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arXiv 2024年

作者： Bu, Weiping Zheng, Xin School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan411105 China

The aim of this paper is to develop a refined error estimate of L1/finite element scheme for a reaction-subdiffusion equation with constant delay τ and uniform time mesh. Under the non-uniform multi-singularity assumption of exact solution in time, the local truncation errors of the L1 scheme with uniform mesh is investigated. Then we introduce a fully discrete finite element scheme of the considered problem. Next, a novel discrete fractional Grönwall inequality with constant delay term is proposed, which does not include the increasing Mittag-Leffler function comparing with some popular other cases. By applying this Grönwall inequality, we obtain the pointwisein-time and piecewise-in-time error estimates of the finite element scheme without the Mittag-Leffler function. In particular, the latter shows that, for the considered interval ((i-1)τ, iτ], although the convergence in time is low for i = 1, it will be improved as the increasing i, which is consistent with the factual assumption that the smoothness of the solution will be improved as the increasing i. Finally, we present some numerical tests to verify the developed theory. Copyright © 2024, The Authors. All rights reserved.

关键词： Finite element method

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New upper and lower eigenvalue bounds for the solution of the continuous algebraic riccati equation

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Asian Journal of Control 2014年第1期16卷 284-291页

作者： Liu, Jianzhou Zhang, Juan Department of Mathematics and Computational Science Xiangtan University Xiangtan Hunan 411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan 411105 China

In this paper, applying majorization inequalities, new upper and lower bounds for summation of eigenvalues (including the trace) of the solution of the continuous algebraic Riccati equation are presented. Corresponding numerical examples illustrate that our new bounds extend some of the recent results. © 2012 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society.

关键词： Eigenvalues and eigenfunctions

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Evolution of the electronic and magnetic properties of zigzag silicene nanoribbon used for hydrogen storage material

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International Journal of Hydrogen Energy 2017年第44期42卷 27184-27205页

作者： Yuliang Mao Chan Tang Gang Guo Jianmei Yuan Jianxin Zhong Hunan Key Laboratory for Micro–Nano Energy Materials and Devices School of Physics and Optoelectronic Engineering Xiangtan University Hunan 411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan 411105 China

Using first-principles calculations, we investigate the evolution of electronic and magnetic properties of zigzag silicene nanoribbon (ZSiNR) along with the concentration of edge adsorbed hydrogen adatoms. Our study shows that the significant covalent bonding helps to stabilize the configurations of hydrogen adsorbed ZSiNRs. The ferro-metallic electronic property of ZSiNR originated from the σ-π mixing effect is suppressed by mono-hydrogenation at the adsorption sites due to the sp 2 bonding. However, bi-hydrogenation at the adsorbed sites will lead to the typical sp 3 bonding, which dominates the electronic property with the increasing of hydrogen adatoms. Under the coexisting situation with mono-hydrogenation and bi-hydrogenation, we find that both the number of adsorption sites and the bonding type of sp 2 or sp 3 have impact on the electronic property of ZSiNR. It is found that symmetry adsorption at the edges changes the stable magnetic state of ZSiNR from ferromagnetic to antiferromagnetic. In contrast, unsymmetrical adsorption along the two edges of ZSiNR keeps its ferromagnetic property.

关键词： Silicene nanoribbon Hydrogen adatom First-principles Edge-adsorption

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The Mean Time to Absorption on Horizontal Partitioned Sierpinski Gasket Networks

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Analysis in Theory and Applications 2024年第1期40卷 1-21页

作者： Zhizhuo Zhang Bo Wu Zuguo Yu School of Mathematics Southeast UniversityNanjingJianagsu 211189China School of Applied Mathematics Nanjing University of Finance and EconomicsNanjingJiangsu 210023China Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education and Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtanHunan 411105China

The random walk is one of the most basic dynamic properties of complex networks,which has gradually become a research hotspot in recent years due to its many applications in actual *** important characteristic of the random walk is the mean time to absorption,which plays an extremely important role in the study of topology,dynamics and practical application of complex *** the mean time to absorption on the regular iterative self-similar network models is an important way to explore the inﬂuence of self-similarity on the properties of random walks on the *** existing literatures have proved that even local self-similar structures can greatly affect the properties of random walks on the global network,but they have failed to prove whether these effects are related to the scale of these self-similar *** this article,we construct and study a class of Horizontal Par-titioned Sierpinski Gasket network model based on the classic Sierpinski gasket net-work,which is composed of local self-similar structures,and the scale of these structures will be controlled by the partition coefﬁcient ***,the analytical expressions and approximate expressions of the mean time to absorption on the network model are obtained,which prove that the size of the self-similar structure in the network will directly restrict the inﬂuence of the self-similar structure on the properties of random walks on the ***,we also analyzed the mean time to absorption of different absorption nodes on the network toﬁnd the location of the node with the highest absorption efﬁciency.

关键词： Mean time to absorption self-similar network Sierpinski Gasket

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Finite Element Analysis of Maxwell’s Equations in Dispersive Lossy Bi-Isotropic Media

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Advances in Applied Mathematics and Mechanics 2013年第4期5卷 494-509页

作者： Yunqing Huang Jichun Li Yanping Lin Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityHunan 411105China Department of Mathematical Sciences University of Nevada Las VegasLas VegasNevada 89154-4020USA Department of AppliedMathematics Hong Kong Polytechnic UniversityHung HomHong Kongand Department of Mathematical and Statistics ScienceUniversity of AlbertaEdmonton ABAlberta T6G 2G1Canada

In this paper,the time-dependent Maxwell’s equations used to modeling wave propagation in dispersive lossy bi-isotropic media are *** and uniqueness of the modeling equations are *** fully discrete finite element sch... 详细信息

关键词： Maxwell’s equations bi-isotropic media finite element method

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Asymptotic stability of many numerical schemes for phase-field modeling

arXiv

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arXiv 2024年

作者： Li, Pansheng Wang, Dongling Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

The numerical stability of nonlinear equations has been a long-standing concern and there is no standard framework for analyzing long-term qualitative behavior. In the recent breakthrough work [XX23], a rigorous numerical analysis was conducted on the numerical solution of a scalar ODE containing a cubic polynomial derived from the Allen-Cahn equation. It was found that only the implicit Euler method converge to the correct steady state for any given initial value u0 under the unique solvability and energy stability. But all the other commonly used second-order numerical schemes exhibit sensitivity to initial conditions and may converge to an incorrect equilibrium state as tn → ∞. This indicates that energy stability may not be decisive for the long-term qualitative correctness of numerical solutions. We found that using another fundamental property of the solution, namely monotonicity instead of energy stability, is sufficient to ensure that many common numerical schemes converge to the correct equilibrium state. This leads us to introduce the critical step size constant h∗ = h∗(u0, ϵ) that ensures the monotonicity and unique solvability of the numerical solutions, where the scaling parameter ϵ ∈ (0, 1). For a given numerical method, if the initial value u0 is given, no matter how large it is, we prove that h∗ > 0. As long as the actual simulation step 0 ∗, the numerical solution preserves monotonicity and converges to the correct equilibrium state. On the other hand, we prove that the implicit Euler scheme h∗ = h∗(ϵ), which is independent of u0 and only depends on ϵ. Hence regardless of the initial value taken, the simulation can be guaranteed to be correct when h ∗. But for various other numerical methods, no mater how small the step size h is in advance, there will always be initial values that cause simulation errors. In fact, for these numerical methods, we prove that infu0∈R h∗(u0, ϵ) = 0. Various numerical experiments are used to confirm the theoretical anal

关键词： Nonlinear equations

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Adaptive Edge Finite Element Method and Numerical Design for Metasurface Cloak

SSRN

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SSRN 2023年

作者： Yang, Wei Wang, Tiancheng Mao, Jiangqiong Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

Metasurfaces are a type of metamaterial that have two-dimensional structures. In comparison to traditional metamaterials, metasurfaces have several advantages such as being lightweight, easily controllable, and simple to design. This paper outlines the phase matching principle and constitutive parameter theory of electromagnetic (EM) metasurfaces, as well as the constitutive parameter analysis and model equations for metasurface cloaks. Due to the singularity of metasurface cloaks, an adaptive edge finite element method (AEFEM) is developed in this paper. Additionally, the convergence theory of time-harmonic Maxwell’s equations for metasurface cloak is established under the T-coercivity assumption. The numerical simulations of arc-shaped metasurface cloaks have confirmed the accuracy of our model and numerical theory. Furthermore, we have utilized our numerical method to design a triangular-shaped metasurface cloak with fewer singularities. © 2023, The Authors. All rights reserved.

关键词： Maxwell equations

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New Upper Solution Bounds of a Class of the Continuous Algebraic Riccati Equations and Their Application in Redundant Control Inputs Systems

SSRN

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SSRN 2023年

作者： Liu, Jianzhou Tang, Fuying School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

In the redundant control inputs problems, the norm of the feedback matrix is smaller, the controlled system is better for control. Therefore, it is the key to solve these problems for studying upper and lower bounds of the symmetric positive definite solutions of the continuous algebra Riccati equations(CAREs) corresponding to the controlled systems. In this paper, we obtain a convergent matrix series as upper bound of the symmetric positive definite solution for a special class of the CAREs. It is the limit of monotonically decreasing upper bound sequence of the symmetric positive definite solution. We use numerous numerical experiments to illustrate that upper bounds obtained are better than the relevant results in the existing literature when the terms are sufficiently large. Moreover, we apply the main theoretical results to the related redundant control inputs systems. Under a wider range of conditions, a class of redundant control inputs problems is solved, and the recent relevant results are improved. © 2023, The Authors. All rights reserved.

关键词： Riccati equations

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SOLVING TIME-DEPENDENT PARAMETRIC PDES BY MULTICLASS CLASSIFICATION-BASED REDUCED ORDER MODEL

arXiv

引用

arXiv 2021年

作者： Cui, Chen Jiang, Kai Shu, Shi School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

In this paper, we propose a network model, the multiclass classification-based reduced order model (MC-ROM), for solving time-dependent parametric partial differential equations (PPDEs). This work is inspired by the observation of applying the deep learning-based reduced order model (DL-ROM) [14] to solve diffusion-dominant PPDEs. We find that the DL-ROM has a good approximation for some special model parameters, but it cannot approximate the drastic changes of the solution as time evolves. Based on this fact, we classify the dataset according to the magnitude of the solutions and construct corresponding subnets dependent on different types of data. Then we train a classifier to integrate different subnets together to obtain the MC-ROM. When subsets have the same architecture, we can use transfer learning techniques to accelerate offline training. Numerical experiments show that the MC-ROM improves the generalization ability of the DL-ROM both for diffusion- and convection-dominant problems, and maintains the DL-ROM’s advantage of good approximation ability. We also compare the approximation accuracy and computational efficiency of the proper orthogonal decomposition (POD) which is not suitable for convection-dominant problems. For diffusion-dominant problems, the MC-ROM has better approximation accuracy than the POD in a small dimensionality reduction space, and its computational performance is more efficient than the POD’s. Copyright © 2021, The Authors. All rights reserved.

关键词： Partial differential equations

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An adaptive high-order surface finite element method for polymeric self-consistent field theory on general curved surfaces

arXiv

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arXiv 2021年

作者： Jiang, Kai Wang, Xin Liu, Jianggang Wei, Huayi School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

In this paper, we develop an adaptive high-order surface finite element method (FEM) incorporating spectral deferred correction method for chain contour discretization to solve polymeric self-consistent field equations on general curved surfaces. The high-order surface FEM is obtained by the high-order surface geometrical approximation and the high-order function space approximation. Numerical results demonstrate that the precision order of these methods is consistent with theoretical prediction. In order to describe the sharp interface in the strongly segregated system more accurately, an adaptive FEM equipped with a new Log marking strategy is proposed. Compared with the traditional strategy, the Log marking strategy can not only label the elements that need to be refined or coarsened, but also give the refined or coarsened times, which can make full use of the information of a posterior error estimator and improve the efficiency of the adaptive algorithm. To demonstrate the power of our approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces. Numerical results illustrate the efficiency of the proposed method, especially for strongly segregated systems with economical discretization nodes. Copyright © 2021, The Authors. All rights reserved.

关键词： Molecular orbitals

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