Computers & Mathematics with Applications最新文献

英文中文

Error estimates of the weak Galerkin mixed finite element method for parabolic interface problems 抛物界面问题的弱Galerkin混合有限元法误差估计

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-04-01 Epub Date: 2026-01-24 DOI: 10.1016/j.camwa.2026.01.024

Amit Kumar Pal , Jhuma Sen Gupta , Rajen Kumar Sinha

This paper aims to study a priori error analysis of the weak Galerkin mixed finite element method (WG-MFEM) for parabolic interface problems in a two-dimensional bounded convex polygonal domain. While discontinuous functions are employed for the approximation of spatial variable, an implicit backward Euler scheme is used for the time variable. Due to the presence of the discontinuous coefficient across the interface, the solution of parabolic interface problems possesses very low global regularity. Using the Stein extension operator and the H¹(div)-extension operator leads to the novel approximation results for the L² projection operators for both the scalar and the vector-valued functions, respectively. With the help of mixed elliptic projection operator and the new approximation properties combined with the standard energy argument, an almost optimal order a priori error bounds are derived for both the solution and the flux variables in the L^∞(L²) norm. Numerical outcomes for some test problems are reported to confirm the theoretical analysis.

本文研究了二维有界凸多边形区域中抛物界面问题的弱Galerkin混合有限元法的先验误差分析。空间变量采用不连续函数逼近，时间变量采用隐式后向欧拉格式逼近。由于界面上存在不连续系数，抛物型界面问题的解具有很低的全局正则性。使用Stein扩展算子和H1(div)-扩展算子分别可以得到标量函数和向量值函数的L2投影算子的新的近似结果。利用混合椭圆投影算子和新的近似性质，结合标准能量参数，导出了L∞(L2)范数上的解和通量变量的几乎最优阶先验误差界。文中还报道了一些试验问题的数值结果，以证实理论分析。

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引用次数: 0

Symmetric direct discontinuous Galerkin method for the biharmonic equation with non-homogeneous boundary condition 非齐次边界条件双调和方程的对称直接不连续Galerkin方法

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-04-01 Epub Date: 2026-01-19 DOI: 10.1016/j.camwa.2026.01.010

Hongying Huang , Huanhuan Wang , Lin Zhang

The direct discontinuous Galerkin method is proposed and analyzed for the biharmonic equation with non-homogeneous boundary conditions. Numerical fluxes on all edges of arbitrary triangles are defined, which are concerned with four parameters (β₁, β₂, β₃, β₄), and then the weak variational form of the original problem is derived. The corresponding bilinear form is proved to be bounded and coercive for sufficiently large values of the parameters β₁ and β₂. The well-posedness of the variational problem in finite element space

V_{h}^{k}

and optimal error estimates of the approximate solution under L²-norm and energy norm are obtained. The accuracy and reliability of the proposed method were verified through numerical experiments. Although the theoretical analysis requires the solution u ∈ H⁴, numerical results indicate that the method remains effective for solutions with low regularity.

提出并分析了具有非齐次边界条件的双调和方程的直接不连续伽辽金方法。定义了任意三角形所有边上与四个参数（β1、β2、β3、β4）有关的数值通量，导出了原问题的弱变分形式。当参数β1和β2的值足够大时，证明了相应的双线性形式是有界的和强制的。得到了有限元空间Vhk中变分问题的适定性，以及在l2范数和能量范数下近似解的最优误差估计。通过数值实验验证了该方法的准确性和可靠性。虽然理论分析需要解u ∈ H4，但数值结果表明，该方法对于低正则性解仍然有效。

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引用次数: 0

Lattice-Boltzmann-inspired finite-difference schemes for the convection-diffusion equation 对流扩散方程的格子-玻尔兹曼激励有限差分格式

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-04-01 Epub Date: 2026-02-24 DOI: 10.1016/j.camwa.2026.02.008

Xingchun Xu , Yurong He , Bing Dai , Jiaqi Zhu

The standard lattice Boltzmann method is typically limited to second-order accuracy for the convection-diffusion equation. In this study, we perform a sixth-order expansion of the lattice Boltzmann model and subsequently derive the optimal relaxation time that eliminates specific high-order error contributions. Inspired by this expansion, we develop three-level and four-level finite-difference schemes expressed solely in terms of the equilibrium distribution. Theoretical analysis demonstrates that both schemes achieve fourth-order accuracy at their respective optimal relaxation times on coarse meshes. Gaussian-hill benchmarks confirm that the optimal relaxation times and convergence rates coincide with the analytical predictions. In addition, high-order extrapolation formulas for Dirichlet and Neumann boundaries enable the four-level finite-difference scheme to achieve the lowest relative error, two orders of magnitude below that of the standard lattice Boltzmann model.

标准晶格玻尔兹曼方法通常限于对流扩散方程的二阶精度。在本研究中，我们对晶格玻尔兹曼模型进行了六阶展开，并随后推导出消除特定高阶误差贡献的最佳松弛时间。受这一展开的启发，我们发展了仅用平衡分布表示的三层和四层有限差分格式。理论分析表明，在粗糙网格上，两种方法在各自的最优松弛时间都能达到四阶精度。高斯-希尔基准证实了最优松弛时间和收敛速率与分析预测一致。此外，Dirichlet和Neumann边界的高阶外推公式使四层有限差分格式的相对误差最低，比标准晶格玻尔兹曼模型低两个数量级。

引用次数: 0

EM propagation analysis of multilayered fully anisotropic media with an efficacious model equivalent approach 基于有效模型等效方法的多层全各向异性介质电磁传播分析

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-04-01 Epub Date: 2026-01-23 DOI: 10.1016/j.camwa.2026.01.022

Huan Wang , Naixing Feng , Chong-Zhi Han , Jinfeng Zhu , Lixia Yang , Atef Z. Elsherbeni

In this paper, the model equivalent approach is developed for full-wave analysis of electromagnetic propagation in multilayered fully anisotropic lossy media. In the process of geometric simulation, the 3D planar layered model is projected onto an axial stratified structure, effectively reducing spatial complexity. Then, the mesh free scheme is adopted for discretization, thus significantly decreasing the resource consumption. To accommodate generalized electromagnetic media, the governing equation for the electric field is formulated based on fully anisotropic media, characterized by full tensor parameters. The Galerkin method is employed to generate weak form partial differential equations (PDEs), then, the FEM is adopted to address the equations. To ensure accuracy in the FEM implementation, the divergence condition is imposed as a constraint on the PDEs, effectively eliminating spurious solutions from the computational domain. Finally, three numerical examples are presented to verify the effectiveness of the proposed method.

本文建立了多层全各向异性有耗介质中电磁传播全波分析的模型等效方法。在几何模拟过程中，将三维平面分层模型投影到轴向分层结构上，有效降低了空间复杂度。然后，采用无网格格式进行离散化，从而大大降低了资源消耗。为了适应广义电磁介质，在全各向异性介质的基础上建立了以全张量参数为特征的电场控制方程。采用伽辽金法生成弱形式偏微分方程，然后采用有限元法求解。为了保证有限元实现的准确性，对偏微分方程施加了发散条件作为约束，有效地消除了计算域中的伪解。最后给出了三个数值算例，验证了所提方法的有效性。

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引用次数: 0

Uncertainty analysis framework of MPS and implementation in the simulation of MCCI phenomenon MPS的不确定性分析框架及其在MCCI现象模拟中的实现

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-04-01 Epub Date: 2026-01-28 DOI: 10.1016/j.camwa.2026.01.031

Xinkun Xiao , Qinghang Cai , Tianrui Li , Ronghua Chen , Guanghui Su

This study establishes the Moving Particle Semi-implicit Plus Uncertainty (MPSPU) framework to enable rigorous uncertainty quantification (UQ) for particle-based simulations in nuclear reactor safety analysis. Designed to extend the Best Estimate Plus Uncertainty (BEPU) methodology, MPSPU addresses the specific challenges of Lagrangian particle methods while maintaining compatibility with existing regulatory assessment protocols. The framework is validated using the SURC-4 experiment, which simulates the Molten Core–Concrete Interaction (MCCI) phenomenon. A critical advancement is the formulation of a time-dependent sensitivity analysis, which reveals that melt temperature is the dominant driver governing early-stage MCCI behavior. Furthermore, a comparative evaluation of surrogate models for MPS time-series data identifies Long Short-Term Memory (LSTM) networks as the optimal architecture, outperforming conventional polynomial and neural network approaches. To demonstrate the framework's practical utility, an end-to-end calculation example is presented, illustrating the complete workflow from raw simulation data to regulatory-grade risk metrics. This example explicitly quantifies the conditional failure probability of concrete ablation depth against safety limits, showcasing the framework's ability to support risk-informed decision-making. Ultimately, this work provides a systematic pathway for integrating particle methods into safety analysis.

本研究建立了移动粒子半隐式加不确定性（MPSPU）框架，为核反应堆安全分析中基于粒子的模拟提供严格的不确定性量化（UQ）。MPSPU旨在扩展最佳估计加不确定性（BEPU）方法，解决拉格朗日粒子方法的特定挑战，同时保持与现有监管评估协议的兼容性。使用SURC-4实验对该框架进行了验证，该实验模拟了熔融核-混凝土相互作用（MCCI）现象。一个关键的进步是制定了一个时间依赖的敏感性分析，这表明熔体温度是控制早期MCCI行为的主要驱动因素。此外，对MPS时间序列数据的替代模型进行了比较评估，发现长短期记忆（LSTM）网络是最优架构，优于传统的多项式和神经网络方法。为了展示该框架的实用性，本文给出了一个端到端计算示例，说明了从原始模拟数据到监管级风险指标的完整工作流程。这个例子明确地量化了混凝土消融深度相对于安全限制的条件失效概率，展示了框架支持风险知情决策的能力。最终，这项工作为将粒子方法整合到安全分析中提供了一个系统的途径。

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引用次数: 0

Unconditional uniform superconvergent error estimates of anisotropic nonconforming energy-dissipative SAV-CN FEM for fourth-order singular perturbation Bi-flux diffusion model 四阶奇异摄动双通量扩散模型各向异性非协调能量耗散SAV-CN有限元的无条件一致超收敛误差估计

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-04-01 Epub Date: 2026-01-20 DOI: 10.1016/j.camwa.2026.01.012

Dongyang Shi , Sihui Zhang

This investigation aims to delve into the unconditional uniform superconvergence behavior of the scalar auxiliary variable (SAV) Crank-Nicolson (CN) fully discrete scheme for the fourth-order singular perturbation Bi-flux diffusion equation with anisotropic constrained nonconforming rotated

Q_{1}

(

C N Q_{1}^{r o t}

) finite element method (FEM). Innovative high-accuracy estimates tailored specifically for the

C N Q_{1}^{r o t}

element on anisotropic meshes are established as the cornerstone for achieving superconvergent outcomes. Following this, a novel SAV-CN scheme is formulated, with the energy dissipation theoretically given to guarantee that the numerical solutions remain bounded in the broken H¹-norm. Then, by leveraging the aforementioned novel estimates and sidestepping the reliance on inverse inequalities, the unconditional uniform superclose result is rigorously derived, which transcends the constraints of the negative power of the perturbation parameter

A = \sqrt{β (1 - β) R}

and the limitations imposed by the ratio between the temporal step τ and spatial partition parameter h. Furthermore, an application of the economic interpolation post-processing technique enables the procurement of superconvergence estimates for the devised numerical schemes. Ultimately, the theoretical analysis is strengthened through the conduct of numerical experiments.

研究了四阶各向异性约束非协调旋转Q1 （CNQ1rot）有限元法四阶奇异摄动双通量扩散方程的标量辅助变量（SAV） Crank-Nicolson （CN）完全离散格式的无条件一致超收敛行为。在各向异性网格上为CNQ1rot元素量身定制的创新高精度估计是实现超收敛结果的基石。在此基础上，提出了一种新的SAV-CN格式，并在理论上给出了能量耗散，以保证数值解在h1 -范数破碎时保持有界。然后，利用上述新估计，避开对逆不等式的依赖，严格推导了无条件均匀超接近结果，该结果超越了扰动参数A=β(1−β)R的负幂约束和时间步长τ与空间分割参数h之比的限制。经济插值后处理技术的应用使所设计的数值格式获得了超收敛估计。最后，通过数值实验加强理论分析。

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引用次数: 0

Reconstruction of shape and impedance of a cavity from two boundary measurements 用两次边界测量重建腔体的形状和阻抗

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-03-24 DOI: 10.1016/j.camwa.2026.03.005

Weifu Fang

引用次数: 0

Preconditioning FEM discretisations of the high-frequency Helmholtz and Maxwell equations by either perturbing the coefficients or adding absorption 通过扰动系数或加入吸收对高频亥姆霍兹方程和麦克斯韦方程的有限元离散进行预处理

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-03-19 DOI: 10.1016/j.camwa.2026.03.009

E. A. Spence

引用次数: 0

A stabilized finite element method for the Navier–Stokes/Darcy coupled problem Navier-Stokes /Darcy耦合问题的稳定有限元方法

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-03-18 DOI: 10.1016/j.camwa.2026.03.002

Rodolfo Araya, Cristian Cárcamo, Abner H. Poza

引用次数: 0

Solving the Monge-Ampère equation via Poisson series physics-informed neural networks and its convergence analysis 利用泊松系列物理信息神经网络求解monge - ampantere方程及其收敛性分析

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-03-15 Epub Date: 2026-01-30 DOI: 10.1016/j.camwa.2026.01.033

Ruibo Zhang , Fengjun Li , Jianqiang Liu

The Monge-Ampère equation is originated from geometric surface theory and is widely applied in optimal transport theory, image processing, optimization problem and so on. The numerical solution of the Monge-Ampère equation has recently attracted more and more attention. Physics-informed neural networks (PINNs), a new paradigm in numerical methods, introduce physical constraints during the training process so that the model not only can learn patterns in the data, but also satisfy the laws of physics. In our work, we try to solve the Monge-Ampère equation with Dirichlet boundary conditions by using the PINNs. To our knowledge, this is the first time that PINNs is applied to solve the Monge-Ampère equation. Unfortunately, the Monge-Ampère equation involves determinant calculation, which leads to calculation failure using the conventional PINNs. For this reason, inspired by the fixed-point method, we construct a Poisson series physics-informed neural networks (PS-PINNs) framework to solve this problem. The Monge-Ampère equation is transformed into a Poisson series using the fixed-point method, which avoids the direct computation of the determinant. As part of our analysis, we prove the convergence of loss function and neural networks in PS-PINNs. Moreover, we study the performance of PS-PINNs with source functions containing singularities and noise, as well as in asymmetric domains. It is worth noting that we can obtain better numerical results using a small number of sampling points and iterations. The data and code accompanying this paper are publicly available at https://github.com/RuiboZhangping/PSPINN.

monge - ampantere方程起源于几何曲面理论，广泛应用于最优输运理论、图像处理、优化问题等领域。monge - ampantere方程的数值解近年来受到越来越多的关注。物理信息神经网络（pinn）是数值方法中的一种新范式，它在训练过程中引入物理约束，使模型不仅能够学习数据中的模式，而且能够满足物理定律。在我们的工作中，我们尝试用pinn来求解具有Dirichlet边界条件的monge - amp方程。据我们所知，这是pin - ns第一次被应用于求解monge - ampante方程。不幸的是，monge - ampantere方程涉及行列式计算，这导致使用传统的pin n计算失败。因此，受不动点法的启发，我们构建了一个泊松系列物理信息神经网络（ps - pinn）框架来解决这个问题。采用不动点法将monge - ampantere方程转化为泊松级数，避免了行列式的直接计算。作为分析的一部分，我们证明了ps - pinn中损失函数和神经网络的收敛性。此外，我们还研究了源函数包含奇异点和噪声以及非对称域的ps - pin的性能。值得注意的是，我们可以使用少量的采样点和迭代获得更好的数值结果。本文附带的数据和代码可在https://github.com/RuiboZhangping/PSPINN上公开获取。

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