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1. 非均匀介质中时谐麦克斯韦方程阻抗边界条件的并行重叠型区域分解方法收敛性

Convergence of parallel overlapping domain decomposition methods with impedance boundary conditions for time-harmonic Maxwell equations in heterogeneous media

作者: Luyu Cen, Shihua Gong, Euan A. Spence, Yue Yu

类别: math.NA

发布日期: 2026-06-03

arXiv ID: 2606.04982v1

摘要翻译: 本文分析非均匀介质中时谐麦克斯韦方程带阻抗边界条件的并行重叠型区域分解方法的收敛性。我们证明并行迭代方法在适当的函数空间中是适定的，并通过描述相邻子区域间相互作用的阻抗-阻抗映射刻画误差传播算子。对于条形区域分解，我们借助阻抗-阻抗映射的范数推导出显式的收敛估计。在离散层面，我们基于 Nédélec 元离散发展出这些结果的有限元对应形式。在网格加密时离散阻抗-阻抗映射逼近其连续对应这一假设下，我们证明离散方法继承了连续方法的收敛行为。我们用条形区域分解的数值实验说明该理论，并展示超出理论分析范围的棋盘形区域分解的数值实验。

Comment: 38 pages

链接: http://arxiv.org/abs/2606.04982v1

1. 研究对象和背景

时谐Maxwell方程是描述电磁波传播的基本模型，其形式为：给定有界Lipschitz区域 $\Omega \subset \mathbb{R}^3$，变系数 $\mu, \epsilon \in L^\infty(\Omega)$ 满足 $0<\mu_{\min}\le \mu\le\mu_{\max}<\infty$ 和 $0<\epsilon_{\min}\le \epsilon\le\epsilon_{\max}<\infty$，以及边界阻抗参数 $\lambda\in L^\infty(\partial\Omega)$ 满足 $0<\lambda_{\min}\le\lambda\le\lambda_{\max}<\infty$，考虑内部阻抗问题：

$$ \begin{cases} \operatorname{curl}(\mu^{-1}\operatorname{curl} u) - k^2\epsilon u = f & \text{在 } \Omega \text{ 内},\\ \mu^{-1}\operatorname{curl} u\times n - i k \lambda u_T = g_T & \text{在 } \partial\Omega \text{ 上}, \end{cases} $$

其中 $u$ 是电场，$k>0$ 是波数，$u_T = (n\times u)\times n$ 是切向分量，$f\in L^2(\Omega)$, $g_T\in L^2_T(\partial\Omega)$ 是已知数据。该方程在雷达、通信、医学成像等领域有广泛应用。高频情形下，数值求解面临巨大挑战：为分辨振荡解，需要足够细的网格，导致离散系统规模庞大；此外系统通常非Hermitian且高度不定，使得迭代求解器设计困难。在非均匀介质中，变系数材料参数进一步加剧了波传播和数值逼近的复杂性。

有限元方法（FEM）在求解Maxwell方程时存在“污染效应”：即固定每个波长的网格点数目，随着波数 $k$ 增大，精度下降。已有工作给出了固定多项式阶次下，为保证精度所需的网格尺寸 $h$ 与 $k$ 的关系，例如采用第一/第二族 Nédélec 元 [8,31]、连续内罚方法 [32,30] 和内罚间断Galerkin方法 [17]；进一步，将多项式阶次按照 $\log k$ 增长可以消除污染效应 [41,34,35]。因此，线性系统规模随 $k$ 增长至少为 $\mathcal{O}(k^d)$，需要高效的求解技术。

区域分解方法（DD）是求解大规模系统的重要途径。对于Maxwell方程，已存在两层重叠方法的严格收敛理论，其中子区域求解与全局粗空间结合，常作为GMRES预条件子。例如，在固定 $k$ 下使用分片多项式粗空间的收敛性 [22]；对含吸收（虚波数 $k$）问题的分析 [5]，近期推广到非均匀介质情形 [4]。当吸收足够小时，含吸收问题可很好地预条件实波数问题 [42]，并被用于构造问题适应的粗空间 [28]。这些两层方法分析的核心是证明预处理系统的数值域不包含原点，从而利用 Elman 估计 [15,14] 及其变种 [3] 得到 GMRES 收敛性。

相比之下，单层重叠区域分解方法（不使用粗空间）的理论远未成熟。对于Maxwell方程，目前尚无收敛理论（无论连续还是离散层次）。单层方法与两层方法性质迥异：两层方法中子区域通常很小（直径 $\sim k^{-1}$），不涉及波的传播；而单层方法子区域较大，必须理解波如何在子区域间传播。事实上，当使用两层方法时，粗问题通常仍然很大，往往采用单层方法迭代求解，因此理解单层方法的收敛行为是完成全两层分析的必经之路。此外，如图1所示，单层方法的预处理系统数值域包含原点，故传统场值分析不能直接使用，必须另辟蹊径。

在Helmholtz方程（常系数）方面，[20,21,27] 发展了并行Schwarz方法配合阻抗边界条件的收敛理论，其核心是将误差传播算子用子域边界之间的“阻抗到阻抗映射”(impedance-to-impedance maps)表示，并对其范数进行估计，从而得到收敛性。本文旨在将这一分析框架推广到时谐Maxwell方程，并从一开始就处理变系数介质，同时建立离散 Nédélec 元逼近的相应理论，填补 Maxwell 单层重叠区域分解方法理论分析的空白。

2. 主要定理或主要结果

本文的主要结果包括以下四个方面：

连续层次并行Schwarz方法的适定性与误差传播算子表示
设 $\Omega$ 被 $N$ 个重叠Lipschitz子区域 $\Omega_\ell$ 覆盖，并存在相应单位分解 $\chi_\ell$。定义并行Schwarz迭代（定义2.9）：给定上一迭代解 $u^n$，在每个子区域上求解
$$ \begin{cases} (\operatorname{curl}(\mu^{-1}\operatorname{curl}) - k^2\epsilon) u_\ell^{n+1} = f & \text{在 } \Omega_\ell \text{ 内},\\ (\mu^{-1}\operatorname{curl} u_\ell^{n+1} - i k\lambda n_\ell\times u_\ell^{n+1})\times n_\ell = g_T & \text{在 } \partial\Omega_\ell\cap\partial\Omega \text{ 上},\\ (\mu^{-1}\operatorname{curl} u_\ell^{n+1} - i k\widetilde{\lambda} n_\ell\times u_\ell^{n+1})\times n_\ell = (\mu^{-1}\operatorname{curl} u^{n} - i k\widetilde{\lambda} n_\ell\times u^{n})\times n_\ell & \text{在 } \partial\Omega_\ell\setminus\partial\Omega \text{ 上}, \end{cases} $$
然后通过 $u^{n+1} = \sum_\ell \chi_\ell u_\ell^{n+1}$ 合成全局解。
定理2.10证明：若每个子区域 $\Omega_\ell$ 为有界Lipschitz且单连通、边界连通、且系数满足适当光滑性，则从 $u^n = (u^n_1,\dots,u^n_N)\in \mathbf{U} = \prod_\ell U(\Omega_\ell)$ 出发，新的各子区域解 $u_\ell^{n+1}\in U(\Omega_\ell)$，从而 $u^{n+1}\in \mathbf{U}$。
定理2.12进一步保证在额外假设（假设2.11）$(\nabla\chi_\ell\cdot\nabla)u_\ell^{n+1}\in L^2(\Omega_\ell)$ 下，合成解 $u^{n+1}\in U(\Omega)$，迭代在全局空间 $U(\Omega)$ 中也适定。
通过子区域误差方程，误差传播算子 $\mathcal{E}$ 可表为作用在子域边界的阻抗映射的组合，具体形式为（参见第3节）：
$$ \mathcal{E} = \mathcal{P} \circ \mathcal{M}, $$
其中 $\mathcal{M}$ 将子域边界上的误差阻抗数据映射为新的误差阻抗数据，$\mathcal{P}$ 为解算子。这一表示将收敛性归结为阻抗映射范数的研究。
条带分解的收敛充分条件
针对 $d=2$ 或 $3$ 维的条带分解 (strip decomposition)，假设子域为 $\Omega_\ell = (a_{\ell}, b_{\ell})\times \Sigma$（$\Sigma$ 为截面），利用阻抗映射的特殊结构，得到显式收敛估计。核心结果（第4节）表明：如果子域间阻抗映射的算子范数满足
$$ \| \mathcal{I}_{i\to j} \| < 1 \quad \text{对所有相邻子域 } i,j, $$
则并行Schwarz迭代以收缩因子 $\rho = \max \|\mathcal{I}_{i\to j}\|$ 收敛。这里 $\mathcal{I}_{i\to j}$ 定义为从界面 $\Gamma_i$ 到界面 $\Gamma_j$ 的阻抗映射，具体为：
$$ \mathcal{I}_{i\to j} g := (\mu^{-1}\operatorname{curl} v - i k\widetilde{\lambda} v_T)\big|_{\Gamma_j}, \quad v\in U_0(\Omega_i) \text{ 且 } (\mu^{-1}\operatorname{curl} v - i k\widetilde{\lambda} v_T)\big|_{\Gamma_i} = g. $$
该条件将全局收敛分析转化为局部界面映射的范数估计，降低了分析复杂性。类似框架可推广到更一般的分解，但需处理更复杂的界面几何。
离散情形：Nédélec元离散与RAS-imp方法
采用 Nédélec 有限元对子问题离散，得到了离散的并行Schwarz迭代，并证明其等价于限制加性Schwarz方法配合阻抗传输条件 (RAS-imp)（§5.2）。定义离散误差传播算子 $\mathcal{E}_h$，它与连续情形具有类似结构，只是阻抗映射替换为离散对应 $\mathcal{I}_{i\to j,h}$。
定理（第6节）：假设离散阻抗映射按网格细化逼近连续映射，即对每个子域界面，当 $h\to 0$ 时，
$$ \|\mathcal{I}_{i\to j,h} - \mathcal{I}_{i\to j}\|_{L^2\to L^2} \to 0, $$
则存在 $h_0>0$，使得对所有 $h\leq h_0$，离散迭代的收敛因子与连续情形任意接近，从而离散方法继承了连续方法的收敛行为。该结果不要求网格拟一致，只依赖于离散阻抗映射的逼近性质。
数值实验验证
二维数值实验分别针对均匀和非均匀介质，采用条带和棋盘 (checkerboard) 分解。条带分解结果验证了误差传播算子的幂收缩性 (power contractivity) 以及阻抗映射范数的有界性，且观察到的收敛速度与理论估计一致。棋盘分解虽超出当前理论范围，但数值显示RAS-imp作为定常迭代或Krylov预条件子依然有效，表明理论有望进一步拓展。

3. 研究方法、关键技术和核心工具

本文证明框架的核心创新在于利用“阻抗到阻抗映射”来刻画子域耦合，并配合调和分析技术处理Maxwell方程对函数空间的特殊要求。主要技术点及障碍克服如下：

函数空间刻画与阻抗映射有界性
单层Schwarz方法的分析需要在子域上具有良好的迹定理和正则性，尤其是阻抗边界条件涉及 $u_T$ 和 $\operatorname{curl} u\times n$ 的 $L^2$ 迹。传统上，$H(\operatorname{curl})$ 只保证 $u_T\in H^{-1/2}(\partial\Omega)$，但阻抗条件需要更强的 $L^2$ 正则性。为此，作者引入了空间 $U(D)$，其包含额外条件 $\operatorname{curl}(\mu^{-1}\operatorname{curl} u)\in L^2(D)$ 和 $\operatorname{curl} u\times n\in L^2(\partial D)$。关键结果是定理2.4：当 $\mu\in W^{1,\infty}(D)$ 且 $D$ 单连通、边界连通时，$U(D) = W(D) := \{u\in H^{1/2}(\operatorname{curl}, D)\cap H(\operatorname{div}, D) \mid \operatorname{curl}(\mu^{-1}\operatorname{curl}u)\in L^2(D)\}$。其证明依赖定理2.6（[10, Theorem 2]），后者利用了Jerison和Kenig关于 $H^{3/2}$ 正则性的深刻结果 [26] 以及Nečas关于 $H^1$ 边界迹的等价刻画 [39]。此等价性允许从 $U(D)$ 推出 $u$ 为 $H^{1/2}$，进而保证了乘积法则和单位分解操作时的收敛性。
借助 $U(D)$ 的刻画，证明了引理2.8：从 $u\in U(D)$ 可得出阻抗迹 $(\mu^{-1}\operatorname{curl} u - i k\lambda u_T)\times n \in L^2(\partial D')$（对任意Lipschitz子域 $D'$），且对“Maxwell-调和”函数（满足齐次方程的 $U_0(D)$）该迹的 $L^2$ 范数被边界上的阻抗迹所控制。这是保证迭代中阻抗交换数据保持在 $L^2$ 的关键。
误差传播算子的阻抗映射表示
与Helmholtz方程工作[20]类似，作者通过消去内部体积源，将子域误差用齐次方程 $u_0\in U_0(\Omega_\ell)$ 表示，并利用阻抗边界条件的传输特性，将子域间误差传播表达为映射的复合。具体地，对于相邻子域，可定义局部阻抗映射 $\mathcal{I}_{\Gamma_1\to\Gamma_2}$，将界面 $\Gamma_1$ 上的阻抗数据映射到 $\Gamma_2$ 上的阻抗数据。通过引入适当的限制算子和粘贴算子，全局误差传播算子 $\mathcal{E}$ 在 $\ell^2$ 范数下可由这些界面映射的范数控制。这种方法将无限维算子范数估计转换为对局部界面映射的估计，本质上是利用波的传播衰减性质。
条带分解下的范数估计
在条带分解中，由于每个子域的横截面不变，波在沿条带方向传播时，阻抗映射可以通过分离变量和模式分析。虽然本文未详细给出具体的解析范数估计（可能留待后续工作），但给出了充分条件架构，并指出类似于Helmholtz的情形[27]，高阶模态可能衰减很快，从而保证低维映射范数小于1。数值实验（§7.4）计算了这些范数，证实了有界性和小于1的性质。
离散方法的逼近继承性
离散分析的关键在于定义合适的离散阻抗映射 $\mathcal{I}_{i\to j,h}$，并证明其收敛到连续映射。这要求构造离散子问题（带有阻抗边界条件的Nédélec元）并定义相应的离散 $U_{0h}$ 空间。论证思路是：若连续误差传播算子 $\mathcal{E}$ 的范数为 $\rho<1$，且 $\mathcal{E}_h$ 可写为类似的阻抗映射复合，则通过三角不等式有
$$ \|\mathcal{E}_h\| \le \|\mathcal{E}\| + \|\mathcal{E}_h - \mathcal{E}\|, $$
而 $\|\mathcal{E}_h - \mathcal{E}\|$ 可被 $\max\|\mathcal{I}_{i\to j,h} - \mathcal{I}_{i\to j}\|$ 控制。该控制需要保证离散解到连续解的光滑恢复，这里依赖Nédélec元的渐进逼近性质以及离散函数空间与连续空间的嵌入。值得注意，该结果不要求网格的拟均匀性，只要求阻抗映射的局部逼近性，这为自适应网格提供了可能。
适定性中额外正则性的处理
相比于Helmholtz方程，Maxwell情形的最大技术障碍在于合成解 $u^{n+1} = \sum\chi_\ell u_\ell^{n+1}$ 是否仍在空间 $U(\Omega)$ 中，这要求控制 $\operatorname{curl}(\mu^{-1}\operatorname{curl}(\chi_\ell u_\ell^{n+1}))$ 的 $L^2$ 范数。展开后出现项 $(\nabla\chi_\ell\cdot\nabla)u_\ell^{n+1}$，而 $U(\Omega_\ell)$ 并不直接包含二阶导数控制。作者将此作为假设2.11，并讨论了其在特定几何（如条带区域且 $\chi_\ell$ 只依赖条带方向且在内边界附近为零）下成立的可能性（猜想2.14）。尽管尚未严格证明，但数值实验未显示异常。这一点的处理反映了Maxwell系统与标量Helmholtz方程的根本区别。

4. 与之前工作的比较

本文与已有工作的联系与区别主要体现在以下几个方面：

与Helmholtz方程单层Schwarz方法理论的对比
[20,21,27]针对常系数Helmholtz方程建立了完整的阻抗传输条件并行Schwarz方法的PDE和离散级收敛理论。本文可视为该框架向Maxwell方程和变系数的直接推广。然而，推广并非平凡，面临如下显著差异：

函数空间与正则性：Helmholtz分析中，解空间 $U_{\text{Helm}} := \{u\in H^1(\Omega): \Delta u\in L^2(\Omega), \partial_n u\in L^2(\partial\Omega)\}$ 自然提供 $H^1$ 正则性，使得单位分解后的二阶导数控制相对容易。Maxwell情形下，对应的空间 $U(\Omega)$ 并不保证 $H^1$，仅能通过 $U(D)=W(D)$ 的深刻结果（依赖于调和分析）获得 $H^{1/2}$ 正则性，从而导致合成解的 $\operatorname{curl}(\mu^{-1}\operatorname{curl})$ 控制需额外假设（假设2.11）。
边界条件：Helmholtz方程使用阻抗条件 $\partial_n u - i k \lambda u = g$，仅涉及法向导数和函数值；Maxwell方程为 $\mu^{-1}\operatorname{curl} u\times n - i k \lambda u_T = g_T$，涉及切向旋度和切向分量，迹算子的处理更为复杂。
变系数：本文从一开始就允许 $\mu,\epsilon$ 为分片光滑变系数，而 [20,21,27] 最初为常系数，后续虽有变系数扩展但未含Maxwell。变系数要求对 $U(D)=W(D)$ 等定理中的 $\mu$ 作 $W^{1,\infty}$ 假设，且需保证唯一延拓性质（引用 [2,29] 结果）。
尽管如此，本文在条带分解下给出了类似Helmholtz情形的收敛充分条件（基于阻抗映射范数），并在备注6.8中总结了Maxwell类比Helmholtz结果的程度，指出核心困难已解决，但高波数渐近估计尚未展开。

与Maxwell两层区域分解方法的对比
已有Maxwell两层方法理论（如 [22,5,4]）基于数值域不包含原点的场值分析，并配合GMRES收敛估计。这些方法要求粗空间充分分辨波的传播，且子区域尺寸通常为波长的倒数量级。本文的单层方法没有粗空间，子区域尺寸可较大，其收敛性由波的衰减和局部传输特性决定。

分析工具的根本不同：图1显示单层方法预处理矩阵的数值域包含原点，因此不能使用Elman型估计，必须直接研究误差传播算子的谱半径或范数。这体现了比两层方法更大的分析难度。
子区域大小与可扩展性：两层方法理论上具有弱依赖于波数的可扩展性（通过增加粗空间维数），而单层方法作为定常迭代通常随子区域数量或波数增长而减慢收敛，但这正是其实际角色的真实反映（常作为内迭代求解粗问题）。本文提供了理解这种行为的理论框架。
对两层方法的互补：本文结果可直接应用于两层方法中粗问题求解的收敛性评估，从而完善两层方法的全分析。

离散分析的特色与推广性
与 [21] 对Helmholtz的离散分析类似，本文通过阻抗映射的收敛性假设将连续收敛性传递到离散。不同之处在于：

采用的是Nédélec旋度协调元，而非标准Lagrange元，需处理de Rham复形和切向迹的恰当逼近。本文未详述 $h\to 0$ 时离散阻抗映射收敛性的证明（可能基于标准逼近理论），但给出了数值验证。
结论对网格非结构化和局部细化具有鲁棒性，因为只依赖局部界面的阻抗映射误差，而不需要全局拟均匀假设。
棋盘分解的数值实验揭示了理论进一步拓展的可能性：此时子域界面方向复杂，阻抗映射不再容易分解为简单模态，但数值显示依然收敛，且作为GMRES预条件子表现良好。

局限性与未来工作
作者在文中明确指出：

仅对条带分解给出了理论的收敛条件，一般分解的充分条件仍待研究（但提供了可比拟Helmholtz的框架）。
未给出阻抗映射范数关于波数 $k$ 的渐近估计，即尚未量化高波数下如何缩放，这是实现实用预条件子设计的关键。
假设2.11仅在特定几何条件下可望成立，一般Lipschitz子域和单位分解下严格论证仍需进一步工作。
数值实验限于二维，三维情形需要更多计算资源。
尽管如此，本文作为Maxwell方程单层阻抗Schwarz方法的开拓性理论工作，为后续的波数显式分析、三维扩展、以及结合粗空间的混合策略奠定了严格的数学基础。

2. 使用无参数间断Galerkin方法对二阶时间依赖鞍点Maxwell系统的数值分析：首个最优 ${\bf L}^{2}$-范数误差估计

Numerical analysis of the second-order time-dependent saddle point Maxwell system via a parameter-free discontinuous Galerkin method: The first optimal ${\bf L}^{2}$-norm error estimates

作者: Achyuta Ranjan Dutta Mohapatra, Bhupen Deka

类别: math.NA

发布日期: 2026-06-03

arXiv ID: 2606.04488v1

摘要翻译: 我们针对作为鞍点问题形式的时间依赖Maxwell系统，提出了一种新颖的无参数间断Galerkin (dG) 有限元法 (FEM)。我们建立了所提出的半离散问题的稳定性，并推导了电场变量在能量范数和 ( {\bf L}^{2} ) 范数下以及位势函数在 ( L^{2} ) 范数下的最优误差估计。据我们所知，本文首次对使用任何有限元方法变体的二阶时间依赖鞍点Maxwell方程组进行了最优 ( {\bf L}^{2} )-范数误差分析。此外，我们提出了几种完全离散的时间积分格式，并通过二维和三维设置的实例验证了最优收敛结果。

Comment: 无

链接: http://arxiv.org/abs/2606.04488v1

1. 研究对象和背景

本文研究时间依赖 Maxwell 方程组的数值解法, 重点关注其作为鞍点问题的二阶含时形式。原始 Maxwell 系统是一阶 curl-div 系统:

$$ \begin{cases} \alpha^{-1} H_t = -\nabla\times E,\\ \nabla\times H = \epsilon E_t - J,\\ \nabla\cdot(\epsilon E) = \rho,\\ \nabla\cdot(\alpha H) = 0, \end{cases} $$

其中 $E$ 为电场, $H$ 为磁场, $J$ 为电流密度, $\rho$ 为电荷密度, 材料参数 $\epsilon$ (介电常数) 与 $\alpha$ (磁导率倒数) 设为常数。区域 $\Omega\subset\mathbb{R}^d$ ($d=2,3$) 为有界凸多面体, 边界 $\partial\Omega$ 为理想电导体 (PEC)。消去磁场 $H$ 后得到关于电场的二阶方程:

$$ \epsilon E_{tt} + \nabla\times(\alpha\nabla\times E) = F \quad \text{in } \Omega\times(0,T), $$

其中 $F = -J_t$, 同时仍需满足散度条件 $\nabla\cdot(\epsilon E)=\rho$。为放松该散度约束, 引入 Lagrange 乘子 $P$ (势函数), 将问题转化为如下鞍点系统:

$$ \begin{cases} \epsilon E_{tt} + \nabla\times(\alpha\nabla\times E) - \epsilon\nabla P = F,\\ \nabla\cdot(\epsilon E) = \rho,\\ \eta\times E = 0,\; P=0 \text{ on } \partial\Omega\times(0,T),\\ E(x,0)=u(x),\; E_t(x,0)=v(x). \end{cases} $$

这一形式显式包含 Gauss 定律, 避免了传统方法中散度条件自动满足的隐含假设, 因而更具一般性, 适用于非均匀介质、界面问题等复杂情况。

该问题的数值分析颇具挑战: 二阶时间导数需要处理双曲型方程的长时间稳定性; 混合结构要求有限元空间满足 inf-sup 条件以保证 $P$ 的稳定逼近; 电场 $E$ 的 $L^2$ 范数最优误差估计通常依赖对偶论证, 但对应的对偶问题耦合了旋度与梯度算子, 其正则性对区域形状敏感。

已有工作大多处理不含 $P$ 的标准二阶 Maxwell 方程, 即预先假设电场精确满足散度条件。Monk (2003) 利用旋度协调 Nédélec 元和连续有限元, 得到能量范数最优估计, 但 $L^2$ 估值需特殊 Nédélec 第二类元; Ciarlet et al. 用向后 Euler 和线性 Nédélec 二类元达到全离散最优收敛; Grote et al. (2007, 2010) 对 IPDG 半离散格式先后建立了能量范数和 $L^2$ 范数的最优收敛率, 但均需要用户选取惩罚参数。对于显式含 $P$ 的鞍点系统, Assous et al. (2010) 采用修正 Taylor-Hood 元但未给出误差分析; 文献 [2] 用协调元得到 $H(\mathrm{curl})$ 最优收敛, 却未估计 $P$ 且无 $L^2$ 结果; Qi et al. (2023) 用弱 Galerkin 方法获得电场能量半范和 $P$ 的 $L^2$ 范数最优估计, 但仍缺少电场 $L^2$ 范数分析。

本文针对二阶含时鞍点 Maxwell 系统, 采用一种无需惩罚参数的间断有限元方法进行空间离散, 旨在填补电场 $L^2$ 范数最优误差估计这一空白。该方法基于修正弱旋度与弱梯度算子, 格式构造简洁, 理论分析上可统一得到电场在两个范数下的最优收敛阶, 并且对电场正则性的要求可降低至 $H^{1+l}$ ($l>0$), 得到 $O(h^{1+l})$ 收敛率。此外, 本文还构造多种时间离散格式并进行数值验证。

2. 主要定理或主要结果

本文的主要成果包括半离散格式的稳定性、电场在能量范数与 $L^2$ 范数下的最优误差估计以及势函数的 $L^2$ 最优误差估计, 并辅以全离散格式及数值验证。

稳定性 (引理 4.7): 对半离散无参数 dG 格式 (4.1)-(4.4), 若初值与源项满足适当正则性 ($u,v\in H^2\cap H_0(\mathrm{curl})$, $F\in H^2(0,T;H^2)$, $\rho\in H^4(0,T;L^2)$), 则对 $t\in[0,T]$ 有
$$ \|E_{ht}(t)\|^2 + |||E_h(t)|||^2 \le C,\qquad \|P_h(t)\|^2 \le C, $$
其中 $|||\cdot|||$ 为离散能量范数 (定义见 (3.6)), $C$ 与网格尺寸 $h$ 无关, 仅依赖于数据。
电场能量范数误差估计: 令 $\mathbf{R}_h$ 为 Ritz 投影 (3.5)。假设精确解满足 $(E,P)\in L^\infty(0,T; H^{1+k}(\Omega)\cap H_0(\mathrm{curl};\Omega))\times L^\infty(0,T; H^k(\Omega)\cap H_0^1(\Omega))$, 且时间导数适当可积, 则有
$$ |||E(t)-E_h(t)||| \le C h^k \big( \|E\|_{L^\infty(H^{1+k})} + \|P\|_{L^\infty(H^k)} + \text{时间导数项} \big). $$
此估计为所用多项式次数 $k$ 下的最优阶, $|||\cdot|||$ 等价于离散 $H(\mathrm{curl})$ 范数。
电场 $L^2$ 范数最优误差估计 (定理 5.2): 在相同的正则性假设下, 有
$$ \|E(t)-E_h(t)\| \le C h^{k+\delta} \big( \|E\|_{L^\infty(H^{1+k})} + \cdots \big), $$
其中 $\delta\in(0,1]$ 为由对偶椭圆问题的正则性所确定的指数; 当区域为凸时, $\delta=1$, 得到 $O(h^{k+1})$ 的超收敛。更重要的是, 若电场仅具有 $H^{1+l}(\Omega)$ 正则性, $0 $$ \|E(t)-E_h(t)\| \le C h^{1+l} \big( \|E\|_{L^\infty(H^{1+l})} + \cdots \big). $$
此定理是文献中首次对鞍点 Maxwell 方程给出最优 $L^2$ 范数误差分析。
势函数 $L^2$ 范数估计:
$$ \|P(t)-P_h(t)\| \le C h^k \big( \|E\|_{L^\infty(H^{1+k})} + \|P\|_{L^\infty(H^k)} + \cdots \big). $$
与电场能量范数一样, 也是 $k$ 次多项式的最优阶。
全离散格式及数值实验: 文中构造了基于向后 Euler 和 Crank-Nicolson 的时间离散方案, 二维和三维算例均显示空间误差按 $O(h^k)$ (能量范数) 和 $O(h^{k+1})$ ($L^2$ 范数) 收敛, 与理论预测一致。

3. 研究方法、关键技术和核心工具

本文核心方法是基于修正弱旋度和弱梯度的无参数间断有限元离散, 并结合 Ritz 投影与对偶论证来获得最优 $L^2$ 误差估计。主要技术障碍及其解决方案如下:

1. 无参数 dG 格式的构造
传统间断有限元需要通过惩罚项 (如 IPDG 中的 $\frac{\gamma}{h_e}$) 保证格式的强制性和稳定性, 其系数 $\gamma$ 需大于某阈值, 且对精度有一定影响。本文采用 Wang et al. [26] 中发展的修正弱微分算子, 彻底避免了可调参数。对矢量函数 $v_h\in V_h$, 其在每个单元 $K$ 上的离散弱旋度 $\nabla_w\times v_h\in[P_{k-1}(K)]^{2d-3}$ 由下式定义:

$$ (\nabla_w\times v_h, w)_K = (v_h, \nabla\times w)_K + \langle n\times \{\!\!\{v_h\}\!\!\}_e, w\rangle_{\partial K},\quad \forall w\in[P_{k-1}(K)]^{2d-3}. $$

类似地, 对标量函数 $q_h\in Q_h$, 其离散弱梯度 $\nabla_w q_h\in P_k(K)$ 满足:

$$ (\nabla_w q_h, r)_K = -(q_h, \nabla\cdot r)_K + \langle \{\!\!\{q_h\}\!\!\}_e, n\cdot r\rangle_{\partial K},\quad \forall r\in P_k(K). $$

基于此, 定义离散双线性形式

$$ \mathcal{A}(v_h,w_h) = \sum_{K\in\mathcal{K}_h} (\alpha\nabla_w\times v_h, \nabla_w\times w_h)_K + S(v_h,w_h), $$

其中稳定化项 $S(v_h,w_h) = \sum_{K} h_K^{-1} (\langle [v_h]_{T,e}, [w_h]_{T,e}\rangle_{\partial K} + \langle [v_h]_{N,e}, [w_h]_{N,e}\rangle_{\partial K\setminus\partial\Omega})$。该稳定化项仅依赖网格尺寸, 不包含任何需人工调整的惩罚系数, 且由逆不等式可证得强制性: $\mathcal{A}(v_h,v_h) \ge C |||v_h|||^2$。

2. 椭圆投影与最优投影误差
为将连续解的时间依赖性与空间离散分离, 引入 Ritz 投影 $(\mathbf{R}_h z, \mathbf{E}_h q)$ 作为如下椭圆问题的数值解:

$$ \begin{cases} \nabla\times(\alpha\nabla\times z) - \epsilon\nabla q = \mathbf{f}_{z,q},\\ \nabla\cdot(\epsilon z) = \mathbf{g}_{z,q}, \end{cases} $$

其对应的离散格式为 (3.5)。引理 3.3 是该投影逼近性质的核心, 尤其是 $L^2$ 范数 $O(h^{k+\delta})$ 的估计, 证明依赖于对偶论证。考虑对偶问题:

$$ \begin{cases} \nabla\times(\alpha\nabla\times \Psi) - \epsilon\nabla \Phi = \Theta,\\ \nabla\cdot(\epsilon \Psi) = 0,\\ \eta\times\Psi = 0,\; \Phi=0 \text{ on } \partial\Omega, \end{cases} $$

在凸区域假设下, 该问题有正则性提升: $\|\Psi\|_{1+\delta} + \|\Phi\|_{\delta} \le C\|\Theta\|$, 其中 $\delta>0$, 对凸多边形通常 $\delta=1$。利用 Galerkin 正交性, 可得 $L^2$ 投影误差 $\|\mathbf{T}_h z - \mathbf{R}_h z\|$ 被 $h^{\delta}$ 乘以能量范数误差所控制, 从而得到 $O(h^{k+\delta})$ 的阶。这一过程虽经典, 但此处的对偶问题因旋度与梯度的耦合并非标准椭圆系统, 其正则性分析依赖于 [15] 中的结论。

3. 半离散误差分析的 Gronwall 框架
令 $e_h = E_h - \mathbf{R}_h E$, $\xi_h = P_h - \mathbf{E}_h P$, 由原方程 (4.1)-(4.2) 与投影定义相减, 可得误差方程:

$$ (\epsilon e_{htt}, v_h) + \mathcal{A}(e_h, v_h) - \mathcal{B}(v_h, \xi_h) = (\epsilon (\mathbf{R}_h E_{tt} - E_{tt}), v_h), \quad \forall v_h\in V_h^0, $$

$$ \mathcal{B}(e_h, q_h) = 0, \quad \forall q_h\in Q_h^0. $$

选取 $v_h = e_{ht}$, 并利用 $\mathcal{B}(e_{ht},\xi_h) = -\frac{d}{dt}\mathcal{B}(e_h,\xi_h) + \mathcal{B}(e_h,\xi_{ht})$ 及第二个方程, 可得

$$ \frac12 \frac{d}{dt}(\|e_{ht}\|^2 + |||e_h|||^2) = (\epsilon (\mathbf{R}_h E_{tt} - E_{tt}), e_{ht}) + \mathcal{B}(e_h,\xi_{ht}) - \mathcal{B}(e_{ht},\xi_h). $$

为控制含 $\xi_h$ 的项, 需要 inf-sup 条件 (引理 3.2): 对任意 $q_h\in Q_h^0$, 存在 $w_h\in V_h^0$ 满足 $\mathcal{B}(w_h,q_h) = \|q_h\|^2$ 且 $|||w_h|||_1 \le C\|q_h\|$。据此可将 $\|\xi_h\|$ 和 $\|\xi_{ht}\|$ 通过 $|||e_h|||$ 和 $|||e_{ht}|||$ 控制, 最终得到关于 $\|e_{ht}\|^2 + |||e_h|||^2$ 的微分不等式。结合 Gronwall 不等式 (引理 4.4-4.6) 即可累积时间积分, 得到误差界, 再叠加上投影误差 (引理 3.3), 完成定理证明。稳定性分析的思路类似, 但处理的是原始变量 $E_h$, $P_h$ 而非误差。

4. 低正则性 $L^2$ 估计的特殊处理
定理 5.2 中当 $E$ 仅有 $H^{1+l}$ ($l\le k$) 正则性时, 不能直接使用 $k$ 次多项式的最优逼近性质。此时利用修正弱旋度的逼近 (引理 4.3 及其推论) 和 $h$ 范数等价性, 可将投影误差界的幂次调整为 $O(h^{1+l})$, 并结合对偶论证中 $\delta$ 的作用, 仍得到 $O(h^{1+l})$ 的 $L^2$ 收敛率。这为实际计算中解的正则性不足提供了理论保障。

4. 与之前工作的比较

本文与已有二阶时间依赖 Maxwell 方程数值方法相比, 在模型普适性、分析方法及结论上均有显著推进:

1. 模型层面——从隐含 Gauss 定律到显式鞍点系统
此前绝大多数有限元工作 (如 Monk [16], Ciarlet et al. [3], Grote et al. [9,10], Radu & Egger [7], 以及 IPDG 相关研究) 处理的是不含势函数 $P$ 的二阶 Maxwell 方程, 即先验地假定电场满足散度条件 $\nabla\cdot(\epsilon E)=\rho$。这等价于在连续层面就已把 Gauss 定律作为强约束, 离散时若能保证散度自由则自然满足, 否则需通过修正求解。然而, 当介质非均匀、存在界面或需同时计算 $P$ 时, 显式引入 $P$ 的鞍点形式 (1.3) 更为一般和自然。针对该鞍点系统, 文献中仅有零星结果: Assous et al. [1] 提出修正 Taylor-Hood 元却无误差分析; 文献 [2] 用协调元得到电场 $H(\mathrm{curl})$ 最优估计, 但未给出势函数 $P$ 的误差, 也未涉及 $L^2$ 范数; Qi et al. [19] 用弱 Galerkin 方法首次对电场能量半范和 $P$ 的 $L^2$ 范数同时给出最优估计, 但仍缺失电场 $L^2$ 范数结果。本文填补了这一理论空白, 不仅给出能量范数的最优收敛, 还首次获得了电场 $L^2$ 范数的最优误差界, 且在同一框架下自然地给出了 $P$ 的 $L^2$ 估计, 使得整套误差分析完备。

2. 离散方法——无参数 dG vs. 传统惩罚 dG 与弱 Galerkin
间断有限元在电磁计算中颇受欢迎, 但经典 IPDG 必须引入形式如 $\frac{\gamma}{h_e}\int_e [u]^2$ 的惩罚项来强制弱连续性, 系数 $\gamma$ 需大于某问题相关的阈值方可保证稳定性。$\gamma$ 若取得太小会导致不稳定, 太大会恶化条件数并影响精度, 最优值的确定常依赖经验或本征值估计。本文采用的无参数 dG 方法仅包含 $h^{-1}_K$ 缩放的跳跃项, 该系数完全由网格决定, 不再作为可调参数存在。这一设计源自弱梯度/弱旋度概念, 与弱 Galerkin (WG) 方法及杂交高阶 (HHO) 方法属同一技术流派。但与 Qi et al. [19] 的 WG-FEM 相比, 本文的弱旋度 $\nabla_w\times$ 定义在 $[P_{k-1}]^{2d-3}$ 空间, 稳定化只含边界跳量, 无需 WG 中常用的额外稳定子; 且本文的误差分析通过 Ritz 投影统一处理了电场与势, 证明路径更为紧凑。相比 IPDG 的结果 (如 Grote et al. 的 $L^2$ 估计), 本文格式不仅去除了参数依赖性, 还将其推广至鞍点系统, 并给出了同样最优的阶。

3. $L^2$ 误差估计与正则性要求
在 Monk 对标准二阶 Maxwell 的分析中, 要得到 $L^2$ 最优估计, 必须使用第二类 Nédélec 元, 这限制了有限元空间的选择; Grote et al. 的 IPDG $L^2$ 分析也依赖对偶论证, 但他们的结论要求 $E$ 具有 $H^{1+k}$ 正则性。而本文在鞍点系统下, 不仅对高正则情形证明了 $O(h^{k+\delta})$ (凸域上 $\delta=1$, 即 $O(h^{k+1})$), 还进一步证明: 当电场仅具有 $H^{1+l}$ ($0

4. 分析技术——椭圆投影与对偶论证的耦合
本文的核心分析框架是将时间依赖问题视为椭圆投影的扰动, 这本身并不新奇, 但在此系统中, 由于势函数 $P$ 的引入, 投影对 $(\mathbf{R}_h,\mathbf{E}_h)$ 需同时逼近电场与势。对偶论证中对偶问题为旋度-梯度耦合系统, 其正则性提升涉及到 Maxwell 算子的性质, 在凸区域的假设下方可获得 $\delta$ 指数。这一技术路线在弱 Galerkin 文献中有迹可循, 但本文成功将其推进到 $L^2$ 范数, 并通过细致的 Gronwall 估计处理了二阶时间导数项带来的额外复杂性。相较于 Qi et al. 的工作, 他们用能量半范估计时避开了 $L^2$ 对偶论证, 而本文选择直面该难点并取得成功。

5. 局限与展望
本文假设区域 $\Omega$ 为凸多面体, 介质均匀 ($\alpha,\epsilon$ 常数), 边界为纯 PEC。对于非凸区域 (如凹角、裂缝), $\delta$ 可能小于 1, $L^2$ 收敛阶将退化; 文中未给出非凸情形下的显式界。此外, 全离散格式的理论分析在提供的页数中未见, 可能仅靠数值实验验证。实际电磁问题中常涉及吸收边界条件 (如 PML)、非齐次介质与曲线界面, 本文方法能否直接拓展尚待研究。尽管如此, 作为首个对二阶鞍点 Maxwell 系统给出 $L^2$ 最优误差的工作, 其无参数 dG 方案和投影分析技术为后续发展 (如自适应网格、时域有限元等) 提供了坚实的基础。

3. 变阶 $hp$ 有限元界面问题的一致 Schwarz 预条件子

Uniform Schwarz Preconditioners for Variable-Degree $hp$ Finite Element Interface Problems

作者: Situan Li, Weiying Zheng

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03141v1

摘要翻译: 我们为变阶 $hp$ 有限元离散的反应扩散问题和贴合界面问题构造了关于 $h$ 和 $p$ 鲁棒的、保次数的空间分解和加性 Schwarz 预条件子。在任意维数的协调单纯形网格上，单区域结果允许任意的单元次数分布，仅要求 $p_K\ge1$。引入了一种极小平均 Falk–Winther 气泡变换，通过对每个子单纯形取固定相邻单元上的平均，该相邻单元具有极小多项式次数。所得分量保持在指定的变阶空间中，并满足 $L^2$ 和 $H^1$ 稳定估计，其中的常数与网格尺寸、多项式次数以及次数在单元间的变化方式无关。与一个稳定的连续分片仿射分量一起，这给出了一个适用于系数局部可比的单区域反应扩散问题的 $hp$ 一致 Schwarz 预条件子。

Comment: 32 pages, 3 figures, 5 tables

链接: http://arxiv.org/abs/2606.03141v1

1. 研究对象和背景

本文研究变阶次 $hp$ 有限元离散下反应扩散问题和拟合界面问题的均匀 Schwarz 预条件子构造与分析。考虑以下两类问题：

(1) 单域反应扩散问题：设 $\Omega\subset\mathbb{R}^d$ 为有界多角域，系数 $\alpha\in L^\infty(\Omega)$ 满足 $\alpha>0$， $\sigma\in L^\infty(\Omega)$ 满足 $\sigma\ge0$。在 Dirichlet 边界 $\Gamma_D\subset\partial\Omega$ 上施加齐次条件。定义连续双线性形式

$$a(u,v)=\int_\Omega (\alpha\nabla u\cdot\nabla v+\sigma uv)\,dx,\qquad u,v\in H^1_D(\Omega).$$

采用变阶次 $hp$ 有限元空间

$$V_p(\mathcal{T}_h)=W_p(\mathcal{T}_h)\cap H^1_D(\Omega),\quad W_p(\mathcal{T}_h)=\{v\in H^1(\Omega):v|_K\in P_{p_K}(K),\,K\in\mathcal{T}_h\},$$

其中 $\mathcal{T}_h$ 是形状正则的共形单纯形网格，多项式阶次向量 $p=(p_K)_{K\in\mathcal{T}_h}$ 仅需满足 $p_K\ge1$，允许相邻单元阶次任意变化。

(2) 拟合界面问题：在三维情形，考虑分片常系数扩散方程，界面 $\Gamma$ 为分片平面，网格 $\mathcal{T}_h$ 与界面匹配。有限元空间在界面两侧允许不连续，界面条件通过对称 Nitsche 方法弱施加。离散双线性形式为

$$a_{\Gamma}(u,v)=\int_{\Omega\setminus\Gamma}\alpha\nabla u\cdot\nabla v\,dx-\int_{\Gamma}\{\!\!\{\alpha\nabla u\}\!\!\}\cdot\![\!\\!v\]\,ds-\int_{\Gamma}\{\!\!\{\alpha\nabla v\}\!\!\}\cdot\![\!\[u\]\]\,ds+\int_{\Gamma}\frac{\gamma}{h}\alpha\![\!\[u\]\]\cdot\![\!\[v\]\]\,ds,$$

其中 $\![\!\[\cdot\]\]$ 表示跳量，$\{\!\!\{\cdot\}\!\!\}$ 为加权平均，$\gamma>0$ 为惩罚参数。对应有限元空间为

$$V_p^\Gamma(\mathcal{T}_h)=\{v\in L^2(\Omega):v|_K\in P_{p_K}(K)\,\forall K\in\mathcal{T}_h,\,v|_{\Gamma_D}=0\},$$

但未显式增加跨界面连续性。

这些问题的高阶离散产生病态刚度矩阵，条件数以 $O(h^{-2}p^4)$ 增长，且系数对比 $\alpha_{\max}/\alpha_{\min}$ 进一步恶化。构造不依赖 $h$、$p$ 以及系数对比的预条件子是计算科学的核心课题。

经典 $p$-和 $hp$-Schwarz 预条件子（如 [1–4,8,12,13]）针对固定阶次或满足一定阶次相容条件的 $hp$ 空间。其核心是 Falk–Winther 标量 bubble 变换 [7]，它通过宏单元平均将函数分解为局部 bubble 分量，具有 $L^2$ 和 $H^1$ 稳定性。然而，当单元阶次可变时，直接使用该变换会破坏阶次相容：宏单元平均涉及相邻单元的高阶迹，导致低阶单元上产生高于容许阶次的多项式，输出不再属于原变阶次空间。因此，现有预条件子要么要求全局一致阶次，要么引入阶次相容性条件。

本文试图填补的空白是：在完全不限制单元阶次分布（仅 $p_K\ge1$）的变阶次 $hp$ 空间中，构造保持空间相容的 Schwarz 分解，并获得对 $h$、$p$ 及其分布均均匀的预条件子；同时，将此框架推广至系数对比任意大的拟合界面问题，且界面单元只需满足一个公共阶次条件。

2. 主要定理或主要结果

本文的主要理论结果分为三部分：

1. 最小平均变阶次 bubble 变换 (定理2.13)

设 $\mathcal{T}$ 为 $\mathbb{R}^n$ 中形状正则的共形单纯形网格，$p=(p_K)$ 满足 $p_K\ge1$。对每个真子单形 $f\in\Delta_m(\mathcal{T})$ ($m $$p_f = \min_{K\in\mathcal{T}_f} p_K,$$

并固定一个拥有者单元 $K_f^\star\in\mathcal{T}_f$ 使得 $p_{K_f^\star}=p_f$。利用该单元定义局部平均

$$A_f^\star v(\lambda) = -\!\!\!\int_{K_f^\star} v(G_f(\lambda,y))\,dy,\quad G_f(\lambda,y)=\sum_{i=0}^m\lambda_i x_i + b(\lambda)y,$$

其中 $\lambda\in S_m^c$，$b(\lambda)=1-\sum_i\lambda_i$。经 cutoff 修正

$$C_f^\star v = K_m A_f^\star v, \quad K_m w(\lambda)=\sum_{I\in\mathcal{I}_m} (-1)^{|I|}\frac{b(\lambda)}{b(P_I\lambda)} w(P_I\lambda),$$

递归地得到 bubble 分量：$u_0=u$，对 $m=0,\ldots,n-1$，

$$B_f^\star u = C_f^\star u_m\;(f\in\Delta_m),\quad u_{m+1}=u_m-\sum_{f\in\Delta_m} B_f^\star u,$$

顶部单元 $K$ 上 $B_K^\star u = u_n|_K$。结论： (i) $u=\sum_{f\in\Delta(\mathcal{T})} B_f^\star u$ 在 $L^2$ 中成立，$\operatorname{supp} B_f^\star u\subset\Omega_f$。 (ii) 对任意 $u\in L^2(\Omega)$，有 $L^2$ 稳定性：

$$\sum_{f\in\Delta(\mathcal{T})} \|B_f^\star u\|_{L^2(\Omega_f)}^2 \le C \|u\|_{L^2(\Omega)}^2.$$

(iii) 对任意 $u\in H^1(\Omega)$，有 $B_f^\star u\in \mathring{H}^1(\Omega_f)$ 且 $H^1$ 稳定性：

$$\sum_{f\in\Delta(\mathcal{T})} \|B_f^\star u\|_{H^1(\Omega_f)}^2 \le C \|u\|_{H^1(\Omega)}^2.$$

(iv) 若 $u\in W_p(\mathcal{T})$，则 $B_f^\star u\in \mathring{W}_p(\mathcal{T}_f)$，即保持变阶次空间。所有常数 $C$ 仅依赖于网格几何和维数，与 $h$、$p$、阶次分布及系数无关。

2. 单域反应扩散预条件子 (定理3.2)

在 $\Omega\subset\mathbb{R}^d$ 上，系数满足顶点片局部可比条件：对每个顶点 $a$，存在常数 $C_\alpha,C_\sigma$ 使得

$$\operatorname{esssup}_{\Omega_a}\alpha \le C_\alpha \operatorname{essinf}_{\Omega_a}\alpha,\quad \operatorname{esssup}_{\Omega_a}\sigma \le C_\sigma \operatorname{essinf}_{\Omega_a}\sigma.$$

取 $V_0$ 为连续分片线性有限元空间，$V_f=\mathring{W}_p(\mathcal{T}_h^f)\cap V_p(\mathcal{T}_h)$。则任意 $u\in V_p(\mathcal{T}_h)$ 可分解为

$$u = u_0 + \sum_{f\in\Delta(\mathcal{T}_h)} B_f^\star w, \quad w=u-T_1 u,$$

其中 $T_1:V_p\to V_0$ 为稳定的低阶提取算子。该分解满足能量稳定性：

$$\|u_0\|_A^2 + \sum_{f\in\Delta(\mathcal{T}_h)} \|B_f^\star w\|_A^2 \le C \|u\|_A^2,$$

$C$ 与 $h$、$p$、阶次分布以及系数幅值无关。据此构造的加法 Schwarz 预条件子 $B^{-1}=\sum_{f} R_f^* A_f^{-1} R_f$ (其中局部求解器 $A_f$ 为 $a(\cdot,\cdot)$ 限制在 $V_f$) 满足 $\kappa(B^{-1}A)\le C$。

3. 三维拟合界面预条件子 (定理6.5)

在拟合界面网格上，假设接触界面 $\Gamma$ 的所有单元具有公共阶次 $p_\Gamma\ge1$，但远离界面的单元阶次可为任意 $p_K\ge1$。界面 Schur 补问题不直接求解，而是先对界面跳量进行迹升力（利用 [11] 的 $p$-鲁棒升力），升力方向由惩罚项中系数加权决定（即选择高扩散侧）。将符合性余量用加权单侧 bubble 变换分解，其中界面宏单元的拥有者单元选在高扩散侧，以避免系数对比影响稳定性。最终的顶点片预条件子 (vertex-patch preconditioner) 的条件数满足

$$\kappa(B_\Gamma^{-1}A_\Gamma) \le C,$$

常数 $C$ 与 $h$、$p_\Gamma$、远离界面的局部阶次、扩散系数对比 $\alpha_{\max}/\alpha_{\min}$ 以及系数绝对大小均无关。数值实验验证了理论，并暗示公共阶次假设可能进一步放宽。

3. 研究方法、关键技术和核心工具

本文的技术核心在于改造经典的 Falk–Winther bubble 变换，使其在变阶次 $hp$ 空间中保持相容性并维持稳定性。主要障碍在于：原变换中的宏单元平均 $A_f v$ 涉及相邻所有单元的多项式迹的线性组合，当单元阶次不同时，在低阶单元上的限制可能超过该单元的容许阶次。作者的创新点在于引入“最小平均”：对每个子单形 $f$，选取一个具有最小阶次 $p_f$ 的相邻单元 $K_f^\star$，仅用该单元上的迹信息定义平均算子 $A_f^\star$。此举确保 $A_f^\star v$ 的多项式阶次不超过 $p_f$，结合 cutoff 引理 $K_m$ 的多项式保持性质，最终输出阶次不超过相邻单元中最小者，从而与变阶次空间相容。

具体证明框架如下：

1. 单元素估计的迁移 原 Fourier–Winther 变换的分析基础是一系列单元素估计（命题2.4）。这些估计最初针对单元素平均算子 $A_{f,K}$ 建立，随后通过求和推广到宏单元平均。注意到 $A_f^\star = A_{f,K_f^\star}$，因此所有单元素估计自动适用于新的最小平均，无需重新证明。这使 $L^2$ 和 $H^1$ 稳定性的证明直接继承 [7] 的论证，仅需将原宏单元平均替换为最小平均即可。

2. 递归余量估计的重用 稳定性证明的核心在于递归构造中余量 $u_m$ 的加权估计（引理2.9）。关键步骤是证明若 $u_m$ 在已处理低维子单形上满足加权 $L^2$ 衰减，则 $C_f^\star u_m$ 不仅具有 $H^1$ 稳定性 (引理2.8)，且余量 $u_{m+1}$ 在下一维子单形上具有类似的加权衰减。这依赖于 (i) Hardy 型估计 (2.4) 控制 $\rho_f^{-1}(u_m-A_f^\star u_m)$；(ii) 加权估计 (2.5) 控制 $\rho_f^{-1}A_f^\star u_m(P_I\lambda_f)$; (iii) 插值项 $C_g^\star u_m (g\ne f)$ 通过顶点坐标函数 $\lambda_i$ 的权重控制。引理2.8和引理2.9的证明与原文几乎一致，因所用估计仅依赖单元素结果。

3. 有限元空间保持性的归纳证明 (定理2.11) 这是全新内容。归纳假设：已知 $u_m\in W_p$ 且在 $j

4. 界面问题的处理 界面问题引入两个额外机制：

迹升力与加权选择：界面跳量 $\![\!\\!u\]\$ 在界面上分解后，利用 [11] 的 $p$-鲁棒不连续升力算子在体单元内延拓。升力方向由惩罚项中的加权平均选择，即若某侧扩散系数大则优先向该侧延拓，以避免小系数侧承受大梯度惩罚。
加权单侧 bubble 变换：对符合性余量，修改最小平均的标准：界面宏单元（包含界面片的宏单元）的拥有者强制选在高扩散侧。这样，在加权 $H^1$ 范数估计中，系数对比不会出现在局部逆不等式常数中，从而实现对比鲁棒性。

5. 顶点片预条件子的组装 将底层的 bubble 分量按最小维子单形的顶点分组，形成顶点片子空间。由于 bubble 分量原本就支撑在子单形的宏单元上，按顶点分组后每个顶点片子空间仅包含与该顶点关联的局部 bubble，重叠有限且形状正则，从而得到实用的加法 Schwarz 预条件子。

4. 与之前工作的比较

本文与前人工作的关键差异和推进主要体现在以下几个方面：

1. 对变阶次 $hp$ 空间的相容性处理 经典 Falk–Winther bubble 变换 [7] 及基于它的 Schwarz 预条件子 (如 [8,12]) 适用于固定阶次或满足“阶次兼容” 条件的 $hp$ 空间 (例如相邻单元阶次之比有界)。这是因为宏单元平均 $A_f v$ 在单形 $S_m^c$ 上为 $\max_{K\supset f} p_K$ 次多项式，通过 cutoff 回代到低阶单元时可能产生超出容许阶次的多项式。本文通过最小平均 $A_f^\star$ 从根本上解决了这一问题：输出阶次被限定在 $p_f$，从而首次实现了对任意阶次分布（仅要求 $p_K\ge1$）的严格保持空间相容的 bubble 变换。这在 $hp$ 自适应与局部阶次剧烈变化的场景中至关重要。

2. 对网格尺寸 $h$、阶次 $p$ 及其分布的完全均匀稳定性 定理2.13 的稳定常数仅依赖网格几何，与 $h$、$p$ 以及 $p$ 在单元间的变化方式完全无关。这优于多数现有结果：即使对于固定阶次空间，一些 $p$-乘法型预条件子也只能提供关于 $p$ 的弱依赖（如对数增长），而本文的加法 Schwarz 框架实现了对 $p$ 的完全鲁棒。此外，对阶次分布的独立性意味着无需任何辅助的阶次平滑过渡假设（例如 [4] 中的 grading 条件），极大扩展了适用范围。

3. 界面问题的对比鲁棒性 在拟合界面问题方面，已有工作 (如 [5,9]) 通常针对低阶有限元或小对比系数设计。对于 $hp$ 离散和大系数对比，常见的对策是引入对角缩放 (diagonal scaling) 或基于子区域分解的对齐预条件子，但其条件数往往随系数对比增长。本文通过将跳量升力导向高系数侧，并在 bubble 分解中显式加权选出优势侧，实现了条件数与扩散系数对比 $\alpha_{\max}/\alpha_{\min}$ 和系数绝对大小的完全独立。这一对比鲁棒性即使在固定阶次空间中也具有新颖性，在现有 $hp$-Nitsche 文献中未见报道。

4. 公共阶次假设的必要性与潜在扩展 界面问题的主要理论结果（定理6.5）要求界面接触单元具有公共阶次 $p_\Gamma$。这一假设出现在何处？原因在于，界面跳量的迹升力依赖于界面上多项式的阶次，若两侧阶次不同，升力算子的 $p$-鲁棒性结论可能不成立。但在数值实验中，作者测试了不满足该假设的情形，观察到条件数仍保持均匀有界。文中明确说明 “Numerical experiments … suggest robustness beyond the common-degree assumption.” 这表明该假设可能仅源于证明技术限制，而非方法本身的本质缺陷；作者也暗示这为未来工作留出了空间。与假设更严的界面预条件器相比，本文至少对界面处局部一致阶次的情形提供了严谨证明，且远离界面的阶次完全自由。

5. 对 Falk–Winther 变换框架的继承与精简 本文的最小平均改造保持了原始变换的所有稳定性估计，且证明几乎可以直接迁移。这归因于将最小平均表达为单元素平均的特例。因此，作者无需重复复杂的不等式推导，仅需关注阶次保持性这一新元素。这种“最小改动”策略既确证了新构造的可靠性，又避免了冗长的新分析。与其他试图通过约束投影或局部插值实现变阶次相容的工作相比（例如某些基于分层基或局部低阶提取的方法），本文的 bubble 分解直接提供了自然的 $hp$-鲁棒正交分解，更利于构造预处理子。

中文标题: 反应扩散问题在各项异性网格上的嵌入式特雷夫茨间断伽辽金方法

摘要翻译: 我们提出并分析了一种用于各项异性网格上反应扩散问题的嵌入式特雷夫茨间断伽辽金方法。该方法通过嵌入到张量积DG空间来施加松弛的局部特雷夫茨条件，从而在保持底层高阶离散逼近性质的同时得到约化的整体系统。我们证明了该方法在各向异性且可能为曲边四边形单元上的稳定性和拟最优性，并推导了各向异性的先验误差估计。$h$-和$hp$-细化的数值实验，包括曲边区域示例，验证了理论结果。

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链接: http://arxiv.org/abs/2606.05131v1

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链接: http://arxiv.org/abs/2606.05088v1

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链接: http://arxiv.org/abs/2606.04942v1

摘要: The escalating accumulation of orbital debris poses a critical threat to future space operations. Space-based lasers leveraging laser ablation have emerged as a promising approach for mitigating debris proliferation and preserving the orbital environment. Current literature, however, treats space-based laser debris remediation as a deterministic problem, assuming that momentum transfer and the resulting debris perturbations are precisely known. In reality, laser-to-debris engagement outcomes are inherently stochastic due to partially known debris characteristics. Compounding this challenge, estimating critical laser-matter parameters in situ, such as the momentum coupling coefficient, requires ablation that consequently perturbs the debris trajectory. This establishes a coupled ablation-and-estimation problem in which the laser platform and target debris encounter geometry influences remediation effectiveness and estimation accuracy. To address this problem, we present a joint ablation-and-estimation methodology that provides insights into the driving factors that make different encounter geometries improve or degrade overall remediation and estimation performance. Results across multiple coplanar and out-of-plane encounter geometries demonstrate how periapsis-lowering capacity, linear system observability, and nonlinear estimation performance evolve as laser parameters and relative orbit geometry vary. By identifying the key drivers behind these metrics, this study highlights critical considerations for the safe and effective operation of space-based lasers under uncertainty.

4. 地震波传播时域有限差分格式的任意高阶成形模板

Arbitrary high order shaped stencils for time domain finite difference schemes in seismic wave propagation

作者: Pedro S. Peixto, Felipe A. G. Silva

类别: math.NA

发布日期: 2026-06-03

arXiv ID: 2606.04882v1

链接: http://arxiv.org/abs/2606.04882v1

摘要: Finite Difference Schemes are widely used in the approximation of different hyperbolic (wave-like) differential equations, and are particularly important for seismic wave modelling and its applications. Classical methods based on Taylor Series are dominant in the literature; however, it is known that these methods can suffer from excessive numerical dispersion. In this paper, we review and extend existing high-order in space finite difference schemes for acoustic wave propagation, featuring different stencil geometries ranging from classical cross stencils to stencils with rhombus or square-like shapes, and propose a general mathematical framework for their derivation. The numerical implementation is performed in a symbolic, high-level framework (Devito), which compiles and runs highly optimized, stencil-based computations, allowing for a low-level interpretation of the methods efficiency. We demonstrate that non-cross stencil shapes, such as rhombus and square-based stencils, do not necessarily provide added accuracy or dispersion reduction in general, despite their increased computational cost. However, results on both idealized and realistic velocity models confirm the benefits of using dispersion-optimized cross-stencils, indicating adequate accuracy with reduced computational cost on more compact stencils compared to classic approaches. Finally, our implementation of the methods provides ease of use for full-scale acoustic seismic inversion problems using Devito.

5. 在GPU上拟合带有可选单调性约束的散乱数据：LipFit包

Fitting scattered data with optional monotonicity constraints on GPU: LipFit package

作者: Gleb Beliakov

类别: math.NA, cs.LG, cs.MS

发布日期: 2026-06-03

arXiv ID: 2606.04670v1

链接: http://arxiv.org/abs/2606.04670v1

摘要: This paper presents a method of multivariate scattered data interpolation and approximation that produces optimal Lipschitz-continuous approximation, subject to the desired monotonicity constraints. This method relies on tight upper and lower approximations to the data, and is similar in its spirit to the nearest-neighbour approximation but does not suffer from discontinuities. Local Lipschitz interpolation and Lipschitz smoothing are also presented. This approach falls under the umbrella of instance-based approximation with no training phase, and it is suitable for GPU-based parallelisation. A Python GPU-friendly package LipFit which implements the methods discussed is discussed.

6. 强磁场下带电粒子动力学的一类新型高阶一致精确局部线性展开指数积分器

A novel class of high-order uniformly accurate exponential integrators with local linear extension for the charged-particle dynamics under strong magnetic field

作者: Lina Wang, Bin Wang, Beibei Zhu

类别: math.NA

发布日期: 2026-06-03

arXiv ID: 2606.04478v1

链接: http://arxiv.org/abs/2606.04478v1

摘要: In this paper, we develop a novel class of high-order uniformly accurate exponential integrators for charged-particle dynamics under a strong magnetic field. The small parameter $0<\varepsilon\ll 1$ induces rapid temporal oscillations, rendering traditional numerical methods prohibitively expensive due to severe step-size restrictions. To address this issue, a linearization technology that introduces auxiliary polynomial variables is employed to recast the original charged-particle dynamics as a higher-dimensional system. Classical exponential integrators are subsequently applied to this augmented formulation, which inherently carries richer structural information, thereby yielding a family of uniformly accurate exponential integrators that can reach arbitrarily high order without requiring any order conditions. For the maximal ordering scaling strong magnetic field, we rigorously demonstrate via algebraic techniques that the proposed schemes with auxiliary polynomial variables of degree $k(k\geq 2)$ achieve an $\mathcal{O}(\varepsilon h^{k+1})$ improved error estimate for the position and a uniform $\mathcal{O}(h^{k+1})$ error estimate for the velocity. Numerical experiments validate the advantages of the methods. The theoretical and numerical in vestigation is finally extended to relativistic charged-particle dynamics in a four-dimensional framework with maximal ordering scaling strong magnetic field.

7. 求解定常MHD问题的解耦迭代格式

Decoupled iterative schemes for solving stationary MHD problems

作者: Aziz Takhirov, Driss Yakoubi

类别: math.NA

发布日期: 2026-06-03

arXiv ID: 2606.04467v1

链接: http://arxiv.org/abs/2606.04467v1

摘要: We develop a novel iterative approach for solving the incompressible magnetohydrodynamics problem. The main idea is to split the velocity-momentum and magnetic induction equations with respect to the diffusive terms. As a result, we get a smaller system that is iteration-level-dependent, along with two Stokes systems that need to be assembled only once. We also extended the scheme to the Els{ä}sser variables reformulation of the equations. For both schemes, we established boundedness and convergence. Several numerical experiments are presented to show the effectiveness of the schemes.

8. MeshTok: 面向可扩展偏微分方程Transformer的高效多尺度令牌化

MeshTok: Efficient Multi-Scale Tokenization for Scalable PDE Transformers

作者: Yanshun Zhao, Xiaoyu Peng, Jiamin Jiang, Congcong Zhu, Jingrun Chen

类别: cs.LG, math.NA

发布日期: 2026-06-03

arXiv ID: 2606.04366v1

链接: http://arxiv.org/abs/2606.04366v1

摘要: Conventional patchified Transformers operate on uniform spatial partitions, distributing computational effort evenly across the domain irrespective of local features. This inflexible tokenization scheme is inherently limited in its ability to efficiently represent and process solutions to complex PDEs. To address this, we propose MeshTok, an adaptive mesh refinement (AMR)-inspired tokenization and sequence modeling framework. This method selectively refines spatial regions exhibiting sharp gradients, transient features, or multiscale structures, generating a heterogeneous set of multiscale tokens defined on a fixed simulation grid. These tokens are processed within a unified Transformer sequence, enabling the model to simultaneously capture coarse-grained global context and fine-grained local details without requiring specialized architectural components. Although adaptive refinement moderately increases token count, it promotes a more targeted allocation of computational resources to physically informative regions, which we view as a practical inductive bias rather than a formal optimality guarantee. Experimental evaluations across multiple PDE families and benchmark datasets demonstrate that MeshTok consistently improves the efficiency-accuracy trade-off compared to uniform-grid baselines. This suggests adaptive multiscale tokenization as a scalable and generalizable design principle for neural PDE modeling. Code is available at https://github.com/SCAILab-USTC/MeshTok.

9. 一种用于谱鲁棒半定规划的扰动q-Tsallis自协调障碍函数

A Perturbed q-Tsallis Self-Concordant Barrier for Spectrally Robust Semidefinite Programming

作者: Sergio Assuncao Monteiro, Fabricio Alves Barbosa da Silva

类别: math.OC, math.NA

发布日期: 2026-06-03

arXiv ID: 2606.04348v1

链接: http://arxiv.org/abs/2606.04348v1

摘要: We introduce and analyse a perturbed $q$-Tsallis barrier for semidefinite programming (SDP), defined as a spectral perturbation of the classical log-det barrier on the cone of positive definite matrices. The barrier introduces eigenvalue-adaptive stiffening through a Tsallis-type matrix-power term controlled by parameters $q>1$ and $η\geq0$. Our main theoretical contribution is a sharp characterisation of the differential self-concordance regime of the barrier. We prove that the barrier is differentially self-concordant on the interior of the positive semidefinite cone for all $η\geq0$ if and only if $q\in(1,2]$, establishing the exact threshold at $q=2$. For $q>2$, uniform self-concordance fails globally, although local sufficient conditions remain valid on compact spectral domains. On compact feasible sets, the effective local barrier parameter remains $O(n)$, preserving the same asymptotic iteration complexity class as the classical log-det barrier. We further establish a spectral robustness result showing that the sensitivity of the central path to perturbations is selectively damped in small-eigenvalue directions according to the scaling $κ(X^*)^{-(q-1)}$, where $κ(X^*)$ denotes the spectral condition number. This yields improved robustness relative to the classical log-det barrier for ill-conditioned SDP solutions. Finally, we develop a Mehrotra-type primal–dual predictor–corrector interior-point method equipped with a Lanczos-based Krylov kernel for evaluating matrix powers efficiently. Numerical experiments validate the theoretical predictions and demonstrate improved robustness together with significant computational acceleration.

10. 多组分膜相场模型的指数时间差分格式

Exponential Time Differencing Schemes for a Phase-Field Model of Multicomponent Membranes

作者: Wangbo Luo, Zhonghua Qiao, Yanxiang Zhao

类别: math.NA, math-ph, nlin.PS

发布日期: 2026-06-03

arXiv ID: 2606.04347v1

链接: http://arxiv.org/abs/2606.04347v1

摘要: In this paper, we develop and analyze exponential time differencing (ETD) schemes for a phase-field model of multicomponent membranes proposed in our previous work \cite{luo2025ohta}, in which membrane deformation is governed by a force-balance phase-field equation and protein segregation is described by a membrane-associated Ohta-Kawasaki (OK) dynamics. For a fixed phase-field membrane, we introduce a geometry-adapted operator splitting method based on the localization function, which reformulates the surface OK dynamics into a form suitable for ETD integration. The resulting first- and second-order ETD schemes, combined with finite-difference spatial discretization, are rigorously proved to satisfy a discrete maximum-bound principle and unconditional energy stability. For the coupled system, we construct stabilized ETD schemes in an FFT-based spectral framework, treating stiff linear terms exactly and nonlinear mechanochemical couplings explicitly. A narrow-band implementation further reduces the computational cost by restricting surface calculations to the diffuse membrane region. Numerical experiments confirm the predicted temporal accuracy, maximum-bound preservation, and energy decay for the fixed-membrane OK problem, and demonstrate stable and efficient three-dimensional simulations of protein-driven pattern formation and membrane deformation.

11. 多孔介质中变饱和流Richards方程的高效求解器

An Efficient Solver for the Richards Equation for Variably Saturated Flows in Porous Media

作者: Xuelong Gu, Qi Wang

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.04292v1

链接: http://arxiv.org/abs/2606.04292v1

摘要: We present a nonlinear multigrid solver for the Richards equation in variably saturated porous media with strongly nonlinear hydraulic conductivity and water-retention relationships. The governing equation is discretized using a second-order conservative finite-difference scheme in space and an implicit backward differentiation formula in time. The core component of the solver is a nonlinear Gauss–Seidel (NGS) smoother based on a triangular splitting of the diffusion operator combined with diagonal stabilization. This construction yields a sequence of locally decoupled scalar nonlinear problems that can be solved efficiently and robustly using only a few Newton iterations. Under suitable monotonicity assumptions, we establish the convergence of the NGS iteration in the $L^\infty$ norm and derive explicit conditions on the stabilization parameters. Numerical experiments for benchmark infiltration, drainage, and root-uptake problems demonstrate that the proposed NGS-based multigrid framework is both computationally efficient and robust.

12. Fisher–von Mises–Langevin 分布中平方浓度的无偏估计与浓度无偏估计的不可能性

Unbiased estimation of squared concentration in the Fisher-von Mises-Langevin distribution and the impossibility of unbiased concentration

作者: Zain Jabbar, Yuqin Jiang, Andrey A. Popov

类别: math.ST, math.NA

发布日期: 2026-06-02

arXiv ID: 2606.04267v1

链接: http://arxiv.org/abs/2606.04267v1

摘要: The estimation of concentration parameter in Fisher-von Mises-Langevin distribution is the directional statistics analogue of the estimation of the precision matrix for the Gaussian distribution. In this work we show that unbiased estimation of this parameter is impossible. With this realization in hand, we provide an alternative parameterization of the Fisher-von Mises-Langevin distribution in terms of the squared concentration, which we term the intensity. We fruther show that unbiased estimation of thereof is possible, and provide (almost) unbiased estimators thereof in terms of a partial sum U-statistic. We showcase our new estimator on synthetic data, New York taxi trip data, and on spherical word embeddings.

13. 具有学习相互作用的非局部平均场薛定谔桥

Nonlocal Mean Field Schrödinger Bridge with Learned Interactions

作者: Daisuke Inoue, Mathieu Laurière, Dante Kalise

类别: math.OC, cs.LG, math.NA

发布日期: 2026-06-02

arXiv ID: 2606.04265v1

链接: http://arxiv.org/abs/2606.04265v1

摘要: The Schrödinger Bridge Problem constructs a stochastic process that connects an initial distribution to a terminal distribution with minimum energy. This work considers its mean-field extension, the Mean-Field Schrödinger Bridge, for interacting particle systems. With nonlocal interactions, evaluating the resulting particle-dependent distributional terms can scale quadratically with the population size, which makes large-scale problems intractable. We address this bottleneck by approximating the nonlocal interactions with neural network surrogates. The resulting four-stage alternating algorithm reduces the per-step cost from quadratic to linear in the population size at inference. We also derive Grönwall-type stability bounds that show how surrogate errors propagate to the generated trajectories. In numerical experiments on navigation and opinion-dynamics tasks, the proposed method reproduces trajectories obtained with analytical evaluation and reduces training time.

14. 一种检测确定Hermitian矩阵对的不定LOBPCG型算法

An indefinite LOBPCG type of algorithm for detecting a definite Hermitian matrix pair

作者: Marija Miloloža Pandur

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.04259v1

链接: http://arxiv.org/abs/2606.04259v1

摘要: A Hermitian matrix pair $(A,B)$ is called definite if some real linear combination of the matrices $A$ and $B$ is a positive definite matrix. Detection of the definiteness is not straightforward. We propose a basic subspace algorithm for detecting a large definite matrix pair $(A,B)$ with indefinite $B$. The proposed subspace algorithm is based on iterative testing of small projected Hermitian matrix pairs formed by using subspaces of small dimensions. Furthermore, we propose a specialized algorithm with parameter $m$, and its preconditioned variant. In the specialized algorithm with $m=3$ we choose the subspaces like in the indefinite locally optimal block preconditioned conjugate gradient (LOBPCG) method. Numerical experiments demonstrate the efficiency of our specialized algorithm, applied on medium-sized pairs, as well as, on large and banded pairs. Our algorithm very quickly detects (in)definiteness; much faster than some other algorithms.

15. Cahn-Hilliard-Navier-Stokes方程的一种能量稳定的隐式凸分裂BDF2格式

An Energy-Stable Implicit Convex-Splitting BDF2 Scheme for the Cahn-Hilliard-Navier-Stokes Equations

作者: Xuelong Gu, Qi Wang

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.04204v1

链接: http://arxiv.org/abs/2606.04204v1

摘要: We develop an energy-stable implicit convex-splitting BDF2 discretization (CS-BDF2) of the Cahn–Hilliard–Navier–Stokes equations. For the Cahn–Hilliard equation, BDF2 analyses can establish energy stability by testing the phase equation in the (H^{-1}) metric. For CHNS, this test is not compatible with the coupled energy estimate: the momentum equation is tested by (\bfu^{n+1}), while the transported phase equation is tested by (μ^{n+1}) so that transport cancels capillary work. The chemical-potential relation must then be paired with the BDF2 phase increment ((3φ^{n+1}-4φ^n+φ^{n-1})/2); its nonlinear part must produce a BDF2 bulk-energy difference, up to nonnegative higher-order history terms. To overcome this difficulty, we introduce a new BDF2-compatible convex-splitting approximation of the nonlinear bulk force that directly yields a discrete bulk-energy identity and enables a discrete energy analysis for the CHNS system. Specifically, we discretize the bulk force (f(φ)=φ^3-φ) by (χ(φ^{\dagger,n+1},φ^{\dagger,n})-φ^{,n+1}), where (χ(a,b)=\tfrac14(a^2+b^2)(a+b)), (φ^{\dagger,n+1}=\tfrac{3φ^{n+1}-φ^n}{2}), (φ^{\dagger,n}=\tfrac{3φ^n-φ^{n-1}}{2}), and (φ^{,n+1}=2φ^n-φ^{n-1}). This discretization is based on the shifted BDF2 identity ((3φ^{n+1}-4φ^n+φ^{n-1})/2=φ^{\dagger,n+1}-φ^{\dagger,n}). With a matching discretization of the reversible coupling terms in CHNS, the scheme is mass conservative, uniquely solvable, and unconditionally energy stable. We prove second-order convergence for the phase variable, chemical potential, velocity, and pressure.

16. 凸函数的采样与重构

Sampling and reconstruction of convex functions

作者: Andrea Bonito, Albert Cohen, Wolfgang Dahmen, Ronald Devore, Guergana Petrova, Jonathan W. Siegel

类别: math.NA, math.FA

发布日期: 2026-06-02

arXiv ID: 2606.04190v1

链接: http://arxiv.org/abs/2606.04190v1

摘要: We discuss optimal recovery for classes of multivariate convex functions from given point samples, as well as the sampling numbers of these classes, corresponding to optimal sample choices. Upper and lower bounds for either variant are established when the reconstruction error is measured in $L_p$ for $1\leq p\leq \infty$. These bounds match, sometimes up to logarithmic factors, and therefore characterize the respective optimal rate of decay. For classical smoothness classes such as Sobolev, Hölder or Besov spaces, it is well known that the optimal decay rate of sampling numbers can be achieved by sampling on uniform tensor product grids and using linear methods of reconstruction, such as piecewise polynomial interpolation. One of the main findings in this paper is that for classes of convex functions, these procedures generally produce suboptimal rates, except when $p=1$ and $p=\infty$, and are outperformed by nonlinear reconstruction methods that do not employ tensor product grids.

17. 基于高斯求积的具有几何误差衰减的短指数和逼近

Approximation by short exponential sums with geometric error decay based on Gauss quadrature

作者: Gerlind Plonka, Yannick Riebe, Annie Cuyt

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03855v1

链接: http://arxiv.org/abs/2606.03855v1

摘要: We present new short exponential sum approximations of length $N$ for $f_1(x)=\frac{1}{a+x}$ with $a>0$ on $[0, \infty)$ and for $f_2(x)= {\mathrm e}^{-x^2/2σ}$ with $σ>0$ on ${\mathbb R}$ with geometric error decay $ρ^{-2N}$ for user-defined $N \ge 2$ and $ρ>1$. The approximations are built over consecutive intervals $[b_j, \, b_{j+1}) \subset [0, \infty)$, $j \in {\mathbb N}_{0}$, with interval lengths that depend on $ρ$ and grow exponentially for $f_1$ and are equidistant for $f_2$. All parameters determining the exponential sum approximations on $[b_j, \, b_{j+1})$ are easily computed from the initial parameters on $[b_0, \, b_{1})$, ensuring numerical stability. Our method is based on Gauss-Laguerre and Gauss-Hermite quadrature, respectively, applied to suitable parametric integral representations of $f_1$ and $f_2$. This technique ensures consistent relative errors across all intervals. Using the obtained exponential sum approximations, we achieve highly accurate approximations of $\log(x)$ on $[1,\infty)$ and of the error function $\mathrm{erf}(x)$ with predictable geometric error decay. Numerical examples for $N=8$ and $N=10$ clearly illustrate the theoretical error estimates.

18. 基于能量的模型的三项递推迭代

Three-term recurrence iterations for energy-based models

作者: R. Altmann, J. Ramme, P. Schulze

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03833v1

链接: http://arxiv.org/abs/2606.03833v1

摘要: It is well-known that the midpoint rule preserves the dissipation inequality if applied to a certain class of energy-based models. We introduce an appropriate scaling of the state variables such that the symmetric part of the resulting iteration matrix is guaranteed to be positive definite. This allows the application of three-term iteration schemes such as the methods of Widlund and Rapoport. Special emphasis is put on examples where the symmetric part is block diagonal such that the computations decouple. This then leads to efficient dissipation-preserving numerical schemes as illustrated in two numerical examples, namely the biharmonic heat equation and linear poroelasticity.

19. 图正则化非负降双四元数矩阵分解用于彩色图像识别

Graph Regularized Non-negative Reduced Biquaternion Matrix Factorization for Color Image Recognition

作者: Hailang Wu, Yonghe Liu, Bingxuan Yu, Chaoqian Li

类别: cs.CV, math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03654v1

链接: http://arxiv.org/abs/2606.03654v1

摘要: Non-negative reduced biquaternion matrix factorization (NRBMF) uses the product of reduced biquaternion (RB) matrices to incorporate the non-negativity constraints of color image pixels into the factorization process. However, NRBMF mainly focuses on reconstruction accuracy and does not exploit the local geometric structure of image data, which may limit the discriminative ability of the learned low-dimensional features. To address this issue, we propose a graph regularized non-negative reduced biquaternion matrix factorization (GNRBMF) model for color image recognition. The proposed model incorporates a graph Laplacian regularizer into the reduced biquaternion coefficient matrix, encouraging nearby samples in the original space to have similar representations in the learned feature space. Meanwhile, GNRBMF retains the non-negativity-preserving property of NRBMF in the reduced biquaternion domain. To solve the optimization problem, a component-wise alternating projected gradient algorithm is derived, and its convergence properties are analyzed. Experimental results demonstrate that the proposed GNRBMF model achieves competitive or superior recognition performance in some tested settings.

20. 多项式代数在非线性方程求解器中的应用

Application of polynomial algebras to non-linear equation solvers

作者: Jordi Canela, Daniel Pérez-Palau

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03651v1

链接: http://arxiv.org/abs/2606.03651v1

摘要: This paper presents a novel application of Jet Transport, a high-order automatic differentiation technique, to enhance classical numerical methods, with a focus on Newton’s method. We prove a central theorem establishing that, under appropriate conditions, applying Jet Transport within a Newton iteration doubles the number of correct coefficients in the Taylor series approximation of the solution. This theoretical result is then extended to the practical case where the exact solution is unknown, demonstrating the expected quadratic convergence (error reduction from ( \varepsilon ) to ( \varepsilon^2 )) while simultaneously doubling the order of accuracy in the series expansion. The efficacy of the resulting Jet-Newton method is demonstrated through three illustrative examples: an academic problem validating the theoretical convergence rates, the solution of Kepler’s equation, and a new continuation algorithm for computing zero-velocity curves in the circular restricted three-body problem. These examples showcase the method’s capability to provide high-order semi-analytical approximations.

21. 关于带不确定性的肿瘤生长模型的多保真度方法

On multi-fidelity methods for a tumor growth model with uncertainties

作者: Huimin Yu, Liu Liu, Yu Feng, Te Qi

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03607v1

链接: http://arxiv.org/abs/2606.03607v1

摘要: We develop a hierarchical multi-fidelity (MF) framework for efficient uncertainty quantification of porous-medium equation (PME) tumor growth models with moving free boundaries. The proposed approach combines coarse-grid PME solvers, level-set approximations of the Hele–Shaw limit, and fine-grid asymptotic-preserving PME discretizations, thereby integrating both discretization-based and asymptotic-model-based fidelity reduction. To guide the selection of high-fidelity samples, we introduce a residual-based farthest-point sampling (RFPS) criterion that combines projection residual information with a distance-based separation term in the low-fidelity snapshot space. Based on this criterion, we construct both bi-fidelity and tri-fidelity approximations, together with empirical error indicators for adaptive refinement. Numerical experiments are conducted in both bi-fidelity and tri-fidelity settings under several uncertainty scenarios, showing that the proposed multi-fidelity approximations achieve accurate results with reduced high-fidelity sampling cost in the reported tests.

22. 一种用于三维各向异性声子晶体能带结构计算的高效奇偶分块方法

An Efficient Parity-Blocked Method for Band-Structure Computation of 3D Anisotropic Phononic Crystals

作者: Jingkai Zhang, Xing-Long Lyu, Tiexiang Li, Wen-Wei Lin

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03599v1

链接: http://arxiv.org/abs/2606.03599v1

摘要: Band-structure calculations for three-dimensional anisotropic phononic crystals require the repeated solution of large elastic generalized eigenvalue problems along Bloch paths. In standard staggered-grid discretizations, anisotropic coupling may involve derivative components located at incompatible grid positions, so additional interpolation or averaging closures are often introduced. This paper proposes a parity-blocked rotated staggered discretization based on four Bloch-periodic body-diagonal differences. The directional derivatives are reconstructed from these diagonal differences, leading to a Hermitian $B_hC_hB_h^H$ generalized eigenvalue formulation that incorporates anisotropic derivative coupling without separate interpolation closures. On even grids, when the stiffness and mass matrices are nodewise local multiplication matrices, the body-diagonal shifts preserve two independent parity invariants. The discrete velocity space is then decomposed exactly into four mutually independent block subspaces, and the full discrete spectrum can be recovered by solving the four smaller eigenvalue problems and merging their spectra. The full and block formulations are further organized in a unified Fourier SVD framework, which supports $Γ$-point zero-mode treatment, shift-invert Krylov iteration, inner PCG solves, and GPU matrix-vector products. Numerical experiments for a three-dimensional two-phase anisotropic phononic crystal show that the block implementation preserves the full-space spectrum while substantially reducing the wall-clock time. The results demonstrate that the proposed method provides a structured and efficient solver for large-scale band-structure computations of three-dimensional anisotropic phononic crystals.

23. 场路耦合问题多速率联合仿真技术比较

A Comparison of Multirate Co-Simulation Techniques for Field-Circuit Coupled Problems

作者: Michael Wiesheu, Sebastian Schöps, Idoia Cortes Garcia

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03594v1

链接: http://arxiv.org/abs/2606.03594v1

摘要: This paper compares three different multirate splitting approaches for the application on field-circuit coupled magnetoquasistatic simulations. For these methods, again three different variants for exchanging values between the field and circuit are tested, namely voltages, currents and flux correction terms. All scenarios are applied on two different benchmark problems, i.e. a coil inductor and transformer model coupled to different circuits. The convergence behavior of different time steppers (Implicit Euler and Trapezoidal Rule) is determined for all possible settings, and guidelines for practical applications are derived.

24. 非自伴不定施图姆-刘维尔问题的左半平面特征值的有界性及其在傅里叶模态方法中的应用

Boundedness of Left Half-Plane Eigenvalues for Non-Selfadjoint Indefinite Sturm–Liouville Problems with Applications to Fourier Modal Methods

作者: Ehsan Faghihifar

类别: math.NA, physics.optics

发布日期: 2026-06-02

arXiv ID: 2606.03537v1

链接: http://arxiv.org/abs/2606.03537v1

摘要: We study a general class of non-selfadjoint indefinite Sturm–Liouville problems of the form

$$ -(p\,y')' + q\,y = λ\, p\, y, $$

on a finite interval with complex-valued coefficients, where $p$ is piecewise in $W^{2,\infty}$, non-vanishing, and satisfies a non-degenerate interface condition, and $q$ is bounded. We prove that all eigenvalues in the open left half-plane are contained in a bounded set, which, by classical Sturm–Liouville theory, implies their finiteness. A prominent instance of this class arises in the lamellar grating diffraction problem with transverse-magnetic (TM) polarization, where $p=ε(x)^{-1}$ is the inverse of a spatially varying permittivity profile. Our result provides a simple and rigorous criterion for identifying non-physical spurious modes in low-loss metallic gratings – a notorious source of instability in Fourier modal methods. Numerical examples illustrate the practical utility of the criterion.

25. 半线性抛物方程Parareal算法的线性收敛性

Linear Convergence of Parareal Algorithm for Semilinear Parabolic Equations

作者: Guanglian Li, Qingle Lin, Shu-lin Wu, Zhi Zhou

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03516v1

链接: http://arxiv.org/abs/2606.03516v1

摘要: Long-time simulations of evolution equations present substantial computational challenges due to the inherently sequential nature of conventional time-stepping schemes. The parareal method, a leading parallel-in-time (PinT) algorithm, offers a promising approach to overcome the challenge by introducing concurrency in the time domain. While its convergence theory is well-established for linear problems, extending the theory to nonlinear problems, particularly when the problem data have only limited regularity, remains a significant challenge. In this work, we provide the convergence analysis of the parareal algorithm for solving semilinear parabolic equations with an $H^2$ initial data. We employ stable rational approximations and first-order linearization as coarse propagators, establish the linear convergence of the parareal algorithm and provide a sharp estimate for the convergence factor. The analysis combines the error-splitting technique from the superlinear convergence analysis of the parareal method, a refined linear convergence theory for linear parabolic equations, and \textsl{a priori} error estimates that are optimal with respect to the regularity of the problem data. The analysis shows the close connection between the convergence behavior of nonlinear models and their linear counterparts. Numerical experiments fully support the theoretical findings.

26. 基于体素的量子计算方法（VBQC）用于固体力学问题

A Voxel-Based Quantum Computing Method (VBQC) for Solid Mechanics Problem

作者: Feng Wu, Yuxiang Yang, Li Zhu, Chen Li, Yansong Guo, Xu Guo

类别: cs.CE, math.NA, quant-ph

发布日期: 2026-06-02

arXiv ID: 2606.03515v1

链接: http://arxiv.org/abs/2606.03515v1

摘要: Quantum computing presents a promising method to overcome the efficiency and memory constraints in large-scale mechanical problems, with numerous successful applications demonstrated in fluid mechanics. However, solid mechanics problems usually require irregular grids for spatial discretization, due to the Lagrange formulations and complex boundaries, which makes the quantum simulation of the system matrix, e.g., the mass or stiffness matrix which is often referred to as the Hamiltonian in quantum computing, difficult to be effectively conducted. This study proposes a voxel-based quantum computing method (VBQC) for the quantum simulation of Hamiltonians in solid mechanics. VBQC applies voxel grids to discretize the spatial domain, thereby enabling the system matrix to exhibit the tridiagonal fractal property. Based on this property, the system matrix can be decomposed into three groups of fundamental matrices, $\mathbf{k}_{n}$, $\mathbf{c}_{n}$, and $\mathbf{q}_{n}$. This decomposition process is referred to as the KCQ decomposition. By integrating the KCQ decomposition with the quantum Fourier transform and the quantum multiplexer, VBQC enables efficient quantum simulation of Hamiltonians in solid mechanics. Three specific solid problems with different dimensions and numbers of variables are applied to preliminarily verify the correctness of the proposed VBQC for solid mechanics problems.

27. 模型误设下算子学习的物理引导修正

Physics-guided correction for operator learning under model misspecification

作者: Lei Ma, Nicolas Boullé, Yu-Sen Yang, Hao Wu, Ling Guo

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03469v1

链接: http://arxiv.org/abs/2606.03469v1

摘要: Physics-informed operator learning provides an efficient framework for approximating solution operators of partial differential equations by combining observational data with governing physical laws. However, most existing methods implicitly assume that the prescribed governing equation is accurate. This assumption may fail in practical applications, where model simplifications, missing physical effects, parameter drift, or incomplete constitutive relations can lead to model misspecification. In this work, we propose a physics-guided operator correction framework for learning solution operators under misspecified governing equations. At the operator level, the target mapping is decomposed into a prior operator induced by an approximate physical model and a learnable correction operator that accounts for the remaining discrepancy. Although the formulation is architecture independent, we realize it using a serial DeepONet architecture, where the first DeepONet provides a prior prediction and the second DeepONet learns an additive correction conditioned on both the input function and the prior prediction. The learned correction is incorporated into the physics residual and trained together with data-consistency constraints, allowing the model to retain useful physical structure while adapting to inaccurate governing equations. Numerical experiments on diffusion-reaction, Burgers, cavity flow, and hyperelastic problems show that the proposed method substantially reduces errors induced by misspecified physics. Additional tests under sparse and noisy observations further demonstrate the robustness of the framework and its ability to provide informative uncertainty estimates through deep ensembles.

28. 保结构量子线方法用于混合边界条件下演化型偏微分方程

Structure-Preserving Quantum Method of Lines for Evolutionary PDEs with Mixed Boundary Conditions

作者: Yixuan Liang, Jin-Peng Liu

类别: quant-ph, math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03407v1

链接: http://arxiv.org/abs/2606.03407v1

摘要: We give detailed analysis and circuit design of structure-preserving quantum algorithms for second-order linear evolutionary PDEs, including parabolic equations and hyperbolic equations with mixed Dirichlet, Neumann, and periodic boundary conditions and source terms. While prior quantum algorithms usually neglect the stability problem from the PDE-to-ODE reduction, our method-of-lines approach investigates the boundary lifting via Coons interpolation and boundary-aware discretization, so that the resulting semi-discrete systems are stable and compatible with efficient quantum ODE primitives. For the parabolic problem, we use a diagonal similarity transform to ensure the semi-discrete generator must have a positive semi-definite Hermitian part, and then solve the resulting ODE system by the optimal linear combination of Hamiltonian simulation (LCHS). For the hyperbolic problem, we rewrite the semi-discrete equation as an equivalent first-order system and solve it by Hamiltonian simulation. We implement our quantum algorithms with explicit block-encoding constructions and circuit implementations, as well as demonstrating the end-to-end complexity bounds together with spatial and quadrature error estimates. We conduct classical numerical experiments on the convection-diffusion equation, inhomogeneous heat equation, and Klein-Gordon equation to validate our structure-preserving analysis and algorithmic constructions.

29. 有界区域上拟线性偏微分方程的边界兼容相互作用逼近

Boundary-compatible interacting approximations of quasilinear PDEs on bounded domains

作者: Thierry Paul, Emmanuel Trélat

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.04049v1

链接: http://arxiv.org/abs/2606.04049v1

摘要: We develop a general operator-theoretic route that turns Kato-type quasilinear evolution systems on a Banach scale $(Z,X)$ into finite-dimensional interacting approximations. The construction proceeds in two steps. First, one introduces a regularized family $(A_\varepsilon,f_\varepsilon)$ indexed by a scale parameter $\varepsilon>0$, for which the drift $A_\varepsilon[t,z]z+f_\varepsilon[t,z]$ takes values in an output space $Y$ suitable for discretization. Second, one discretizes this regularized dynamics by a sampling-reconstruction pair $(P_N,R_N)$ and obtains an interacting ODE on a finite-dimensional state space $V_N\simeq\R^{dN}$. Our main abstract theorem provides a quantitative estimate of the discrepancy $y_\varepsilon^N-y$ between the lifted discrete solution and the exact one, separating the regularization error $χ(\varepsilon)$ from the discretization error $(1+L_\varepsilon)N^{-γ}$, where $L_\varepsilon$ measures the size of the regularized drift in the output norm. This makes explicit the trade-off between the regularization scale $\varepsilon$, the discretization scale $N$, and the possible deterioration of $L_\varepsilon$ as $\varepsilon\to 0$. As a running example, we focus on quasilinear PDEs on bounded Lipschitz domains with boundary conditions. We show that Burenkov’s variable-step mollifiers provide a boundary-compatible kernelization: they regularize differential operators into explicit integral-interaction operators supported inside the domain and preserve boundary traces of sufficiently regular fields. In this setting one can choose an output space $Y$ for which $L_\varepsilon$ remains uniformly bounded, leading to algebraic convergence rates in $N$ for quasi-uniform discretizations.

30. 基于黑塞矩阵恢复的C0有限元方法求解非散度型椭圆方程

Hessian-recovery-based C0 finite element methods for non-divergence form elliptic equations

作者: Minqiang Xu, Boying Wu, Lei Zhang

类别: math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03276v1

链接: http://arxiv.org/abs/2606.03276v1

摘要: A Hessian-recovery-based C0 finite element framework is proposed for second-order elliptic equations in non-divergence form. The construction is based on a direct approximation of the strong non-divergence operator: the Hessian D2u is replaced by a recovered Hessian Hhuh, so that A : D2u is approximated by A : Hhuh. The resulting discretizations include a nodal formulation and a Galerkin-type formulation for general Lagrange finite element spaces, as well as a biorthogonal Petrov-Galerkin formulation for linear elements. The analysis focuses on the recovered nodal matrix and identifies two verifiable algebraic solvability mechanisms. The first is a globally monotone regime leading to a discrete maximum principle, and the second is a localized Schur-complement criterion for sign-violating rows. A uniform inverse bound and a condition-number estimate are derived in the globally monotone case. Residual consistency estimates are obtained from the Hessian recovery error. In the globally monotone regime, these estimates combine with the uniform inverse bound to give a nodal L-error estimate for the nodal formulation. Numerical experiments with nonsmooth and discontinuous coefficients support the predicted algebraic diagnostics and show the accuracy of the proposed recovered-residual discretizations. A Monge-Ampere type test further illustrates the use of the recovered Hessian in a Newton iteration for a fully nonlinear problem.

31. 要有光：神经算子的反射、折射与散射

Let There Be Light: Reflection, Refraction and Scattering for Neural Operators

作者: Keke Wu, Yixuan Zhang, Jingrun Chen

类别: cs.LG, math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03262v1

链接: http://arxiv.org/abs/2606.03262v1

摘要: Neural operators learn mappings between infinite-dimensional function spaces and provide a data-driven surrogate modeling paradigm for parametric partial differential equations (PDEs). Existing architectures typically obtain expressivity by parameterizing integral kernels in prescribed transform domains or by applying attention-like interactions over discretized spatial points. While these approaches have achieved substantial progress, they often face a persistent trade-off among physical interpretability, nonlocal spatial communication, mesh scalability, and computational cost. We propose a Light-inspired neural operator(LiNO), an operator-learning architecture whose latent evolution is decomposed into three mechanisms motivated by elementary light transport: reflection, refraction, and scattering. Reflection and refraction act as adaptive pointwise transformations in latent feature space, enabling local feature reorientation and anisotropic modulation, whereas scattering performs input-dependent nonlocal propagation over the physical domain. We first formulate scattering as a normalized pairwise kernel with relative positional bias, and then develop an efficient scattering variant that replaces explicit pairwise interactions with positive-feature global propagation and a local diffusion branch, reducing the dominant spatial complexity from quadratic to linear. This yields a structured neural operator that separates local feature modulation from global spatial communication while retaining a modular and interpretable latent evolution.

32. 用于FWD反分析的PINN与作为替代方案的可微分有限元法的批判性评估

Critical evaluation of PINN for FWD inverse analysis and differentiable FEM as an alternative

作者: Yongjin Choi, Hyeonbin Moon, Seunghwa Ryu

类别: cs.CE, cs.LG, math.NA

发布日期: 2026-06-02

arXiv ID: 2606.03210v1

链接: http://arxiv.org/abs/2606.03210v1

摘要: Automatic-differentiation-based inverse analysis methods, including physics-informed neural networks (PINNs) and differentiable programming, have recently shown great promise due to their ability to compute accurate gradients and convergence efficiency. However, their applicability to falling weight deflectometer (FWD) backcalculation remains unexplored. This study critically evaluates PINN-based inverse analysis for a multilayer pavement system and investigates differentiable finite element method (DiffFEM) as an alternative based on a synthetic benchmark. The standard PINN does not recover layer moduli because of the sharp domain discontinuities inherent to layered pavement systems. Although we use an extended PINN with domain decomposition (XPINN), which shows better performance on discontinuous domains, its performance remains highly sensitive to loss weighting and network architecture, and degrades under measurement noise. By contrast, DiffFEM consistently achieves more accurate, stable, and computationally efficient inversion results. These results indicate that DiffFEM, which enforces the governing physics as a hard constraint, yields better accuracy, robustness, and computational efficiency than PINN-based approaches, in which the governing physics is imposed as a soft constraint through the loss function. More broadly, the findings suggest that the choice between PINN- and DiffFEM-based inverse analysis needs careful consideration, with DiffFEM offering practical advantages when an efficient and robust differentiable forward solver is available.

33. 基于对数网格隐式步进的快速张量网络虚时间演化

Fast Tensor Network Imaginary Time Evolution by Implicit Stepping on Logarithmic Grids

作者: John P. Zima, E. Miles Stoudenmire, Steven R. White, Olivier Parcollet, Jason Kaye

类别: cond-mat.str-el, math.NA, quant-ph

发布日期: 2026-06-01

arXiv ID: 2606.02930v1

链接: http://arxiv.org/abs/2606.02930v1

摘要: We present a new method for the efficient imaginary time evolution of quantum many-body wavefunctions represented by matrix product states (MPS). We first show that logarithmic time grids are sufficient to resolve long imaginary time dynamics, yielding an exponential reduction in the number of time steps compared with standard approaches. We then show that A-stable implicit time-stepping methods for ordinary differential equations allow stable propagation for any time step size. The resulting scheme requires only matrix-vector products and linear solves, standard operations in the MPS toolbox. We validate our approach with two examples: a Heisenberg spin chain, which we use to demonstrate a speedup of several orders of magnitude over the standard time-dependent variational principle method with uniform time steps, and a single-site Anderson impurity model with a metallic bath, for which propagation to large imaginary times allows one to observe the exponential dependence of the Kondo temperature on the interaction strength.

34. 一种用于对称非负矩阵分解和图聚类的非单调梯度算法

A Nonmonotone Gradient-Based Algorithm for Symmetric Nonnegative Matrix Factorization and Graph Clustering

作者: Ryan Swart, Johannes Brust

类别: cs.LG, math.NA, math.OC

发布日期: 2026-06-01

arXiv ID: 2606.02887v1

链接: http://arxiv.org/abs/2606.02887v1

摘要: Symmetric nonnegative matrix factorization (Symmetric NMF) approximates a matrix as $WW^T$ with nonnegative rectangular factor $W$. It has broad applications in graph clustering and machine learning. In contrast to the NMF, projected gradient methods for the symmetric problem had been associated with slow convergence. To address this, we introduce SNMPBB, the first adaptation of nonmonotone projected Barzilai-Borwein methods to Symmetric NMF, demonstrating that gradient algorithms are significantly more effective than previously understood. We further extend SNMPBB to graph clustering using the graph Laplacian regularization (Graph-SNMPBB) and to large problems with low-rank approximations (LAI-SNMPBB). For all variants we prove global convergence to first-order stationary points and also that Barzilai-Borwein curvature information is preserved with randomized approximations. On synthetic data, SNMPBB achieves 6 times speedup over the alternative SymANLS for similar residuals, with advantages growing at higher ranks. Across six real-world clustering benchmarks, Graph-SNMPBB matches or exceeds SymANLS accuracy. Lastly, LAI-SNMPBB outperforms state-of-the-art LAI-SymPGNCG on 34 SuiteSparse matrices in both runtime and residual quality.

35. 扩散Lotka-Volterra方程时空模式逼近的降阶建模

Reduced order modeling for spatio-temporal pattern approximation in diffusive Lotka-Volterra equations

作者: Bülent Karasözen, Murat Uzunca

类别: math.NA

发布日期: 2026-06-01

arXiv ID: 2606.04030v1

链接: http://arxiv.org/abs/2606.04030v1

摘要: This paper presents an efficient reduced order modeling (ROM) framework for simulating spatio-temporal pattern formation in three-species diffusive Lotka-Volterra systems. To alleviate the high computational cost associated with long-time simulations of the high-dimensional full order model (FOM), we apply proper orthogonal decomposition (POD) to project the solution onto a low-dimensional subspace. Further efficiency is achieved through tensorial POD (TPOD), which preserves the quadratic nonlinear structure and enables offline-online decomposition. Numerical experiments demonstrate that both POD and TPOD accurately replicate the key features of spatial segregation patterns while substantially reducing computation time, whereas the TPOD is faster. Additionally, we demonstrate accurate long-time pattern prediction using limited training data.

重点关注论文（完整分析）#

1. 非均匀介质中时谐麦克斯韦方程阻抗边界条件的并行重叠型区域分解方法收敛性#

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#

2. 使用无参数间断Galerkin方法对二阶时间依赖鞍点Maxwell系统的数值分析：首个最优 ${\bf L}^{2}$-范数误差估计#

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#

3. 变阶 $hp$ 有限元界面问题的一致 Schwarz 预条件子#

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. 奇异曲线坐标系下一阶双曲系统的高阶分部求和格式#

摘要翻译#

2. 针对Poisson-Nernst-Planck-Navier-Stokes方程的一种解耦能量稳定混合有限元法#

摘要翻译#

3. 一种用于大型稀疏对称正定矩阵的代数多重尺度预处理器#

摘要翻译#

4. 静电剪切阿尔芬波方程的适定性与有限元逼近#

摘要翻译#

5. 反应扩散问题在各项异性网格上的嵌入式特雷夫茨间断伽辽金方法#

摘要翻译#

不相关论文（基本信息）#

1. 用于代数保持库普曼学习的深度嵌入乘法动态模式分解#

2. Bregman 散度与基于凸对偶的误差控制#

3. 交会几何对天基激光碎片清除与估计的影响#

4. 地震波传播时域有限差分格式的任意高阶成形模板#

5. 在GPU上拟合带有可选单调性约束的散乱数据：LipFit包#

6. 强磁场下带电粒子动力学的一类新型高阶一致精确局部线性展开指数积分器#

7. 求解定常MHD问题的解耦迭代格式#

8. MeshTok: 面向可扩展偏微分方程Transformer的高效多尺度令牌化#

9. 一种用于谱鲁棒半定规划的扰动q-Tsallis自协调障碍函数#

10. 多组分膜相场模型的指数时间差分格式#

11. 多孔介质中变饱和流Richards方程的高效求解器#

12. Fisher–von Mises–Langevin 分布中平方浓度的无偏估计与浓度无偏估计的不可能性#

13. 具有学习相互作用的非局部平均场薛定谔桥#

14. 一种检测确定Hermitian矩阵对的不定LOBPCG型算法#

15. Cahn-Hilliard-Navier-Stokes方程的一种能量稳定的隐式凸分裂BDF2格式#

16. 凸函数的采样与重构#

17. 基于高斯求积的具有几何误差衰减的短指数和逼近#

18. 基于能量的模型的三项递推迭代#

19. 图正则化非负降双四元数矩阵分解用于彩色图像识别#

20. 多项式代数在非线性方程求解器中的应用#

21. 关于带不确定性的肿瘤生长模型的多保真度方法#

22. 一种用于三维各向异性声子晶体能带结构计算的高效奇偶分块方法#

23. 场路耦合问题多速率联合仿真技术比较#

24. 非自伴不定施图姆-刘维尔问题的左半平面特征值的有界性及其在傅里叶模态方法中的应用#

25. 半线性抛物方程Parareal算法的线性收敛性#

26. 基于体素的量子计算方法（VBQC）用于固体力学问题#

27. 模型误设下算子学习的物理引导修正#

28. 保结构量子线方法用于混合边界条件下演化型偏微分方程#

29. 有界区域上拟线性偏微分方程的边界兼容相互作用逼近#

30. 基于黑塞矩阵恢复的C0有限元方法求解非散度型椭圆方程#

31. 要有光：神经算子的反射、折射与散射#

32. 用于FWD反分析的PINN与作为替代方案的可微分有限元法的批判性评估#

33. 基于对数网格隐式步进的快速张量网络虚时间演化#

34. 一种用于对称非负矩阵分解和图聚类的非单调梯度算法#

35. 扩散Lotka-Volterra方程时空模式逼近的降阶建模#

重点关注论文（完整分析）

1. 非均匀介质中时谐麦克斯韦方程阻抗边界条件的并行重叠型区域分解方法收敛性

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

2. 使用无参数间断Galerkin方法对二阶时间依赖鞍点Maxwell系统的数值分析：首个最优 ${\bf L}^{2}$-范数误差估计

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

3. 变阶 $hp$ 有限元界面问题的一致 Schwarz 预条件子

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. 奇异曲线坐标系下一阶双曲系统的高阶分部求和格式

摘要翻译

2. 针对Poisson-Nernst-Planck-Navier-Stokes方程的一种解耦能量稳定混合有限元法

摘要翻译

3. 一种用于大型稀疏对称正定矩阵的代数多重尺度预处理器

摘要翻译

4. 静电剪切阿尔芬波方程的适定性与有限元逼近

摘要翻译

5. 反应扩散问题在各项异性网格上的嵌入式特雷夫茨间断伽辽金方法

摘要翻译

不相关论文（基本信息）

1. 用于代数保持库普曼学习的深度嵌入乘法动态模式分解

2. Bregman 散度与基于凸对偶的误差控制

3. 交会几何对天基激光碎片清除与估计的影响

4. 地震波传播时域有限差分格式的任意高阶成形模板

5. 在GPU上拟合带有可选单调性约束的散乱数据：LipFit包

6. 强磁场下带电粒子动力学的一类新型高阶一致精确局部线性展开指数积分器

7. 求解定常MHD问题的解耦迭代格式

8. MeshTok: 面向可扩展偏微分方程Transformer的高效多尺度令牌化

9. 一种用于谱鲁棒半定规划的扰动q-Tsallis自协调障碍函数

10. 多组分膜相场模型的指数时间差分格式

11. 多孔介质中变饱和流Richards方程的高效求解器

12. Fisher–von Mises–Langevin 分布中平方浓度的无偏估计与浓度无偏估计的不可能性

13. 具有学习相互作用的非局部平均场薛定谔桥

14. 一种检测确定Hermitian矩阵对的不定LOBPCG型算法

15. Cahn-Hilliard-Navier-Stokes方程的一种能量稳定的隐式凸分裂BDF2格式

16. 凸函数的采样与重构

17. 基于高斯求积的具有几何误差衰减的短指数和逼近

18. 基于能量的模型的三项递推迭代

19. 图正则化非负降双四元数矩阵分解用于彩色图像识别

20. 多项式代数在非线性方程求解器中的应用

21. 关于带不确定性的肿瘤生长模型的多保真度方法

22. 一种用于三维各向异性声子晶体能带结构计算的高效奇偶分块方法

23. 场路耦合问题多速率联合仿真技术比较

24. 非自伴不定施图姆-刘维尔问题的左半平面特征值的有界性及其在傅里叶模态方法中的应用

25. 半线性抛物方程Parareal算法的线性收敛性

26. 基于体素的量子计算方法（VBQC）用于固体力学问题

27. 模型误设下算子学习的物理引导修正

28. 保结构量子线方法用于混合边界条件下演化型偏微分方程

29. 有界区域上拟线性偏微分方程的边界兼容相互作用逼近

30. 基于黑塞矩阵恢复的C0有限元方法求解非散度型椭圆方程

31. 要有光：神经算子的反射、折射与散射

32. 用于FWD反分析的PINN与作为替代方案的可微分有限元法的批判性评估

33. 基于对数网格隐式步进的快速张量网络虚时间演化

34. 一种用于对称非负矩阵分解和图聚类的非单调梯度算法

35. 扩散Lotka-Volterra方程时空模式逼近的降阶建模