Ladyzhenskaya模型的非协调有限元逼近

§1.引言 Navier-Stokes方程是描述粘性不可压缩流体运动的偏微分方程,它是研究这类非线性问题的很好的数学模型。但是,当速度梯度较大时,方程的整体解是否唯一可解,这个问题尚未得到解决。为此,对于不可压缩粘性流体的定常情形,Ladyzhenskaya提出利用下面的模型来代替Navier-Stokes方程。 设Ω是R~n(n=2或3)中的有界区域,边界?ΩLipshitz连续,u是流体速度,p是     

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.     

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for semisingularly perturbed reaction–diffusion problems

The numerical approximation by a lower‐order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving semisingular perturbation problems. The quasi‐optimal‐order error estimates are proved in the ε‐weighted H¹‐norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε‐weighted H¹‐norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

抛物问题非协调元多重网格法

抛物问题非协调元多重网格法周叔子,文承标（湖南大学应用数学系）ＮＯＮＣＯＮＦＯＲＭＩＮＧＥＬＥＭＥＮＴＭＵＬＴＩＧＲＩＤＭＥＴＨＯＤＦＯＲＭＲＡＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＺｈｏｕＳｈｕ－ｚｉ;ＷｅｎＣｈｅｎｇ－ｂｉａｏ（ＨｕｎａｎＵｎｉｖｅｒｓｉ...     

二阶椭圆问题的各向异性非协调Crouzeix-Raviart型有限元方法

对满足最大角条件和坐标系条件的二维区域中的各向异性一般三角形网格,研究了二阶椭圆问题的非协调Crouzeix-Raviart型线性三角形有限元逼近,得到了最优的能量模和L2-模误差估计结果．     

非定常Navier-Stokes方程的质量集中各向异性非协调有限元逼近

该文将一个低阶Crouzeix-Raviart型非协调三角形元应用到非定常Navier-Stokes方程,给出了其质量集中有限元逼近格式.在不需要传统Ritz-Volterra投影下,通过引入两个辅助有限元空间对边界进行估计的技巧,在各向异性网格下导出了速度的L~2模和能量模及压力的L~2模的误差估计.     

Navier－Stokes方程的变网格非协调有限元法

本文通过所谓的速度－压力型公式讨论了Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程的变网格非协调有限元逼近，得到了在确定模意义下的速度、压力误差估计，且在一定条件下，某些误差估计能达到最优。     

Two-level Additive Schwarz Preconditioners for Nonconforming Finite Element Methods

Two-level additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar second-order symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergence-free nonconforming P1 finite element approximation of the stationary Stokes equations. The condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap.

     

A superconvergent nonconforming mixed finite element method for the Navier–Stokes equations

The superconvergence for a nonconforming mixed finite element approximation of the Navier–Stokes equations is analyzed in this article. The velocity field is approximated by the constrained nonconforming rotated Q₁ (CNRQ₁) element, and the pressure is approximated by the piecewise constant functions. Under some regularity assumptions, the superconvergence estimates for both the velocity in broken H¹‐norm and the pressure in L²‐norm are obtained. Some numerical examples are presented to demonstrate our theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 646–660, 2016     

HOW TO PROVE THE DISCRETE RELIABILITY FOR NONCONFORMING FINITE ELEMENT METHODS

Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity.One key ingredient is the discrete reliability of a residualbased a posteriori error estimator,which controls the error of two discrete finite element solutions based on two nested triangulations.In the error analysis of nonconforming finite element methods,like the Crouzeix-Raviart or Morley finite element schemes,the difference of the piecewise derivatives of discontinuous approximations to the distributional gradients of global Sobolev functions plays a dominant role and is the object of this paper.The nonconforming interpolation operator,which comes natural with the definition of the aforementioned nonconforming finite element in the sense of Ciarlet,allows for stability and approximation properties that enable direct proofs of the reliability for the residual that monitors the equilibrium condition.The novel approach of this paper is the suggestion of a right-inverse of this interpolation operator in conforming piecewise polynomials to design a nonconforming approximation of a given coarse-grid approximation on a refined triangulation.The results of this paper allow for simple proofs of the discrete reliability in any space dimension and multiply connected domains on general shape-regular triangulations beyond newest-vertex bisection of simplices.Particular attention is on optimal constants in some standard discrete estimates listed in the appendices.     

非线性抛物方程的质量集中非协调元分析

将一个低阶Crouzeix-Raviart型非协调三角形元应用到一类非线性抛物方程,并建立了质量集中的半离散和向后Euler全离散逼近格式,在一般各向异性网格上利用插值算子导出了L²-模的最优误差估计.     

A nonconforming exponentially fitted finite element method for two‐dimensional drift‐diffusion models in semiconductors

A new nonconforming exponentially fitted finite element for a Galerkin approximation of convection–diffusion equations with a dominating advective term is considered. The attention is here focused on the drift‐diffusion current continuity equations in semiconductor device modeling. The scheme extends to the two‐dimensional case, the well known Scharfetter–Gummel method, by imposing a divergence‐free current over each element of the triangulation. Convergence of the method in the energy norm is proved and some numerical results are included. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 133–150, 1999     

Recovery type a posteriori estimates and superconvergence for nonconforming FEM of eigenvalue problems

The main goal of this paper is to present recovery type a posteriori error estimators and superconvergence for the nonconforming finite element eigenvalue approximation of self-adjoint elliptic equations by projection methods. Based on the superconvergence results of nonconforming finite element for the eigenfunction we derive superconvergence and recovery type a posteriori error estimates of the eigenvalue. The results are based on some regularity assumption for the elliptic problem and are applicable to the lowest order nonconforming finite element approximations of self-adjoint elliptic eigenvalue problems with quasi-regular partitions. Therefore, the results of this paper can be employed to provide useful a posteriori error estimators in practical computing under unstructured meshes.     

A nonconforming mixed finite element for second-order elliptic problems

In a recent work, Hiptmair [Mathematisches Institut, M9404, 1994] has constructed and analyzed a family of nonconforming mixed finite elements for second-order elliptic problems. However, his analysis does not work on the lowest order elements. In this article, we show that it is possible to construct a nonconforming mixed finite element for the lowest order case. We prove the convergence and give estimates of optimal order for this finite element. Our proof is based on the use of the properties of the so-called nonconforming bubble function to control the consistency terms introduced by the nonconforming approximation. We further establish an equivalence between this mixed finite element and the nonconforming piecewise quadratic finite element of Fortin and Soulie [J. Numer. Methods Eng., 19, 505–520, 1983]. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 445–457, 1997     

各向异性EQ1rot非协调元高精度分析的一般格式

本文在各向异性网格下讨论了一般二阶椭圆方程的EQ₁^rot非协调有限元逼近. 利用Taylor展开, 积分恒等式和平均值技巧导出了一些关于该元新的高精度估计. 再结合该元所具有的二个特殊性质: (a)当精确解属于H³时, 其相容误差为O(h²)阶比它的插值误差高一阶; (b)插值算子与Ritz投影算子等价,得到了在能量模意义下O(h²)阶的超逼近性质. 进而,借助于插值后处理技术给出了整体超收敛的一般估计式.     

Patch‐wise local projection stabilized finite element methods for convection–diffusion–reaction problems

In this article, we develop patch‐wise local projection‐stabilized conforming and nonconforming finite element methods for the convection–diffusion–reaction problems. It is a composition of the standard Galerkin finite element method, the patch‐wise local projection stabilization, and weakly imposed Dirichlet boundary conditions on the discrete solution. In this paper, a priori error analysis is established with respect to a patch‐wise local projection norm for the conforming and the nonconforming finite element methods. The numerical experiments confirm the efficiency of the proposed stabilization technique and validate the theoretical convergence rates.     

位移障碍下四阶变分不等式问题的非协调有限元一般误差估计式

1.引 言 关于二阶变分不等式问题的非协调有限元逼近已有大量研究[1-5].但是,对于四阶变分不等式的研究相对而言较少[6-7].[8,9,10]给出了位移障碍问题的非协调有限元,包括C0元(如Zienkiewicz元及Adini元)和非C0元(如Morley元及De Veubeke元)逼近的理论分析及最优误差估计.经过仔细分析发现,其成功的关键技巧是充分利用上述单元的一个     

带补偿C^0非协调板元的多重网格法

本文考虑重调和方程的Ｃ０非协调元逼近．通过双线性型ｃｋ（ｕ，ｖ）引入的补偿和将多重网格法应用到Ｃ０非协调板元，给出了更精确的逼近．     

A class of random variational inequalities and simple random unilateral boundary value problems - existence,discretization, finite element approximation

The paper deals with a class of random variational inequalities and simple random elliptic boundary value problems with unilateral conditions. Here randomness enters in the coefficient of the elliptic operator and in the right hand side of the p.d.e. In addition to existence and uniqueness results a theory of combined probabilistic deterministic discretization is developped that includes nonconforming approxima¬tion of unilateral constraints. Without any regularity assumptions on the solution, norm convergence of the full approximation process is established. The theory is applied to a Helmholtz like elliptic equation with Signorini boundary conditions as a simple model problem, where Galerkin discretization is realized by finite element approximation