A Nonconforming Arbitrary Quadrilateral Finite Element Method for Approximating Maxwell's Equations

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.

Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations

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.

P_1-nonconforming triangular finite element method for elliptic and parabolic interface problems

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.

A new streamline diffusion finite element method for the generalized Oseen problem

This paper aims to present a new streamline diffusion method with low order rectangular Bernardi-Raugel elements to solve the generalized Oseen equations. With the help of the Bramble-Hilbert lemma, the optimal errors of the velocity and pressure are estimated, which are independent of the considered parameter e. With an interpolation postprocessing approach, the superconvergent error of the pressure is obtained. Finally, a numerical experiment is carried out to confirm the theoretical results.

CONVERGENCE ANALYSIS OF NONCONFORMING QUADRILATERAL FINITE ELEMENT METHODS FOR NONLINEAR COUPLED SCHRODINGER-HELMHOLTZ EQUATIONS

The focus of this paper is on two novel linearized Crank-Nicolson schemes with nonconforming quadrilateral finite element methods(FEMs)for the nonlinear coupled Schrodinger-Helmholtz equations.Optimal L^(2) and H^(1) estimates of orders O(h^(2)+τ^(2))and O(h^(2)+τ^(2))are derived respectively without any grid-ratio condition through the following two keys.One is that a time-discrete system is introduced to split the error into the temporal error and the spatial error,which leads to optimal temporal error estimates of order O(τ^(2))in L^(2) and the broken H^(1)-norms,as well as the uniform boundness of numerical solutions in L^(∞) norm.The other is that a novel projection is utilized,which can iron out the difficulty of the existence of the consistency errors.This leads to derive optimal spatial error estimates of orders O(h^(2))in L^(2)-norm and O(h)in the broken H^(1)-norm under the H^(2) regularity of the solutions for the time-discrete system.At last,two numerical examples are provided to confirm the theoretical analysis.Here,h is the subdivision parameter,and τ is the time step.

UNIFORM SUPERCONVERGENCE ANALYSIS OF A TWO-GRID MIXED FINITE ELEMENT METHOD FOR THE TIME-DEPENDENT BI-WAVE PROBLEM MODELING D-WAVE SUPERCONDUCTORS

In this paper,a two-grid mixed finite element method(MFEM)of implicit Backward Euler(BE)formula is presented for the fourth order time-dependent singularly perturbed Bi-wave problem for d-wave superconductors by the nonconforming EQ_(1)^(rot) element.In this approach,the original nonlinear system is solved on the coarse mesh through the Newton iteration method,and then the linear system is computed on the fine mesh with Taylor’s expansion.Based on the high accuracy results of the chosen element,the uniform superclose and superconvergent estimates in the broken H^(1)-norm are derived,which are independent of the negative powers of the perturbation parameter appeared in the considered problem.Numerical results illustrate that the computing cost of the proposed two-grid method is much less than that of the conventional Galerkin MFEM without loss of accuracy.

SUPERCONVERGENCE ANALYSIS OF A BDF-GALERKIN FEM FOR THE NONLINEAR KLEIN-GORDON-SCHRODINGER EQUATIONS WITH DAMPING MECHANISM

The focus of this paper is on a linearized backward differential formula(BDF)scheme with Galerkin FEM for the nonlinear Klein-Gordon-Schrödinger equations(KGSEs)with damping mechanism.Optimal error estimates and superconvergence results are proved without any time-step restriction condition for the proposed scheme.The proof consists of three ingredients.First,a temporal-spatial error splitting argument is employed to bound the numerical solution in certain strong norms.Second,optimal error estimates are derived through a novel splitting technique to deal with the time derivative and some sharp estimates to cope with the nonlinear terms.Third,by virtue of the relationship between the Ritz projection and the interpolation,as well as a so-called"lifting"technique,the superconvergence behavior of order O(h^(2)+τ^(2))in H^(1)-norm for the original variables are deduced.Finally,a numerical experiment is conducted to confirm our theoretical analysis.Here,h is the spatial subdivision parameter,andτis the time step.

A LOW ORDER NONCONFORMING MIXED FINITE ELEMENT METHOD FOR NON-STATIONARY INCOMPRESSIBLE MAGNETOHYDRODYNAMICS SYSTEM

A low order nonconforming mixed finite element method(FEM)is established for the fully coupled non-stationary incompressible magnetohydrodynamics(MHD)problem in a bounded domain in 3D.The lowest order finite elements on tetrahedra or hexahedra are chosen to approximate the pressure,the velocity field and the magnetic field,in which the hydrodynamic unknowns are approximated by inf-sup stable finite element pairs and the magnetic field by H^(1)(Ω)-conforming finite elements,respectively.The existence and uniqueness of the approximate solutions are shown.Optimal order error estimates of L^(2)(H^(1))-norm for the velocity field,L^(2)(L^(2))-norm for the pressure and the broken L^(2)(H^(1))-norm for the magnetic field are derived.

AN ANISOTROPIC NONCONFORMING FINITE ELEMENT METHOD FOR APPROXIMATING A CLASS OF NONLINEAR SOBOLEV EQUATIONS

An anisotropic nonconforming finite element method is presented for a class of nonlinear Sobolev equations. The optimal error estimates and supercloseness are obtained for both semi-discrete and fully-discrete approximate schemes, which are the same as the traditional finite element methods. In addition, the global superconvergence is derived through the postprocessing technique. Numerical experiments are included to illustrate the feasibility of the proposed method.

APPROXIMATION OF NONCONFORMING QUASI-WILSON ELEMENT FOR SINE-GORDON EQUATIONS

In this paper, nonconforming quasi-Wilson finite element approximation to a class of nonlinear sine-Gordan equations is discussed. Based on the known higher accuracy results of bilinear element and different techniques from the existing literature, it is proved that the inner product △↓（u - Ih^1u）, △↓vh） and the consistency error can be estimated as order O（h^2） in broken H^1 - norm/L^2 - norm when u ∈ H^3（Ω）/H^4（Ω）, where Ih^1u is the bilinear interpolation of u, Vh belongs to the quasi-Wilson finite element space. At the same time, the superclose result with order O（h^2） for semi-discrete scheme under generalized rectangular meshes is derived. Furthermore, a fully-discrete scheme is proposed and the corresponding error estimate of order O（h^2 ＋ τ^2） is obtained for the rectangular partition when u ∈ H^4（Ω）, which is as same as that of the bilinear element with ADI scheme and one order higher than that of the usual analysis on nonconforming finite elements.

ACCURACY ANALYSIS FOR QUASI-CAREY ELEMENT

In this paper, a new triangular element （Quasi-Carey element） is constructed by the idea of Specht element. It is shown that this Quasi-Carey element possesses a very special property, i.e., the consistency error is of order O（h^2）, one order higher than its interpolation error when the exact solution belongs to H^3（Ω）. However, the interpolation error and consistency error of Carey element are of order O（h）. It seems that the above special property has never been seen for other triangular elements for the second order problems.

AN ANISOTROPIC NONCONFORMING FINITE ELEMENT SCHEME WITH MOVING GRIDS FOR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS

A Crank-Nicolson scheme based on nonconforming finite element with moving grids is dis- cussed for a class of parabolic integro-differential equations under anisotropic meshes. The corresponding convergence analysis is presented and the error estimates are obtained by using the interpolation operator instead of the conventional elliptic projection which is an indispensable tool in the convergence analysis of traditional finite element methods in previous literature.

SUPERCONVERGENCE ANALYSIS OF THE STABLE CONFORMING RECTANGULAR MIXED FINITE ELEMENTS FOR THE LINEAR ELASTICITY PROBLEM

In this paper, we consider the linear elasticity problem based on the Hellinger-Reissner variational principle. An O（h2） order superclose property for the stress and displacement and a global superconvergence result of the displacement are established by employing a Clement interpolation, an integral identity and appropriate postprocessing techniques.

SUPERCONVERGENCE ANALYSIS OF A NONCONFORMING TRIANGULAR ELEMENT ON ANISOTROPIC MESHES

The class of anisotropic meshes we conceived abandons the regular assumption. Some distinct properties of Carey＇s element are used to deal with the superconvergence for a class of two- dimensional second-order elliptic boundary value problems on anisotropic meshes. The optimal results are obtained and numerical examples are given to confirm our theoretical analysis.

HIGH ACCURACY ANALYSIS OF THE FINITE ELEMENT METHOD FOR NONLINEAR VISCOELASTIC WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

The standard finite elements of degree p over the rectangular meshes are applied to solve a kind of nonlinear viscoelastic wave equations with nonlinear boundary conditions, and the superclose property of the continuous Galerkin approximation is derived without using the nonclassical elliptic projection of the exact solution of the model problem. The global superconvergence of one order higher than the traditional error estimate is also obtained through the postprocessing technique.

AN ANISOTROPIC LOCKING-FREE NONCONFORMING TRIANGULAR FINITE ELEMENT METHOD FOR PLANAR LINEAR ELASTICITY PROBLEM

The main aim of this paper is to study the nonconforming linear triangular Crouzeix- Raviart type finite element approximation of planar linear elasticity problem with the pure displacement boundary value on anisotropic general triangular meshes satisfying the maximal angle condition and coordinate system condition. The optimal order error estimates of energy norm and L2-norm are obtained, which are independent of lame parameter λ. Numerical results are given to demonstrate the validity of our theoretical analysis.Mathematics subject classification： 65N30, 65N15.

A LOW ORDER NONCONFORMING ANISOTROPIC FINITE ELEMENT APPROXIMATION TO PARABOLIC PROBLEM

A low order nonconforming finite element is applied to the parabolic problem with anisotropicmeshes.Both the semidiscrete and fully discrete forms are studied.Some superclose properties andsuperconvergence are obtained through some novel approaches and techniques.

NONCONFORMING FINITE ELEMENT METHOD FOR NONLINEAR PARABOLIC EQUATIONS

A nonconforming finite element method for the nonlinear parabolic equations is studied inthis paper.The convergence analysis is presented and the optimal error estimate in L^2(‖·‖_h)norm isobtained through Ritz projection technique,where ‖·‖_h is a norm over the finite element space.

UNCONDITIONALLY SUPERCLOSE ANALYSIS OF A NEW MIXED FINITE ELEMENT METHOD FOR NONLINEAR PARABOLIC EQUATIONS

This paper develops a framework to deal with the unconditional superclose analysis of nonlinear parabolic equation.Taking the finite dement pair Q11/Q01×Q10 as an example, a new mixed finite element method (FEM)is established and the r-independent superclose results of the original variable u in Hi-norm and the flux variable q=-a(u)■u in L^2- norm are deduced (τ is the temporal partition parameter).A key to our analysis is all error splitting technique,with which the time-discrete and the spatial-discrete systems are constructed,respectively.For the first system,tile boundedness of the temporal errors are obtained.For the second system,the spatial superclose results are presented unconditionally.while the previous literature always only obtain the convergent estimates or require certain time step conditions.Finally,some numerical results are provided to confirm the theoretical analysis,and show the efficiency of the proposed method.

ANISOTROPIC CROUZEIX-RAVIART TYPE NONCONFORMING FINITE ELEMENT METHODS TO VARIATIONAL INEQUALITY PROBLEM WITH DISPLACEMENT OBSTACLE

In this paper, anisotropic Crouzeix-Raviart type nonconforming finite element meth- ods are considered for solving the second order variational inequality with displacement obstacle. The convergence analysis is presented and the optimal order error estimates are obtained under the hypothesis of the finite length of the free boundary. Numerical results are provided to illustrate the correctness of theoretical analysis.