This paper proposes a modified Morley element method for a fourth order elliptic singular perturbation problem. The method also uses Morley element or rectangle Morley element, but linear or bilinear approximation of ...This paper proposes a modified Morley element method for a fourth order elliptic singular perturbation problem. The method also uses Morley element or rectangle Morley element, but linear or bilinear approximation of finite element functions is used in the lower part of the bilinear form. It is shown that the modified method converges uniformly in the perturbation parameter.展开更多
This paper presents an optimal solver for the Morley element problem for the boundaryvalue problem of the biharmonic equation by decomposing it into several subproblems and solving these subproblems optimally. The opt...This paper presents an optimal solver for the Morley element problem for the boundaryvalue problem of the biharmonic equation by decomposing it into several subproblems and solving these subproblems optimally. The optimality of the proposed method is mathematically proved for general shape-regular grids.展开更多
The main aim of this paper is to study tile convergence of a nonconforming triangular plate element-Morley element under anisotropic meshes. By a novel approach, an explicit bound for the interpolation error is derive...The main aim of this paper is to study tile convergence of a nonconforming triangular plate element-Morley element under anisotropic meshes. By a novel approach, an explicit bound for the interpolation error is derived for arbitrary triangular meshes （which even need not satisfy the maximal angle condition and the coordinate system condition ）, the optimal consistency error is obtained for a family of anisotropically graded finite element meshes.展开更多
We present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of （D（h） order...We present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of （D（h） order in energy norm and of O（h2） order in L2 norm on general d-rectangular triangulations. Moreover, when the triangulation is uniform, the convergence rate can be of O（h2） order in energy norm, and the convergence rate in L2 norm is still of O（h2） order, which cannot be improved. Numerical examples are presented to demonstrate our theoretical results.展开更多
In this paper, we consider some multigrid algorithms for the biharmonic problem discretized by Morley element on nonnested meshes. Through taking the averages of the nodal variables we construct an intergrid transfer ...In this paper, we consider some multigrid algorithms for the biharmonic problem discretized by Morley element on nonnested meshes. Through taking the averages of the nodal variables we construct an intergrid transfer operator that satisfies a certain stable approximation property. The so-called regularity-approximation assumption is then established. Optimal convergence properties of the W-cycle and a uniform condition number estimate for the variable V-cycle preconditioner are presented. This technique is applicable to other nonconforming plate elements.展开更多
In this paper, following our original ideas[9], we first consider a weakly overlapping additive Schwarz preconditioner according to the framework of [2] for Morley element and show that its condition number is quasi-o...In this paper, following our original ideas[9], we first consider a weakly overlapping additive Schwarz preconditioner according to the framework of [2] for Morley element and show that its condition number is quasi-optimal; we then analyze in detail the structure of this preconditioner, and after proper choices of the inexact solvers, we obtain a quasi-optimal nonoverlapping domain decomposition preconditioner in the last. Compared with [12], [13], it seems that according to this paper's procedure we can make out more thoroughly the relationship between overlapping and nonoverlapping domain decomposition methods for nonconforming plate elements, and certainly, we have also proposed another formal and simple strategy to construct nonoverlapping domain decomposition preconditioners for nonconforming plate elements.展开更多
In this paper,we analyze a nonconforming finite element method for the computation of transmission eigenvalues and the corresponding eigenfunctions.The error estimates of the eigenvalue and eigenfunction approximation...In this paper,we analyze a nonconforming finite element method for the computation of transmission eigenvalues and the corresponding eigenfunctions.The error estimates of the eigenvalue and eigenfunction approximation are given,respectively.Finally,some numerical examples are provided to validate the theoretical results.展开更多
基金The work of the first author was supported by the National Natural Science Foundation of China （10571006）. The work of the second author was supported by National Science Foundation DMS-0209479 and DMS-0215392 and the Changjiang Professorship through Peking University.
文摘This paper proposes a modified Morley element method for a fourth order elliptic singular perturbation problem. The method also uses Morley element or rectangle Morley element, but linear or bilinear approximation of finite element functions is used in the lower part of the bilinear form. It is shown that the modified method converges uniformly in the perturbation parameter.
基金Acknowledgments. The authors would like to thank Professor Jinchao Xu for his valuable suggestions and comments. The author C. Feng is partially supported by the NSFC Grants NO. 11571293 and 11201398, the Project of Scientific Research Fund of Hunan Provincial Education Department （14B044）, Specialized research Fund for the Doctoral Program of Higher Education of China Grant 20124301110003 and Hunan Provincial Natural Science Foundation of China （14JJ2063） S. Zhang is partially supported by the NSFC Grants NO. 11101415 and 11471026, and the SRF for ROCS, SEM.
文摘This paper presents an optimal solver for the Morley element problem for the boundaryvalue problem of the biharmonic equation by decomposing it into several subproblems and solving these subproblems optimally. The optimality of the proposed method is mathematically proved for general shape-regular grids.
文摘The main aim of this paper is to study tile convergence of a nonconforming triangular plate element-Morley element under anisotropic meshes. By a novel approach, an explicit bound for the interpolation error is derived for arbitrary triangular meshes （which even need not satisfy the maximal angle condition and the coordinate system condition ）, the optimal consistency error is obtained for a family of anisotropically graded finite element meshes.
基金supported by National Natural Science Foundation of China (Grant Nos. 11471026, 11271035, 91430213, 11421101 and 11101415)
文摘We present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of （D（h） order in energy norm and of O（h2） order in L2 norm on general d-rectangular triangulations. Moreover, when the triangulation is uniform, the convergence rate can be of O（h2） order in energy norm, and the convergence rate in L2 norm is still of O（h2） order, which cannot be improved. Numerical examples are presented to demonstrate our theoretical results.
文摘In this paper, we consider some multigrid algorithms for the biharmonic problem discretized by Morley element on nonnested meshes. Through taking the averages of the nodal variables we construct an intergrid transfer operator that satisfies a certain stable approximation property. The so-called regularity-approximation assumption is then established. Optimal convergence properties of the W-cycle and a uniform condition number estimate for the variable V-cycle preconditioner are presented. This technique is applicable to other nonconforming plate elements.
文摘In this paper, following our original ideas[9], we first consider a weakly overlapping additive Schwarz preconditioner according to the framework of [2] for Morley element and show that its condition number is quasi-optimal; we then analyze in detail the structure of this preconditioner, and after proper choices of the inexact solvers, we obtain a quasi-optimal nonoverlapping domain decomposition preconditioner in the last. Compared with [12], [13], it seems that according to this paper's procedure we can make out more thoroughly the relationship between overlapping and nonoverlapping domain decomposition methods for nonconforming plate elements, and certainly, we have also proposed another formal and simple strategy to construct nonoverlapping domain decomposition preconditioners for nonconforming plate elements.
基金Xia Ji is supported by the National Natural Science Foundation of China(No.11271018,No.91230203)the Special Funds for National Basic Research Program of China(973 Program 2012CB025904 and 863 Program 2012AA01A309)+1 种基金the national Center for Mathematics and Interdisciplinary Science,CAS.Hehu Xie is supported in part by the National Natural Science Foundations of China(NSFC 91330202,11001259,11371026,11031006,2011CB309703)the national Center for Mathematics and Interdisciplinary Science,CAS,the President Foundation of AMSS-CAS。
文摘In this paper,we analyze a nonconforming finite element method for the computation of transmission eigenvalues and the corresponding eigenfunctions.The error estimates of the eigenvalue and eigenfunction approximation are given,respectively.Finally,some numerical examples are provided to validate the theoretical results.
