展球面行走算法

Extended Walk-on-Spheres Algorithm for Linear and Nonlinear Elliptic Problems of Divergence-type

作者: Iulian Cîmpean, Andreea Grecu, Arghir Zarnescu

类别: math.NA, math.AP, math.PR

发布日期: 2026-08-07

arXiv ID: 2608.07017v1

链接: http://arxiv.org/abs/2608.07017v1

摘要: The Walk-on-Spheres algorithm, introduced by M. E. Muller in 1956, is a well known Monte Carlo method that leverages Brownian exit distributions from spheres to solve the Laplace equation with Dirichlet boundary conditions. Its mesh-free nature, robustness on complex geometries, favorable scaling with dimension, and intrinsic parallelism distinguish it from mesh-based solvers. However, its efficient applicability has been essentially limited to operators that admit explicit probabilistic exit laws, excluding most variable-coefficient and nonlinear elliptic operators. We propose a general framework that aims to overcome this limitation by using the classical Dirichlet Laplacian and harmonic extension as universal building blocks. Rather than seeking a custom stochastic representation for each operator, we employ Walk-on-Spheres to precompute a reusable numerical operator toolbox that approximates the inverse Dirichlet Laplacian, the harmonic extension operator, and their gradients. These precomputed operators are then used to represent candidate solutions and to transform arbitrary Dirichlet boundary value problems into finite-dimensional algebraic system for an unknown source term. Solving the resulting algebraic system and substituting back yields an approximate solution to the original PDE. Even more, for a general linear second order elliptic operator, the above mentioned precomputed toolbox can be directly used to obtain not just an approximation of a certain solution of the corresponding generalized Dirichlet problem, but an estimator of both the Green’s integral operator and the elliptic measure operator. Numerical experiments on a range of benchmarks, including non-symmetric and anisotropic linear elliptic equations, semilinear and quasilinear problems, demonstrate the method’s flexibility and efficiency.

13. Poisson问题中非拟合谱元上基于最小化的广义高阶多项式校正

Generalized high-order minimization-based polynomial corrections on unfitted spectral elements for the Poisson problem

作者: Mirco Ciallella, Jens Visbech

类别: math.NA

发布日期: 2026-08-07

arXiv ID: 2608.06999v1

链接: http://arxiv.org/abs/2608.06999v1

摘要: Higher-order finite element methods are effective for solving partial differential equations. However, applying them in complex curved domains is often difficult due to challenges in creating high-quality curvilinear meshes. Unfitted, or embedded, methods provide a valid alternative by avoiding complete mesh generation and curved element integration, but their accuracy can suffer from geometric errors introduced during embedding. In this work, we explore a new family of polynomial corrections obtained by solving a local minimization problem to improve the consistency of embedded boundary methods. This approach generalizes existing techniques such as the shifted boundary method (SBM) and the reconstruction for off-site data (ROD) method. Unlike the SBM, which depends on truncated Taylor expansions, the proposed family of polynomial corrections is derived from a constrained minimization problem, similar to the ROD method. We demonstrate that this approach yields better system conditioning than the original SBM and provides a flexible framework that can be applied pointwise, without solving the full ROD linear system for each boundary element. This paper presents four formulations based on different functionals and extends the family of minimization-based polynomial corrections to handle Neumann and Robin boundary conditions, demonstrating that an elegant formulation is possible in this context. Several numerical experiments for the Poisson problem are presented to show that the generalized polynomial corrections achieve high-order accuracy across all boundary conditions.

14. 一种混合维度扩散问题的模型及其有限元近似

A model and a finite element approximation of the mixed-dimensionality diffusion problem

作者: Ignacio Romero, David Portillo

类别: math.NA, math-ph

发布日期: 2026-08-07

arXiv ID: 2608.06976v1

链接: http://arxiv.org/abs/2608.06976v1

摘要: We present the formulation of a boundary value problem that models the coupled behavior of a three-dimensional diffusive solid with one-dimensional diffusive fibers embedded inside it. We introduce a variational statement of the problem that identifies the linked diffusive fields as energy minimizers under a coupling constraint. This saddle-point problem is proved to be well posed. Then, we introduce a finite element discretization of the proposed boundary value problem, and we prove the convergence of the finite element solution to the exact one. The most significant feature of this approximation is that the meshes of the bodies need not be conforming. Numerical examples confirm the theoretical results

15. 面向高维问题的迭代阈值低秩时间积分方法

Iterative thresholding low-rank time integration for high-dimensional problems

作者: Markus Bachmayr, Tianyu Jin, Polina Sachsenmaier, Federico Vismara

类别: math.NA

发布日期: 2026-08-07

arXiv ID: 2608.06905v1

链接: http://arxiv.org/abs/2608.06905v1

摘要: This work analyzes a method for time integration of high-dimensional linear Schrödinger-type problems based on hierarchical tensor approximations. In particular, this method provides a balance between error bounds and associated approximation ranks, using a scheme for iterative refinement with soft thresholding of tensors. The practical performance of the method is illustrated by numerical tests on coupled oscillators.

16. 关于浅层神经网络的广义Barron空间

On Generalized Barron Spaces for Shallow Neural Networks

作者: Shuai Lu, Leqi Zhang

类别: math.NA

发布日期: 2026-08-07

arXiv ID: 2608.06843v1

链接: http://arxiv.org/abs/2608.06843v1

摘要: Classical Barron spaces are function spaces specifically designed for shallow neural networks mostly with ReLU, $\mathrm{ReLU}^k$ (RePU) or Lipschitz continuous activation functions. In the present work, we introduce a generalized Barron space ( B_σ^{\varphi} ) for shallow neural networks with a generic activation function possessing certain smoothness properties. The subscript ( σ) denotes the activation function, while the superscript ( \varphi ) controls the smoothness of the generalized Barron spaces, defined via a ( \varphi )-weighted integral norm imposed on neural network parameter measures. Under certain assumptions on ( \varphi ) and ( σ), we show that ( B_σ^{\varphi} ) can be continuously embedded into Sobolev spaces. We also explore the relationships among various function spaces for shallow neural networks, demonstrating that our definition encompasses most conventional ones. As applications of the proposed generalized Barron spaces, we derive approximation rates within these spaces and establish error bounds for numerical differentiation with regularization penalized by the newly introduced generalized Barron norm. Numerical examples confirm that the proposed spaces allow the construction of neural networks with varying degrees of smoothness while using the same activation function.

17. 基于确定性样本的经验最小二乘的最优神经网络逼近

Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samples

作者: Xinliang Liu, Tong Mao, Jinchao Xu

类别: math.NA, cs.LG

发布日期: 2026-08-07

arXiv ID: 2608.06687v1

链接: http://arxiv.org/abs/2608.06687v1

摘要: We develop a rigorous theory of discrete residual least-squares approximation for elliptic spectral equations $\mathfrak L_βu=f$ using linearized ReLU$^k$ neural networks on the sphere, where $\mathfrak L_β$ is a positive elliptic spectral multiplier of order $β$. Given a parameter set $Θ_n=\{θ_{j}^*\}_{j=1}^n\subset\mathbb S^d$, we approximate $u$ in the linearized network space $L_n^k(Θ_n)$ by the discrete residual on the collocation points $\{η_i^*\}_{i=1}^m$ \begin{equation*} u_{n,m}\in\arg\min_{v_n\in L_n^k(Θ_n)}\frac1m\sum_{i=1}^m\left(f(η_i^)-\mathfrak L_βv_n(η_i^)\right)^2. \end{equation*} With $k>\frac{d-1}{2}+β$, for antipodally quasi-uniform network parameter sets and any quasi-uniform collocation points with $m\gtrsim n$, we prove that \begin{equation*} |u-u_{n,m}|{\mathcal H^β(\mathbb S^d)}\eqsim|f-\mathfrak L_βu{n,m}|{\mathcal L^2(\mathbb S^d)}\lesssim n^{-\frac{r}{d}} \begin{cases} |f|{\mathcal W^{r,p}(\mathbb S^d)},&\frac{d}{p}<r\leq \frac{d}{2},~p>2,\ |f|{\mathcal H^r(\mathbb S^d)},&r>\frac{d}{2}. \end{cases} \end{equation*} We also establish a high-probability residual estimate, up to a logarithmic factor and an arbitrarily small smoothness loss, for i.i.d.\ uniformly distributed collocation points. The key analytical ingredient is a Bernstein inequality for linearized ReLU$^k$ network spaces. If $\underline h$ denotes the antipodal separation distance of the network parameters, then \begin{equation*} |v_n|{\mathcal H^r(\mathbb S^d)}\lesssim\underline h^{-(r-s)}|v_n|_{\mathcal H^s(\mathbb S^d)},\qquad 0\leq s<r<k+\tfrac12. \end{equation*}

18. 低分数阶光滑性的加权Sobolev空间中的多元积分与近似

Multivariate integration and approximation in weighted Sobolev spaces of low fractional smoothness

作者: Mou Cai, Takashi Goda

类别: math.NA

发布日期: 2026-08-07

arXiv ID: 2608.06685v1

链接: http://arxiv.org/abs/2608.06685v1

摘要: The weighted half-period cosine space has often been employed in the theory of quasi-Monte Carlo methods for multivariate integration and approximation of non-periodic functions. For integer-order smoothness, its norm equivalence to certain weighted unanchored Sobolev spaces has been established in the literature. In this work, we extend this equivalence to fractional-order smoothness up to $2$. By introducing an explicit representation via Slobodeckij-type seminorms, we prove a norm equivalence between the half-period cosine spaces and the corresponding weighted unanchored Sobolev spaces. Our Sobolev norm representation clarifies how the fractional regularity dictates the presence or absence of boundary constraints and (non-)periodic structures. Furthermore, we investigate the limiting behavior of these fractional spaces as the smoothness parameter approaches integer boundaries, establishing a continuous bridge to the classical integer-order Sobolev spaces. These equivalence results enable us to transfer the near-optimal error bounds and tractability results for multivariate integration and function approximation from the half-period cosine settings to our newly introduced fractional Sobolev spaces.

19. 用于二维Navier–Stokes方程涡团动力学的随机批处理法的平均场误差估计

Mean field error estimate of the random batch method for vortex blob dynamics for the 2D Navier–Stokes Equation

作者: Zhenyu Huang

类别: math.NA

发布日期: 2026-08-06

arXiv ID: 2608.06533v1

链接: http://arxiv.org/abs/2608.06533v1

摘要: We propose and analyze the random batch vortex blob method for the 2D Navier–Stokes equation in vorticity form on the whole plane. The vortex blob method is based on an interacting particle system of $N$ particles with computational complexity of $O(N^2)$, which is reduced to $O(N)$ by the random batch method \cite{JinLiLiu2020}. Our main result is a quantitative law level mean field error estimate whose dependence on the blob radius remains algebraic. We treat the two main error mechanisms separately. The random batch error is controlled through a locally coupled auxiliary partition, a symmetric law comparison, and Fisher-information dissipation. The mean field fluctuation is estimated by exploiting the oddness and divergence-free structure of the Biot–Savart kernel. For smooth, strictly positive initial vorticity, we prove on every finite time interval a normalized relative-entropy bound of order $ O\!\left(\varepsilon^{-4}τ^2+N^{-1} \right), $ with constants independent of $N$, $τ$, and $varepsilon$. Here $τ$ is the batch refreshing interval and $\varepsilon$ is the blob radius. As a consequence, the fixed-particle marginals converge strongly in $L^1$ to tensor products of the regularized vorticity solution when $N\to\infty$ and $\varepsilon^{-2}τ\to 0$.

重点关注论文（完整分析）#

1. 一种适用于Stokes界面问题的压力稳健非协调浸入有限元方法#

1. 研究对象和背景#

2. 主要定理或主要结果#

3. 研究方法、关键技术和核心工具#

4. 与之前工作的比较#

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摘要翻译#

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3. 用于Willmore和Helfrich流的参数化逼近的约束Onsager变分框架#

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4. 多面体域中抛物问题的一类 HDG 方法最大模估计的统一分析#

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6. 求解玻尔兹曼方程的弱对抗神经推进方法#

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7. CT-PIKAN：用于曲线域偏微分方程求解的坐标变换物理知情 Kolmogorov-Arnold 网络与基于自动微分的度量评估#

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8. 双曲守恒律的熵稳定间断伽辽金方法的误差估计#

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1. 活性 Cahn-Hilliard 方程的数值分析与粗化动力学#

2. 通过求解多个GPU并行化重复系统加速共轭梯度法#

3. 非协调Helmholtz离散化的区域分解预条件子比较#

4. GPU加速的轨道传播与高保真太阳辐射压建模#

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6. 用于三维流体-二维平板作用系统的舒尔补区域分解：分析与预条件#

7. 渐近保持微观-宏观格式用于准中性和低马赫数区等离子体模拟的动力学升级#

8. 用于矩阵函数的Lanczos方法的最优近最优性界#

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14. 一种混合维度扩散问题的模型及其有限元近似#

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重点关注论文（完整分析）

1. 一种适用于Stokes界面问题的压力稳健非协调浸入有限元方法

1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

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1. 论分裂形式在间断 Galerkin 格式混淆误差与熵稳定性中的作用

摘要翻译

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摘要翻译

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