A LOCKING-FREE ANISOTROPIC NONCONFORMING FINITE ELEMENT FOR PLANAR LINEAR ELASTICITY PROBLEM

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.

A NEW NONCONFORMING MIXED FINITE ELEMENT SCHEME FOR THE STATIONARY NAVIER-STOKES EQUATIONS

In this article, a new stable nonconforming mixed finite element scheme is proposed for the stationary Navier-Stokes equations, in which a new low order Crouzeix- Raviart type nonconforming rectangular element is taken for approximating space for the velocity and the piecewise constant element for the pressure. The optimal order error estimates for the approximation of both the velocity and the pressure in L2-norm are established, as well as one in broken H1-norm for the velocity. Numerical experiments are given which are consistent with our theoretical analysis.

A lumped mass nonconforming finite element method for nonlinear parabolic integro-differential equations on anisotropic meshes

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.

Superconvergence of nonconforming finite element penalty scheme for Stokes problem using L^2 projection method

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.

A locking-free anisotropic nonconforming rectangular finite element approximation for the planar elasticity problem

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.

EXPLICIT ERROR ESTIMATE FOR THE NONCONFORMING WILSON'S ELEMENT

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.

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.

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

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.

Anisotropic rectangular nonconforming finite element analysis for Sobolev equations

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.

Anisotropic nonconforming Crouzeix-Raviart type FEM forsecond-order elliptic problems

The nonconforming Crouzeix-Raviart type linear triangular finite element approximate to second-order elliptic problems is studied on anisotropic general triangular meshes in 2D satisfying the maximal angle condition and the coordinate system condition. The optimal-order error estimates of the broken energy norm and L2-norm are obtained.

New nonconforming finite element method for solving transient Naiver-Stokes equations

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.

A Locking-free Nonconforming Finite Element for Planar Linear Elasticity

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.

AN OPTIMAL V-CYCLE MULTIGRID METHOD FOR CONFORMING AND NONCONFORMING PLATE ELEMENTS

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.

CONVERGENCE OF THE NONCONFORMING WILSON'S BRICK FOR ARBITRARY HEXAHEDRAL MESHES

The nonconforming Wilson’s brick classically is restricted to regular hexahedral meshes. Lesaint and Zlamal[6] relaxed this constraint for the two-dimensional analonue of this element In this paper we extend their results to three dimensions and prove that and where u is the exact solution, u_h is the approximate solution and is the usual norm for the Sobolev space H^1(?).

Nonconforming stabilized combined finite element method for Reissner-Mindlin plate

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.

Quadratic nonconforming finite element method for 3D Stokes equations on cuboid meshes

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.

NEW SECOND ORDER NONCONFORMING TRIANGULAR ELEMENT FOR PLANAR ELASTICITY PROBLEMS

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.

Nonconforming finite elements for the equation of planar elasticity

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.

Nonconforming Finite Element Approximation on Anisotropic Meshes with Numerical Quadrature

In this paper we mainly discuss the nonconforming fimte element method for second order elliptic boundary value problems on anisotropic meshes. By changing thediscretization form（i.e., by use of numerical quadrature in the procedure of computing the left load）, we obtain the optimal estimate O（h）, which is as same as in the traditionalfinite element analysis when the load f ∈ H1 （Ω）η Co（Ω） which is weaker than the previousstudies. The results obtained in this paper are also valid to the conforming triangular elementand nonconforming Carey＇s element.

Monolithic approach to thermal fluid-structure interaction withnonconforming interfaces

This paper presents a monolithic approach to the thermal fluidstructure interaction （FSI） with nonconforming interfaces. The thermal viscous flow is governed by the Boussinesq approximation and the incompressible NavierStokes equations. The motion of the fluid domain is accounted for by an arbitrary LagrangianEulerian （ALE） strategy. A pseudosolid formulation is used to manage the deformation of the fluid do main. The structure is described by the geometrically nonlinear thermoelastic dynamics. An efficient data transfer strategy based on the Gauss points is proposed to guarantee the equilibrium of the stresses and heat along the interface. The resulting strongly coupled set of nonlinear equations for the fluid, solution procedure. A numerical example efficiency of the methodology. structure, and heat is solved by a monolithic is presented to demonstrate the robustness and