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CONVERGENCE ANALYSIS OF SOME FINITE ELEMENT PARALLEL ALGORITHMS FOR THE STATIONARY INCOMPRESSIBLE MHD EQUATIONS

Journal of computational Mathematics 2024年第1期42卷 49-70页

作者： Xiaojing Dong Yinnian He 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 Institute of Applied Physics and Computational Mathematics Beijing 100088China School of Mathematics and Statistics Xi'an Jiaotong UniversityXi'an 710049China

By combination of iteration methods with the partition of unity method(PUM),some finite element parallel algorithms for the stationary incompressible magnetohydrodynamics(MHD)with different physical parameters are presented and *** algorithms are highly *** first,a global solution is obtained on a coarse grid for all approaches by one of the iteration *** parallelized residual schemes,local corrected solutions are calculated on finer meshes with overlapping *** subdomains can be achieved flexibly by a class of *** proposed algorithm is proved to be uniformly stable and ***,one numerical example is presented to confirm the theoretical findings.

关键词： Partition of unity method Local and parallel algorithm Finite element method Iteration methods Magnetohydrodynamics

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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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Multiscale Modeling with Online Correction for Darcy–Forchheimer Flow

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Lobachevskii Journal of Mathematics 2024年第11期45卷 5424-5436页

作者： Spiridonov, D.A. Huang, J. Laboratory of Computational Technologies and Artificial Intelligence North-Eastern Federal University Yakutsk 677000 Russian Federation School of Mathematics and Computational Science Xiangtan University National Center for Applied Mathematics in Hunan and Hunan Key Laboratory for Computation and Simulation in Science and Engineering Hunan Xiangtan 411105 China

Abstract: In this research, we present an algorithm of the Online generalized multiscale finite element method (Online GMsFEM) for Darcy–Forchheimer model in fractured media. The mathematical model describes a nonlinear Darcy flow with a high inertial effect and flow speed. The fine grid approximation is carried out by the finite element method (FEM) and to approximate lower dimensional fractures we use a discrete fracture model (DFM). We use a model reduction approach called Online GMsFEM, that is based on local residuals of coarse neighborhoods. The online multiscale basis functions are constructed in each local domain based on residuals. Such online multiscale basis functions make it possible take into account the effects of nonlinear coefficients in the Darcy–Forchheimer equation. We present numerical results in two-dimensional heterogeneous fractured domain. We investigate the accuracy of the method, depending on the influence of nonlinearity. The numerical experiments show that the Online GMsFEM has high accuracy for such nonlinear problems. The accuracy weakly depends on the magnitude of the nonlinear part of the equation. © Pleiades Publishing, Ltd. 2024.

关键词： Darcy–Forchheimer model discrete fracture model finite element method fractured domain multiscale model reduction nonlinear flow problem Online generalized multiscale finite element method

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Superconvergence of Rectangular Mixed Finite Element Methods for Constrained Optimal Control Problem

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Advances in Applied Mathematics and Mechanics 2010年第1期2卷 56-75页

作者： Yanping Chen Li Dai Zuliang Lu School of Mathematical Sciences South China Normal UniversityGuangzhouGuangdongChina Mathematics Teaching and Research Group Xiguan Foreign Language SchoolGuangzhouGuangdongChina Hunan Key Laboratory for Computation and Simulation in Science and Engineering Department of MathematicsXiangtan UniversityXiangtanHunanChina

We investigate the superconvergence properties of the constrained quadratic elliptic optimal control problem which is solved by using rectangular mixed finite element *** use the lowest order Raviart-Thomas mixed finite element spaces to approximate the state and co-state variables and use piecewise constant functions to approximate the control *** obtain the superconvergence of O(h^(1.s))(01.for the control ***,we present two numerical examples to confirm our superconvergence results.

关键词： Constrained optimal control problem linear elliptic equation mixed finite element methods rectangular partition superconvergence properties

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Mathematical simulation of Metamaterial Solar Cells

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Advances in Applied Mathematics and Mechanics 2011年第6期3卷 702-715页

作者： Jichun Li Yitung Chen Yang Liu 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 Mechanical Engineering University of Nevada Las VegasLas VegasNevada 89154-4027USA

In this paper,we propose several solar cell designs based on *** numerical simulations of various designs with different materials are carried *** tests show that metamaterial solar cells are quite efficient,and over ... 详细信息

关键词： Maxwell’s equations metamaterial solar cell finite element method

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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 *** advantage of this method is that it can deal with the singularly perturbed reaction-diffusion *** advantage of this method is that it produces fewer degrees of freedom than the traditional WG method by eliminating the element boundaries *** 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 *** error estimates are established for the corresponding numerical approximation in various *** numerical results are reported to confirm the theory.

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

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Jacobi Spectral Collocation Method Based on Lagrange Interpolation Polynomials for Solving Nonlinear Fractional Integro-Differential Equations

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Advances in Applied Mathematics and Mechanics 2018年第6期10卷 1440-1458页

作者： Xingfa Yang Yin Yang Yanping Chen Jie Liu State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body Hunan UniversityChangsha 410082HunanChina 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 411105HunanChina School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongChina Center for Research on Leading Technology of Special Equipment School of Mechanical and Electric EngineeringGuangzhou UniversityGuangzhou 510006GuangdongChina

In this paper,we study a class of nonlinear fractional integro-differential equations,the fractional derivative is described in the Caputo *** the properties of the Caputo derivative,we convert the fractional integro-differential equations into equivalent integral-differential equations of Volterra type with singular kernel,then we propose and analyze a spectral Jacobi-collocation approximation for nonlinear integro-differential equations of Volterra *** provide a rigorous error analysis for the spectral methods,which shows that both the errors of approximate solutions and the errors of approximate fractional derivatives of the solutions decay exponentially in L^(∞)-norm and weighted L^(2)-norm.

关键词： Spectral method nonlinear fractional derivative Volterra integro-differential equations Caputo derivative

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Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel

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Advances in Applied Mathematics and Mechanics 2017年第6期9卷 1506-1524页

作者： Xiong Liu Yanping Chen Hunan Key Laboratory for Computation and Simulation in Science and Engineering Department of MathematicsXiangtan UniversityXiangtanHunan 411105China School of Mathematical Sciences South China Normal UniversityGuangzhouGuangdong 510631China School of Mathematics and Statistics Lingnan Normal UniversityZhanjiangGuangdong 524048China

In this paper,a Chebyshev-collocation spectral method is developed for Volterra integral equations(VIEs)of second kind with weakly singular *** first change the equation into an equivalent VIE so that the solution of the new equation possesses better *** integral term in the resulting VIE is approximated by Gauss quadrature formulas using the Chebyshev collocation *** convergence analysis of this method is based on the Lebesgue constant for the Lagrange interpolation polynomials,approximation theory for orthogonal polynomials,and the operator *** spectral rate of convergence for the proposed method is established in the L^(∞)-norm and weighted L^(2)-*** results are presented to demonstrate the effectiveness of the proposed method.

关键词： Chebyshev collocation method Volterra integral equations spectral rate of convergence H¨older continuity

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Spectral Galerkin Approximation of Fractional Optimal Control Problems with Fractional Laplacian

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Advances in Applied Mathematics and Mechanics 2023年第6期15卷 1631-1654页

作者： Jiaqi Zhang Yin Yang Zhaojie Zhou School of Mathematics and Computational Science Xiangtan UniversityHunan Key Laboratory for Computation and Simulation in Science and EngineeringKey Laboratory of Intelligent Computing and Information Processing of Ministry of EducationXiangtan 411105HunanChina School of Mathematics and Computational Science Xiangtan UniversityNational Center for Applied Mathematics in HunanHunan International Scientific and Technological Innovation Cooperation Base of Computational ScienceXiangtan 411105HunanChina School of Mathematics and Statistics Shandong Normal UniversityJinan 250014ShandongChina

In this paper spectral Galerkin approximation of optimal control problem governed by fractional elliptic equation is *** deal with the nonlocality of fractional Laplacian operator the Caffarelli-Silvestre extension is *** first order optimality condition of the extended optimal control problem is derived.A spectral Galerkin discrete scheme for the extended problem based on weighted Laguerre polynomials is developed.A priori error estimates for the spectral Galerkin discrete scheme is *** experiments are presented to show the effectiveness of our methods and to verify the theoretical findings.

关键词： Fractional Laplacian optimal control problem Caffarelli-Silvestre extension weighted Laguerre polynomials

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RECOVERY A POSTERIORI ERROR ESTIMATES FOR GENERAL CONVEX ELLIPTIC OPTIMAL CONTROL PROBLEMS SUBJECT TO POINTWISE CONTROL CONSTRAINTS

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Journal of computational Mathematics 2009年第4期27卷 543-560页

作者： Yanping Chen Yao Fu Huanwen Liu Yongquan Dai Huayi Wei School of Mathematical Sciences South China Normal University Guangzhou 510631 China College of Mechanical Engineering Jiangxi Yu Zhou Scientific and Technological Institute Xinyu 338029 China School of Mathematics and Computer Science Guangxi University for Nationalities Nanning 530006 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University

Superconvergence and recovery a posteriori error estimates of the finite element ap- proximation for general convex optimal control problems are investigated in this paper. We obtain the superconvergence properties of finite element solutions, and by using the superconvergence results we get recovery a posteriori error estimates which are asymptotically exact under some regularity conditions. Some numerical examples are provided to verify the theoretical results.

关键词： General convex optimal control problems Finite element approximation Control constraints Superconvergence Recovery operator.

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