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1. 几何相容有限元方法：适配与未适配网格上的界面问题
Geometry-Conforming Finite Element Methods for Interface Problems on Fitted and Unfitted Meshes
作者: Yuanhui Lin, Tao Lin, Xu Zhang, Minfu Feng
类别: math.NA
发布日期: 2026-08-14
arXiv ID: 2608.14484v1
摘要翻译: 我们为二维曲域上的椭圆边值与界面问题发展了一个任意次的几何相容有限元（GC-FE）框架。利用Frenet–Serret变换，曲边界和界面适配的单元段被精确表示，而Frenet坐标下的多项式在物理坐标下生成一般的非多项式局部形函数。对于界面未适配网格，曲边界单元上的GC-FE空间与界面切割单元上的几何相容浸没有限元（GC-IFE）空间耦合，其余位置使用标准多项式空间。我们建立了GC-FE空间的最优逼近、逆和迹估计。对于适配网格，我们证明了一个对称内罚间断Galerkin离散格式在能量范数和$L^2$范数下的适定性和最优误差估计。通过精确保留给定的曲线，该方法避免了与曲几何近似相关的几何变分犯罪，并且不需要相应的几何一致性估计。数值实验确认了预测的阶数，表明在报告的测试中全局精度与节点等参有限元相当，且真实界面迹误差更小，并展示了在界面未适配网格上的耦合GC-FE-GC-IFE方法。
Comment: 无
链接: http://arxiv.org/abs/2608.14484v1
1. 研究对象和背景
本文研究二维有界光滑区域上的椭圆边值问题与椭圆界面问题. 边值问题为: 求 $u$ 满足
$$-\nabla\cdot(\beta\nabla u)=f \quad \text{in }\Omega,\quad u=g \quad \text{on }\partial\Omega,$$其中 $\Omega\subset\mathbb{R}^2$ 是有界区域, $\beta$ 为正函数. 界面问题为: 设 $\Gamma_i$ 是一条光滑曲线, 将 $\Omega$ 分为 $\Omega^+$ 与 $\Omega^-$, $\beta$ 分别在两个子区域内光滑且在 $\Gamma_i$ 上可能有跳跃, 求 $u$ 满足
$$-\nabla\cdot(\beta\nabla u)=f \quad \text{in }\Omega^+\cup\Omega^-,\quad u=g \quad \text{on }\partial\Omega,$$以及界面跳跃条件
$$\llbracket u\rrbracket_{\Gamma_i}=0,\quad \llbracket\beta\partial_n u\rrbracket_{\Gamma_i}=0,$$其中 $n$ 是 $\Gamma_i$ 的单位法向, $\llbracket\cdot\rrbracket_{\Gamma_i}$ 表示跨过界面的跳跃. 这类问题出现在热传导, 多孔介质流动, 电磁学以及复合/多相材料建模中. 在这些应用中, 弯曲边界 $\partial\Omega$ 或界面 $\Gamma_i$ 携带重要的物理和几何信息. 然而, 当用直边网格逼近曲边区域时, 多边形边界与物理边界通常相差 $O(h^2)$, 对于高阶有限元空间, 这种几何一致性误差可能主导离散误差, 除非几何也被逼近到相应的高阶. 这种区域扰动是 Strang 意义下的经典变分犯罪. 类似地, 在界面问题中, 跳跃条件应当理想地施加在物理界面 $\Gamma_i$ 上, 而不是其多边形逼近上. 经典的等参有限元方法通过多项式映射构造曲边单元, 但除非物理几何恰好位于所选多项式映射空间内, 计算几何仍是一种近似; 而且大曲率下构造有效的曲边网格可能很困难, 需要额外的几何一致性分析. 对于界面问题, 拟合网格必须适应界面或其足够精确的近似, 界面几何变化可能需要昂贵的网格重建; 浸入有限元方法虽然可以使用与界面无关的网格, 但通常将界面近似为直线段. 最近, 文献中利用 Frenet–Serret 装置精确表示弯曲界面, 构造了几何一致浸入有限元 (GC-IFE) 空间, 并在非结构化三角形网格上得到了推广. 本文的核心观察是: Frenet 变换不仅可用于浸入有限元, 也提供了等参映射的一种内在的、几何一致的替代方案, 用于弯曲边界和拟合界面. 本文试图填补的空白是: 建立一套系统的任意次几何一致有限元 (GC-FE) 框架, 在拟合与非拟合网格上统一处理曲边区域上的边界值问题和界面问题, 同时避免几何变分犯罪和相应的几何一致性估计.
2. 主要定理或主要结果
本文的主要结果如下. 第一, 建立了任意次几何一致有限元 (GC-FE) 空间的构造框架. 对于每个曲边或拟合界面单元 $K\in\mathcal{T}_h^c$, 利用 Frenet 变换 $P_\Gamma$ 将弯曲单元映射到参数域, 在参数域上取多项式空间 $P_m(\widehat{K})$, 然后通过 $P_\Gamma$ 复合拉回得到物理单元上的形状函数空间, 即 $V_h|_K = \{ v: v = \widehat{v}\circ R_\Gamma, \widehat{v}\in P_m(\widehat{K}) \}$. 这些形状函数在物理坐标下一般是非多项式函数. 文中证明了该 GC-FE 空间具有最优阶逼近性质: 对足够光滑的精确解 $u$, 存在 $u_h \in V_h$ 使得
$$\|u-u_h\|_{L^2(K)} + h_K |u-u_h|_{H^1(K)} \leq C h_K^{m+1} \|u\|_{H^{m+1}(K)}$$对每个曲边单元 $K$ 成立, 同时建立了逆不等式 $|v|_{H^1(K)} \leq C h_K^{-1} \|v\|_{L^2(K)}$ 和迹不等式等. 第二, 对于界面拟合网格, 提出了基于 GC-FE 空间的对称内部惩罚不连续 Galerkin (SIPDG) 离散格式. 由于物理边界和界面被精确保留, 分析中无需单独的几何一致性估计. 证明了该格式适定, 并给出了能量范数与 $L^2$ 范数下的最优误差估计: 设 $u$ 是问题 (1.1)–(1.2) 或 (1.3)–(1.5) 的精确解, $u_h$ 是数值解, 则
$$\|u-u_h\|_{\text{energy}} \leq C h^m \left( \sum_K \|u\|_{H^{m+1}(K)}^2 \right)^{1/2}, \quad \|u-u_h\|_{L^2(\Omega)} \leq C h^{m+1} \left( \sum_K \|u\|_{H^{m+1}(K)}^2 \right)^{1/2},$$其中常数 $C$ 不依赖于 $h$ 与网格. 第三, 对于界面非拟合网格, 将曲边边界单元上的 GC-FE 空间与界面切割单元上的 GC-IFE 空间耦合, 在其它单元上使用标准多项式空间, 得到完整的离散空间, 并数值验证了该方法. 第四, 数值实验表明, 与结点等参有限元相比, GC-FE 方法在全局精度上相当, 但在所报告的算例中, 真实界面迹误差更小, 且不需要对几何进行多项式逼近.
3. 研究方法、关键技术和核心工具
本文的核心技术工具是 Frenet–Serret 变换及其逆变换. 对于一条正则 $C^2$ 曲线 $\Gamma$, 定义 $P_\Gamma(\eta,\xi)=g(\xi)+\eta n(\xi)$, 其中 $g$ 是弧长参数化或正则参数化, $n$ 是单位法向, 则曲线附近的点可用 $\eta$ 表示到曲线的有向距离, $\xi$ 表示曲线参数. 该变换的雅可比行列式为 $\det DP_\Gamma = \|g'(\xi)\|(1+\eta\kappa(\xi))$, 在管状邻域内一致有界 (引理 2.1). 利用该变换, 曲边单元被拉回到一个参数矩形, 其中曲线变成平直线. 文中建立了若干几何估计: 引理 2.2 表明元素上曲线参数 $\xi$ 的极值在顶点达到, 因此投影曲线段与顶点生成的曲线段一致; 引理 2.3 和 2.4 证明该投影曲线段的长度与元素直径 $h_K$ 互为上下界; 引理 2.5 证明虚构元素的覆盖具有有限重叠性质, 即对任意点 $X$, 包含它的虚构元素个数有界, 该性质用于局部化估计; 引理 2.6 给出函数在物理域与参数域之间的 $L^2$ 与 $H^1$ 范数等价性. 这些估计为将参数域上的标准多项式逼近结果转移到物理域上的非多项式空间奠定了基础. 在拟合网格的 SIPDG 分析中, 作者利用几何精确性避免了等参方法中常见的几何扰动项, 从而无需估计几何一致性误差. 与 GC-IFE 的耦合中, 界面单元上的空间直接采用文献 [29] 的 GC-IFE 空间, 该空间在精确界面上施加跳跃条件, 因此无需对界面进行分段线性近似. 本文的真正创新点在于将 Frenet 变换从浸入有限元推广到曲边边界和拟合界面, 并建立了完整的逼近理论, 使得几何保形成为可能. 主要障碍在于非多项式形状函数的逼近性质和逆/迹估计不能直接套用标准理论, 需要借助上述范数等价和覆盖性质来克服.
4. 与之前工作的比较
与经典的等参有限元方法相比, 本文的 GC-FE 方法不引入任何多项式代理几何. 等参方法通过多项式映射 $F_K(\widehat{K})$ 定义曲边单元, 一般只能近似逼近真实几何, 除非真实几何本身是多项式曲线. 因此等参分析必须量化物理几何与计算几何之间的差异, 并估计由此引起的变分形式扰动, 即几何一致性误差. 本文由于直接保留精确曲线作为单元边, 从根源上消除了这类误差, 不需要额外的几何一致性估计. 此外, 等参映射在大曲率或大变形下可能导致单元翻转, 而 GC-FE 基于管状邻域上的 Frenet 变换, 只要曲线正则且单元尺寸足够小, 映射的正则性有保证. 其代价是形状函数一般不再是多项式, 这要求重新建立逼近和逆/迹理论, 并且在数值积分时需要考虑非多项式函数的精确积分或适当的高阶求积. 与经典的浸入有限元 (IFE) 方法相比, 传统 IFE 方法在网格切割单元上修改基函数以满足跳跃条件, 但界面本身仍被直线边近似, 跳跃条件只能近似满足. 而 GC-IFE 方法使用 Frenet 变换将界面拉直, 在精确界面上施加跳跃条件, 从而获得更好的界面迹误差. 本文的贡献在于将这一思想同时用于拟合网格的曲边边界和界面, 并与不拟合网格上的 GC-IFE 实现耦合, 形成一个统一框架. 与文献 [4,5,29] 中的 GC-IFE 工作相比, 本文着重分析了拟合网格上的 GC-FE 空间的逼近、逆和迹性质, 并给出了 SIPDG 格式的最优误差估计, 这是以往工作未涉及的. 在数值表现上, 作者报告 GC-FE 的全局精度与结点等参有限元相当, 但在测试的真实界面迹误差上更小. 需要指出的是, 本文的分析要求边界和界面曲线具有 $C^{m+2}$ 光滑性, 且所有曲边单元需要位于管状邻域内, 这隐含着 $h$ 相对于曲率半径足够小的条件; 对于大曲率或几何奇点, 方法可能需要额外的处理, 文中未展开讨论.
2. 分析二阶谐波产生散射的协调有限元方法
Analysis of a Conforming Finite Element Method for Second-Harmonic Generation Scattering
作者: Ansh Desai, Peter Monk
类别: math.NA
发布日期: 2026-08-14
arXiv ID: 2608.14300v1
摘要翻译: 在频域中，非线性声学和电磁波传播在某些情况下可以由二阶谐波产生系统建模，该系统由两个耦合的非线性亥姆霍兹方程组成。我们分析了使用精确狄利克雷到诺伊曼边界条件截断的标量二阶谐波产生散射问题的协调有限元逼近。离散化使用次数为$p$的连续分段多项式有限元。在涉及入射场和非线性磁化率的小数据条件下，我们证明了在给定小球中连续和离散非线性解的存在性和唯一性。在同一条件下，我们通过非线性Céa型论证推导了拟最优的先验$H^1$-误差估计，该论证将线性Galerkin拟最优性与耦合非线性的小数据稳定性估计相结合。我们还分析了用于计算离散非线性解的不动点迭代的收敛性，并量化了有限元逼近和非线性迭代误差的组合效应。一个制造解实验说明了预测的有限元收敛速率，而额外的基于PML的散射计算展示了二维和三维中非线性不动点求解器的行为。
Comment: 无
链接: http://arxiv.org/abs/2608.14300v1
1. 研究对象和背景
本文研究二次谐波产生（Second-Harmonic Generation, SHG）散射问题的保形有限元离散与误差分析。在频域中，SHG由两个耦合的非线性亥姆霍兹方程描述：一个基频场 $u_1$ 与一个二次谐波场 $u_2$ 相互耦合，非线性源项分别为 $\chi_1 u_1 \overline{u_2}$ 和 $\chi_2 u_1^2$。设 $D \subset \mathbb{R}^d$（$d\in\{2,3\}$）为非线性材料占据的有界 Lipschitz 域，背景折射率在 $\mathbb{R}^d\setminus D$ 中为 1。给定入射平面波 $u^i(\mathbf{x})=\exp(i\kappa_1 \mathbf{d}_0\cdot\mathbf{x})$，其中 $\kappa_1=\omega/c_0$ 为基频波数，$\kappa_2=2\kappa_1$ 为二次谐波波数。记总基频场 $u_1=u^i+u_1^s$（$u_1^s$ 为散射场），$u_2$ 为二次谐波场。全空间散射问题为：求 $(u_1,u_2)\in H^1_{\mathrm{loc}}(\mathbb{R}^d)\times H^1_{\mathrm{loc}}(\mathbb{R}^d)$ 满足
$$ \Delta u_1 + \kappa_1^2 n_1 u_1 = -\chi_1 u_1 \overline{u_2} \quad \text{in } \mathbb{R}^d, \qquad \Delta u_2 + \kappa_2^2 n_2 u_2 = -\chi_2 u_1^2 \quad \text{in } \mathbb{R}^d, $$以及 Sommerfeld 辐射条件。为在有界区域上进行有限元分析，选取半径足够大的球 $B_R$ 使得 $D\subset B_R$，并采用精确 Dirichlet-to-Neumann (DtN) 映射 $T_k$ 截断外域。截断后的问题化为：求 $u_1,u_2\in V:=H^1(B_R;\mathbb{C})$ 满足
$$ \Delta u_1 + \kappa_1^2 n_1 u_1 = -\chi_1 u_1 \overline{u_2} \quad \text{in } B_R, \qquad \Delta u_2 + \kappa_2^2 n_2 u_2 = -\chi_2 u_1^2 \quad \text{in } B_R, $$并施加边界条件
$$ \partial_\nu u_1 = T_{\kappa_1}\gamma(u_1-u^i)+\partial_\nu u^i,\qquad \partial_\nu u_2 = T_{\kappa_2}\gamma u_2. $$其弱形式（2.13）-（2.14）为：求 $(u_1,u_2)\in V\times V$ 使得
$$ a_1(u_1,\xi)=\int_{B_R} \chi_1 u_1 \overline{u_2}\,\overline{\xi}\,dx+\ell(\xi),\qquad a_2(u_2,\eta)=\int_{B_R} \chi_2 u_1^2\,\overline{\eta}\,dx, $$对一切 $\xi,\eta\in V$ 成立，其中双线性形式 $a_j$ 由（2.15）-（2.16）定义。该模型的重要性在于 SHG 是非线性光学的基本过程，其数值模拟需要严格的误差控制。已有工作如 Bao-Dobson [4] 仅分析一维模型，Jiang-Li [13] 使用 PML 和连续内罚法，但尚未有保形有限元在精确 DtN 截断下的完整误差分析。本文旨在填补这一空白，证明连续和离散问题的小数据适定性，并推导拟最优的先验误差估计。
2. 主要定理或主要结果
论文的主要结果在定理 3.5 和定理 3.8 中给出。设 $c\widetilde{M}_j=\max\{M_j,M_{\mathrm{lin}}\}$，其中 $M_j$ 为连续线性算子 $A_j$ 的稳定性常数，$M_{\mathrm{lin}}$ 为离散线性问题的稳定性常数。令 $C_{\mathrm{prod}}$ 为乘积估计常数，并定义（见 (3.6)）
$$ \alpha_\rho = c\widetilde{M}_1 C_{\mathrm{prod}}\|\chi_1\|_{L^\infty}\rho,\qquad \beta_\rho = 2c\widetilde{M}_2 C_{\mathrm{prod}}\|\chi_2\|_{L^\infty}\rho, $$$$ \sigma_\rho = \beta_\rho \rho^2,\qquad \theta_\rho = \alpha_\rho\beta_\rho/2,\qquad q_\rho = \frac{3}{2}\alpha_\rho\beta_\rho. $$在以下小数据条件下：存在 $\rho>0$ 使得
$$ c\widetilde{M}_1\|\ell\|_{V'}+\theta_\rho\rho \le \rho,\qquad q_\rho < 1, $$（等价于 $a+\mu\rho^3\le\rho$ 且 $3\mu\rho^2<1$，进一步等价于显式阈值
$$ c\widetilde{M}_1^3 c\widetilde{M}_2 C_{\mathrm{prod}}^2\|\ell\|_{V'}^2\|\chi_1\|_{L^\infty}\|\chi_2\|_{L^\infty} < \frac{4}{27}, $$见注 3.6），定理 3.5 同时证明了连续问题 (2.13)-(2.14) 在球 $B_\rho = \{(u,v)\in V\times V: \|u\|_V\le\rho,\ \|v\|_V\le\sigma_\rho\}$ 中存在唯一解 $(u_1,u_2)$，以及对每个 $0 其中 $C_{\mathrm{FE}}$ 与 $h$ 无关。结合 Assumption 3.1 的逼近性质，若解具有 $H^{1+s}$ 正则性（$0 离散迭代有类似估计，因此迭代以线性速率收敛，且有限元逼近误差与非线性迭代误差的联合效应被定量刻画。 论文的核心技术障碍是 SHG 耦合系统的非线性项在消去 $u_2$ 后呈现非局部特性：$u_1$ 的约化方程包含 Helmholtz 预解式 $A_2^{-1}$，因此不能直接利用针对 Kerr 型点态非线性的标量框架。作者的创新点在于采用统一收缩论证同时处理连续和离散问题，并利用非线性 Céa 型引理直接对有限元解进行误差估计。关键工具包括： 线性算子可逆性与稳定性：命题 2.3 利用 Gårding 不等式、紧性、外 DtN 恒等式和边界唯一延拓引理（引理 2.1）证明 $A_j:V\to V'$ 是构影，从而存在连续稳定性常数 $M_j$。命题 3.3 给出离散情形下对 $0 乘积估计：引理 2.2 利用 Sobolev 嵌入 $H^1(B_R)\hookrightarrow L^4(B_R)$ 得到 $\|wz\|_{V'}\le C_{\mathrm{prod}}\|w\|_V\|z\|_V$ 及其 Lipschitz 型差分估计 (2.18)，这是估计非线性项 $\chi_1 u_1\overline{u_2}$ 和 $\chi_2 u_1^2$ 的关键。 抽象定点映射：定义 $X$ 为 $V$（连续情形）或 $V_h$（离散情形），构造映射 其中 $J_X$ 是 $L^2$ 到 $X'$ 的自然映射。利用 (3.13)-(3.14) 和常数定义，证明 $F_X$ 在 $B_{\rho,X}^1=\{u\in X:\|u\|_V\le\rho\}$ 上为压缩映射（压缩系数 $q_\rho<1$），从而同时获得连续和离散解的存在唯一性。这一方法避免了分别构造迭代，使连续与离散分析统一。 非线性 Céa 型误差估计：定理 3.8 的证明不依赖迭代解的中间误差，而是直接比较连续解 $(u_1,u_2)$ 与离散解 $(u_{1,h},u_{2,h})$。关键在于利用线性 Galerkin 拟最优性、乘积估计的 Lipschitz 连续性和球 $B_\rho$ 中的一致有界性，将非线性误差项转化为 $\|u_1-u_{1,h}\|_V+\|u_2-u_{2,h}\|_V$ 的倍率，最终通过小数据条件 $q_\rho<1$ 封闭估计。这种方法比 Jiang-Li [13] 中先分析迭代解再取极限的途径更直接。 精确 DtN 边界：使用精确 DtN 映射消除了截断误差，使误差分析聚焦于非线性耦合与有限元逼近。对于实际实现中的 DtN 截断，文中说明需额外添加类似 [3,2] 中的逼近项。 证明框架的特点在于“先统一、后线性化”：先用收缩论证获得连续和离散解的存在唯一性，再通过非线性 Céa 引理将误差估计归结为线性问题的最优逼近。 与已有工作相比，本文的推进体现在多个维度： 与一维模型相比：Bao-Dobson [4] 仅分析一维 SHG 模型的存在性，未涉及有限元离散。本文将分析推广到 $d=2,3$，并给出完整的误差估计。Cakoni 等人 [7] 从 Maxwell 方程推导了 SHG 散射模型，但关注的是二次谐波不向外传播的条件，而本文关注数值逼近。Acosta-Uhlmann [1] 的时域 Westervelt 模型是声学背景，本文的频域耦合系统与其相关但方程不同。 与一维数值模拟相比：Yuan [23] 和 Bao-Xu-Yuan [5] 对一维 SHG 系统采用有限元加定点迭代，但没有严格的误差分析。本文提供了多维情形下的收敛阶证明，并量化了有限元误差与迭代误差的联合效应，为实际计算提供了理论保证。 与标量非线性 Helmholtz 方程相比：对 Kerr 型点态非线性（如 [22,21,3,2]），非线性项是 $|u|^2u$ 型，局部结构简单。SHG 系统在消去 $u_2$ 后，约化方程含非局部算子 $A_2^{-1}$，导致非线性不满足点态性质，因此不能直接套用标量框架。本文处理的耦合系统更具挑战性，且证明了在类似小数据条件下的适定性。 与 Jiang-Li [13] 最接近：Jiang-Li 使用 PML 截断和线性连续内罚有限元，分析了迭代方案的收敛性并给出预渐近误差估计。本文的差异在于：(a) 使用标准保形 Galerkin 离散和精确 DtN 边界，便于分离有限元误差与边界截断误差；(b) 采用统一收缩论证同时处理连续和离散，而 Jiang-Li 的分析针对迭代过程；(c) 误差估计直接作用于有限元解，避免了迭代解的中间误差，技术更简洁。代价是本文的理论分析基于精确 DtN，若实际使用截断 DtN 或 PML，需补充相应的逼近误差项；而 Jiang-Li 直接分析了 PML 截断，更贴近实际计算。此外，本文明确指出不处理表面 SHG 模型（如 [10]），这是适用范围的限制。 小数据条件的显式形式：本文给出了显式的阈值条件（注 3.6），将入射场、非线性磁化率、线性稳定性常数 $\widetilde{M}_j$ 和乘积常数 $C_{\mathrm{prod}}$ 联系起来，且成立 $c\widetilde{M}_1^3 c\widetilde{M}_2 C_{\mathrm{prod}}^2\|\ell\|_{V'}^2\|\chi_1\|\|\chi_2\| < 4/27$。这种条件比抽象的“足够小”更可验证，也显示出频率依赖性（通过 $\widetilde{M}_j$ 对 $\kappa_j$ 的依赖）。不过，作者未显式展开与 Jiang-Li 在常数取舍上的定量比较，也未给出 PML 情形的误差上界，这些可从提供内容中确认的只是定性优势。 Nodal discontinuous Galerkin methods for non-ideal equations of state: pressure equilibrium preservation and entropy correction 作者: Jesse CHan, Hendrik Ranocha, Raymond Park, Joshua Lampert, Eric Ching, Ayaboe Edoh 类别: math.NA, cs.CE 发布日期: 2026-08-14 arXiv ID: 2608.14506v1 链接: http://arxiv.org/abs/2608.14506v1 中文标题: 非理想状态方程的节点间断伽辽金方法：压力平衡保持与熵修正 摘要翻译: 保结构间断伽辽金（DG）方法通常能提高真实流体高阶模拟的鲁棒性。除守恒性外，关键结构还包括压力平衡的保持以及至少一个熵不等式的满足。在本工作中，我们研究了使用精确压力平衡保持（EPEC）和近似压力平衡保持（APEC）通量差分DG格式的守恒离散，以及通过使用最小耗散修正针对非理想状态方程（EOS）的熵稳定格式。
我们引入了对EPEC格式的分析，并基于Tadmor混合条件的推广，提出了一种设计此类通量的新方法。我们还分析了APEC DG格式，并表明耗散界面惩罚项的引入不会显著增加压力平衡误差，尤其是在高阶近似下。最后，我们观察到，当与APEC通量差分格式结合时，熵修正可提高欠分辨解和长时间模拟的鲁棒性。 Full-field and Bloch-periodic-factor discretizations: Accuracy and phantom modes 作者: Igor Tsukerman 类别: physics.optics, math.NA 发布日期: 2026-08-14 arXiv ID: 2608.14348v1 链接: http://arxiv.org/abs/2608.14348v1 中文标题: 全场与Bloch周期因子离散化：精度与幽灵模式 摘要翻译: 在线性周期介质中，可以区分Bloch模式问题的四种主要表述。首先，有两种常见的互补方式来切割Bloch簇（Bloch波矢与相应频率配对）：频率对波矢，或者反过来，波矢对频率。本文仅讨论后者，它直接适用于有损介质、倏逝波和复能带结构。另一个交叉路口是全场（FF）与晶格周期Bloch因子（PF）表述。它们在连续层面上完全等价，但常见的离散化技术可能会破坏这种等价性，不仅近似地，而且定性地。在FF问题中，主要未知特征值是Bloch相位因子（而不是波数），它仅通过对边界耦合进入。另一方面，PF表述将Bloch波数注入微分算子，并导致体积二次束。因此，这里考虑的此类标准PF离散化，与相应的FF离散化相反，违反了倒格矢（布里渊区平移）协变性——正如论文中的理论和数值例子所示，这可能导致不准确甚至非物理的模式。从数学角度看，论文强调了保结构算法的作用。结果和建议的实际重要性源于PF表述在光学、光子学及其他领域的广泛采用——例如，在光子与声学结构中计算传播和倏逝Bloch模式、拓扑光子学、有效质量与拓扑绝缘体能带理论。 Vanishing Clutter: Fast and Accurate Shape Imaging via Active Cloaking 作者: Haoqiang Xiao, Guang-Hui Zheng 类别: math.NA 发布日期: 2026-08-14 arXiv ID: 2608.13839v1 链接: http://arxiv.org/abs/2608.13839v1 中文标题: 消除杂散：基于主动隐身的快速精确形状成像 摘要翻译: 近场扫描光学显微镜（NSOM）中目标样品的成像从根本上受到测量伪影和多次探针-样品散射造成的数据污染的限制。探针是实现亚波长分辨率所必需的，但它固有地扰动局域场，降低信噪比，并使定量形状重构的反问题高度不适定。我们首先建立相应正模型的适定性，为后续成像提供严格基础。然后，我们将主动隐身——通常作为成像的对立面——重新用作消除探针诱导干扰的实现机制。我们不直接从受污染数据中重构样品，而是通过构造一个最优控制问题来主动隐身探针，并证明其极小元的存在性和稳定性。利用层状等离子体结构中局域异常共振的理论，我们推导出精确的闭式极小元，从而避免了最优控制问题的计算上不可行的迭代求解。由此产生的隐身驱动的干扰消除产生了几乎无探针的测量环境，实现了快速、无伪影的形状重构。大量数值实验证明了形状重构的准确性，并且相对于直接应用于未经隐身预处理、受探针污染的数据的常规迭代方法，实现了显著加速——通常达数量级——验证了所提方法在高保真亚波长成像中的鲁棒性和变革潜力。 An Efficient Solver for Finite Element-based Constrained Transport in 3D Magnetohydrodynamics Applied to Magnetic Confinement Fusion 作者: Golo A. Wimmer, Konstantin Lipnikov, Ben S. Southworth, Xian-Zhu Tang 类别: math.NA 发布日期: 2026-08-13 arXiv ID: 2608.13829v1 链接: http://arxiv.org/abs/2608.13829v1 中文标题: 面向磁约束聚变的三维磁流体动力学中基于有限元的约束输运的高效求解器 摘要翻译: 我们提出了一种高效求解器框架，用于在真实托卡马克几何中电阻磁流体动力学（MHD）中出现的刚性磁波耦合。该方法基于隐式-隐式（IMIM）时间分裂，将快磁波和各向异性热输运与较慢的声学动力学分开，同时保持完全耦合（Krzysik et al. 2026）。在该公式中，磁波子系统表现为各向异性的curl-curl算子，从而能够使用可扩展的辅助空间Maxwell（AMS）多重网格求解器。为了在离散层面利用这一结构，我们对磁场采用curl-相容有限元空间，并设计速度空间以保持由洛伦兹力耦合引起的curl-curl结构。所得到的相容离散格式保持离散磁散度约束，同时产生直接适用于基于AMS的高效求解器的线性系统。我们在全非线性三维托卡马克测试案例中展示了求解器的效率以及所得结构保持离散格式的精度和稳定性。 Comment on arXiv:2106.08363v3 [math.NA], E. Abreu, A. Espirito Santo, W. Lambert, and J. Perez, Convergence of a Lagrangian–Eulerian scheme by a weak asymptotic analysis for one-dimensional hyperbolic problems 作者: Frederico Furtado, Felipe Pereira, Li-Ming Yeh 类别: math.NA 发布日期: 2026-08-14 arXiv ID: 2608.14520v1 链接: http://arxiv.org/abs/2608.14520v1 摘要: This Comment concerns arXiv:2106.08363v3 [math.NA] by E. Abreu, A. Espirito Santo, W. Lambert and J. Perez, published in Numer. Methods Partial Differential Equations 39 (2023) 2400-2443. That article builds its scheme on space-time control volumes whose lateral boundaries, called “no-flow curves”, solve dsigma/dt = H(u)/u and are presented as new. We show, equation by equation, that this object is the space-time integral curve of the locally conservative Eulerian-Lagrangian method of Douglas, Pereira and Yeh [Comput. Geosci. 4 (2000) 1-40], that the no-flow region is the tube those curves bound, and that its scalar forward-tracked form appears in Mancuso, Pereira and de Souza [TEMA 8 (2007) 269-276, 277-286]. At issue is not moving a control volume, which is generic, but which curve moves it: H’(u) and H(u)/u coincide identically only for a linear flux, and only the latter makes lateral mass flux vanish. The commented article itself calls the 2000 paper the first to introduce space-time local conservation, with an integral tube bounded by integral curves, while its abstract calls the same object introduced by the authors. A full-text corpus documents the changing terminology and attribution. Papers published in 2025 and 2026 use the same construction while citing later work but not DPY. Another 2026 paper gives mixed attribution: it calls the no-flow curve an “extension” of the DPY integral curve, although its own zero-flux definition and ratio ODE show identity, and it states that diffusion and dispersion do not modify the defining vector field. It extends the model, scheme and analysis, not the continuous curve. The later discrete contributions are not challenged. Hybrid Dealiasing and Implicit Packing for Real Convolutions 作者: Noel Murasko, John C. Bowman 类别: math.NA 发布日期: 2026-08-14 arXiv ID: 2608.14497v1 链接: http://arxiv.org/abs/2608.14497v1 摘要: Hybrid dealiasing is an FFT-based method for computing linear convolutions of complex-valued data that reduces the cost of dealiasing by performing zero padding implicitly. We develop two new algorithms that extend hybrid dealiasing to real-valued convolutions.
The first algorithm exploits conjugate symmetries in the transformed data and computes each residue contribution directly. The second algorithm employs complex-valued hybrid dealiasing via a new implicit packing technique, which packs real-valued data into complex-valued arrays and performs all unpacking and packing operations implicitly in the transformed domain. Multidimensional versions of both algorithms are obtained by recursive decomposition into lower-dimensional convolutions.
Based on these algorithms, efficient routines to compute convolutions of equal-length input arrays are implemented in the open-source FFTW++ library. These routines outperform the standard method of explicit zero padding in one, two, and three dimensions. In many cases, implicit packing without hybrid dealiasing is nearly as fast as the general hybrid dealiasing algorithm, particularly when multithreading. Reduced Order Modeling and Applications to Inverse State Estimation 作者: Olga Mula, Agustín Somacal 类别: math.NA 发布日期: 2026-08-14 arXiv ID: 2608.14328v1 链接: http://arxiv.org/abs/2608.14328v1 摘要: Solving parametric partial differential equations (PDEs) repeatedly for many parameter values,as needed in optimal control, uncertainty quantification, and inverse problems, is often prohibitive with classical discretizations. This chapter introduces Reduced Order Modeling (ROM), which builds compressed yet accurate representations of parametric solution sets for fast online evaluation. We formulate the approximation of parametric PDEs as a supervised learning task and review linear approximation, very effective for elliptic and parabolic problems, before turning to nonlinear methods needed when solutions exhibit discontinuities or steep gradients. We then show how to use reduced-order models to efficiently recover the state of a physical system from limited measurements, which is an inverse problem known as state estimation. Throughout, we favor results with theoretical guarantees while also discussing practical methods, and provide a companion Jupyter notebook implementing the main algorithms. Analytic inverse problems with finitely many random measurements 作者: Giovanni S. Alberti, Damiano Poletti, Simone Sanna, Matteo Santacesaria 类别: math.AP, math.NA, math.PR 发布日期: 2026-08-14 arXiv ID: 2608.14324v1 链接: http://arxiv.org/abs/2608.14324v1 摘要: While infinite-dimensional inverse problems are traditionally analyzed assuming continuous data, practical applications rely on finitely many discrete measurements. Recent deterministic approaches establish that unknowns belonging to a $d$-dimensional model class can be stably recovered from finitely many measurements. However, for severely ill-posed problems, such as the Calderón problem and inverse scattering, the known constructions may require a number of measurements that is exponential in $d$. We show that random sampling reduces this count dramatically if one asks only for exact identifiability. By exploiting the analytic geometry of the forward maps, we prove that, whenever the infinite-data problem is injective on the model class, $2d+1$ random scalar measurements determine the unknown uniquely, almost surely. Applications are given to the Calderón problem, with both infinite- and finite-dimensional boundary sampling, and to inverse medium scattering from randomly sampled far-field values. Coupling-Aware Vanka Smoothing for Multigrid Preconditioning of the Implicit Immersed Boundary Equations 作者: Cole Gruninger, Boyce E. Griffith 类别: math.NA, physics.comp-ph, physics.flu-dyn 发布日期: 2026-08-14 arXiv ID: 2608.14310v1 链接: http://arxiv.org/abs/2608.14310v1 摘要: The immersed boundary (IB) method models fluid–structure interaction using the natural Lagrangian and Eulerian formulations of structural mechanics and fluid dynamics, respectively, but explicit time discretization of the IB force imposes a stiffness-dependent upper bound on the time-step size. Treating these forces implicitly removes this restriction, but the resulting coupled linear systems become increasingly difficult to solve as the structural stiffness increases. Algebraically eliminating the Lagrangian degrees of freedom yields a reduced Eulerian velocity–pressure IB system. Here we introduce a coupling-aware Vanka (CAV) smoothing strategy to enable effective multigrid preconditioning of this system. CAV patches are built as unions of standard pressure-centered Vanka patches, with the graph of the Eulerian elasticity matrix determining which patches are combined. Under grid refinement, CAV patch sizes remain bounded, and the computational cost of each multigrid cycle thereby grows linearly with the number of Eulerian degrees of freedom. Tests using target-point, membrane, and beam force laws show that CAV-preconditioned FGMRES reduces the relative residual by ten orders of magnitude in $9--15$ iterations, with little growth under grid refinement. In a nonlinear benchmark modeling flow past a flexible fiber, the average number of FGMRES iterations per Newton solve increases only from $8.6$ to $9.5$ as the Eulerian grid is refined from $32\times32$ to $256\times256$ cells. To our knowledge, CAV provides the first robust multigrid strategy for time-dependent implicit IB formulations. TOGEARI: Interaction-Space Preconditioning for Condensed Finite-Element Systems with IPC Contact 作者: Yanlin Liu, Chao Huang, Kaixiang Yao, Yao Shen 类别: math.NA, cs.CE 发布日期: 2026-08-14 arXiv ID: 2608.14162v1 链接: http://arxiv.org/abs/2608.14162v1 摘要: Condensed finite-element contact systems combine a large material-volumetric core with a thin factorized contact update. We formulate Truncated Operator-Gram Eigenspace Approximation of Relevant Interactions (TOGEARI), a factor-space compression of that update. It selects a low-dimensional space of contact combinations and installs their contribution as a Woodbury right preconditioner. The assembled Newton equation and independent residual test remain fixed. The inverse action requires an invertible core and reduced Woodbury system. Core-response selection additionally assumes a symmetric core and remains available when that core is indefinite. For a positive definite core, the response values have an energy ordering. The same analysis then gives the exact generalized spectrum, a scaled inverse-error identity, and the optimal worst omitted interaction at each dimension. A raw contact-factor selector provides a lower-setup empirical alternative. We derive the split for a quadratic-tetrahedral displacement formulation with locally condensed elementwise-constant pressure and a factorized positive-semidefinite IPC normal tangent. In a frozen system with 325,260 free coordinates, an eight-dimensional subspace of a 94-row contact factor with registered numerical rank 42 reduces the Arnoldi basis from 11 vectors to 5. The core-response and raw spaces are closely aligned on the primary and near-repeat extractions; row-norm and deterministic-random controls require 12 vectors. The retained-dimension sweep reduces the warm median from 2.849 s to 1.629 s. Post-load setup plus first solve changes from 53.885 s to 61.958 s. These results identify a compact interaction correction in one frozen, frictionless regime. Distinct Newton states, evolving contact, contact-rank and mesh scaling, friction, and nested pressure-space performance remain open experimental questions. Adaptive multigrid for high-order discontinuous Galerkin methods based on the full approximation scheme 作者: Jörg Stiller 类别: math.NA 发布日期: 2026-08-14 arXiv ID: 2608.14123v1 链接: http://arxiv.org/abs/2608.14123v1 摘要: We propose an adaptive multigrid (MG) method for discontinuous Galerkin formulations of elliptic problems using Brandt’s full approximation scheme (FAS). Unlike common approaches, this method achieves local $hp$-refinement of hexahedral meshes without the need for hanging nodes. The core component of the FAS-MG method is an overlapping Schwarz smoother, which is optionally accelerated by a Krylov method. This smoother is designed for unstructured curvilinear meshes but maintains a tensor-product structure for fast diagonalization. Numerical experiments demonstrate the exceptional efficiency of the FAS-MG method. Dedicated studies confirm its robustness against high aspect ratios, element deformation, and irregular mesh topology. We also verify its capability for dynamic parallel mesh adaptation using the wave-front benchmark of Červený, Dobrev, and Kolev (SIAM J. Sci. Comp. 41, 2019). Finally, we present preliminary results of extending the method to incompressible Navier-Stokes problems. Unconditionally stable and energy conserving discretization of the dynamic von Kármán equations 作者: Carsten Carstensen, Neela Nataraj, Ricardo Ruiz-Baier, Aamir Yousuf 类别: math.NA 发布日期: 2026-08-14 arXiv ID: 2608.14087v1 链接: http://arxiv.org/abs/2608.14087v1 摘要: A fully discrete approximation of the dynamic von Kármán equations combines nonconforming Morley finite element methods for spatial discretization with an energy conserving modified unconditionally stable Newmark second- order time-stepping scheme. Brouwer’s fixed-point theorem establishes existence of a solution to the fully discrete scheme and further uniqueness and stability estimates follow for small loads. Optimal order a priori error estimates in the piecewise energy norm with quadratic convergence in time are derived for the fully discrete scheme. The results of the numerical experiments validate the theoretical error bounds. A counterexample to Kusner’s conjecture on equilateral sets 作者: Logan R. Chalmers 类别: math.MG, math.CO, math.NA 发布日期: 2026-08-14 arXiv ID: 2608.14013v1 链接: http://arxiv.org/abs/2608.14013v1 摘要: We disprove Kusner’s 1983 conjecture that every equilateral set in $\ell_p^n$ with $2 An $h$-adaptive Tetrahedral Spectral Element Method with Applications to Kohn-Sham Density Functional Theory 作者: Zeyu Wang, Hongfei Zhan, Guanghui Hu 类别: math.NA 发布日期: 2026-08-14 arXiv ID: 2608.14006v1 链接: http://arxiv.org/abs/2608.14006v1 摘要: High-order $h$-adaptive spectral element methods on tetrahedral meshes provide an effective framework for resolving localized singularities and multiscale structures in complex three-dimensional geometries. However, their development is often hindered by difficulties in maintaining $C^0$ continuity across refinement interfaces and efficiently transferring solutions between adaptive meshes. Such limitations are particularly relevant in demanding applications such as all-electron Kohn-Sham density functional theory, which place stringent requirements on the accurate resolution of both nuclear singularities and multiple physical scales. In this paper, we present an efficient $h$-adaptive tetrahedral spectral element framework. To address the continuity challenge, we develop an adaptive strategy that combines element orientation alignment with geometric red-green refinement, thereby eliminating the need for algebraic hanging-node constraints while preserving inter-element continuity. Furthermore, an efficient topology-based point-location algorithm is introduced to accelerate interpolation between adaptive meshes. Numerical experiments on Poisson and Laplacian eigenvalue problems confirm the spectral convergence of the proposed method. Applications to all-electron Kohn-Sham equations further demonstrate its capability to accurately resolve nuclear singularities. Moreover, parallel performance studies exhibit excellent scalability, with matrix assembly and adaptivity modules generally achieving speedups above 15 times and the proposed interpolation algorithm attaining speedups ranging from 25 to 35 on 64-core configurations compared to the single-core performance. These results indicate that the proposed framework provides an accurate, robust, and efficient solution for large-scale, high-resolution simulations. Nonparametric Schrödinger Bridge Time Series Generator: Algorithm, Convergence Analysis and Applications 作者: Daili Sheng, Minghui Song, Hui Sun 类别: math.NA 发布日期: 2026-08-14 arXiv ID: 2608.13968v1 链接: http://arxiv.org/abs/2608.13968v1 摘要: We conduct a convergence analysis for the Schrödinger Bridge Time Series (SBTS) data generator. Starting from a regularized formulation in which the data ensemble is mixed with a standard multivariate Gaussian distribution with a prescribed probability, we prove that the Euler-Maruyama discretization converges to the mixed target distribution with half-order convergence rate, provided that the ensemble size and kernel bandwidth are chosen appropriately. We further show that the regularized distribution converges to the original target distribution as the mixing probability tends to zero. The analysis simultaneously accounts for the ensemble approximation error, kernel approximation error, and time-discretization error, and therefore provides a full distributional convergence result for the SBTS generator. Empirically, we further examine the flexibility of the method by replacing the Wiener reference measure with the path measure induced by a more general SDE. The numerical experiments show that the schemes based on both the original Wiener reference measure and the SDE-induced reference measure achieve comparable performance, demonstrating the robustness and stability of the SBTS framework. Semi-supervised Concordance Learning for Optimal Individual Treatment Regimes 作者: Mengjiao Peng, Yong Zhou, Wenbin Lu 类别: stat.ME, math.NA 发布日期: 2026-08-14 arXiv ID: 2608.13945v1 链接: http://arxiv.org/abs/2608.13945v1 摘要: Finding the optimal individualized treatment rule that maps individual characteristics or contextual information to treatment assignments has been extensively investigated in existing literature, with widespread practical applications. This paper considers the estimation of optimal treatment regimes within a semi-supervised data framework (exemplified by electronic medical record data). In such settings, only a tiny proportion of observations have observed outcome labels, owing to high labeling costs, time limitations, data privacy concerns, and other constraints, while covariates and treatment assignments are available for all study subjects. We develop a semi-parametric inference method for optimal treatment regimes, which leverages outcome- unlabeled samples with complete covariate and treatment information to enhance estimation efficiency. The proposed estimation framework consists of two key steps: first, flexible nonparametric imputation via single-index kernel smoothing; second, subsequent estimation of the optimal treatment regime based on concordance-assisted learning. We establish the consistency and asymptotic normality of our proposed estimators. Numerical simulation studies demonstrate that our method achieves higher efficiency and stronger robustness relative to fully supervised estimators under finite-sample settings. We further validate the practical value of our proposed framework using the MIMIC-III and ACTG175 datasets. PDE Realization and Structure-Aware Solvers for a Compact Two-Stage Fourth-Order IMEX Method 作者: Zhixin Huo 类别: math.NA 发布日期: 2026-08-14 arXiv ID: 2608.13875v1 链接: http://arxiv.org/abs/2608.13875v1 摘要: This paper develops a PDE realization and solver framework for a compact two-stage fourth-order two-derivative implicit–explicit time discretization for stiff split problems. The key difficulty is a consistency–cost coupling: full-field temporal differentiation is required to preserve the mixed explicit–implicit interactions of the time integrator, but the same interactions widen the implicit stage operators and can make each solve substantially more expensive. We show that a second-order ADER/Cauchy–Kowalevski local evolution is sufficient for the fourth-order outer composition, establish smooth reconstruction consistency and a fully discrete error balance, and derive the leading Lie-bracket defect produced by self differentiation of the split fields. An inexact-stage analysis gives asymmetric midpoint and endpoint residual tolerances that preserve fourth-order accuracy. For the widened stages, the complete mixed action is retained matrix-free while only dominant stiff physics is approximated in the inverse. This yields quadratic and shifted preconditioners, Fourier and multilevel realizations, a semilinear reaction–diffusion reduction, and source-local elimination for relaxation systems. Exact quadratic cancellation, a diffusion-dominated Fourier estimate, and an $\varepsilon/Δx$ bound for Jin–Xin relaxation explain the main solver mechanisms. Numerical ablations verify the consistency and tolerance results, while a two-dimensional Brusselator study on two grids shows favorable error-versus-wall-time behavior over a useful accuracy range. A classical stiff-front benchmark also identifies the separate spatial shock–source limitation. The results support a regime-dependent efficiency claim rather than a universal speedup. Approximating matrix functions by block Krylov methods with randomized vectors 作者: Josh Kane, Lucas Onisk, Lothar Reichel, Giuseppe Rodriguez 类别: math.NA 发布日期: 2026-08-13 arXiv ID: 2608.13714v1 链接: http://arxiv.org/abs/2608.13714v1 摘要: The need to evaluate expressions of the form $f(A)\mathbf{b}$, where $A$ is a square matrix, $f$ is a function, and $\mathbf{b}$ is a vector, arises in several areas of applied mathematics. When the matrix $A$ is very large, it is usually not attractive to evaluate $f(A)$. Instead, $f(A)\mathbf{b}$ often is approximated by computing an estimate in a Krylov subspace that depends on $A$ and $\mathbf{b}$, and only requires that $f$ be evaluated at a small matrix. This paper explores the application of several variants of randomized block Krylov methods to the approximation of $f(A)\mathbf{b}$. Computed examples suggest that block Krylov methods with an initial block vector that contains $\mathbf{b}$ as well as a few randomly generated vectors may require less computing time and reduce the number of Krylov steps than standard Krylov methods. An adaptive superconvergent mixed finite element method for exactly symmetric linear elasticity based on local residual minimization 作者: Ernesto Cáceres, Patrick Vega 类别: math.NA 发布日期: 2026-08-13 arXiv ID: 2608.13701v1 链接: http://arxiv.org/abs/2608.13701v1 摘要: We introduce an a posteriori error estimator for exactly symmetric mixed finite element discretizations of linear elasticity, based on recasting a Stenberg-type postprocessing scheme as a local residual minimization problem in a discrete dual norm. This construction yields, as a dual variable and at no additional computational cost, a Riesz representative of the associated local residual, from which we build an a posteriori error indicator. We establish a reliability estimate, with the dependence on the Lamé parameters tracked explicitly, and a local efficiency estimate, without auxiliary bubble functions, in the standard compressible regime, as well as an alternative reliability estimate, based on a robust stability property and an Oswald averaging operator, with a constant that remains bounded as $λ\to\infty$; the local efficiency estimate holds, with the same bounded behavior, uniformly in both regimes. Notably, a single indicator and a single comparison norm serve both regimes, in contrast with existing hypercircle-based estimators, which require a distinct construction for the incompressible limit. Numerical examples, including a benchmark with a known singular solution and one without an analytical solution, validate the theoretical findings.3. 研究方法、关键技术和核心工具
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