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two-grid method FOR CHARACTERISTICS FINITE-ELEMENT SOLUTION OF 2D NONLINEAR CONVECTION-DOMINATED DIFFUSION PROBLEM

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Applied Mathematics and Mechanics(English Edition) 2005年第11期26卷 1506-1514页

作者：秦新强School of Sciences Xi’an Jiaotong University Xi’an 710049 P.R.China School of Sciences Xi’an University of Technology Xi’an 710048 P.R.China) 马逸尘章胤 School of Sciences Xi’an Jiaotong UniversityXi’an 710049 P.R.China School of Sciences Xi’an University of TechnologyXi’an 710048 P.R.China

For two-dimension nonlinear convection diffusion equation, a two-grid method of characteristics finite-element solution was constructed. In this method the nonlinear iterations is only to execute on the coarse grid and the fine-grid solution can be obtained in a single linear step. For the nonlinear convection-dominated diffusion equation, this method can not only stabilize the numerical oscillation but also accelerate the convergence and improve the computational efficiency. The error analysis demonstrates if the mesh sizes between coarse-grid and fine-grid satisfy the certain relationship, the two-grid solution and the characteristics finite-element solution have the same order of accuracy. The numerical is more efficient than that of characteristics example confirms that the two-grid method finite-element method.

关键词： convection-diffusion equation characteristics finite-element two-grid method convergence

Analysis of a two-grid method for semiconductor device problem

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Applied Mathematics and Mechanics(English Edition) 2021年第1期42卷 143-158页

作者： Ying LIU Yanping CHEN Yunqing HUANG Qingfeng LI School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and EngineeringXiangtan UniversityXiangtan 411105Hunan ProvinceChina College of Information and Intelligence Hunan Agricultural UniversityChangsha 410128China School of Mathematical Sciences South China Normal UniversityGuangzhou 510631China

The mathematical model of a semiconductor device is governed by a system of quasi-linear partial differential *** electric potential equation is approximated by a mixed finite element method,and the concentration equations are approximated by a standard Galerkin *** estimate the error of the numerical solutions in the sense of the *** linearize the full discrete scheme of the problem,we present an efficient two-grid method based on the idea of Newton *** main procedures are to solve the small scaled nonlinear equations on the coarse grid and then deal with the linear equations on the fine *** estimation for the two-grid solutions is analyzed in *** is shown that this method still achieves asymptotically optimal approximations as long as a mesh size satisfies H=O(h^1/2).Numerical experiments are given to illustrate the efficiency of the two-grid method.

关键词： two-grid method semiconductor device mixed finite element method Galerkin method L^q error estimate

A two-grid method with Expanded Mixed Element for Nonlinear Reaction-diffusion Equations

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Acta Mathematicae Applicatae Sinica 2011年第3期27卷 495-502页

作者： Wei Liu Hong-xing Rui Hui Guo School of Mathematics Shandong University Jinan 250100 China School of Mathematics and Information Ludong University Yantai 264025 China ~School of Mathematics and Computational Science China University of Petroleum Dongying 257061 China

Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O（△t ＋ h k＋1 ＋ H 2k＋2 d/2 ）（k ≥ 1）, where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.

关键词： two-grid method expanded mixed finite element reaction-diffusion equation nonlinear problem

two-grid Crank-Nicolson FiniteVolume Element method for the Time-Dependent Schrodinger Equation

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Advances in Applied Mathematics and Mechanics 2022年第6期14卷 1357-1380页

作者： Chuanjun Chen Yuzhi Lou Tong Zhang School of Mathematics and Information Sciences Yantai UniversityYantaiShandong 264005China

In this paper,we construct a Crank-Nicolson finite volume element scheme and a two-grid decoupling algorithm for solving the time-dependent Schr¨odinger *** the idea of two-grid discretization,the decoupling algorithm involves solving a small coupling system on a coarse grid space and a decoupling system with two independent Poisson problems on a fine grid space,which can ensure the accuracy while the size of coarse grid is much coarser than that of fine *** further provide the optimal error estimate of these two schemes rigorously by using elliptic projection ***,numerical simulations are provided to verify the correctness of the theoretical analysis.

关键词： Finite volume element method two-grid method Crank-Nicolson scheme error estimates Schr¨odinger equation.

two-grid partition of unity method for second order elliptic problems

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Applied Mathematics and Mechanics(English Edition) 2008年第4期29卷 527-533页

作者：王琤黄自萍李立康 Department of Mathematics Tongji University Chinese-German College Tongji University Department of Mathematics Fudan University

A two-grid partition of unity method for second order elliptic problems is proposed and analyzed. The standard two-grid method is a local and parallel method usually leading to a discontinuous solution in the entire computational domain. Partition of unity method is employed to glue all the local solutions together to get the global continuous one, which is optimal in HI-norm. Furthermore, it is shown that the L^2 error can be improved by using the coarse grid correction. Numerical experiments are reported to support the theoretical results.

关键词： second order elliptic problems two-grid method partition of unity

two-grid Finite-Element method for the two-Dimensional Time-Dependent Schrodinger Equation

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Advances in Applied Mathematics and Mechanics 2013年第2期5卷 180-193页

作者： Hongmei Zhang Jicheng Jin Jianyun Wang School of Mathematics and Computational Science Xiangtan UniversityXiangtan 411105HunanChina School of Science Hunan University of TechnologyZhuzhou 412007HunanChina

In this paper,we construct semi-discrete two-grid finite element schemes and full-discrete two-grid finite element schemes for the two-dimensional timedependent Schrodinger *** semi-discrete schemes are proved to be convergent with an optimal convergence order and the full-discrete schemes,verified by a numerical example,work well and are more efficient than the standard finite element method.

关键词： Schrodinger equation two-grid method finite element 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 *** 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,***,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

THE ACCELERATED SEARCH-EXTENSION method FOR COMPUTING MULTIPLE SOLUTIONS OF SEMILINEAR PDEs

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Acta Mathematica Scientia 2009年第4期29卷 803-816页

作者：刘跃武谢资清陈传淼 Key Laboratory of Computational and Stochastic Mathematics and Its Applications Universities of Hunan Province Hunan Normal University College of Science Hunan Agricultural University

In this paper, we propose an accelerated search-extension method （ASEM） based on the interpolated coefficient finite element method, the search-extension method （SEM） and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in I-D and 2-D cases will show the efficiency of our approach.

关键词： semilinear PDEs multiple solutions accelerated search-extension method (ASEM) two-grid method

A two-Level method for Nonsymmetric Eigenvalue Problems

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Acta Mathematicae Applicatae Sinica 2005年第1期21卷 1-12页

作者： Karel Kolman Mathematical Institute Academy of Sciences of the Czech Republic ■itná 25 11567 Praha 1 Czech Republic

A two-level discretization method for eigenvalue problems is *** to the standard Galerkin finite element discretization technique performed on a fine gridthis method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector(eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problemsfor the case of eigenvalue approximation of nonsymmetric problems). The improved solution has theasymptotic accuracy of the Galerkin discretization solution. The link between the method and theiterated Galerkin method is established. Error estimates for the general nonsymmetric case arederived.

关键词： eigenvalue problems finite elements postprocessing two-level method two-grid method iterated Galerkin method