A NONCONFORMING ANISOTROPIC FINITE ELEMENT APPROXIMATION WITH MOVING GRIDS FOR STOKES PROBLEM

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.

CONVERGENCE ANALYSIS FOR A NONCONFORMING MEMBRANE ELEMENT ON ANISOTROPIC MESHES

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.

Superconvergence Analysis and Extrapolation of Quasi-Wilson Nonconforming Finite Element Method for Nonlinear Sobolev Equations

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.

EQ^(rot)_1 Nonconforming Finite Element Method for Nonlinear Dual Phase Lagging Heat Conduction Equations

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.

Superconvergence Analysis of Splitting Positive Definite Nonconforming Mixed Finite Element Method for Pseudo-hyperbolic Equations

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.

The Crouzeix-Raviart Type Nonconforming Finite Element Method for the Nonstationary Navier-Stokes Equations on Anisotropic Meshes

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.

High Accuracy Analysis of the Lowest Order H1-Galerkin Mixed Finite Element Method for Nonlinear Sine-Gordon Equations

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.

A Second Order Nonconforming Rectangular Finite Element Method for Approximating Maxwell's Equations

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.

Nonconforming Finite Element Methods for the Constrained Optimal Control Problems Governed by Nonsmooth Elliptic Equations

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.

Streamline-Diffusion Method of a Lowest Order Nonconforming Rectangular Finite Element for Convection-Diffusion Problem

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.