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TWO-GRID ALGORITHM OF H^(1)-GALERKIN MIXED FINITE ELEMENT METHODS FOR SEMILINEAR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS

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Journal of computational Mathematics 2022年第5期40卷 667-685页

作者： Tianliang Hou Chunmei Liu Chunlei Dai Luoping Chen Yin Yang School of Mathematics and Statistics Beihua UniversityJilin 132013China College of Science Hunan University of Science and EngineeringYongzhou 425199China School of Mathematics Southwest Jiaotong UniversityChengdu 611756China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing&Information Processing of Ministry of EducationSchool of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105China

In this paper,we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by H1-Galerkin mixed finite element *** use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization,and backward Euler scheme for temporal ***,a priori error estimates and some superclose properties are ***,a two-grid scheme is presented and its convergence is *** the proposed two-grid scheme,the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal ***,a numerical experiment is implemented to verify theoretical results of the proposed *** theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice h=H^(2).

关键词： Semilinear parabolic integro-differential equations H^(1)-Galerkin mixed finite element method A priori error estimates Two-grid Superclose.

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Local convergence analysis of L1/finite element scheme for a constant delay reaction-subdiffusion equation with uniform time mesh

arXiv

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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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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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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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Analytical solutions for Earth discontinuous coverage of satellite constellation with repeating ground tracks

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Chinese Journal of Aeronautics 2022年第10期35卷 275-291页

作者： Xiangyue HE Haiyang LI College of Aerospace Science and Engineering National University of Defense TechnologyChangsha 410073China Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions Changsha 410073China

This paper presents an analytical model for calculating the Earth discontinuous coverage of satellite constellation with repeating ground tracks by integrating and extending the application of coverage region and route ***,the visibility condition for a ground point is represented as a coverage region in the two-dimension map of visibility properties,and the trajectories of satellites with circular orbits and repeating ground tracks are converted to several inclined lines in the *** analyzing the intersections of the lines and the edge of the coverage region,the coverage durations for the ground point can be *** on the point coverage,the variations of coverage characteristics along the parallel are analyzed,and the regional or global coverage characteristics of constellations can be *** examples show that the proposed method can accurately and rapidly calculate the coverage characteristics,*** time and coverage *** calculated results are extremely close to those of the Satellite Tool Kit(STK)and are also superior to the existing research *** proposed analytical model can be a useful tool for constellation design and coverage performance analysis.

关键词： Analytical model Coverage region Earth discontinuous coverage Repeating ground track Satellite constellation Two-dimension map of visibility properties

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A Convergence Analysis of a Structure-Preserving Gradient Flow Method for the All-Electron Kohn-Sham Model

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Numerical Mathematics(Theory,Methods and Applications) 2023年第3期16卷 597-621页

作者： Yedan Shen Ting Wang Jie Zhou Guanghui Hu School of Mathematics and Information Science Guangzhou UniversityGuangzhouChina Faculty of Science and Technology University of MacaoMacao SARChina Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtanChina School of Mathematical Computational Sciences Xiangtan UniversityXiangtanChina Zhuhai UM Science&Technology Research Institute ZhuhaiChina Guangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications University of MacaoMacao SARChina

In[Dai et al.,***.18(4)(2020)],a structure-preserving gradient flow method was proposed for the ground state calculation in Kohn-Sham density functional theory,based on which a linearized method was developed in[Hu et al.,EAJAM.13(2)(2023)]for further improving the numerical *** this paper,a complete convergence analysis is delivered for such a linearized method for the all-electron Kohn-Sham ***,the convergence,the asymptotic stability,as well as the structure-preserving property of the linearized numerical scheme in the method is discussed following previous works,while spatially,the convergence of the h-adaptive mesh method is demonstrated following[Chen et al.,***.12(2014)],with a key study on the boundedness of the Kohn-Sham potential for the all-electron Kohn-Sham *** examples confirm the theoretical results very well.

关键词： Kohn-Sham density functional theory gradient flow model structure-preserving linear scheme convergence analysis

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On the vanishing dissipation limit for the incompressible MHD equations on bounded domains

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science China Mathematics 2022年第1期65卷 31-50页

作者： Qin Duan Yuelong Xiao Zhouping Xin Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105China College of Mathematics and Statistics Shenzhen UniversityShenzhen 518060China The Institute of Mathematical Sciences The Chinese University of Hong KongHong Kong 999077China

In this paper,we investigate the solvability,regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magnetohydrodynamic(MHD)equations in bounded *** the boundary,the velocity field fulfills a Navier-slip condition,while the magnetic field satisfies the insulating *** is shown that the initial boundary value problem has a global weak solution for a general smooth *** importantly,for a flat domain,we establish the uniform local well-posedness of the strong solution with higher-order uniform regularity and the asymptotic convergence with a rate to the solution of the ideal MHD equation as the dissipations tend to zero.

关键词： MHD equations initial boundary value problem vanishing dissipation limit

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Design of Multi-Coupled Laminates with Extension-Twisting Coupling for Application in Adaptive Structures

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Computer Modeling in engineering & sciences 2022年第1期130卷 415-441页

作者： Da Cui Daokui Li Shiming Zhou College of Aerospace Science and Engineering National University of Defense TechnologyChangsha410073China Hunan Key Laboratory of Intelligent Planning and Simulation for AerospaceMissions Changsha410073China

The multiple coupling of composite laminates has a unique advantage in improving the macro mechanical properties of composite structures.A total of three hygro-thermally stablemulti-coupled laminates with extensiontwisting coupling were presented,which were conducive to the formation of passive adaptive ***,the multi-coupled laminates were used to design the bending-twisting coupled box structure,in which the configuration of laminate and box structure could be extended to variable cross-section *** optimal design of stacking sequence was realized,the optimization objectives of which were to maximize bending-twisting coupling of box structure and extension-twisting coupling of laminate,*** effects of multiple coupling on hygro-thermal stability,coupling,failure strength,buckling load,robustness and other comprehensive mechanical properties of laminates and box structures were analyzed by parametric modeling *** results show that the extension-twisting coupling of laminate and the bending-twisting coupling of box structures can be greatly improved by 450%and 260%at maximum,***,it would have a negative impact on the failure strength and buckling load,which,however,can be minimized by a reasonable paving *** laminates have good robustness,and the bending-twisting coupling helps improve ***,the hygro-thermal stability and mechanical properties were verified by numerical simulation with finite element method.

关键词： Multi-coupled laminate passive adaptive structures optimal design mechanical properties numerical simulation

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Kernel-learning parameter prediction and evaluation in algebraic multigrid method for several PDEs

arXiv

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arXiv 2025年

作者： Zhang, Juan Luo, Junyue Zhang, Fangfang Yue, Xiaoqiang 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

This paper explores the application of kernel learning methods for parameter prediction and evaluation in the Algebraic Multigrid Method (AMG), focusing on several Partial Differential Equation (PDE) problems. AMG is an efficient iterative solver for large-scale sparse linear systems, particularly those derived from elliptic and parabolic PDE discretizations. However, its performance heavily relies on numerous parameters, which are often set empirically and are highly sensitive to AMG’s effectiveness. Traditional parameter optimization methods are either computationally expensive or lack theoretical support. To address this, we propose a Gaussian Process Regression (GPR)-based strategy to optimize AMG parameters and introduce evaluation metrics to assess their effectiveness. Trained on small-scale datasets, GPR predicts nearly optimal parameters, bypassing the time-consuming parameter sweeping process. We also use kernel learning techniques to build a kernel function library and determine the optimal kernel function through linear combination, enhancing prediction accuracy. In numerical experiments, we tested typical PDEs such as the constant-coefficient Poisson equation, variable-coefficient Poisson equation, diffusion equation, and Helmholtz equation. Results show that GPR-predicted parameters match grid search results in iteration counts while significantly reducing computational time. A comprehensive analysis using metrics like mean squared error, prediction interval coverage, and Bayesian information criterion confirms GPR’s efficiency and reliability. These findings validate GPR’s effectiveness in AMG parameter optimization and provide theoretical support for AMG’s practical application. © 2025, CC BY.

关键词： Poisson equation

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Shock wave and bubble characteristics of underwater array explosion of charges

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Defence Technology（防务技术） 2022年第8期18卷 1445-1453页

作者： Hong-Wei Hu Pu Song Shuang-feng Guo Hai-yun Feng Dao-kui Li College of Aerospace Science and Engineering National University of Defense TechnologyChangsha410073HunanChina Xi'an Modern Chemistry Research Institute Xi'an710065ShaanxiChina Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions Changsha410073HunanChina

In this study, we focused on the effect of the underwater explosion parameters of multi-point array explosion. The shock wave and bubble parameters of aggregate charge, two charges, and four charges were measured through an underwater explosion test, and their influence on the explosion power field of charge quantity and array distance was analyzed. Results show that the multi-shock wave collision of array explosion can be approximated to a linear superposition, and the interaction of delayed shock wave can be deemed as the increase of the shock wave baseline. Shock wave focusing and delayed superposition increase the shock wave peak pressure. Compared with the aggregate charge, the greater the number of array explosion points is, the higher the impulse and the gain of the bubble peak pressure are. At the same array distance, the smaller the charge quantity is, the higher the bubble impulse will be. At the same charge quantity, the smaller the array distance is, the higher the bubble impulse will be. The bubble period decreases gradually with the increase of the charge quantity, but the test orientation has little effect on the bubble period.

关键词： Explosion mechanics Underwater explosion Array explosion Shock waveBubble

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