生成时间: 2026年08月12日 12:28:32

重点关注论文数量: 1

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不相关论文数量: 13

重点关注论文（完整分析）

1. 解耦势积分方程的高阶 Galekin 格式

A High-Order Galerkin Scheme for the Decoupled Potential Integral Equations

作者: David Scott Winterrose

类别: math.NA

发布日期: 2026-08-11

arXiv ID: 2608.10799v1

摘要翻译: 我们证明了解耦势积分方程在 Sobolev 空间中的适定性。利用基本的伪微分技巧，在标准 $L^2$ 配对下获得了离散稳定性，进而我们推导出采用间断基函数的高阶格式的谱收敛性，或者当对系统的向量部分施加额外的逐单元切向连续性时亦然。该格式被证明比成熟的商业 EFIE/CFIE 求解器具有更好的条件数，在很多情况下几乎与波数、网格密度和阶数无关。

Comment: 19 pages

链接: http://arxiv.org/abs/2608.10799v1

1. 研究对象和背景

本文研究三维空间中有界良导体的时谐电磁波散射问题. 设散射体边界 $\Gamma\subset\mathbb{R}^3$ 是光滑的定向紧曲面, 允许有多个连通分支, 外部介质为波数 $k\ge0$ 的均匀无界区域. 传统上, 电磁散射的边界积分方程方法依赖于电场积分方程 (EFIE)、磁场积分方程 (MFIE) 或组合场积分方程 (CFIE). 然而这些方程都存在固有缺陷: EFIE 和 MFIE 在内部共振频率处有解不唯一, EFIE 在稠密网格下出现低频崩溃, MFIE 在 $k\to0$ 时存在灾难性抵消, 而且当曲面亏格非零时, 所有方程在静态极限下都会出现拓扑崩溃. 在算子层面, 这导致未离散的系统就已经条件数很差, 任何离散化后的技巧都无法根除这一方程本质带来的问题.

解耦势积分方程 (Decoupled Potential Integral Equations, DPIE) [19,21] 能够避开上述困难. DPIE 本质上是对修正积分方程 [6] 的一种变形, 并添加了零均值约束以消除静态极限下算子的核. 其物理模型为理想电导体 (Perfect Electric Conductor, PEC) 上的散射问题.

设 $\nu$ 为 $\Gamma$ 的单位外法向量, $J$ 为近复结构 $J(a)=\nu\times a$. 记 $S_k$ 为单层位势, $D_k$ 为双层位势, $D_k^t$ 为其转置, $W_k$ 为超奇异算子, $M_k$ 为磁算子:

$$\begin{aligned} S_k\rho(x)&=\int_\Gamma \Phi_k(x,y)\rho(y)d\sigma(y),\\ D_k\rho(x)&=\int_\Gamma \left[\frac{\partial}{\partial\nu(y)}\Phi_k(x,y)\right]\rho(y)d\sigma(y),\\ D_k^t\rho(x)&=\int_\Gamma \left[\frac{\partial}{\partial\nu(x)}\Phi_k(x,y)\right]\rho(y)d\sigma(y),\\ W_k\rho(x)&=\lim_{\varepsilon\to0}\int_{\Gamma\setminus\{z\in\Gamma\mid |z-x|<\varepsilon\}} \left[\frac{\partial}{\partial\nu(x)}\frac{\partial}{\partial\nu(y)}\Phi_k(x,y)\right]\rho(y)d\sigma(y),\\ M_k a(x)&=\int_\Gamma \nu(x)\times(\nabla_x\Phi_k(x,y)\times a(y))d\sigma(y), \end{aligned}$$

其中 $\Phi_k$ 为 Helmholtz 方程的格林函数. 向量 DPIE 系统为:

$$\begin{pmatrix} \frac12 I+M_k & L_k \\ 0 & \frac12 I+D_k \end{pmatrix} \begin{pmatrix} a \\ \rho \end{pmatrix} + i\eta \begin{pmatrix} \nu\times S_k J & \operatorname{curl}_\Gamma S_k \\ S_k\operatorname{curl}_\Gamma & -k^2 S_k \end{pmatrix} \begin{pmatrix} a \\ \rho \end{pmatrix} - \begin{pmatrix} 0 \\ \sum_{j=1}^N v_j\chi_{\Gamma_j} \end{pmatrix} = \begin{pmatrix} f \\ h \end{pmatrix},$$

$$\int_{\Gamma_j} \left[-\nu\cdot S_k(\nu\rho) + i\eta\left(\nu\cdot S_k J a - \frac12\rho + D_k^t\rho\right)\right] d\sigma = q_j,$$

其中 $L_k\rho = -\nu\times S_k(\nu\rho)$, $a$ 为切向电流, $\rho$ 为标量势, $v_j$ 为 Lagrange 乘子, $\eta\in\mathbb C$ 为自由参数. 标量 DPIE 系统为:

$$\left(\frac12 I + D_k - i\eta S_k\right)\rho - \sum_{j=1}^N v_j\chi_{\Gamma_j} = h,\quad \int_{\Gamma_j} \left[(W_k-W_0)\rho + i\eta\left(\frac12 - D_k^t\right)\rho\right] d\sigma = q_j.$$

已有工作主要在 Hölder 空间框架下证明了 DPIE 的适定性 [19,21], 但尚未在 Sobolev 空间尺度上建立系统的泛函分析理论. 此外, 已有的伽辽金边界元实现均为低阶格式, 基于 RWG 屋顶基函数 [10,3,15], 而高阶求解器大多采用 Nyström 型配置法 [20,21]. 本文的目标正是填补这一空白: 在光滑曲面假设下, 证明 DPIE 在整个 Sobolev 空间尺度上的适定性, 并构造高阶伽辽金离散格式, 同时分析其稳定性和收敛性.

2. 主要定理或主要结果

本文的主要理论结果分为适定性与离散稳定性两方面, 均要求参数 $\eta\in\mathbb{R}\setminus\{0\}$ 且散射体边界 $\Gamma$ 是光滑的紧曲面.

定理 2.4 (Sobolev 空间适定性): 向量与标量 DPIE 系统对任意 $s\in\mathbb{R}$ 都是适定的, 即混合算子 $V:\bigl(H^s(\Gamma,T\Gamma\oplus\mathbb{C})\bigr)\oplus\mathbb{C}^N\to\bigl(H^s(\Gamma,T\Gamma\oplus\mathbb{C})\bigr)\oplus\mathbb{C}^N$ 和 $S: H^s(\Gamma)\oplus\mathbb{C}^N\to H^s(\Gamma)\oplus\mathbb{C}^N$ 均为同构.

定理 2.5 ($L^2$-Galerkin 稳定性): 设 $\{\mathcal{H}_n\}$ 为 $L^2(\Gamma,T\Gamma\oplus\mathbb{C})\oplus\mathbb{C}^N$ (或 $L^2(\Gamma)\oplus\mathbb{C}^N$) 的一族 Galerkin 子空间序列, 则当 $n$ 充分大时, 离散 inf-sup 条件成立:

$$\inf_{u\in\mathcal{H}_n}\sup_{w\in\mathcal{H}_n} \frac{|\langle V u, w\rangle|}{\|u\|\|w\|} \ge \alpha >0,\quad \inf_{u\in\mathcal{H}_n}\sup_{w\in\mathcal{H}_n} \frac{|\langle S u, w\rangle|}{\|u\|\|w\|} \ge \beta >0.$$

由此可导出拟最优误差估计:

$$\|V^{-1}g - u_n\|_{L^2\oplus\mathbb{C}^N} \le \frac{\|V\|}{\alpha}\inf_{w\in\mathcal{H}_n}\|V^{-1}g - w\|_{L^2\oplus\mathbb{C}^N}.$$

推论 3.1 (高阶收敛性): 在形状正则的网格剖分和分片张量积 Legendre 多项式基下, 若真解具有局部 $H^m$ 正则性, 则有不连续伽辽金解的误差估计:

$$\|V^{-1}g - u_n\|_{L^2(\Gamma,T\Gamma\oplus\mathbb{C})\oplus\mathbb{C}^N} \le \frac{\|V\|}{\alpha} \sum_j \sum_{Q} \varepsilon(Q) \|f\|_{H^m(\tau_j(Q), T\tau_j(Q)\oplus\mathbb{C})},$$

$$\|S^{-1}g - u_n\|_{L^2(\Gamma)\oplus\mathbb{C}^N} \le \frac{\|S\|}{\beta} \sum_j \sum_{Q} \varepsilon(Q) \|f\|_{H^m(\tau_j(Q))},$$

其中 $\varepsilon(Q) = C_m \frac{h(Q)^{\min\{p(Q)+1,m\}}}{p(Q)^m}$, $h(Q)$ 为单元直径, $p(Q)$ 为多项式阶数, $C_m$ 仅依赖于几何与基函数选取. 若采用切向连续的向量基, 亦可得到类似估计.

3. 研究方法、关键技术和核心工具

本文的核心分析工具是伪微分算子理论, 通过该框架将 DPIE 系统中的积分算子视为 $\Gamma$ 上的经典伪微分算子, 并充分利用主象征的代数结构.

1. 伪微分算子框架下的适定性

首先确认各积分算子的阶数: $S_k, D_k, D_k^t\in\Psi^{-1}(\Gamma)$, $W_k\in\Psi^1(\Gamma)$ 但 $W_k-W_0\in\Psi^{-1}(\Gamma)$, $M_k\in\Psi^{-1}(\Gamma, T_\mathbb{C}\Gamma)$. 由此可将向量 DPIE 的主算子 $A$ 写为

$$A = \begin{pmatrix} \frac12 I & i\eta\operatorname{curl}_\Gamma S_k \\ i\eta S_k\operatorname{curl}_\Gamma & \frac12 I \end{pmatrix} + K_k,\qquad K_k\in\Psi^{-1}(\Gamma; T_\mathbb{C}\Gamma\oplus\mathbb{C}),$$

进而通过象征计算得到 $A$ 的主象征:

$$\sigma^0(A)(x,\xi)\begin{pmatrix}\delta a \\ \delta\rho\end{pmatrix} = \begin{pmatrix} \frac12\delta a - \frac12\eta(\nu(x)\times \xi^\sharp/|\xi|_g)\delta\rho \\ +\frac12\eta(\nu(x)\times \xi^\sharp/|\xi|_g)\cdot \delta a + \frac12\delta\rho \end{pmatrix}.$$

在切平面的正交分解下, 该象征等价于矩阵

$$\frac12\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & -\eta \\ 0 & +\eta & 1 \end{pmatrix},$$

当 $\eta\in\mathbb{C}\setminus\{-i,+i\}$ 时可逆, 当 $\eta\in\mathbb{R}$ 时强椭圆. 由此可知 $A$ 是强椭圆算子, 存在 Gårding 不等式:

$$\operatorname{Re}\langle A\boldsymbol{f},\boldsymbol{f}\rangle_{L^2} \ge \alpha_0\|\boldsymbol{f}\|_{L^2}^2 - \alpha_{-1/2}\|\boldsymbol{f}\|_{H^{-1/2}}^2,\quad \boldsymbol{f}=(a,\rho).$$

进一步结合 Fredholm 性质与指标同伦不变性, 证明 $\operatorname{ind}A=0$. 加入零均值约束后, 通过紧扰动论证可得混合算子 $V$ 和 $S$ 也是 Fredholm 指标零. 再利用光滑函数类的唯一性 (来自 [19,6]) 导出核空间的平凡性, 从而得到可逆性. 这一过程完全在 $H^s$ 尺度上进行, 不依赖 Hölder 空间.

2. 离散稳定性与拟最优估计

在 $L^2$ 框架下, 使用 Schatz 型的矛盾论证从 Gårding 不等式导出离散 inf-sup 条件. 具体来说, 假设不满足离散 inf-sup, 则存在一列单位长度的离散函数 $u_n$, 使得 $\langle V u_n, w\rangle\to0$ 对所有 $w$ 成立, 即 $V u_n\rightharpoonup0$ 弱收敛于 $L^2$. 利用 Gårding 不等式和范数关系, 结合 $H^{-1/2}$ 中的紧性, 可证得弱极限 $u$ 非零; 另一方面, 由弱收敛推知 $V u=0$, 与可逆性矛盾. 由此得到与网格无关的稳定性常数, 并保证离散系统的条件数一致有界.

3. 高阶基函数的设计与积分处理

离散部分选用分片张量积 Legendre 多项式. 为了对 $L^2$ 配点直接应用伽辽金法, 需处理 $\operatorname{curl}_\Gamma S_k$ 和 $S_k\operatorname{curl}_\Gamma$ 中的 Cauchy 主值积分. 通过分部积分将包含旋度的项转化为弱奇异核积分与线积分之和:

$$\langle S_k\operatorname{curl}_\Gamma a,\varphi\rangle = \sum_{U_j} \left( \int_{U_j} (\operatorname{curl}_{U_j} a|_{U_j}) S_k\varphi\,d\sigma + \int_{\partial U_j} (t_{\partial U_j}\cdot a|_{U_j}) S_k\varphi\,d\ell \right).$$

若基函数在单元边界上切向连续, 则线积分相互抵消, 避免计算奇异贡献. 据此, 文中提出两种基函数选取:

不连续正交基: 直接在参考单元上取 $L^2$-正交的 Legendre 多项式, 并利用度量平方根 $\Lambda$ 进行协变变换, 得到局部正交向量基. 此方案保持全局正交性, 能得到极好的条件数 (几乎不依赖于波数、网格尺寸和多项式阶数), 但需要计算线积分.
切向连续基: 采用修正的 Legendre 多项式 $\tilde P_n$ 构造满足单元边连续性条件的 Nedelec 型函数, 从而消去线积分, 实现更简便, 但条件数略逊于不连续基.

误差估计基于 Babuška-Suri 逼近引理, 仅需形状正则网格 (不强求拟均匀), 且 $h$ 和 $p$ 的误差常数明确. 对于切向连续基, 可通过 Scott-Zhang 型算子得到类似结果.

4. 与之前工作的比较

本文与已有 DPIE 及边界元方法的研究相比, 在以下方面做了实质性推进:

1. 函数空间框架的推广 文献 [19, 21] 在 Hölder 空间框架下建立了 DPIE 的适定性. 尽管 Hölder 空间可涵盖较低正则性, 但现代数值分析通常在 Sobolev 空间进行误差估计与稳定性分析. 本文首次给出 DPIE 在整个 Sobolev 尺度 $H^s$ 上的完整适定性证明 (定理 2.4), 这使离散误差的 $L^2$ 和负范数估计有了理论基础. 同时, 证明过程完全基于伪微分算子理论, 比原 Hölder 空间的位势论证更具代数清晰性, 且自然导出强椭圆性, 为后续离散稳定性铺平道路.

2. 离散化策略: 从低阶到高阶, 从 Nyström 到伽辽金

与现有伽辽金实现 [10, 3, 15] 对比: 这些工作均采用低阶 RWG 屋顶基函数, 虽然几何灵活性好, 但仅有一阶精度, 且条件数随网格加密增长较快. 本文设计了基于张量积 Legendre 多项式的高阶格式, 可以达到谱收敛速度, 并且条件数表现出对 $h$、$p$ 和 $k$ 的高度独立性.
与 Nyström 型求解器 [20, 21] 对比: 此类方法通过配置点实现高阶精度, 但不直接具有最小二乘或伽辽金方法的对称性和超收敛特性. 本文的伽辽金格式在 $L^2$ 配对下自然诱导出对称正定的部分, 且稳定性由强椭圆性保证, 理论更简洁.

3. 条件数优势 数值实验 (尽管文中未给出具体数据, 但摘要和引言中强调) 表明, 本文的格式相比成熟的商用 EFIE/CFIE 求解器具有显著更好的条件数. 特别地, 对于不连续基方案, “在许多情形下条件数几乎不依赖于波数、网格密度和多项式阶数”, 这是现有伽辽金边界元难以达到的. 其原因在于 $L^2$ 正交基的选取保留了算子强椭圆性在离散层的对应性质, 而传统低阶基往往破坏这一结构.

4. 与解耦场积分方程 (DFIE) 的区别 文献 [18] 提出针对传输问题的 DFIE, 同样在 Hölder 空间下讨论且未涉及 Sobolev 空间. 本文明确聚焦于 PEC 问题的 DPIE, 两者方程形式不同, 但分析思路有相通之处. 本文的 Sobolev 空间处理为未来在 DFIE 或其它类似系统上推广提供了范本.

中文标题: 离散几何上非线性 Naghdi 壳的形状优化

摘要翻译: 薄壳通过其形状承担荷载，在有限挠度下，其刚度随荷载本身而变化，因此基于线性模型得到的形状最优解可能远非最优。我们提出了一套自动化框架，将完全几何非线性的壳体响应嵌入形状优化循环中。正模型采用五参数非线性 Naghdi 壳，并通过部分选择性缩减积分来抑制锁定，它直接作用于离散（多边形）三角剖分上，并带有数值重构的指向场——从而省去了等几何方法中精确的中面参数化，而一旦几何本身成为设计变量，这种参数图便不复存在。该模型在 Firedrake 中实现，自动生成残差、一致切线与伴随方程；通过全加载-连续求解的算法微分计算出的形状导数，驱动 Fireshape/ROL 信赖域优化器。正求解器重现了 Sze/Abaqus 基准算例——一端夹支的半圆柱壳受点荷载的情形，捕捉到几何线性模型无法再现的渐进刚化现象。优化结果对照 COMSOL 基准算例——受弯的钣金托架——进行了验证：该框架演化出相同的中面离面褶皱机制，并在给定的位移预算内使弹性应变能降低了 87%，且与基准算例的幅值和面积变化一致。将其应用于弯曲半圆柱时，框架形成了一条光滑的刚化折痕，使壳体在荷载下的平均挠度减小了 78%。

不相关论文（基本信息）

1. 有理逼近与分段多项式逼近有多大不同？

How different is rational approximation from piecewise polynomial approximation?

作者: Daan Huybrechs

类别: math.NA

发布日期: 2026-08-11

arXiv ID: 2608.11120v1

链接: http://arxiv.org/abs/2608.11120v1

摘要: The first aim of this paper is to show that there is merit to the question posed in the title. Indeed, for certain function classes, approximation by rational functions and by piecewise polynomials are surprisingly similar. Between these two, polynomials are more widely studied and more widely used. One setting in which both approaches are equally well understood, and equally practical in use, is that of approximating univariate functions with point singularities. In this context we can fully address the question. We review classical literature on the topic which shows that both approaches do indeed achieve similar convergence rates. However, rational approximations come with significantly smaller constants. Owing to recent advances in practical rational approximation, we can augment the discussion with a comparison of modern numerical techniques that achieve the optimal rates. We end by showing numerically that the difference becomes even more pronounced in several variables.

2. 随机化Tucker草图GMRES

Randomized Tucker-Sketched GMRES

作者: Alberto Bucci, Martina Iannacito, Mirjeta Pasha, Rudi Smith

类别: math.NA

发布日期: 2026-08-11

arXiv ID: 2608.11091v1

链接: http://arxiv.org/abs/2608.11091v1

摘要: We address the problem of solving large-scale tensor-structured linear systems in the Tucker format. In this setting, standard iterative solvers such as GMRES face a fundamental bottleneck: the multilinear ranks of the Krylov basis vectors grow with the iteration count, leading to rapidly increasing tensor operation costs and memory requirements. To overcome these challenges, we propose two randomized algorithms within the sketched GMRES framework that replace full Arnoldi orthogonalization with short recurrences. The first, RHOSVD-Tucker sGMRES, uses randomized HOSVD with per-iteration rank selection, providing robustness across a wide range of problems. The second method, MLN-Tucker sGMRES, leverages the multilinear Nyström approximation with a fixed rank, enabling streaming computations; the streamability of the approximation further allows, at no additional cost, a memory-efficient reconstruction of the solution from a compact sketched representation of the Krylov basis. Both methods outperform standard low-rank Tucker solvers in symmetric and non-symmetric settings. Applied to inverse problems, the low-rank Tucker constraint acts as an implicit regularizer; combined with adaptive projected Tikhonov penalization and automatic regularization parameter selection, the methods yield stable reconstructions.

3. 扩散拟蒙特卡罗

Diffusion Quasi-Monte Carlo

作者: Jianlong Chen, Yifeng Yu

类别: math.NA

发布日期: 2026-08-11

arXiv ID: 2608.11055v1

链接: http://arxiv.org/abs/2608.11055v1

摘要: We study high-dimensional numerical integration with respect to complex target measures using diffusion-based transport maps and randomized quasi-Monte Carlo (RQMC). Score-based diffusion models induce a deterministic probability flow ODE that transports a simple prior to the target, suggesting a principled way to transform low-discrepancy points on the unit cube into informative samples. We construct a cube-to-target map by composing a Gaussian base transformation (the component-wise inverse Gaussian CDF) with an Euler-discretized probability flow ODE. To retain unbiasedness under transport approximation, we formulate integration as importance sampling (IS) on the cube. Our main result provides verifiable conditions under which the resulting IS integrand satisfies the boundary growth condition, implying an $O(N^{-1+ε})$ RMSE for scrambled nets. We then establish these conditions for diffusion probability-flow transport under mild bounded-derivative assumptions on the learned vector field, explicitly controlling the boundary singularities introduced by the inverse Gaussian CDF. Experiments range from a 2D mixture to 784D images and a 40,960D conditional vorticity-assimilation task; in the latter, blocked scrambled Sobol’ sampling reduces the randomization standard deviation of nonlinear accuracy metrics at essentially unchanged online denoising cost. Together, these results give a theoretical and empirical foundation for combining diffusion generative modeling with high-precision RQMC integration.

4. 一种用于真三维重力反演的谱域伪逆方法

A Spectral-Domain Pseudo-Inverse Method for True 3D Gravity Inversion

作者: Shengchang Chen

类别: physics.geo-ph, math.NA

发布日期: 2026-08-11

arXiv ID: 2608.11014v1

链接: http://arxiv.org/abs/2608.11014v1

摘要: The main difficulty in 3D gravity inversion is that surface observations lack vertical wavenumber information, making the problem underdetermined and depth resolution poor. Building on the author’s spectral-domain pseudo-inverse theory for unitary diagonalizable systems, this paper presents a true 3D inversion method. Surface data are analytically continued upward via Laplace’s equation to form a 3D data volume, providing the vertical wavenumber sampling for the 3D Fourier transform. A general analytical expression for the half-space spectrum is derived, separating the horizontal spectrum from the vertical propagation kernel. The Green’s function of the 3D Poisson equation is unitarily diagonalized, yielding the forward spectral response $λ(\kk) = -i 4πG k_z / K^2$, and a stable inverse filter $q_α(λ) = \barλ/(|λ|^2 + α)$ is constructed. This filter is proved to have bounded stability and consistency (reducing to the exact inverse as $α\to 0^+$). Validation with a homogeneous sphere model shows correct recovery of the anomaly location and singular behavior at the source. The contributions are twofold: (1) upward continuation constructs a 3D volume from 2D surface data, mitigating underdetermination; (2) the inversion is reduced to one forward 3D Fourier transform, one spectral scaling, and one inverse transform.

5. 具有非全局Lipschitz系数的随机微分方程的自适应时间步长Euler–Maruyama格式：一致收敛性、稳定性与遍历性

Adaptive Time-Stepping Euler–Maruyama Scheme for SDEs with Non-Globally Lipschitz Coefficients: Uniform Convergence, Stability and Ergodicity

作者: Xueqi Wen, Shan Huang, Xiaoyue Li

类别: math.NA

发布日期: 2026-08-11

arXiv ID: 2608.10855v1

链接: http://arxiv.org/abs/2608.10855v1

摘要: This paper develops an adaptive time-stepping Euler–Maruyama scheme for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. By dynamically adjusting the timestep at each iteration, the proposed scheme effectively prevents numerical instability. We prove the moment boundedness of the numerical solution and establish a $1/2$-order strong convergence rate both on finite-time intervals and uniformly in time. Furthermore, the scheme faithfully inherits the $p$th moment exponential stability of the underlying SDE. For long-time ergodic dynamics, we establish the polynomial ergodicity of the numerical invariant measure. Moreover, we show that the numerical invariant measure converges to the invariant measure of the underlying SDE at an optimal rate of $1/2$ in the $L^q$-Wasserstein distance. Numerical experiments confirm our theoretical results and indicate the superior accuracy and computational performance of the proposed scheme over several fixed-step and existing adaptive methods.

6. 三维电多孔弹性方程的锁定分析与高阶无锁定有限元方法

Locking analysis and high-order locking-free finite element method for three-dimensional electroporoelasticity equations

作者: Xuan Liu, Yongkui Zou, Yanzhao Cao

类别: math.NA

发布日期: 2026-08-11

arXiv ID: 2608.10830v1

链接: http://arxiv.org/abs/2608.10830v1

摘要: Electroporoelasticity equations couple Maxwell’s equations with Biot’s poroelasticity model and admit severe Poisson locking in standard conforming finite element discretizations when the Lamé constant is large. In this paper, we provide a rigorous analysis of the Poisson locking phenomenon for three-dimensional quasi-static electroporoelasticity equations and show that the spatial convergence order of conforming finite element approximations is reduced in the nearly incompressible regime. To eliminate this locking effect, we introduce a five-field formulation and a fully discrete high-order finite element method. We prove uniform stability with respect to large Lamé constant. Based on this, we establish a locking-free scheme by deriving its uniform error estimates with respect to the Lamé coefficient. The analysis covers the fully coupled electromagnetic-poroelastic system and applies to high-order elements in three dimensions. Extensive numerical experiments are presented to verify the theoretical convergence rates and demonstrate robustness with respect to the Lamé parameter.

7. 用于逼近的连续Schur-Riesz分析

Successive Schur-Riesz Analysis for Approximation

作者: Matthew Francis Dixon

类别: math.NA, math.FA

发布日期: 2026-08-11

arXiv ID: 2608.10757v1

链接: http://arxiv.org/abs/2608.10757v1

摘要: Many approximation methods enlarge a trial space by adjoining function blocks generated by different operators. Exact redundancy and strong cross-level interaction can make coefficients nonunique and render pairwise or diagonal-dominance tests needlessly pessimistic. For (V_m=\sum_{\ell\leq m}S_\ell(E_\ell)) in a Hilbert space $\mathcal H$, we quotient coefficients representing the same function and control successive orthogonal innovations to obtain Riesz bounds independent of (m). The setting includes factored operators (S_\ell=T_\ell\circ\cdots\circ T_1:E_\ell\to\mathcal H), with compatible intermediate spaces, but the theorem allows arbitrary bounded (S_\ell). The same constants control approximation, truncation, perturbation, and levelwise error. A block Schur complement identifies the intrinsic new dimension and gives the exact reduction in squared best-approximation error, leading to a constructive enrichment procedure. Nonstationary and lifted examples give positive intrinsic bounds where diagonal-dominance estimates are negative or labelled Gram matrices are singular; adaptive and recycled-subspace calculations illustrate the distinct roles of representation stability and application-specific utility.

8. 通过SA-NODEs的长时间轨迹近似：模型预测与Floquet策略

Long-Time Trajectory Approximation via SA-NODEs: Model Predictive and Floquet Strategies

作者: Ziqian Li, Nikolaos M. Matzakos

类别: cs.LG, math.NA

发布日期: 2026-08-11

arXiv ID: 2608.10738v1

链接: http://arxiv.org/abs/2608.10738v1

摘要: We study the approximation of dynamical systems by semi-autonomous neural ordinary differential equations (SA-NODEs) over long time horizons. For a single network trained on the whole horizon, the available error bound deteriorates double exponentially in the horizon length. We develop two training strategies that avoid this barrier, each built on a reset of the state. The model predictive strategy partitions the horizon adaptively and restarts every window from observed data: when training meets a prescribed tolerance on every window, the composite model meets it uniformly in time, with a parameter budget linear in the horizon for targets with a bounded, uniformly regular reachable tube. The Floquet strategy addresses autonomous targets with a stable limit cycle and uses no data at deployment: a certified contraction of the learned return map confines the error to linear growth in the number of elapsed periods. For the time-periodic architecture we deploy, the scalar certificate degenerates; we prove instead a uniform-in-time orbital guarantee whose hypotheses are measured on the trained model, and an obstruction showing that, for an exactly periodic learned field, small one-period error and a contracting stroboscopic map cannot hold at once. Numerical experiments on four benchmarks confirm the predicted error laws and measure the hypotheses of every guarantee.

9. 一种用于可压缩Navier–Stokes方程组的带梯度重构的高效稳健五阶HWENO格式

An efficient and robust fifth-order HWENO scheme with gradient reconstruction for compressible Navier–Stokes equations

作者: Zijun Guo, Chuan Fan, Zhuang Zhao

类别: math.NA, math-ph

发布日期: 2026-08-11

arXiv ID: 2608.10569v1

链接: http://arxiv.org/abs/2608.10569v1

摘要: In this paper, we propose an efficient and robust fifth-order finite-volume Hermite weighted essentially non-oscillatory (HWENO) scheme with gradient reconstruction for the compressible Navier–Stokes equations. The key idea is to construct weak-derivative moments for each dissipative variable by taking the arithmetic average of two HWENO interface traces, which are then processed componentwise through the standard scalar nonlinear HWENO reconstruction formula, with only the input moments differing from those used in the convective reconstruction, thereby supplying the gradients needed for the viscous fluxes. This reconstruction achieves fifth-order accuracy using only candidate polynomials of degree at most four, avoids the order reduction that would result from direct differentiation, and introduces negligible additional computational cost. To ensure robustness, especially in extreme test cases involving strong shocks or low densities, we apply a positivity-preserving limiter to the conservative states while retaining the nonlinear HWENO reconstruction for the viscous gradients. This combination is essential for stable computations in flows with strong gradients. Both the conservative and gradient reconstructions share the same compact stencils and candidate-polynomial structure, enabling a single implementation routine without introducing additional algorithmic complexity. Numerical results confirm that the proposed scheme delivers fifth-order accuracy and high resolution, offers competitive computational efficiency, and robustly resolves challenging finite-Reynolds-number Navier–Stokes flows in which the positivity-preserving limiter is actively engaged.

10. 求解 von Kármán 障碍问题的二次 $C^0$ 内部惩罚方法

A Quadratic $C^0$ Interior Penalty Method for the von Kármán Obstacle Problem

作者: Sharat Gaddam

类别: math.NA

发布日期: 2026-08-11

arXiv ID: 2608.10507v1

链接: http://arxiv.org/abs/2608.10507v1

摘要: This article proposes and analyses a quadratic $C^0$ interior penalty method for the displacement obstacle problem of the von Kármán plate. The discrete space consists of Lagrange $P_2$ finite elements and the obstacle constraint is imposed at the vertices. The trilinear form of the von Kármán bracket is modified by terms on the edges so that it is bounded in the discrete energy norm. The well-posedness of the discrete problem, namely the existence of a discrete solution and its uniqueness under a smallness condition on the data, is established. The Sobolev and Friedrichs constants of the discrete energy norm are quantified with an explicit dependence on the mesh size. The main result is an error estimate of order $\mathcal{O}(h^α)$ in the discrete energy norm, where $1/2<α\le1$ is the index of elliptic regularity of the biharmonic operator on the polygonal domain. Numerical experiments on a square and on an L-shaped domain confirm the predicted rates. The coincidence set has positive measure in one example and empty interior in another. The experiments also show how the penalty parameter affects the rates and identify a threshold in the size of the obstacle beyond which the iterative solver fails on fine meshes.

11. 用于理想磁流体力学的一种五阶无散度有限差分Hermite WENO格式

A fifth-order divergence-free finite difference Hermite WENO scheme for ideal magnetohydrodynamics

作者: Peiwen Chen, Shi Jin, Jianxian Qiu, Zhuang Zhao

类别: math.NA

发布日期: 2026-08-11

arXiv ID: 2608.10390v1

链接: http://arxiv.org/abs/2608.10390v1

摘要: In this paper, we present a fifth-order finite difference divergence-free Hermite weighted essentially non-oscillatory (HWENO) scheme for the ideal magnetohydrodynamics (MHD) equations. In this framework, both the solution and its spatial partial derivatives are evolved in time and jointly employed in the spatial reconstruction procedure. A major challenge in MHD simulations is preserving the divergence-free constraint of the magnetic field, which is generally violated by standard numerical methods designed solely for hyperbolic conservation laws. To address this issue, we first solve the MHD equations within the HWENO framework for hyperbolic conservation laws, yielding a magnetic field divergence that remains zero up to high-order accuracy in smooth regions. Subsequently, we apply a correction that evenly distributes the divergence error among the partial derivatives involved in the divergence-free constraint, thereby rendering the magnetic field discretely divergence-free at the new time level. This approach offers several advantages. First, the scheme retains the conservation property, as only the partial derivatives of the numerical solution are corrected, leaving the conserved variables unchanged. Second, the correction applied to the partial derivatives of the magnetic field components introduces only a high-order perturbation, thereby preserving the overall accuracy. Third, the divergence-free treatment significantly enhances robustness, as most benchmark test cases cannot be run stably without such a correction. Fourth, the correction is a simple linear operation applied at each stage of the time integration, incurring negligible additional computational cost. Extensive numerical experiments demonstrate the accuracy, resolution, efficiency, effectiveness, and robustness of the proposed scheme.

12. 用于求解双曲守恒律的高效弱熵PINN

Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Laws

作者: Qi Gao, Kuang Huang, Xuan Di

类别: math.NA, cs.LG

发布日期: 2026-08-11

arXiv ID: 2608.10389v1

链接: http://arxiv.org/abs/2608.10389v1

摘要: In recent years, neural networks have significantly advanced numerical solutions of partial differential equations (PDEs). However, solving PDEs with discontinuous solutions, such as hyperbolic conservation laws, remains challenging for neural network-based methods such as physics-informed neural networks (PINNs). Existing methods often rely on strong prior assumptions such as knowledge of discontinuity locations, or they introduce artificial smoothing terms that degrade accuracy. However, accurately solving these conservation laws and predicting the formation and propagation of discontinuities in solutions is crucial in many practical applications, including gas dynamics and traffic flow modeling. In this paper, we introduce a novel Weak-Entropy PINN (WEPINN) framework for hyperbolic conservation laws with discontinuous solutions. The method enforces the governing equations in their weak (integral) formulation and incorporates the entropy condition to select the physically admissible solution, while employing the discrete fast Fourier transform (DFFT) for efficient numerical integration. Our method is tested through extensive numerical experiments on a variety of scalar conservation laws and systems of conservation laws in one and two dimensional spaces. These experiments demonstrate that our method can accurately resolve sharp discontinuities while effectively capturing interactions between multiple shock and rarefaction waves.

13. 用于完全不可压缩超弹性固体流固耦合的有限元浸没边界法的体积稳定化混合格式

A Volumetrically Stabilized Mixed Formulation of the Finite Element Immersed Boundary Method for Fluid Structure Interaction with Fully Incompressible Hyperelastic Solids

作者: Abhishek Mishra, Paul T. Bauman

类别: math.NA, physics.comp-ph

发布日期: 2026-08-10

arXiv ID: 2608.10217v1

链接: http://arxiv.org/abs/2608.10217v1

摘要: The finite element immersed boundary method (FE IBM) with a distributed Lagrange multiplier is an attractive framework for fluid structure interaction (FSI) because it couples an Eulerian description of an incompressible fluid to a Lagrangian description of an immersed solid on two independent, nonconforming meshes, avoiding the costly remeshing required by body fitted arbitrary Lagrangian Eulerian methods. When the immersed solid is modelled as a fully incompressible hyperelastic material, however, a direct finite element discretization of the deviatoric solid stress fails to enforce the Lagrangian incompressibility constraint, and the computed structure exhibits spurious volumetric instabilities and locking. In this work we present a mixed formulation of the distributed Lagrange multiplier FEIBM that restores volumetric stability. Following the theory of nearly incompressible hyperelasticity, we augment the solid stress with a volumetric contribution derived from a dilatational strain energy and introduce an additional solid pressure field, enforced weakly, that plays the role of the Lagrange multiplier for the Lagrangian incompressibility constraint J = 1. The resulting formulation is discretized in space by finite elements and in time by an unconditionally stable semi implicit scheme, and is implemented as a reusable Immersed Boundary physics kernel within the GRINS multiphysics framework, built on the libMesh finite element library. The method is verified on three FSI benchmarks, an elliptically displaced thick ring returning to equilibrium, a radially stretched incompressible ring, and a disk falling under gravity in a viscous fluid, for which the mixed formulation removes the volumetric failure observed with the unstabilized formulation and reproduces the analytical terminal velocity of the falling disk to within 1%.

重点关注论文（完整分析）#

1. 解耦势积分方程的高阶 Galekin 格式#

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#

了解领域论文（摘要翻译）#

1. 物理信息神经网络中的导数计算：自动微分、有限差分及超越#

摘要翻译#

2. 多面体网格上Hybrid High-Order微分形式的一致离散庞加莱不等式#

摘要翻译#

3. 一种求解时间分数阶移动-不动输运方程的数值方法#

摘要翻译#

4. 三维变密度不可压无感磁流体动力学方程的高阶能量稳定且电荷守恒的拉格朗日有限元法#

摘要翻译#

5. 离散几何上非线性 Naghdi 壳的形状优化#

摘要翻译#

不相关论文（基本信息）#

1. 有理逼近与分段多项式逼近有多大不同？#

2. 随机化Tucker草图GMRES#

3. 扩散拟蒙特卡罗#

4. 一种用于真三维重力反演的谱域伪逆方法#

5. 具有非全局Lipschitz系数的随机微分方程的自适应时间步长Euler–Maruyama格式：一致收敛性、稳定性与遍历性#

6. 三维电多孔弹性方程的锁定分析与高阶无锁定有限元方法#

7. 用于逼近的连续Schur-Riesz分析#

8. 通过SA-NODEs的长时间轨迹近似：模型预测与Floquet策略#

9. 一种用于可压缩Navier–Stokes方程组的带梯度重构的高效稳健五阶HWENO格式#

10. 求解 von Kármán 障碍问题的二次 $C^0$ 内部惩罚方法#

11. 用于理想磁流体力学的一种五阶无散度有限差分Hermite WENO格式#

12. 用于求解双曲守恒律的高效弱熵PINN#

13. 用于完全不可压缩超弹性固体流固耦合的有限元浸没边界法的体积稳定化混合格式#

重点关注论文（完整分析）

1. 解耦势积分方程的高阶 Galekin 格式

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

了解领域论文（摘要翻译）

1. 物理信息神经网络中的导数计算：自动微分、有限差分及超越

摘要翻译

2. 多面体网格上Hybrid High-Order微分形式的一致离散庞加莱不等式

摘要翻译

3. 一种求解时间分数阶移动-不动输运方程的数值方法

摘要翻译

4. 三维变密度不可压无感磁流体动力学方程的高阶能量稳定且电荷守恒的拉格朗日有限元法

摘要翻译

5. 离散几何上非线性 Naghdi 壳的形状优化

摘要翻译

不相关论文（基本信息）

1. 有理逼近与分段多项式逼近有多大不同？

2. 随机化Tucker草图GMRES

3. 扩散拟蒙特卡罗

4. 一种用于真三维重力反演的谱域伪逆方法

5. 具有非全局Lipschitz系数的随机微分方程的自适应时间步长Euler–Maruyama格式：一致收敛性、稳定性与遍历性

6. 三维电多孔弹性方程的锁定分析与高阶无锁定有限元方法

7. 用于逼近的连续Schur-Riesz分析

8. 通过SA-NODEs的长时间轨迹近似：模型预测与Floquet策略

9. 一种用于可压缩Navier–Stokes方程组的带梯度重构的高效稳健五阶HWENO格式

10. 求解 von Kármán 障碍问题的二次 $C^0$ 内部惩罚方法

11. 用于理想磁流体力学的一种五阶无散度有限差分Hermite WENO格式

12. 用于求解双曲守恒律的高效弱熵PINN

13. 用于完全不可压缩超弹性固体流固耦合的有限元浸没边界法的体积稳定化混合格式