Based on an asymptotic expansion of （bi）linear finite elements, a new extrapolation formula and extrapolation cascadic multigrid method （EXCMG） are proposed. The key ingredients of the proposed methods are some ne...Based on an asymptotic expansion of （bi）linear finite elements, a new extrapolation formula and extrapolation cascadic multigrid method （EXCMG） are proposed. The key ingredients of the proposed methods are some new extrapolations and quadratic interpolations, which are used to provide better initial values on the refined grid. In the case of triple grids, the errors of the new initial values are analyzed in detail. The numerical experiments show that EXCMG has higher accuracy and efficiency.展开更多
In this paper, three n-rectangle nonconforming elements are proposed with n ≥ 3. They are the extensions of well-known Morley element, Adini element and Bogner-Fox-Schmit element in two spatial dimensions to any high...In this paper, three n-rectangle nonconforming elements are proposed with n ≥ 3. They are the extensions of well-known Morley element, Adini element and Bogner-Fox-Schmit element in two spatial dimensions to any higher dimensions respectively. These elements are all proved to be convergent for a model biharmonic equation in n dimensions.展开更多
In this paper we study the convergence of adaptive finite element methods for the gen- eral non-attine equivalent quadrilateral and hexahedral elements on 1-irregular meshes with hanging nodes. Based on several basic ...In this paper we study the convergence of adaptive finite element methods for the gen- eral non-attine equivalent quadrilateral and hexahedral elements on 1-irregular meshes with hanging nodes. Based on several basic ingredients, such as quasi-orthogonality, estimator reduction and D6fler marking strategy, convergence of the adaptive finite element methods for the general second-order elliptic partial equations is proved. Our analysis is effective for all conforming Qm elements which covers both the two- and three-dimensional cases in a unified fashion.展开更多
In this paper, we discuss the finite volume element method of P1-nonconforming quadrilateral element for elliptic problems and obtain optimal error estimates for general quadrilateral partition. An optimal cascadic mu...In this paper, we discuss the finite volume element method of P1-nonconforming quadrilateral element for elliptic problems and obtain optimal error estimates for general quadrilateral partition. An optimal cascadic multigrid algorithm is proposed to solve the non-symmetric large-scale system resulting from such discretization. Numerical experiments are reported to support our theoretical results.展开更多
The main aim of this paper is to show that the quadrilateral mesh condition RDP（N, ψ） is only sufficient but not necessary for the optimal order error estimate of the Q isoparametric element in the Hi norm.
This is the second part of the paper for the mathematical study of nonconforming rotated Q1 element (NRQ1 hereafter) on arbitrary quadrilateral meshes. Some Poincare Inequalities are proved without assuming the quasi-...This is the second part of the paper for the mathematical study of nonconforming rotated Q1 element (NRQ1 hereafter) on arbitrary quadrilateral meshes. Some Poincare Inequalities are proved without assuming the quasi-uniformity of the mesh subdivision. A discrete trace inequality is also proved.展开更多
In the paper, we analyze the L2 norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the L2 norm of the finite element solution is of second ...In the paper, we analyze the L2 norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the L2 norm of the finite element solution is of second order, which can not be improved generally. The main ingredients are the saturation condition established for these elements and an identity for the error in the energy norm of the finite element solution. The result holds for most of the popular lower order finite element methods in the literature including： the Powell-Sabin C1 -P2 macro element, the nonconforming Morley element, the C1 -Q2 macro element, the nonconforming rectangle Morley element, and the nonconforming incomplete biquadratic element. In addition, the result actually applies to the nonconforming Adini element, the nonconforming Fraeijs de Veubeke elements, and the nonconforming Wang- Xu element and the Wang-Shi-Xu element provided that the saturation condition holds for them. This result solves one long standing problem in the literature： can the L2 norm error estimate of lower order finite element methods of the fourth order problem be two order higher than the error estimate in the energy norm？展开更多
Nonconforming grids with hanging nodes are frequently used in adaptive finite element comput at ions.In all earlier works on such methods,proper cons train ts should be enforced on degrees of freedom on edges/faces wi...Nonconforming grids with hanging nodes are frequently used in adaptive finite element comput at ions.In all earlier works on such methods,proper cons train ts should be enforced on degrees of freedom on edges/faces with hanging nodes to keep continuity,which yield numerical computations much complicated.In 2014,Zhao et al.(2014)presented quadrilateral constraint-free finite element methods on quadrilateral grids with hanging nodes.This paper further develops a hexahedral constraint-free finite element method on hexahedral grids with hanging nodes,which is of greater challenge than the two-dimensional case.Residual-based a posteriori error reliability and efficiency are also established in this paper.展开更多
The Eighth National Conference on Computational Mathematics was held at Sichuan University from October25to29,2007.About400researchers attended the conference,exchanging ideas and introducing their most recent work.Pl...The Eighth National Conference on Computational Mathematics was held at Sichuan University from October25to29,2007.About400researchers attended the conference,exchanging ideas and introducing their most recent work.Plenary addresses were given by eleven leading computational mathematicians.Conference sessions covered such diverse topics in展开更多
This paper provides a simplified derivation for error estimates of the TRUNC plate element. The error analysis for the problem with mixed boundary conditions is also discussed.
We consider the quadrilateral Q1 isoparametric element and establish an optimal error estimate in H^1 norm for the interpolation operator under a weaker mesh condition which admits anisotropic quadrilaterals and allow...We consider the quadrilateral Q1 isoparametric element and establish an optimal error estimate in H^1 norm for the interpolation operator under a weaker mesh condition which admits anisotropic quadrilaterals and allows the quadrilateral to become a regular triangle in the sense of maximum angle condition [5, 11].展开更多
The special issue grows from the international workshop on Recent Mathematical and Com- putational Developments of Maxwell＇s Equations： Challenges and Frontiers held in Weihai, China, July 24-28, 2006. The workshop ...The special issue grows from the international workshop on Recent Mathematical and Com- putational Developments of Maxwell＇s Equations： Challenges and Frontiers held in Weihai, China, July 24-28, 2006. The workshop was successful to bring together researchers in展开更多
We present in this special issue of Communications in Computational Physics selected papers from the seventh ICOSAHOM(International Conference On Spectral And High Order Methods)which was held at the Institute for Com...We present in this special issue of Communications in Computational Physics selected papers from the seventh ICOSAHOM(International Conference On Spectral And High Order Methods)which was held at the Institute for Computational Mathematics and Scientific/Engineering Computing,Chinese Academy of Sciences,Beijing,China,during June 18-22,2007.These papers were refereed by members of the scientific committee of ICOSAHOM,editors of Communications in Computational Physics and other scientists following the same refereeing procedure as that for regular CiCP manuscripts.展开更多
In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining fu...In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.展开更多
基金The research is supported by the National Natural Science Foundation of China （No. 11071067） and the Key Laboratory of Education Ministry.
文摘Based on an asymptotic expansion of （bi）linear finite elements, a new extrapolation formula and extrapolation cascadic multigrid method （EXCMG） are proposed. The key ingredients of the proposed methods are some new extrapolations and quadratic interpolations, which are used to provide better initial values on the refined grid. In the case of triple grids, the errors of the new initial values are analyzed in detail. The numerical experiments show that EXCMG has higher accuracy and efficiency.
基金The work of the first author was supported by the National Natural Science Fbundation of china(10571006)The work of the shird author was supperted by the Changjiang Professorship of the Ministry of Education of China through Peking University
文摘In this paper, three n-rectangle nonconforming elements are proposed with n ≥ 3. They are the extensions of well-known Morley element, Adini element and Bogner-Fox-Schmit element in two spatial dimensions to any higher dimensions respectively. These elements are all proved to be convergent for a model biharmonic equation in n dimensions.
基金supported by the Special Funds for Major State Basic Research Project (No. 2005CB321701)
文摘In this paper we study the convergence of adaptive finite element methods for the gen- eral non-attine equivalent quadrilateral and hexahedral elements on 1-irregular meshes with hanging nodes. Based on several basic ingredients, such as quasi-orthogonality, estimator reduction and D6fler marking strategy, convergence of the adaptive finite element methods for the general second-order elliptic partial equations is proved. Our analysis is effective for all conforming Qm elements which covers both the two- and three-dimensional cases in a unified fashion.
文摘In this paper, we discuss the finite volume element method of P1-nonconforming quadrilateral element for elliptic problems and obtain optimal error estimates for general quadrilateral partition. An optimal cascadic multigrid algorithm is proposed to solve the non-symmetric large-scale system resulting from such discretization. Numerical experiments are reported to support our theoretical results.
基金This research is supported by the National Science Fbundation of China(No.10371113).
文摘The main aim of this paper is to show that the quadrilateral mesh condition RDP（N, ψ） is only sufficient but not necessary for the optimal order error estimate of the Q isoparametric element in the Hi norm.
基金The work of P.-B.Ming was partially supported by the National Natural Science Foundation of China 10201033
文摘This is the second part of the paper for the mathematical study of nonconforming rotated Q1 element (NRQ1 hereafter) on arbitrary quadrilateral meshes. Some Poincare Inequalities are proved without assuming the quasi-uniformity of the mesh subdivision. A discrete trace inequality is also proved.
文摘In the paper, we analyze the L2 norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the L2 norm of the finite element solution is of second order, which can not be improved generally. The main ingredients are the saturation condition established for these elements and an identity for the error in the energy norm of the finite element solution. The result holds for most of the popular lower order finite element methods in the literature including： the Powell-Sabin C1 -P2 macro element, the nonconforming Morley element, the C1 -Q2 macro element, the nonconforming rectangle Morley element, and the nonconforming incomplete biquadratic element. In addition, the result actually applies to the nonconforming Adini element, the nonconforming Fraeijs de Veubeke elements, and the nonconforming Wang- Xu element and the Wang-Shi-Xu element provided that the saturation condition holds for them. This result solves one long standing problem in the literature： can the L2 norm error estimate of lower order finite element methods of the fourth order problem be two order higher than the error estimate in the energy norm？
基金supported by National Natural Science Foundation of China(Grant No.11671390)supported by National Natural Science Foundation of China(Grant No.11371359)
文摘Nonconforming grids with hanging nodes are frequently used in adaptive finite element comput at ions.In all earlier works on such methods,proper cons train ts should be enforced on degrees of freedom on edges/faces with hanging nodes to keep continuity,which yield numerical computations much complicated.In 2014,Zhao et al.(2014)presented quadrilateral constraint-free finite element methods on quadrilateral grids with hanging nodes.This paper further develops a hexahedral constraint-free finite element method on hexahedral grids with hanging nodes,which is of greater challenge than the two-dimensional case.Residual-based a posteriori error reliability and efficiency are also established in this paper.
文摘The Eighth National Conference on Computational Mathematics was held at Sichuan University from October25to29,2007.About400researchers attended the conference,exchanging ideas and introducing their most recent work.Plenary addresses were given by eleven leading computational mathematicians.Conference sessions covered such diverse topics in
基金The work was partly supported by NNSFC under the grant no. 10371076, E-Institutes of Shanghai Municipal Education Commission, N. E03004 and The Science Foundation of Shanghai under the grant no. 04JC14062. The second author is also engaged with Division of Computational Science, E-Institute of Shanghai Universities, Shanghai Normal University, China.
文摘This paper provides a simplified derivation for error estimates of the TRUNC plate element. The error analysis for the problem with mixed boundary conditions is also discussed.
文摘We consider the quadrilateral Q1 isoparametric element and establish an optimal error estimate in H^1 norm for the interpolation operator under a weaker mesh condition which admits anisotropic quadrilaterals and allows the quadrilateral to become a regular triangle in the sense of maximum angle condition [5, 11].
文摘The special issue grows from the international workshop on Recent Mathematical and Com- putational Developments of Maxwell＇s Equations： Challenges and Frontiers held in Weihai, China, July 24-28, 2006. The workshop was successful to bring together researchers in
文摘We present in this special issue of Communications in Computational Physics selected papers from the seventh ICOSAHOM(International Conference On Spectral And High Order Methods)which was held at the Institute for Computational Mathematics and Scientific/Engineering Computing,Chinese Academy of Sciences,Beijing,China,during June 18-22,2007.These papers were refereed by members of the scientific committee of ICOSAHOM,editors of Communications in Computational Physics and other scientists following the same refereeing procedure as that for regular CiCP manuscripts.
基金Supported by the Scientific Research Foundation for the Doctor,Nanjing University of Aeronautics and Astronautics(No.1008-907359)
文摘In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.
