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Journal of Computational Mathematics 2024年第4期42卷 911-931页

作者： Yuping Zeng Mingchao Cai liuqiang zhong School of Mathematics Jiaying UniversityMeizhou 514015China Department of Mathematics Morgan State University1700 E Cold Spring LnBaltimoreMD 21251USA School of Mathematical Science South China Normal UniversityGuangzhou 520631China

A mixed finite element method is presented for the Biot consolidation problem in *** precisely,the displacement is approximated by using the Crouzeix-Raviart nonconforming finite elements,while the fuid pressure is approximated by using the node conforming finite *** well-posedness of the fully discrete scheme is established,and a corresponding priori error estimate with optimal order in the energy norm is also *** experiments are provided to validate the theoretical results.

关键词： Poroelasticity Nonconforming finite element Inf-sup condition A priori error estimate

Quasi-optimal complexity of adaptive finite element method for linear elasticity problems in two dimensions

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Applied Mathematics and Mechanics(English Edition) 2016年第2期37卷 151-168页

作者： Chunmei LIU liuqiang zhong Shi SHU Yingxiong XIAO Institute for Computational Mathematics College of Science Hunan University of Science and Engineering School of Mathematical Sciences South China Normal University School of Mathematics and Computational Science Xiangtan University College of Civil Engineering and Mechanics Xiangtan University

This paper introduces an adaptive finite element method （AFEM） using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination of the global lower bound and the localized upper bound of the posteriori error estimator, perturbation of oscillation, and cardinality of the marked element set, it is proved that the AFEM is quasi-optimal for linear elasticity problems in two dimensions, and this conclusion is verified by the numerical examples.

关键词： linear elasticity problem adaptive finite element method (AFEM) quasioptimal complexity

Discontinuous Galerkin Methods for Semilinear Elliptic Boundary Value Problem

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Advances in Applied Mathematics and Mechanics 2023年第2期15卷 450-467页

作者： Jiajun Zhan liuqiang zhong Jie Peng School of Mathematical Sciences South China Normal UniversityGuangzhouGuangdong 510631China School of Mathematics and Systems Science Guangdong Polytechnic Normal UniversityGuangzhouGuangdong 510665China

A discontinuous Galerkin(DG)scheme for solving semilinear elliptic problem is developed and analyzed in this *** DG finite element discretization is first established,then the corresponding well-posedness is provided by using Brouwer’s fixed point *** optimal priori error estimates under both DG norm and L^(2)norm are presented,*** results are given to illustrate the efficiency of the proposed approach.

关键词： Semilinear elliptic problem discontinuous Galerkin method error estimates

OPTIMAL ERROR ESTIMATES FOR NEDELEC EDGE ELEMENTS FOR TIME-HARMONIC MAXWELL'S EQUATIONS

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Journal of Computational Mathematics 2009年第5期27卷 563-572页

作者： liuqiang zhong Shi Shu Gabriel Wittum Jinchao Xu School of Mathematical and Computational Sciences Xiangtan University Xiangtan 411105 China Simulation and Modelling Goethe- Center for Scientic Computing Goethe-University Kettenhofweg 139 60325 Frankfurt am Main Germany Department of Mathematics Pennsylvania State University University Park PA 16802 USA

In this paper, we obtain optimal error estimates in both L^2-norm and H（curl）-norm for the Nedelec edge finite element approximation of the time-harmonic Maxwell＇s equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the L^2 error estimates into the L^2 estimate of a discrete divergence-free function which belongs to the edge finite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For Nedelec＇s second type elements, we present an optimal convergence estimate which improves the best results available in the literature.

关键词： Edge finite element Time-harmonic Maxwell's equations.

An Iterative Two-Grid Method of A Finite Element PML Approximation for the Two Dimensional Maxwell Problem

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Advances in Applied Mathematics and Mechanics 2012年第2期4卷 175-189页

作者： Chunmei Liu Shi Shu Yunqing Huang liuqiang zhong Junxian Wang Hunan Key Laboratory for Computation&Simulation in Science and Engineering and Key Laboratory of Intelligent Computing&Information Processing of Ministry of Education Xiangtan UniversityHunan 411105China School of Mathematical Sciences South China Normal UniversityGuangzhou 510631China

In this paper,we propose an iterative two-grid method for the edge finite element discretizations(a saddle-point system)of Perfectly Matched Layer(PML)equations to the Maxwell scattering problem in two ***,we use a fine space to solve a discrete saddle-point system of H(grad)variational problems,denoted by auxiliary system ***,we use a coarse space to solve the original saddle-point ***,we use a fine space again to solve a discrete H(curl)-elliptic variational problems,denoted by auxiliary system ***,we develop a regularization diagonal block preconditioner for auxiliary system 1 and use H-X preconditioner for auxiliary system *** we essentially transform the original problem in a fine space to a corresponding(but much smaller)problem on a coarse space,due to the fact that the above two preconditioners are efficient and *** with some existing iterative methods for solving saddle-point systems,such as PMinres,numerical experiments show the competitive performance of our iterative two-grid method.

关键词： Maxwell scattering edge finite element PML iterative two-grid method

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一类拟线性椭圆方程的两网格混合有限元方法

高等学校计算数学学报 2018年第2期40卷 180-192页

作者：贡文波钟柳强华南师范大学数学科学学院中山大学数据科学与计算机学院

1引言令Ω是R;中带有Lipshcitz连续边界?Ω的一个有界区域,考虑如下Hamilton-JacobiBellman方程（以下简称HJB方程）-div(A（x）▽p)+b（x,p,▽p）=0,在Ω内,（1.1）p（x）=—g,在?Ω上.（1.2）该模型问题最早出现于最优随机控制中（可... 详细信息

1引言令Ω是R;中带有Lipshcitz连续边界?Ω的一个有界区域,考虑如下Hamilton-JacobiBellman方程（以下简称HJB方程）-div(A（x）▽p)+b（x,p,▽p）=0,在Ω内,（1.1）p（x）=—g,在?Ω上.（1.2）该模型问题最早出现于最优随机控制中（可参见[1,2]）,之后在科学、工程和经济领域中得到广泛应用,如在人口动态学中,它描述了某些自然机制的生物学的人口密度的稳态分布（可参见[3]）.由于HJB方程的复杂性,该方程一般没有解析解,从而对其高效数值解的研

关键词： Two-grid method Mixed finite element method Quasi-linear elliptic equation Error analysis

H(curl)-椭圆问题不连续Galerkin法的后验误差估计

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华南师范大学学报（自然科学版） 2012年第3期44卷 18-21页

作者：邢小青钟柳强华南师范大学数学科学学院广东广州510631

针对Lipschitz多面体区域上H(curl)-椭圆问题的不连续Galerkin法,提出了一种新的基于残量型的后验误差估计,并证明了该后验误差的一个上界估计.其中问题的最困难性在于如何处理跳跃项中出现的局部网格尺寸的负次幂.

关键词：不连续Galerkin法后验误差估计 H(curl)-椭圆问题

H(curl)-椭圆问题自适应内罚间断有限元方法的收敛性分析

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华南师范大学学报（自然科学版） 2016年第5期48卷 92-98页

作者：钟柳强程婷邢小青华南师范大学数学科学学院广州510631

针对H(curl)-椭圆问题的自适应内罚间断有限元方法,给出了相应的收敛性证明:把间断有限元空间分解成棱有限元空间及其正交补空间,然后结合误差的整体上界估计、关于加密网格之间的网格尺寸的2个条件以及后验误差指示子的单调性等性质,... 详细信息

针对H(curl)-椭圆问题的自适应内罚间断有限元方法,给出了相应的收敛性证明:把间断有限元空间分解成棱有限元空间及其正交补空间,然后结合误差的整体上界估计、关于加密网格之间的网格尺寸的2个条件以及后验误差指示子的单调性等性质,证明了在连续迭代过程中,关于误差函数的能量范数与尺度化的误差指示子之和是压缩的,即自适应内罚间断有限元方法是收敛的.

关键词：内罚间断有限元方法自适应算法后验误差指示子收敛性