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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 method.The 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 constructed.The developed method's spectral rate of convergence is determined using the L^(∞)-norm and the weighted L^(2)-norm.Numerical 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

Parallel finite element computation of incompressible magnetohydrodynamics based on three iterations

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Applied Mathematics and Mechanics(English Edition) 2022年第1期43卷 141-154页

作者： Qili TANG yunqing huang 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 UniversityXiangtanHunan Province411105China

Based on local algorithms,some parallel finite element(FE)iterative methods for stationary incompressible magnetohydrodynamics(MHD)are presented.These approaches are on account of two-grid skill include two major phases:find the FE solution by solving the nonlinear system on a globally coarse mesh to seize the low frequency component of the solution,and then locally solve linearized residual subproblems by one of three iterations(Stokes-type,Newton,and Oseen-type)on subdomains with fine grid in parallel to approximate the high frequency component.Optimal error estimates with regard to two mesh sizes and iterative steps of the proposed algorithms are given.Some numerical examples are implemented to verify the algorithm.

关键词： local and parallel algorithm finite element(FE)method iteration stationary incompressible magnetohydrodynamics(MHD)

A Robust Modified Weak Galerkin Finite Element Method for Reaction-Diffusion Equations

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Numerical Mathematics(Theory,Methods and Applications) 2022年第1期15卷 68-90页

作者： Guanrong Li Yanping Chen yunqing huang School of Mathematics and Statistics Lingnan Normal UniversityZhanjiang 524048GuangdongP.R.China School of Mathematical Science South China Normal UniversityGuangzhou 510631GuangdongP.R.China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105HunanP.R.China

In this paper,a robust modified weak Galerkin(MWG)finite element method for reaction-diffusion equations is proposed and investigated.An advantage of this method is that it can deal with the singularly perturbed reaction-diffusion equations.Another advantage of this method is that it produces fewer degrees of freedom than the traditional WG method by eliminating the element boundaries freedom.It is worth pointing out that,in our method,the test functions space is the same as the finite element space,which is helpful for the error analysis.Optimalorder error estimates are established for the corresponding numerical approximation in various norms.Some numerical results are reported to confirm the theory.

关键词： Reaction-diffusion equations singular perturbation modified weak Galerkin finite element methods discrete gradient

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 equations.We propose two a posteriori error estimators,one is the recovery-type estimator,and the other is the residual-type estimator.We 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 constructed.The residual-type a posteriori error estimator is also proposed,and it's reliability and effectiveness are proved for variable coefficients time-harmonic Maxwell's equations.The 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

Piecewise Spectral Collocation Method for Second Order Volterra Integro-Differential Equations with Nonvanishing Delay

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Advances in Applied Mathematics and Mechanics 2022年第6期14卷 1333-1356页

作者： Zhenrong Chen Yanping Chen yunqing huang School of Mathematical Sciences South China Normal UniversityGuangzhouGuangdong 510631China School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and EngineeringXiangtan UniversityXiangtanHunan 411105China

In this paper,the piecewise spectral-collocation method is used to solve the second-order Volterra integral differential equation with nonvanishing delay.In this collocation method,the main discontinuity point of the solution of the equation is used to divide the partitions to overcome the disturbance of the numerical error convergence caused by the main discontinuity of the solution of the equation.Derivative approximation in the sense of integral is constructed in numerical format,and the convergence of the spectral collocation method in the sense of the L¥and L2 norm is proved by the Dirichlet formula.At the same time,the error convergence also meets the effect of spectral accuracy convergence.The numerical experimental results are given at the end also verify the correctness of the theoretically proven results.

关键词： Second-order Volterra type integro-differential equation delay function piecewise spectral-collocation method.

A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS

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Journal of Computational Mathematics 2022年第6期40卷 936-954页

作者： Yanping Chen Qiling Gu Qingfeng Li yunqing huang School of Mathematical Sciences South China Normal UniversityGuangzhou 510631China School of Mathematics and Computational Science Xiangtan UniversityXiangtan 411199China

In this paper,we develop a two-grid method(TGM)based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations.A two-grid algorithm is proposed for solving the nonlinear system,which consists of two steps:a nonlinear FE system is solved on a coarse grid,then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution.The fully discrete numerical approximation is analyzed,where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with orderα∈(1,2)andα1∈(0,1).Numerical stability and optimal error estimate O(h^(r+1)+H^(2r+2)+τ^(min{3−α,2−α1}))in L^(2)-norm are presented for two-grid scheme,where t,H and h are the time step size,coarse grid mesh size and fine grid mesh size,respectively.Finally,numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.

关键词： Two-grid method Finite element method Nonlinear time fractional mixed sub-diffusion and diffusion-wave equations L1-CN scheme Stability and convergence

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 constraints.The control problem is approxi-mated by the hp spectral element method with high accuracy and geometricﬂexibil-ity.Optimality conditions of the continuous and discrete optimal control problems are presented,respectively.The a posteriori error estimates both for the control and state variables are established in detail.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.

A Finite Element Variational Multiscale Method for Stationary Incompressible Magnetohydrodynamics Equations

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Advances in Applied Mathematics and Mechanics 2023年第5期15卷 1142-1165页

作者： Huayi huang yunqing huang Qili Tang Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of IntelligentComputing&Information Processing of Ministry of EducationSchool of Mathematics and Computational ScienceXiangtan UniversityXiangtanHunan 411105China

In this paper,we propose a variational multiscale method(VMM)for the stationary incompressible magnetohydrodynamics equations.This method is defined by large-scale spaces for the velocity field and the magnetic field,which aims to solve flows at high Reynolds numbers.We provide a new VMM formulation and prove its stability and convergence.Finally,some numerical experiments are presented to indicate the optimal convergence of our method.

关键词： Variational multiscale method stationary incompressible magnetohydrodynamics large-scale spaces stability and convergence high Reynolds numbers

Mathematical Simulation of Cloaking Metamaterial Structures

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Advances in Applied Mathematics and Mechanics 2012年第1期4卷 93-101页

作者： Jichun Li yunqing huang Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105China Department of Mathematical Sciences University of Nevada Las VegasLas VegasNevada 89154-4020USA

In this paper we present a rigorous derivation of the material parameters for both the cylinder and rectangle cloaking structures.Numerical results using these material parameters are presented to demonstrate the cloa... 详细信息

关键词： Maxwell’s equations metamaterial finite element method cloaking structures

AN EFFICIENT MOVING MESH METHOD FOR A MODEL OF TURBULENT FLOW IN CIRCULAR TUBES

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Journal of Computational Mathematics 2009年第2期27卷 388-399页

作者： Yin Yang Yanping Chen yunqing huang Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan 411105 China School of Mathematical Sciences South China Normal University Guangzhou China

This paper presents an efficient moving problem with an optimal control constrained mesh method to solve a nonlinear singular condition. The physical problem is governed by a new model of turbulent flow in circular tubes proposed by Luo et al. using Prandtl＇s mixing-length theory. Our algorithm is formed by an outer iterative algorithm for handling the optimal control condition and an inner adaptive mesh redistribution algorithm for solving the singular governing equations. We discretize the nonlinear problem by using a upwinding approach, and the resulting nonlinear equations are solved by using the Newton- Raphson method. The mesh is generated and the grid points are moved by using the arc-length equidistribution principle. The numerical results demonstrate that proposed algorithm is effective in capturing the boundary layers associated with the turbulent model.

关键词： Eddy viscosity Turbulent pipe flow Boundary layer Optimal control Moving mesh