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A lowest order divergence-free finite element on rectangular grids

中国数学前沿 2011年第2期6卷 253-270页

作者： Yunqing HUANG Shangyou ZHANG Department of Mathematical Sciences University of Delaware Newark DE 19716 USA Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan 411105 China

It is shown that the conforming Q2,1.1.2-Q'1.mixed element is stable, and provides optimal order of approximation for the Stokes equations on rectangular grids. Here, Q2,1.1.2 = Q2,1.× Q1.2, and Q2,1.denotes the space of continuous piecewise-polynomials of degree 2 or less in the x direction but of degree 1.in the y direction. Q'1.is the space of discontinuous bilinear polynomials, with spurious modes filtered. To be precise, Q'1.is the divergence of the discrete velocity space Q2,1.1.2. Therefore, the resulting finite element solution for the velocity is divergence-free pointwise, when solving the Stokes equations. This element is the lowest order one in a family of divergence-free element, similar to the families of the Bernardi-Raugel element and the RaviartThomas element.

关键词： Mixed finite element Stokes divergence-free element quadrilateral element rectangular grid

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Invariant-preserving schemes with arbitrarily high-order accuracy for the two-component Camassa–Holm system via Hamiltonian boundary value methods

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Communications in Nonlinear science and Numerical {1. 2025年 150卷

作者： Qian Luo Aiguo Xiao Min Li Xiaoqiang Yan School of Mathematics and Computational Science & Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan 411105 China Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan 411105 China School of Mathematics and Physics China University of Geosciences Wuhan 430074 China National Center for Applied Mathematics in Hunan Xiangtan University Hunan 411105 China

In this paper, we present an arbitrarily high-order numerical scheme for the two-component Camassa–Holm system, ensuring the preservation of three invariants: energy and two Casimir functions. The spatial discretization is achieved using Fourier–Galerkin methods, resulting in a semi-discrete system which retains a Hamiltonian structure and approximates the invariants of the original continuous system. Subsequently, an energy-preserving Runge–Kutta method, such as Hamiltonian boundary value methods, is applied to the semi-discrete system, yielding a fully discrete scheme of arbitrarily high order. The proposed scheme is validated through numerical simulations, demonstrating its accuracy and capability in capturing different types of solutions of the two-component Camassa–Holm equation.

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OPTIMAL ERROR ESTIMATES OF THE PARTITION OF UNITY METHOD WITH LOCAL POLYNOMIAL APPROXIMATION SPACES

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Journal of computational Mathematics 2006年第3期24卷 365-372页

作者： Yun-qing Huang Wei Li Fang Su Hunan Key Laboratory for Computation and Simulation in Science and Engineering Institute for Computational and Applied Mathematics Xiangtan University Xiangtan 4 11105 China Department of Computational Science Xiangtan University Xiangtan 4 11105 China

In this paper, we provide a theoretical method（PUFEM）, which belongs to the analysis of the partition of unity finite element family of meshfree methods. The usual error analysis only shows the order of error estimate to the same as the local approximations[1.]. Using standard linear finite element base functions as partition of unity and polynomials as local approximation space, in l-d case, we derive optimal order error estimates for PUFEM interpolants. Our analysis show that the error estimate is of one order higher than the local approximations. The interpolation error estimates yield optimal error estimates for PUFEM solutions of elliptic boundary value problems.

关键词： Meshless methods Partition of unity finite element method（PUFEM） Error estimate.

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Dependence Analysis of the Solutions on the Parameters of Fractional Delay Differential Equations

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Advances in Applied Mathematics and Mechanics 2011年第5期3卷 586-597页

作者： Shuiping Yang Aiguo Xiao Xinyuan Pan School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and EngineeringXiangtan UniversityHunan 411105China Department of Mathematics Huizhou UniversityGuangdong 516007China

In this paper,we investigate the dependence of the solutions on the parameters(order,initial function,right-hand function)of fractional delay differential equations(FDDEs)with the Caputo fractional *** results includi... 详细信息

关键词： Fractional delay differential equation Caputo fractional derivative dependence

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Proximity point algorithm for low-rank matrix recovery from sparse noise corrupted data

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Applied Mathematics and Mechanics(English Edition) 2014年第2期35卷 259-268页

作者：朱玮舒适成礼智 School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Department of Mathematics and Computational Science College of ScienceNational University of Defense Technology

The method of recovering a low-rank matrix with an unknown fraction whose entries are arbitrarily corrupted is known as the robust principal component analysis （RPCA）. This RPCA problem, under some conditions, can be exactly solved via convex optimization by minimizing a combination of the nuclear norm and the 1. norm. In this paper, an algorithm based on the Douglas-Rachford splitting method is proposed for solving the RPCA problem. First, the convex optimization problem is solved by canceling the constraint of the variables, and ~hen the proximity operators of the objective function are computed alternately. The new algorithm can exactly recover the low-rank and sparse components simultaneously, and it is proved to be convergent. Numerical simulations demonstrate the practical utility of the proposed algorithm.

关键词： low-rank matrix recovery sparse noise Douglas-Rachford splitting method,proximity operator

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PRECONDITIONING HIGHER ORDER FINITE ELEMENT SYSTEMS BY ALGEBRAIC MULTIGRID METHOD OF LINEAR ELEMENTS

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Journal of computational Mathematics 2006年第5期24卷 657-664页

作者： Yun-qing Huang Shi Shu Xi-jun Yu Hunan Key Laboratory for Computation and Simulation in Science and Engineering Institute for Computational and Applied Mathematics Xiangtan University Xiangtan 411105 China Laboratory of Computational Physics Institute of Applied physics and Computational Mathematics Beijing 100088 China

We present and analyze a robust preconditioned conjugate gradient method for the higher order Lagrangian finite element systems of a class of elliptic problems. An auxiliary linear element stiffness matrix is chosen to be the preconditioner for higher order finite elements. Then an algebraic multigrid method of linear finite element is applied for solving the preconditioner. The optimal condition number which is independent of the mesh size is obtained. Numerical experiments confirm the efficiency of the algorithm.

关键词： Finite element Algebraic multigrid methods Preconditioned Conjugate Gradient Condition number.

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A Priori and a Posteriori Error Analysis of the Discontinuous Galerkin Methods for Reissner-Mindlin Plates

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Advances in Applied Mathematics and Mechanics 2011年第6期3卷 649-662页

作者： Jun Hu Yunqing Huang LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105HunanChina

In this paper,we apply an a posteriori error control theory that we develop in a very recent paper to three families of the discontinuous Galerkin methods for the Reissner-Mindlin plate *** derive robust a posteriori ... 详细信息

关键词： A posteriori error analysis Reissner-Mindlin plate finite element reduction integration

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NUMERICAL SOLUTIONS OF NONAUTONOMOUS STOCHASTIC DELAY DIFFERENTIAL EQUATIONS BY DISCONTINUOUS GALERKIN METHODS

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Journal of computational Mathematics 2019年第3期37卷 419-436页

作者： Xinjie Dai Aiguo Xiao School of Mathematics and Computational Science Xiangtan UniversityXiangtan 411105China School of Mathematics and Computational Science & Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105China

This paper considers a class of discontinuous Galerkin method,which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis,for numerically solving nonautonomous Stratonovich stochastic delay differential *** prove that the discontinuous Galerkin scheme is strongly convergent:globally stable and analogously asymptotically stable in mean square *** addition,this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential *** tests indicate that the method has first-order and half-order strong mean square convergence,when the diffusion term is without delay and with delay,respectively.

关键词： Discontinuous Galerkin method Wong-Zakai approximation Nonautonomous Stratonovich stochastic delay differential equation

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THE A POSTERIORI ERROR ESTIMATOR OF SDG METHOD FOR VARIABLE COEFFICIENTS TIME-HARMONIC MAXWELL'S EQUATIONS

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Journal of computational Mathematics 2023年第2期41卷 263-286页

作者： Wei Yang Xin Liu Bin He Yunqing Huang Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105China Department of Mathematics Lanzhou Jiaotong UniversityLanzhou 730070China

In this paper,we study the a posteriori error estimator of SDG method for variable coefficients time-harmonic Maxwell's *** propose two a posteriori error estimators,one is the recovery-type estimator,and the other is the residual-type *** first propose the curl-recovery method for the staggered discontinuous Galerkin method(SDGM),and based on the super-convergence result of the postprocessed solution,an asymptotically exact error estimator is *** residual-type a posteriori error estimator is also proposed,and it's reliability and effectiveness are proved for variable coefficients time-harmonic Maxwell's *** efficiency and robustness of the proposed estimators is demonstrated by the numerical experiments.

关键词： Maxwell’s equations A posteriori error estimation Staggered discontinuous Galerkin

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GENERALIZED JACOBI SPECTRAL GALERKIN METHOD FOR FRACTIONAL-ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS

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Journal of computational Mathematics 2024年第2期42卷 355-371页

作者： Yanping Chen Zhenrong Chen Yunqing Huang School of Mathematical Sciences South China Normal UniversityGuangzhou 510631China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105China

For fractional Volterra integro-differential equations(FVIDEs)with weakly singular kernels,this paper proposes a generalized Jacobi spectral Galerkin *** basis functions for the provided method are selected generalized Jacobi functions(GJFs),which can be utilized as natural basis functions of spectral methods for weakly singular FVIDEs when appropriately *** developed method's spectral rate of convergence is determined using the L^(∞)-norm and the weighted L^(2)-*** results indicate the usefulness of the proposed method.

关键词： Generalized Jacobi spectral Galerkin method Fractional-order Volterra integ-ro-differential equations Weakly singular kernels Convergence analysis

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