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arXiv 2024年

作者： Zhang, Juan Luo, Xiao Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China

In this paper, we focus on using optimization methods to solve matrix equations by transforming the problem of solving the Sylvester matrix equation or continuous algebraic Riccati equation into an optimization problem. Initially, we use a constrained convex optimization method (CCOM) to solve the Sylvester matrix equation with 2.1-norm, where we provide a convergence analysis and numerical examples of CCOM;however, the results show that the algorithm is not efficient. To address this issue, we employ classical quasi-Newton methods such as DFP and BFGS algorithms to solve the Sylvester matrix equation and present the convergence and numerical results of the algorithm. Additionally, we compare these algorithms with the CG algorithm and AR algorithm, and our results demonstrate that the presented algorithms are effective. Furthermore, we propose a unified framework of the alternating direction multiplier method (ADMM) for directly solving the continuous algebraic Riccati equation (CARE), and we provide the convergence and numerical results of ADMM. Our experimental results indicate that ADMM is an effective optimization algorithm for solving CARE. Finally, to improve the effectiveness of the optimization method for solving Riccati equation, we propose the Newton-ADMM algorithm framework, where the outer iteration of this method is the classical Newton method, and the inner iteration involves using ADMM to solve Lyapunov matrix equations inexactly. We also provide the convergence and numerical results of this algorithm, which our results demonstrate are more efficient than ADMM for solving CARE. Copyright © 2.2., The Authors. All rights reserved.

关键词： Newton-Raphson method

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The optimal strong convergence rates of the truncated EM and logarithmic truncated EM methods for multi-dimensional nonlinear stochastic differential equations

arXiv

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arXiv 2025年

作者： Hu, Xingwei Dai, Xinjie Xiao, Aiguo School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Yunnan University Yunnan Kunming650504 China

The truncated Euler–Maruyama (EM) method, developed by Mao (2.15), is used to solve multi-dimensional nonlinear stochastic differential equations (SDEs). However, its convergence rate is suboptimal due to an unnecessary infinitesimal factor. The primary goal of this paper is to demonstrate the optimal convergence of the truncated EM method without infinitesimal factors. Besides, the logarithmic truncated EM method has not been studied in multi-dimensional cases, which is the other goal of this paper. We will show the optimal strong convergence order of the positivity-preserving logarithmic truncated EM method for solving multi-dimensional SDEs with positive solutions. Numerical examples are given to support our theoretical conclusions. Copyright © 2.2., The Authors. All rights reserved.

关键词： Stochastic systems

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High-Order Interior Penalty Finite Element Methods for Fourth-Order Phase-Field Models in Fracture Analysis

arXiv

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arXiv 2025年

作者： Tian, Tian Chunyu, Chen Wei, Huayi School of Mathematics and Statistics Xiangtan University Hunan Xiangtan411105 China National Center of Applied Mathematics in Hunan Hunan Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Hunan Xiangtan411105 China

This paper presents a novel approach for solving fourth-order phase-field models in brittle fracture mechanics using the Interior Penalty Finite Element Method (IP-FEM). The fourth-order model improves numerical stability and accuracy compared to traditional second-order phase-field models, particularly when simulating complex crack paths. The IP-FEM provides an efficient framework for discretizing these models, effectively handling nonconforming trial functions and complex boundary conditions. In this study, we leverage the FEALPy framework to implement a flexible computational tool that supports high-order IP-FEM discretizations. Our results show that as the polynomial order increases, the mesh dependence of the phase-field model decreases, offering improved accuracy and faster convergence. Additionally, we explore the trade-offs between computational cost and accuracy with varying polynomial orders and mesh sizes. The findings offer valuable insights for optimizing numerical {2.s of brittle fracture in practical engineering applications. Copyright © 2.2., The Authors. All rights reserved.

关键词： Brittle fracture

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A Posteriori Error Estimates for hp Spectral Element Approximation of Elliptic Control Problems with Integral Control and State Constraints

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Advances in Applied Mathematics and Mechanics 2022年第2期14卷 469-493页

作者： Jinling Zhang Yanping Chen Yunqing Huang Fenglin Huang Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational ScienceXiangtan UniversityXiangtanHunan 411105China School of Mathematical Sciences South China Normal UniversityGuangzhouGuangdong 510631China School of Mathematics and Statistics Xinyang Normal UniversityNo.237 Nanhu RoadShihe DistrictXinyangHenan 464000China

This paper investigates an optimal control problem governed by an elliptic equation with integral control and state *** control problem is approxi-mated by the hp spectral element method with high accuracy and geometricﬂ*** conditions of the continuous and discrete optimal control problems are presented,*** a posteriori error estimates both for the control and state variables are established in *** addition,illustrative numerical examples are carried out to demonstrate the accuracy of theoretical results and the validity of the proposed method.

关键词： Elliptic equations optimal control control-state constraints a posteriori error esti-mates hp spectral element method.

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A PRIORI AND A POSTERIORI ERROR ESTIMATES OF A REALLY PRESSURE-ROBUST VIRTUAL ELEMENT METHOD FOR THE INCOMPRESSIBLE BRINKMAN PROBLEM

arXiv

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arXiv 2024年

作者： Xiong, Yu Chen, Yanping School of Science Nanjing University of Posts and Telecommunications Nanjing210023 China School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

This paper presents both a priori and a posteriori error analyses for a really pressure-robust virtual element method to approximate the incompressible Brinkman problem. We construct a divergence-preserving reconstruction operator using the Raviart–Thomas element for the discretization on the right-hand side. The optimal priori error estimates are carried out, which imply the velocity error in the energy norm is independent of both the continuous pressure and the viscosity. Taking advantage of the virtual element method’s ability to handle more general polygonal meshes, we implement effective mesh refinement strategies and develop a residual-type a posteriori error estimator. This estimator is proven to provide global upper and local lower bounds for the discretization error. Finally, some numerical experiments demonstrate the robustness, accuracy, reliability and efficiency of the method. Copyright © 2.2., The Authors. All rights reserved.

关键词： Mesh generation

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Investigation of double-gyroid grain boundaries beyond twinning

arXiv

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arXiv 2024年

作者： Chen, Jing Sun, Zhangpeng Jiang, Kai Xu, Jie Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Chinese Academy of Sciences Beijing China

We study four double-gyroid (DG) grain boundaries (GBs) with different orientations numerically using the Landau–Brazovskii free energy, including the (42.) twin boundary studied recently, a network switching GB, and two tilt GBs. Topological variations and geometric deformations are investigated. It is found that deviations in strut lengths and dihedral angles from the bulk DG substantially exceed changes in strut angles and nodal coplanarity. We also examine the spectra along the contact plane of two grains and utilize them to evaluate the GB widths. Of the four GBs we study, the network switching GB changes to the least extent topologically and geometrically, meanwhile has the lowest energy and the smallest GB width. Copyright © 2.2., The Authors. All rights reserved.

关键词： Grain boundaries

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A LINEAR, UNCONDITIONALLY STABLE, SECOND ORDER DECOUPLED METHOD FOR THE NEMATIC LIQUID CRYSTAL FLOWS WITH SAV APPROACH

arXiv

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arXiv 2025年

作者： Cao, Ruonan Yi, Nianyu School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China

In this paper, we present a second order, linear, fully decoupled, and unconditionally energy stable scheme for solving the Erickson-Leslie model. This approach integrates the pressure correction method with a scalar auxiliary variable technique. We rigorously demonstrate the unconditional energy stability of the proposed scheme. Furthermore, we present several numerical experiments to validate its convergence order, stability, and computational efficiency. © 2.2., CC BY.

关键词： computational efficiency

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AN EFFICIENT NULLSPACE-PRESERVING SADDLE SEARCH METHOD FOR PHASE TRANSITIONS INVOLVING TRANSLATIONAL INVARIANCE

arXiv

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arXiv 2023年

作者： Cui, Gang Jiang, Kai Zhou, Tiejun Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

In this work, we propose an efficient nullspace-preserving saddle search (NPSS) method for a class of phase transitions involving translational invariance, where the critical states are often degenerate. The NPSS method includes two stages, escaping from the basin and searching for the index-1 generalized saddle point. The NPSS method climbs upward from the generalized local minimum in segments to overcome the challenges of degeneracy. In each segment, an effective ascent direction is ensured by keeping this direction orthogonal to the nullspace of the initial state in this segment. This method can escape the basin quickly and converge to the transition states. We apply the NPSS method to the phase transitions between crystals, and between crystal and quasicrystal, based on the Landau-Brazovskii and Lifshitz-Petrich free energy functionals. Numerical results show a good performance of the NPSS *** Codes 37M05, 49K35, 37N30 © 2.2., CC BY-NC-ND.

关键词： Free energy

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On the stability and exponential decay of the 3D MHD system with mixed partial dissipation near a equilibrium state

arXiv

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arXiv 2024年

作者： Deng, Xuemin Xiao, Yuelong Zang, Aibin Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411100 China The Center of Applied Mathematics Yichun University Jiangxi Yichun336000 China

A main result of this paper establishes the global stability of the 3D MHD equations with mixed partial dissipation near a background magnetic field in the domain Ω = T2.&#2.5; R with T2.= [0, 1]2. More precisely, eac... 详细信息

A main result of this paper establishes the global stability of the 3D MHD equations with mixed partial dissipation near a background magnetic field in the domain Ω = T2.× R with T2.= [0, 1]2. More precisely, each velocity equation lacks its own directional dissipation, and the magnetic equation lacks vertical dissipation in the MHD system. The key point to obtain the stability result is that we decompose the solution (u, b) into the zeroth horizontal mode and the non-zeroth modes and complete the desired bound with the strong Poincaré type inequalities in the treatment of several nonlinear terms. Then we focus on the large-time behavior of the solution, where the non-zeroth modes decay exponentially in H2. and the solution converges to its zeroth horizontal mode. Copyright © 2.2., The Authors. All rights reserved.

关键词： Magnetohydrodynamics

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FETD Study of theWave Propagation in Chiral Metamaterials

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Advances in Applied Mathematics and Mechanics 2021年第1期13卷 191-202页

作者： Lei Yang Fuhao Liu Wei Yang Faculty of Information Technology Macao University of Science and TechnologyMacaoChina School of Mathematics and Computational Science Xiangtan UniversityXiangtanHunan 411105China Hunan Key Laboratory for Computation and Simulation in Science and Engineering and School of Mathematics and Computational Science Xiangtan UniversityXiangtanHunan 411105China

In this paper,we propose a finite element time-domain(FETD)method for the Maxwell’s equations in chiral metamaterials(CMMs).The time-domain model equations are constructed by the auxiliary differential equations(ADEs)*** source excitation method entitled total-field and scattered-field(TF/SF)decomposition technique is applied to FETD method for the first time in simulating the propagation of electromagnetic wave in CMMs,based on which a unified ADE-FETD-UPMLTF/SF scheme is proposed to simulate the wave in *** following properties of CMMs can be observed successfully from the numerical experiments based on our method,i.e.,the ability of the polarization rotation,and the negative phase *** amplitude of reflected wave can effectively be controlled by the physical parameters of CMMs.

关键词： Maxwell’s equations finite element time-domain(FETD) chiral metamaterials(CMMs).

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