This paper is devoted to the five parameters nonconforming finite element schemes with moving grids for velocity-pressure mixed formulations of the nonstationary Stokes problem in 2-D. We show that this element has an...This paper is devoted to the five parameters nonconforming finite element schemes with moving grids for velocity-pressure mixed formulations of the nonstationary Stokes problem in 2-D. We show that this element has anisotropic behavior and derive anisotropic error estimations in some certain norms of the velocity and the pressure based on some novel techniques. Especially through careful analysis we get an interesting result on consistency error estimation, which has never been seen for mixed finite element methods in the previously literatures.展开更多
Regular assumption of finite element meshes is a basic condition of most analysis of finite element approximations both for conventional conforming elements and nonconforming elements. The aim of this paper is to pres...Regular assumption of finite element meshes is a basic condition of most analysis of finite element approximations both for conventional conforming elements and nonconforming elements. The aim of this paper is to present a novel approach of dealing with the approximation of a four-degree nonconforming finite element for the second order elliptic problems on the anisotropic meshes. The optimal error estimates of energy norm and L^2-norm without the regular assumption or quasi-uniform assumption are obtained based on some new special features of this element discovered herein. Numerical results are given to demonstrate validity of our theoretical analysis.展开更多
Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equa- tions is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches....Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equa- tions is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that (Vh(U -- Ihu),VhVh)h may be estimated as order O(h2) when u E H3(Ω), where Iuu denotes the bilinear interpolation of u, vh is a polynomial belongs to quasi-Wilson finite element space and △h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O(h2)/O(h3) in broken Hi-norm, which is one/two order higher than its interpolation error when u ε Ha(Ω)/H4 ((1). Then we derive the optimal order error estimate and su- perclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapola- tion result of order O(h3), two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.展开更多
EQrot nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is, t...EQrot nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is, the consistency error of this element is of order O(h2) one order higher than its interpolation error O(h), the superclose results of order O(h2) in broken Hi-norm are obtained. At the same time, the global superconvergence in broken Hi-norm is deduced by interpolation postprocessing technique. Moreover, the extrapolation result with order O(h4) is derived by constructing a new interpolation postprocessing operator and extrapolation scheme based on the known asymptotic expansion formulas of EQrot element. Finally, optimal error estimate is gained for a proposed fully-discrete scheme by different approaches from the previous literature.展开更多
In this paper, a new splitting positive definite nonconforming mixed finite element method is proposed for pseudo-hyperbolic equations, in which a quasi-Wilson quadrilateral element is used for the flux p, and the bil...In this paper, a new splitting positive definite nonconforming mixed finite element method is proposed for pseudo-hyperbolic equations, in which a quasi-Wilson quadrilateral element is used for the flux p, and the bilinear element is used for u. Superconvergence results in ||·||div,h norm for p and optimal error estimates in L2 norm for u are derived for both semi-discrete and fully discrete schemes under almost uniform meshes.展开更多
This paper is devoted to study the Crouzeix-Raviart (C-R) type nonconforming linear triangular finite element method (FEM) for the nonstationary Navier-Stokes equations on anisotropic meshes. By intro- ducing auxi...This paper is devoted to study the Crouzeix-Raviart (C-R) type nonconforming linear triangular finite element method (FEM) for the nonstationary Navier-Stokes equations on anisotropic meshes. By intro- ducing auxiliary finite element spaces, the error estimates for the velocity in the L2-norm and energy norm, as well as for the pressure in the L2-norm are derived.展开更多
The lowest order H1-Galerkin mixed finite element method (for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order Raviart- Thomas elem...The lowest order H1-Galerkin mixed finite element method (for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order Raviart- Thomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O(h2)/O(h2 + r2) in Hi-norm and H(div; Ω)-norm axe deduced for the semi-discrete and the fully-discrete schemes, where h, r- denote the mesh size and the time step, respectively, which improve the results in the previous literature.展开更多
Abstract The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell's equations. Then the correspondin...Abstract The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell's equations. Then the corresponding optimal error estimates are derived. The difficulty in construction of this finite element scheme is how to choose a compatible pair of degrees of freedom and shape function space so as to make the consistency error due to the nonconformity of the element being of order O(h^3), properly one order higher than that of its interpolation error O(h^2) in the broken energy norm, where h is the subdivision parameter tending to zero.展开更多
In this paper,nonconforming finite element methods(FEMs)are proposed for the constrained optimal control problems(OCPs)governed by the nonsmooth elliptic equations,in which the popular EQr1 ot element is employed to a...In this paper,nonconforming finite element methods(FEMs)are proposed for the constrained optimal control problems(OCPs)governed by the nonsmooth elliptic equations,in which the popular EQr1 ot element is employed to approximate the state and adjoint state,and the piecewise constant element is used to approximate the control.Firstly,the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary problem.Secondly,the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error estimate.Lastly,the methods are extended to some other well-known nonconforming elements.展开更多
The streamline-diffusion method of the lowest order nonconforming rectangular finite element is proposed for convection-diffusion problem. By making full use of the element's special property, the same convergence or...The streamline-diffusion method of the lowest order nonconforming rectangular finite element is proposed for convection-diffusion problem. By making full use of the element's special property, the same convergence order as the previous literature is obtained. In which, the jump terms on the boundary are added to bilinear form with simple user-chosen parameter δKwhich has nothing to do with perturbation parameter εappeared in the problem under considered, the subdivision mesh size hKand the inverse estimate coefficient μin finite element space.展开更多
基金This research is supported by the National Science Foundation of China(No.10371113).The authors would like to thank the anonymous referees for their helpful suggestions.
文摘This paper is devoted to the five parameters nonconforming finite element schemes with moving grids for velocity-pressure mixed formulations of the nonstationary Stokes problem in 2-D. We show that this element has anisotropic behavior and derive anisotropic error estimations in some certain norms of the velocity and the pressure based on some novel techniques. Especially through careful analysis we get an interesting result on consistency error estimation, which has never been seen for mixed finite element methods in the previously literatures.
文摘Regular assumption of finite element meshes is a basic condition of most analysis of finite element approximations both for conventional conforming elements and nonconforming elements. The aim of this paper is to present a novel approach of dealing with the approximation of a four-degree nonconforming finite element for the second order elliptic problems on the anisotropic meshes. The optimal error estimates of energy norm and L^2-norm without the regular assumption or quasi-uniform assumption are obtained based on some new special features of this element discovered herein. Numerical results are given to demonstrate validity of our theoretical analysis.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10971203 11101381+4 种基金 11271340)Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20094101110006)Tianyuan Mathematics Foundation of the National Natural Science Foundation of China(Grant No. 11026154)the Natural Science Foundation of Henan Province (Grant Nos. 112300410026 122300410266)
文摘Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equa- tions is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that (Vh(U -- Ihu),VhVh)h may be estimated as order O(h2) when u E H3(Ω), where Iuu denotes the bilinear interpolation of u, vh is a polynomial belongs to quasi-Wilson finite element space and △h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O(h2)/O(h3) in broken Hi-norm, which is one/two order higher than its interpolation error when u ε Ha(Ω)/H4 ((1). Then we derive the optimal order error estimate and su- perclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapola- tion result of order O(h3), two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.
基金Supported by the National Natural Science Foundation of China (Nos. 10971203 11101381)+3 种基金Tianyuan Mathe-matics Foundation of National Natural Science Foundation of China (No. 11026154)Natural Science Foundation of Henan Province (No. 112300410026)Natural Science Foundation of the Education Department of Henan Province (Nos. 2011A110020 12A110021)
文摘EQrot nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is, the consistency error of this element is of order O(h2) one order higher than its interpolation error O(h), the superclose results of order O(h2) in broken Hi-norm are obtained. At the same time, the global superconvergence in broken Hi-norm is deduced by interpolation postprocessing technique. Moreover, the extrapolation result with order O(h4) is derived by constructing a new interpolation postprocessing operator and extrapolation scheme based on the known asymptotic expansion formulas of EQrot element. Finally, optimal error estimate is gained for a proposed fully-discrete scheme by different approaches from the previous literature.
基金Supported by the National Natural Science Foundation of China(No.10971203,11271340,11101384)the Specialized Research Fund for the Doctoral Program of Higher Education(No.20094101110006)
文摘In this paper, a new splitting positive definite nonconforming mixed finite element method is proposed for pseudo-hyperbolic equations, in which a quasi-Wilson quadrilateral element is used for the flux p, and the bilinear element is used for u. Superconvergence results in ||·||div,h norm for p and optimal error estimates in L2 norm for u are derived for both semi-discrete and fully discrete schemes under almost uniform meshes.
基金Supported by National Science Foundation of China(No.10971203No.11271340)Research Fund for the Doctoral Program of Higher Education of China(No.20094101110006)
文摘This paper is devoted to study the Crouzeix-Raviart (C-R) type nonconforming linear triangular finite element method (FEM) for the nonstationary Navier-Stokes equations on anisotropic meshes. By intro- ducing auxiliary finite element spaces, the error estimates for the velocity in the L2-norm and energy norm, as well as for the pressure in the L2-norm are derived.
基金Supported in part by the National Natural Science Foundation of China under Grant Nos.11671369,11271340the Natural Science Foundation of the Education Department of Henan Province under Grant Nos.14A110009,16A110022
文摘The lowest order H1-Galerkin mixed finite element method (for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order Raviart- Thomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O(h2)/O(h2 + r2) in Hi-norm and H(div; Ω)-norm axe deduced for the semi-discrete and the fully-discrete schemes, where h, r- denote the mesh size and the time step, respectively, which improve the results in the previous literature.
基金Supported by the National Natural Science Foundation of China (No. 10971203)the Doctor Foundationof Henan Institute of Engineering (No. D09008)
文摘Abstract The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell's equations. Then the corresponding optimal error estimates are derived. The difficulty in construction of this finite element scheme is how to choose a compatible pair of degrees of freedom and shape function space so as to make the consistency error due to the nonconformity of the element being of order O(h^3), properly one order higher than that of its interpolation error O(h^2) in the broken energy norm, where h is the subdivision parameter tending to zero.
基金supported by the National Natural Science Foundation of China(Nos.11501527,11671369).
文摘In this paper,nonconforming finite element methods(FEMs)are proposed for the constrained optimal control problems(OCPs)governed by the nonsmooth elliptic equations,in which the popular EQr1 ot element is employed to approximate the state and adjoint state,and the piecewise constant element is used to approximate the control.Firstly,the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary problem.Secondly,the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error estimate.Lastly,the methods are extended to some other well-known nonconforming elements.
基金Supported by the National Natural Science Foundation of China(No.11271340)
文摘The streamline-diffusion method of the lowest order nonconforming rectangular finite element is proposed for convection-diffusion problem. By making full use of the element's special property, the same convergence order as the previous literature is obtained. In which, the jump terms on the boundary are added to bilinear form with simple user-chosen parameter δKwhich has nothing to do with perturbation parameter εappeared in the problem under considered, the subdivision mesh size hKand the inverse estimate coefficient μin finite element space.