The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value problem of planar linear elasticity. The optimal error est...The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value problem of planar linear elasticity. The optimal error estimates are obtained by using some novel approaches and techniques. The method proposed in this article is robust in the sense that the convergence estimates in the energy and L^2-norms are independent-of the Lame parameter λ.展开更多
A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is d...A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.展开更多
In this article, we study the explicit expressions of the constants in the error estimate of the nonconforming finite element method. We explicitly obtain the approximation error estimate and the consistency error est...In this article, we study the explicit expressions of the constants in the error estimate of the nonconforming finite element method. We explicitly obtain the approximation error estimate and the consistency error estimate for the Wilson＇s element without the regular assumption, respectively, which implies the final finite element error estimate. Such explicit a priori error estimates can be used as computable error bounds.展开更多
A modified penalty scheme is discussed for solving the Stokes problem with the Crouzeix-Raviart type nonconforming linear triangular finite element. By the L^2 projection method, the superconvergence results for the v...A modified penalty scheme is discussed for solving the Stokes problem with the Crouzeix-Raviart type nonconforming linear triangular finite element. By the L^2 projection method, the superconvergence results for the velocity and pressure are obtained with a penalty parameter larger than that of the classical penalty scheme. The numerical experiments are carried out to confirm the theoretical results.展开更多
This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this elem...This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.展开更多
This paper studies a low order mixed finite element method （FEM） for nonstationary incompressible Navier-Stokes equations. The velocity and pressure are approximated by the nonconforming constrained Q1^4ot element a...This paper studies a low order mixed finite element method （FEM） for nonstationary incompressible Navier-Stokes equations. The velocity and pressure are approximated by the nonconforming constrained Q1^4ot element and the piecewise constant, respectively. The superconvergent error estimates of the velocity in the broken H^1-norm and the pressure in the L^2-norm are obtained respectively when the exact solutions are reasonably smooth. A numerical experiment is carried out to confirm the theoretical results.展开更多
An anisotropic rectangular nonconforming finite element method for solving the Sobolev equations is discussed under semi-discrete and full discrete schemes. The corresponding optimal convergence error estimates and su...An anisotropic rectangular nonconforming finite element method for solving the Sobolev equations is discussed under semi-discrete and full discrete schemes. The corresponding optimal convergence error estimates and superclose property are derived, which are the same as the traditional conforming finite elements. Furthermore, the global superconvergence is obtained using a post-processing technique. The numerical results show the validity of the theoretical analysis.展开更多
A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretizatio...A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.展开更多
For transient Naiver-Stokes problems, a stabilized nonconforming finite element method is presented, focusing on two pairs inf-sup unstable finite element spaces, i.e., pNC/pNC triangular and pNQ/pNQ quadrilateral fin...For transient Naiver-Stokes problems, a stabilized nonconforming finite element method is presented, focusing on two pairs inf-sup unstable finite element spaces, i.e., pNC/pNC triangular and pNQ/pNQ quadrilateral finite element spaces. The semi- and full-discrete schemes of the stabilized method are studied based on the pressure projection and a variational multi-scale method. It has some attractive features： avoiding higher-order derivatives and edge-based data structures, adding a discrete velocity term only on the fine scale, being effective for high Reynolds number fluid flows, and avoiding increased computation cost. For the full-discrete scheme, it has second-order estimations of time and is unconditionally stable. The presented numerical results agree well with the theoretical results.展开更多
The main aim of this paper is to provide convergence analysis of Quasi-Wilson nonconforming finite element to Maxwell's equations under arbitrary quadrilateral meshes.The error estimates are derived,which are the ...The main aim of this paper is to provide convergence analysis of Quasi-Wilson nonconforming finite element to Maxwell's equations under arbitrary quadrilateral meshes.The error estimates are derived,which are the same as those for conforming elements under conventional regular meshes.展开更多
In this article, a nonconforming quadrilateral element （named modified quasi- Wilson element） is applied to solve the nonlinear schrSdinger equation （NLSE）. On the basis of a special character of this element, tha...In this article, a nonconforming quadrilateral element （named modified quasi- Wilson element） is applied to solve the nonlinear schrSdinger equation （NLSE）. On the basis of a special character of this element, that is, its consistency error is of order O（ha） for broken Ha-norm on arbitrary quadrilateral meshes, which is two order higher than its interpolation error, the optimal order error estimate and superclose property are obtained. Moreover, the global superconvergence result is deduced with the help of interpolation postprocessing technique. Finally, some numerical results are provided to verify the theoretical analysis.展开更多
We propose a locking-free nonconforming finite element method to solve for the displacement variation in the pure displacement boundary value problem of planar linear elasticity. The method proposed in this paper is r...We propose a locking-free nonconforming finite element method to solve for the displacement variation in the pure displacement boundary value problem of planar linear elasticity. The method proposed in this paper is robust and optimal, in the sense that the convergence estimate in the energy is independent of the Lame parameter λ.展开更多
In this paper we study the theoretical properties of multigrid algorithm for discretization of the Poisson equation in 2D using a mortar element method under the assumption that the triangulations on every subdomain a...In this paper we study the theoretical properties of multigrid algorithm for discretization of the Poisson equation in 2D using a mortar element method under the assumption that the triangulations on every subdomain are uniform.We prove the convergence of the W-cycle with a sufficiently large number of smoothing steps.The variable V-cycle multigrid preconditioner are also available.展开更多
The lowest order Pl-nonconforming triangular finite element method （FEM） for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optima...The lowest order Pl-nonconforming triangular finite element method （FEM） for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optimal order error estimates are obtained in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.展开更多
In this paper, an optimal V-cycle multigrid method for some conforming and nonconforming plate elements are constructed. A new method dealing with nonnested multigrid methods is presented.
In this paper, the quadratic nonconforming brick element （MSLK element） intro- duced in [10] is used for the 3D Stokes equations. The instability for the mixed element pair MSLK-P1 is analyzed, where the vector-valu...In this paper, the quadratic nonconforming brick element （MSLK element） intro- duced in [10] is used for the 3D Stokes equations. The instability for the mixed element pair MSLK-P1 is analyzed, where the vector-valued MSLK element approximates the velocity and the piecewise P1 element approximates the pressure. As a cure, we adopt the piecewise P1 macroelement to discretize the pressure instead of the standard piecewise P1 element on cuboid meshes. This new pair is stable and the optimal error estimate is achieved. Numerical examples verify our theoretical analysis.展开更多
Based on combination of two variational principles, a nonconforming stabilized finite element method is presented for the Reissner-Mindlin plates. The method is convergent when the finite element space is energy-compa...Based on combination of two variational principles, a nonconforming stabilized finite element method is presented for the Reissner-Mindlin plates. The method is convergent when the finite element space is energy-compatible. Error estimates are derived. In particular, three finite element spaces are applied in the computation. Numerical results show that the method is insensitive to the mesh distortion and has better performence than the MITC4 and DKQ methods. With properly chosen parameters, high accuracy can be obtained at coarse meshes.展开更多
Two new locking-free nonconforming finite elements for the pure displacement planar elasticity problem are presented. Convergence rates of the elements are uniformly optimal with respect to A. The energy norm and L2 n...Two new locking-free nonconforming finite elements for the pure displacement planar elasticity problem are presented. Convergence rates of the elements are uniformly optimal with respect to A. The energy norm and L2 norm errors are proved to be O（h2） and O（h3）, respectively. Numerical tests confirm the theoretical analysis.展开更多
In the use of finite element methods to the planar elasticity problems,one diffculty is to overcome locking when elasticity constant λ→∞.In the case of traction boundary condition,another diffculty is to make the d...In the use of finite element methods to the planar elasticity problems,one diffculty is to overcome locking when elasticity constant λ→∞.In the case of traction boundary condition,another diffculty is to make the discrete Korn＇s second inequality valid.In this paper,a triangular element is presented.We prove that this element is locking-free,the discrete Korn＇s second inequality holds and the convergence order is two.展开更多
A nonconforming mixed finite element method for nonlinear hyperbolic equations is discussed. Existence and uniqueness of the solution to the discrete problem are proved. Priori estimates of optimal order are derived f...A nonconforming mixed finite element method for nonlinear hyperbolic equations is discussed. Existence and uniqueness of the solution to the discrete problem are proved. Priori estimates of optimal order are derived for both the displacement and the stress.展开更多
基金The research is supported by NSF of China (10371113 10471133)
文摘The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value problem of planar linear elasticity. The optimal error estimates are obtained by using some novel approaches and techniques. The method proposed in this article is robust in the sense that the convergence estimates in the energy and L^2-norms are independent-of the Lame parameter λ.
基金Supported by the National Natural Science Foundation of China (10671184)
文摘A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.
基金supported by National Natural Science Foundation of China (11071226 11201122)
文摘In this article, we study the explicit expressions of the constants in the error estimate of the nonconforming finite element method. We explicitly obtain the approximation error estimate and the consistency error estimate for the Wilson＇s element without the regular assumption, respectively, which implies the final finite element error estimate. Such explicit a priori error estimates can be used as computable error bounds.
基金supported by the National Natural Science Foundation of China (Nos. 10971203 and 11271340)the Research Fund for the Doctoral Program of Higher Education of China (No. 20094101110006)
文摘A modified penalty scheme is discussed for solving the Stokes problem with the Crouzeix-Raviart type nonconforming linear triangular finite element. By the L^2 projection method, the superconvergence results for the velocity and pressure are obtained with a penalty parameter larger than that of the classical penalty scheme. The numerical experiments are carried out to confirm the theoretical results.
基金Supported by the National Natural Science Foundation of China(10371113,10671184)
文摘This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.
基金Project supported by the National Natural Science Foundation of China(No.11271340)
文摘This paper studies a low order mixed finite element method （FEM） for nonstationary incompressible Navier-Stokes equations. The velocity and pressure are approximated by the nonconforming constrained Q1^4ot element and the piecewise constant, respectively. The superconvergent error estimates of the velocity in the broken H^1-norm and the pressure in the L^2-norm are obtained respectively when the exact solutions are reasonably smooth. A numerical experiment is carried out to confirm the theoretical results.
基金the National Natural Science Foundation of China (No.10671184)
文摘An anisotropic rectangular nonconforming finite element method for solving the Sobolev equations is discussed under semi-discrete and full discrete schemes. The corresponding optimal convergence error estimates and superclose property are derived, which are the same as the traditional conforming finite elements. Furthermore, the global superconvergence is obtained using a post-processing technique. The numerical results show the validity of the theoretical analysis.
基金supported by the National Natural Science Foundation of China(No.10771150)the National Basic Research Program of China(No.2005CB321701)+1 种基金the Program for New Century Excellent Talents in University(No.NCET-07-0584)the Natural Science Foundation of Sichuan Province(No.07ZB087)
文摘A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.
基金supported by the National Natural Science Foundation of China(No.11271273)
文摘For transient Naiver-Stokes problems, a stabilized nonconforming finite element method is presented, focusing on two pairs inf-sup unstable finite element spaces, i.e., pNC/pNC triangular and pNQ/pNQ quadrilateral finite element spaces. The semi- and full-discrete schemes of the stabilized method are studied based on the pressure projection and a variational multi-scale method. It has some attractive features： avoiding higher-order derivatives and edge-based data structures, adding a discrete velocity term only on the fine scale, being effective for high Reynolds number fluid flows, and avoiding increased computation cost. For the full-discrete scheme, it has second-order estimations of time and is unconditionally stable. The presented numerical results agree well with the theoretical results.
基金The work was supported by the Chinese National Science Foundation Project （10671184）.
文摘The main aim of this paper is to provide convergence analysis of Quasi-Wilson nonconforming finite element to Maxwell's equations under arbitrary quadrilateral meshes.The error estimates are derived,which are the same as those for conforming elements under conventional regular meshes.
基金supported by the National Natural Science Foundation of China(11271340,11101381)
文摘In this article, a nonconforming quadrilateral element （named modified quasi- Wilson element） is applied to solve the nonlinear schrSdinger equation （NLSE）. On the basis of a special character of this element, that is, its consistency error is of order O（ha） for broken Ha-norm on arbitrary quadrilateral meshes, which is two order higher than its interpolation error, the optimal order error estimate and superclose property are obtained. Moreover, the global superconvergence result is deduced with the help of interpolation postprocessing technique. Finally, some numerical results are provided to verify the theoretical analysis.
基金Supported by the NSF of the Education Henan(200510078005)
文摘We propose a locking-free nonconforming finite element method to solve for the displacement variation in the pure displacement boundary value problem of planar linear elasticity. The method proposed in this paper is robust and optimal, in the sense that the convergence estimate in the energy is independent of the Lame parameter λ.
基金This research was supported by the National Natural Science Foundation of China under grant 10071015
文摘In this paper we study the theoretical properties of multigrid algorithm for discretization of the Poisson equation in 2D using a mortar element method under the assumption that the triangulations on every subdomain are uniform.We prove the convergence of the W-cycle with a sufficiently large number of smoothing steps.The variable V-cycle multigrid preconditioner are also available.
基金Project supported by the National Natural Science Foundation of China(No.11271340)
文摘The lowest order Pl-nonconforming triangular finite element method （FEM） for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optimal order error estimates are obtained in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.
基金The rescarch was supported by the Doctoral Point Foundation of chinese Universities and NSF
文摘In this paper, an optimal V-cycle multigrid method for some conforming and nonconforming plate elements are constructed. A new method dealing with nonnested multigrid methods is presented.
基金Supported by the National Natural Science Foundation of China(11171052,11301053,61328206 and 61272371)the Fundamental Research Funds for the Central Universities
文摘In this paper, the quadratic nonconforming brick element （MSLK element） intro- duced in [10] is used for the 3D Stokes equations. The instability for the mixed element pair MSLK-P1 is analyzed, where the vector-valued MSLK element approximates the velocity and the piecewise P1 element approximates the pressure. As a cure, we adopt the piecewise P1 macroelement to discretize the pressure instead of the standard piecewise P1 element on cuboid meshes. This new pair is stable and the optimal error estimate is achieved. Numerical examples verify our theoretical analysis.
基金supported by the Key Technologies R&D Program of Sichuan Province of China(No. 05GG006-006-2)
文摘Based on combination of two variational principles, a nonconforming stabilized finite element method is presented for the Reissner-Mindlin plates. The method is convergent when the finite element space is energy-compatible. Error estimates are derived. In particular, three finite element spaces are applied in the computation. Numerical results show that the method is insensitive to the mesh distortion and has better performence than the MITC4 and DKQ methods. With properly chosen parameters, high accuracy can be obtained at coarse meshes.
基金Project supported by the National Natural Science Foundation of China (Nos. 10771198 and 11071226)the Foundation of International Science and Technology Cooperation of Henan Province
文摘Two new locking-free nonconforming finite elements for the pure displacement planar elasticity problem are presented. Convergence rates of the elements are uniformly optimal with respect to A. The energy norm and L2 norm errors are proved to be O（h2） and O（h3）, respectively. Numerical tests confirm the theoretical analysis.
文摘In the use of finite element methods to the planar elasticity problems,one diffculty is to overcome locking when elasticity constant λ→∞.In the case of traction boundary condition,another diffculty is to make the discrete Korn＇s second inequality valid.In this paper,a triangular element is presented.We prove that this element is locking-free,the discrete Korn＇s second inequality holds and the convergence order is two.
文摘A nonconforming mixed finite element method for nonlinear hyperbolic equations is discussed. Existence and uniqueness of the solution to the discrete problem are proved. Priori estimates of optimal order are derived for both the displacement and the stress.
