Stokes问题的各向异性平行四边形有限元逼近

1 引言 Stokes问题是标准的混合问题,速度与压力同时计算,关于该问题有限元求解的文章很多(见文献[1-5])但大多都是基于对区域的正则剖分或拟一致剖分,即要求网格剖分满足hk/pK≤C,(A)K∈Jh,其中C>0为一常数,hk,pK分别为单元K的直径及内切园直径,在实际应用问题中,由于边界层或区域的拐角处需考虑物质的各向异性特征,此时对空间区域Q的剖分不再满足正则性或拟一致条件,而需要用各向异性网格剖分,才能更贴切地描述其真实情形.     

Stokes问题的一种新的混合有限元逼近

为了求解Ｓｔｏｋｅｓ问题，本文构造出了一类新的满足ＢＢ－条件的有限元空间对，并给出了相应的超收敛分析．     

CONVERGENCE ANALYSIS OF MORLEY ELEMENT ON ANISOTROPIC MESHES

The main aim of this paper is to study tile convergence of a nonconforming triangular plate element-Morley element under anisotropic meshes. By a novel approach, an explicit bound for the interpolation error is derived for arbitrary triangular meshes （which even need not satisfy the maximal angle condition and the coordinate system condition ）, the optimal consistency error is obtained for a family of anisotropically graded finite element meshes.     

Stokes问题的一个非协调有限元

<正>1引言定常Stokes问题是流体力学中的一种重要问题,是标准的混合问题,速度与压力同时计算.关于该问题有限元求解的文章很多([1],[2],[4],[5],[6],[8],[9]),分析的难点在于单元必须满足离散的Babuska-Brezzi([2])条件.在([4])中提出了著名的非协调Crouzeix-Raviart     

流线扩散有限元方法在分层网格上的收敛性分析

本文在分层网格上分析了采用线性元的流线扩散有限元方法求解一维对流扩散型奇异摄动问题的一致收敛性.在ε≤N~(-1)的前提下,可以证明在SD范数下的一致误差估计为O(N~(-1)(log 1/ε)~2)在数值算例部分对理论结果进行了验证.     

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 offinite element approximations both for conventional conforming elements and nonconform-ing elements.The aim of this paper is to present a novel approach of dealing with theapproximation of a four-degree nonconforming finite element for the second order ellipticproblems 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 onsome new special features of this element discovered herein.Numerical results are givento demonstrate validity of our theoretical analysis.     

各向异性网格下抛物方程一个新的非协调混合元收敛性分析

本文将 Crouzeix-Raviart 型非协调线性三角形元应用到抛物方程,建立了一个新的混合元格式.在抛弃传统有限元分析的必要工具 Ritz 投影算子的前提下,直接利用单元的插值性质和导数转移技巧, 分别得到了各向异性剖分下关于原始变量u 的H^-1-模和积分意义下L²-模以及通量p=-▽u 在L²-模下的最优阶误差估计.数值结果与我们的理论分析是相吻合的.     

ON THE ANISOTROPIC ACCURACY ANALYSIS OF ACM''''S NONCONFORMING FINITE ELEMENT

The main aim of this paper is to study the superconvergence accuracy analysis of thefamous ACM's nonconforming finite element for biharmonic equation under anisotropicmeshes. By using some novel approaches and techniques, the optimal anisotropic inter-polation error and consistency error estimates are obtained. The global error is of orderO(h~2). Lastly, some numerical tests are presented to verify the theoretical analysis.     

A MIXED FINITE ELEMENT METHOD FOR THE CONTACT PROBLEM IN ELASTICITY

Based on the analysis of [7] and [10], we present the mixed finite element approximation of the variational inequality resulting from the contact problem in elasticity. The convergence rate of the stress and displacement field are both improved from O（h3/4） to quasi-optimal O（h│logh│^1/4）. If stronger but reasonable regularity is available, the convergence rate can be optimal O（h）.     

ERROR ANALYSIS OF A STABILIZED FINITE ELEMENT METHOD FOR THE GENERALIZED STOKES PROBLEM

This paper is devoted to the establishment of sharper $a$ $priori$stability and error estimates of a stabilized finite element method proposed by Barrenechea and Valentin for solving the generalized Stokes problem, which involves a viscosity $\nu$ and a reaction constant $\sigma$. With the establishment of sharper stability estimates and the help of $ad$ $hoc$finite element projections, we can explicitly establish the dependence of error bounds of velocity and pressure on the viscosity $\nu$, the reaction constant $\sigma$, and the mesh size $h$. Our analysis reveals that the viscosity $\nu$ and the reaction constant $\sigma$ respectively act in the numerator position and the denominator position in the error estimates of velocity and pressure in standard norms without any weights. Consequently, the stabilization method is indeed suitable for the generalized Stokes problem with a small viscosity $\nu$ and a large reaction constant $\sigma$. The sharper error estimates agree very well with the numerical results.     

Stokes问题混合元方法中数值积分的影响

１引言用混合有限元方法求解Ｓｔｏｋｅｓ问题的近似解时,有两点重要事项．其一是众所周知的稳定化条件,这就导致在一阶及二阶格式中引人水泡（ｂｕｂｂｌｅ）函数（参见［１］,［２］,［４］和［６］）．其二是在实际计算中,含有体力密度ｆ的积分,需代之以数值积分,而保持原有的逼近阶不变．值得指出的是。该数值积分与水泡（ｂｕｂｂｌｅ）函数项无关．考虑数值积分下的Ｓｔｏｋｅｓ问题混合元逼近：设给定空间其中,是有界连通的多边形区域．考虑Ｓｔｏｋｅｓ问题混合变分形式：求Ｖ,ｐ　Ｍ,使得其中表示Ｌ２内积．众所周知,（…     

混合有限元非嵌套网格型预处理法

1 引言 众所周知,二阶椭圆型问题混合有限元离散以后的矩阵是不定的,所以对混合法很难形成一种有效的区域分解法,在文[9]、[10]、[11]中提出了一些混合有限元方法的区域分解法,但在实际计算中有很多局限性。最近Chen对混合有限元法提出一种全新的解释并把它应用到多重网格法中,他的基本思想是混合有限元离散的代数系统实际上等价于某个非协调有限元离散的代数系统,这样可把一个不定问题转化为一个正定问题,本文将基于这种思想考虑混合有限元的区域分解法。 若按传统的Dryia-widlund两水平加性Schwarz方法,要求两层网格间具有嵌套关系,这样在应用中将带来很大的不便。本文将不要求粗网格嵌入细网格中,减少两层网格间的     

A POSTERIORI ERROR ESTIMATION OF THE NEW MIXED ELEMENT SCHEMES FOR SECOND ORDER ELLIPTIC PROBLEM ON ANISOTROPIC MESHES

This paper presents a posteriori residual error estimator for the new mixed el-ement scheme for second order elliptic problem on anisotropic meshes. The reliability and efficiency of our estimator are established without any regularity assumption on the mesh.     

A SUPERONVERGENCE ANALYISIS FOR FINITE ELEMENT SOLUTION BY THE INTERPOLANT POSTPROCESSING ON IRREGULAR MESHES FOR SMOOTH PROBLEM

1. IntroductionThe results in this paper are based on the idea of interpolation postprocessing in [11 andthe techniques of LZ projection processing in [2]For simlicity) we consider the model problem: Finds e Hi(fl),such thatSuppose that Jh and JH are irregular triangulations (or quadrilateral partitions). Theirsizes satisfy h << H, (H - 0). Construct piecewise k-order and r-order finite element spaceSh and SH respectively. Let ah E Sh be the Galerkin approximation of u E HJ(fl), andbe …     

MULTIGRID METHODS FOR THE GENERALIZED STOKES EQUATIONS BASED ON MIXED FINITE ELEMENT METHODS

1. IntroductionIn this paper, we consider the fOllowing generalized stationary Stokes equations:where fl is a bounded convex domain in R', u represents the velocity of fluid, p its pressure; Fand G are external fOrce and source terms. Note that the source…     

A MIXED VIRTUAL ELEMENT METHOD FOR THE BOUSSINESQ PROBLEM ON POLYGONAL MESHES

In this work we introduce and analyze a mixed virtual element method(mixed-VEM)for the two-dimensional stationary Boussinesq problem.The continuous formulation is based on the introduction of a pseudostress tensor depending nonlinearly on the velocity,which allows to obtain an equivalent model in which the main unknowns are given by the aforementioned pseudostress tensor,the velocity and the temperature,whereas the pressure is computed via a postprocessing formula.In addition,an augmented approach together with a fixed point strategy is used to analyze the well-posedness of the resulting continuous formulation.Regarding the discrete problem,we follow the approach employed in a previous work dealing with the Navier-Stokes equations,and couple it with a VEM for the convection-diffusion equation modelling the temperature.More precisely,we use a mixed-VEM for the scheme associated with the fluid equations in such a way that the pseudostress and the velocity are approximated on virtual element subspaces of H(div)and H¹,respectively,whereas a VEM is proposed to approximate the temperature on a virtual element subspace of H¹.In this way,we make use of the L²-orthogonal projectors onto suitable polynomial spaces,which allows the explicit integration of the terms that appear in the bilinear and trilinear forms involved in the scheme for the fluid equations.On the other hand,in order to manipulate the bilinear form associated to the heat equations,we define a suitable projector onto a space of polynomials to deal with the fact that the diffusion tensor,which represents the thermal conductivity,is variable.Next,the corresponding solvability analysis is performed using again appropriate fixed-point arguments.Further,Strang-type estimates are applied to derive the a priori error estimates for the components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure.The corresponding rates of convergence are also established.Finally,several numerical examples illustrating the performance of the mixed-VEM scheme and confirming these theoretical rates are presented.     

ON THE P1 POWELL-SABIN DIVERGENCE-FREE FINITE ELEMENT FOR THE STOKES EQUATIONS

The stability of the P1-P0 mixed-element is established on general Powell-Sabin triangular grids. The piecewise linear finite element solution approximating the velocity is divergence-free pointwise for the Stokes equations. The finite element solution approximating the pressure in the Stokes equations can be obtained as a byproduct if an iterative method is adopted for solving the discrete linear system of equations. Numerical tests are presented confirming the theory on the stability and the optimal order of convergence for the P1 Powell-Sabin divergence-free finite element method.     

鱼骨形网格上二阶方程混合元的超收敛

In this paper, superconvergence of the lowest order Raviart-Thomas mixed finite element approximation for second order Neumann boundary value problem on fishbone shape meshes is analyzed. The main term of the error between the exact solution and the finite element interpolating function is determined by Bramble-Hilbert lemma on the individual finite element. A part of the main term of the error on two adjacent finite elements can be cancelled along the special direction, and thus the higher order error estimate is obtained on the whole domain by summation. Compared with the general finite element error estimate,the convergence rate can be increased from order one to order two in L2-norm by postprocessing superconvergence technique.     

GLOBAL SUPERCONVERGENCE OF THE MIXED FINITEELEMENT METHODS FOR 2-D MAXWELL EQUATIONS

Superconvergence of the mixed finite element methods for 2-d Maxwell equations is studied in this paper. Two order of superconvergent factor can be obtained for the k-th Nedelec elements on the rectangular meshes.