UNIFORMLY-STABLE FINITE ELEMENT METHODS FOR DARCY-STOKES-BRINKMAN MODELS

In this paper, we consider 2D and 3D Darcy-Stokes interface problems. These equations are related to Brinkman model that treats both Darcy＇s law and Stokes equations in a single form of PDE but with strongly discontinuous viscosity coefficient and zerothorder term coefficient. We present three different methods to construct uniformly stable finite element approximations. The first two methods are based on the original weak formulations of Darcy-Stokes-Brinkman equations. In the first method we consider the existing Stokes elements. We show that a stable Stokes element is also uniformly stable with respect to the coefficients and the jumps of Darcy-Stokes-Brinkman equations if and only if the discretely divergence-free velocity implies almost everywhere divergence-free one. In the second method we construct uniformly stable elements by modifying some well-known H（div）-conforming elements. We give some new 2D and 3D elements in a unified way. In the last method we modify the original weak formulation of Darcy-Stokes- Brinkman equations with a stabilization term. We show that all traditional stable Stokes elements are uniformly stable with respect to the coefficients and their jumps under this new formulation.     

Sine-Gordon方程的一类低阶非协调有限元分析

本文讨论了Sine-Gordon方程的一类低阶非协调有限元一般逼近格式,直接利用插值技巧和单元的特殊性质导出了相应未知量的最优误差估计.     

Darcy-Stokes问题的统一稳定化有限体积法分析

本文对Darcy-Stokes问题提出了一种统一的稳定化有限体积法.在离散问题中,采用两种剖分,一种为三角形剖分,一种为其对偶四边形剖分.速度及压力分别采用非协调线性元及分片常数元来做逼近.经证明,文中的统一格式,具有稳定性及最优误差估计.最后用数值算例验证了本文的理论结果.     

WELL-CONDITIONED FRAMES FOR HIGH ORDER FINITE ELEMENT METHODS

The purpose of this paper is to discuss representations of high order C⁰finite element spaces on simplicial meshes in any dimension.When computing with high order piecewise polynomials the conditioning of the basis is likely to be important.The main result of this paper is a construction of representations by frames such that the associated L²condition number is bounded independently of the polynomial degree.To our knowledge,such a representation has not been presented earlier.The main tools we will use for the construction is the bubble transform,introduced previously in[1],and properties of Jacobi polynomials on simplexes in higher dimensions.We also include a brief discussion of preconditioned iterative methods for the finite element systems in the setting of representations by frames.     

SUPERCONVERGENCE ANALYSIS OF LOW ORDER NONCONFORMING MIXED FINITE ELEMENT METHODS FOR TIME-DEPENDENT NAVIER-STOKES EQUATIONS

In this paper,the superconvergence properties of the time-dependent Navier-Stokes equations are investigated by a low order nonconforming mixed finite element method(MFEM).In terms of the integral identity technique,the superclose error estimates for both the velocity in broken H-norm and the pressure in L2-norm are first obtained,which play a key role to bound the numerical solution in Lx-norm.Then the corresponding global superconvergence results are derived through a suitable interpolation postprocessing approach.Finally,some numerical results are provided to demonstrated the theoretical analysis.     

平面线弹性纯位移边值问题低阶非协调矩形元的Robust多重网格方法

1引 言 对于各向同性,均匀介质的平面线弹性问题,当Lamé常数λ→∞(泊松率v→0.5)时,即对于几乎不可压介质,通常的协调有限元格式的解往往不再收敛到原问题的解,或者达不到最优收敛阶,这就是所谓的闭锁现象(见[3],[7],[8]及[10]).究其原因,在通常的有限元分析中,其误差估计的系数与λ有关,当λ→∞时,该系数将趋于无穷大.因此为克服闭锁现象就需要构造特殊的有限元格式,使得当λ→∞时,有限元逼近解仍然收敛到原问题的解.     

MIXED FINITE ELEMENT METHODS FOR A STRONGLY NONLINEAR PARABOLIC PROBLEM

1.IntroductionFOrsecondorderellipticproblems,themixedmethodwasdescribedandanalyzedbymanyauthors[1--3]inthecaseoflinearequationsindivergenceform,aswellasin[4,5]forquasilinearornonlinearproblemsindivergenceform.JohnsonandThorn6e[6]consideredalternativeproofSofthepreviouslyknownerrorestimatesforsuchmethodsintheellipticcase.Theyalsoanalyzedthemixedfiniteelementmethodforthepaxabolicequationgivenbypt--ho~f.Garcia[7]studiedtheconvergenceofmixedfiniteelementapproximationstoquasilinearparabolicequati…     

基于Riesz-表示算子非协调稳定化有限元法

Following the framework of the finite element methods based on Riesz-representing operators developed by Duan Huoyuan in 1997,through discrete Riesz representing-operators on some virtual(non-) conforming finite-dimensional subspaces,a stabilization formulation is presented for the Stokes problem by employing nonconforming elements.This formulation is uniformly coercive and not subject to the Babu ka-Brezzi condition,and the resulted linear algebraic system is positive definite with the spectral condition number O(h-2). Quasi-optimal error bounds are obtained,which is consistent with the interpolation properties of the finite elements used.     

A FINITE VOLUME ELEMENT METHOD FOR THERMAL CONVECTION PROBLEMS

Consider the finite volume element method for the thermal convection problem with the infinite Prandtl number. The author uses a conforming piecewise linear function on a fine triangulation for velocity and temperature, and a piecewise constant function on a coarse triangulation for pressure. For general triangulation the optimal order H^1 norm error estimates are given.     

THE OPTIMAL ORDER ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS TO HYPERBOLIC PROBLEMS

In this paper, the linear finite element approximation to the positive and symmetric,linear hyperbolic systems is analyzed and an O(h^2) order error estimate is established under the conditions of strongly regular triangulation and the H^3-regularity for the exact solutions. The convergence analysis is based on some superclose estimates derived in this paper. Our method and result here are also applicable to general hyperbolic problems.Finally, we discuss the linearized shallow water system of equations.     

THE NONCONFORMING FINITE ELEMENT METHOD FOR SIGNORINI PROBLEM

We present the Crouzeix-Raviart linear nonconforming finite element approximation of the variational inequality resulting from Signorini problem. We show if the displacement field is of H2 regularity, then the convergence rate can be improved from O（h3/4） to quasi-optimal O（h｜log h｜1/4） with respect to the energy norm as that of the continuous linear finite element approximation. If stronger but reasonable regularity is available, the convergence rate can be improved to the optimal O（h） as expected by the linear approximation.     

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     

SUPERCONVERGENCE ANALYSIS OF FINITE ELEMENT METHODS FOR OPTIMAL CONTROL PROBLEMS OF THE STATIONARY BENARD TYPE

In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Benard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in L^∞-norm and optimal error estimates in L^2-norm.     

RICHARDSON EXTRAPOLATION AND DEFECT CORRECTION OF FINITE ELEMENT METHODS FOR OPTIMAL CONTROL PROBLEMS

Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied. The higher order numerical approximations are used to generate a posteriori error estimators for the finite element approximation.     

Stokes问题非协调有限元逼近的最大模估计

本文考察了二维稳态和非稳态Stokes问题的基于速度—压力形式的非协调C-R逼近格式,利用Sobolev权模技巧和权模LBB条件,得到了稳态问题速度(包括它的梯度)和压力逼近解的拟最优的最大模估计,利用稳态问题结果和Stokes投影技巧,得到了非稳态问题速度(包括它的梯度)和压力的半离散逼近解的拟最优的最大模估计。     

A POSTERIORI ESTIMATOR OF NONCONFORMING FINITE ELEMENT METHOD FOR FOURTH ORDER ELLIPTIC PERTURBATION PROBLEMS

In this paper, we consider the nonconforming finite element approximations of fourth order elliptic perturbation problems in two dimensions. We present an a posteriori error estimator under certain conditions, and give an h-version adaptive algorithm based on the error estimation. The local behavior of the estimator is analyzed as well. This estimator works for several nonconforming methods, such as the modified Morley method and the modified Zienkiewicz method, and under some assumptions, it is an optimal one. Numerical examples are reported, with a linear stationary Cahn-HiUiard-type equation as a model problem.     

NONCONFORMING FINITE ELEMENT APPROXIMATIONS TOTHE UNILATERAL PROBLEM

1.IntroductionTherehavebeennumerousworkintheanalysisoffiniteelementmethodsfortheunilateralproblem(c.f.[4]andthereferencestherein).ItshouldbementionedthatinF.Scarpiniet.al.[6],I.Hlavaceket.al.[5]andF.Brezziet.al.[1],theconforminglinearelementapproximationtotheunilateralproblemhavebeenconsidered,andthevariouserrorboundshavebeenobtainedinthedifferentassumptionofregularityofsolutionoftheproblem.Inthispaper3weconsiderthreenonconformingfiniteelement(i.e.twoCrouzrixRaviartlinearelementsandWilsone…