Stokes方程的稳定化间断有限元法

本文对定常的Stokes方程提出了一种新的间断有限元法,通过对通常的间断Galerkin有限元法应用稳定化思想,建立了一个相容的稳定间断有限元格式,对速度和压力的任意分片多项式空间Pl(K),Pm(K)的间断有限元逼近证明了解的存在唯一性,给出了关于速度和压力的L2 范数的最优误差估计．     

INTERIOR ERROR ESTIMATES FOR NONCONFORMING FINITE ELEMENT METHODS OF THE STOKES EQUATIONS

1.IntroductionInteriorerrorestimatesforfiniteelementdiscretizations(conforming)werefirstintroducedbyNitscheandSchatz[14]forsecondorderscalarellipticequationsin1974.Theyprovedthatthelocalaccuracyofthefiniteelementapproximationisboundedintermsoftwofact...     

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.     

ON THE ERROR ESTIMATES FOR THE ROTATIONAL PRESSURE-CORRECTION PROJECTION SPECTRAL METHODS FOR THE UNSTEADY STOKES EQUATIONS

This paper provides an analysis of the rotational form of the pressure-correction methods by spectral approximations for the unsteady Stokes equations. Error estimates in finite time for the fully discrete case are given. Numerical experiences using both spectral and spectral element methods are carried out to confirm the theoretical results.     

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.     

Stokes方程的压力梯度局部投影间断有限元法

本文对定常的Stokes方程提出了一种新的间断有限元法,通过将通常的间断Galerkin有限元法与压力梯度局部投影相结合,建立了一个稳定的间断有限元格式,对速度和压力的任意分片多项式空间P_l(K),P_m(K)的间断有限元逼近证明了解的存在唯一性,给出了关于速度和压力的L~2范数的最优误差估计.     

CONVERGENCE OF A MIXED FINITE ELEMENT FOR THE STOKES PROBLEM ON ANISOTROPIC MESHES

The main aim of this paper is to study the convergence properties of a low order mixed finite element for the Stokes problem under anisotropic meshes. We discuss the anisotropic convergence and superconvergence independent of the aspect ratio. Without the shape regularity assumption and inverse assumption on the meshes, the optimal error estimates and natural superconvergence at central points are obtained. The global superconvergence for the gradient of the velocity and the pressure is derived with the aid of a suitable postprocessing method. Furthermore, we develop a simple method to obtain the superclose properties which improves the results of the previous works .     

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.     

IMPROVED ERROR ESTIMATES FOR MIXED FINITE ELEMENT FOR NONLINEAR HYPERBOLIC EQUATIONS: THE CONTINUOUS-TIME CASE

1. IntroductionLet n be a bounded domain in RZ with Lipschitz boundary an, and unit Outward normalu. FOr ford 0 < T < co j J = (0, p, we discuss ~ face element aPProtoations of secondorder nonlinear hyperbolic equationwith hatiai conditionsand Dinchiet bOUndary conditionWe shall assume that the functions c(x,u), a(x,u), f(x,u,t), g(x,t) and solution u(z,t)have sufficient regular Additionally, we assume that there each constants c*, c*, a*, aama* such thatOpt~ tales Of convergence for Gal…     

无界区域上 Stokes 方程组的混合有限元方法

本文讨论无界区域上 Stokes 方程组边值问题的有限元近似解.为了克服区域的无界性所造成的困难,本文采用“局部化”技巧,首先将问题化为一个等价的有界区域上的边值问题,然后求解这个等价问题的混合有限元近似解,最后给出了有限元近似解的误差分析.     

MULTIGRID FOR THE MORTAR FINITE ELEMENT FOR PARABOLIC PROBLEM

In this paper, a mortar finite element method for parabolic problem is presented. Multi-grid method is used for solving the resulting discrete system. It is shown that the multigrid method is optimal, i.e, the convergence rate is independent of the mesh size L and the time step parameter τ.     

MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS

This paper gives the detailed numerical analysis of mixed finite element method for fractional Navier-Stokes equations.The proposed method is based on the mixed finite element method in space and a finite difference scheme in time.The stability analyses of semi-discretization scheme and fully discrete scheme are discussed in detail.Furthermore,We give the convergence analysis for both semidiscrete and flly discrete schemes and then prove that the numerical solution converges the exact one with order O(h2+k),where h and k:respectively denote the space step size and the time step size.Finally,numerical examples are presented to demonstrate the effectiveness of our numerical methods.     

A MIXED FINITE ELEMENT METHOD ON A STAGGERED MESH FOR NAVIER-STOKES EQUATIONS

In this paper, we introduce a mixed finite element method on a staggered mesh for the numerical solution of the steady state Navier-Stokes equations in which the two components of the velocity and the pressure are defined on three different meshes. This method is a conforming quadrilateral Q1 × Q1 - P0 element approximation for the Navier-Stokes equations. First-order error estimates are obtained for both the velocity and the pressure. Numerical examples are presented to illustrate the effectiveness of the proposed method.     

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．     

HAMILTON-JACOBI-BELLMAN方程的混合元方法

0　引　　言考虑下列随机微分方程 :ｄＸｔｄｔ =σ(Ｘｔ,αｔ) ζｔ+ｂ(Ｘｔ,αｔ)　ｔ≥ 0 ( 1 .1ａ)　　　　Ｘ0 =ｘ ∈ＲＮ ( 1 .1ｂ)　　此处 ,σ和ｂ分别是定义在ＲＮ×Ａ上的矩阵值和向量值函数 ,Ａ是给定的可分空间 ,αｔ值位于Ａ中的随机过程 更严格地 ,方程 ( 1 .1ａ)可写成下列形式 :Ｘｔ =ｘ + ∫ｔ0 σ(Ｘｓ,αｓ)ｄＢｓ+ ∫ｔ0 ｂ(Ｘｓ,αｓ)ｄｓ　ｔ≥ 0 ( 1 .2 )此外 ,Ｂｔ 是ｍ 维Ｂｒｏｗｎ运动 ,使得 :Ｅ(Ｂｔ) =0 ,Ｅ(Ｂ2 ｔ) =ｔ;∫ｔ0 σ(Ｘｓ,αｓ)ｄＢｓ 是随机积分 我们知道 ,σ ,ｂ若足够光滑 ,则保证了对每一…     

LEAST-SQUARES MIXED FINITE ELEMENT METHODS FOR THE INCOMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS

Least-squares mixed finite element methods are proposed and analyzed for the incompressible magnetohydrodynamic equations, where the two vorticities are additionally introduced as independent variables in order that the primal equations are transformed into the first-order systems. We show that there hold the coerciveness and the optimal error bound in appropriate norms for all variables under consideration, which can be approximated by all kinds of continuous element. Consequently, the Babuska-Brezzi condition(i.e. the inf-sup condition) and the indefiniteness are avoided which are essential features of the classical mixed methods.     

MULTIGRID METHODS FOR MORLEY ELEMENT ON NONNESTED MESHES

1.IntroductionWeconsidersomemultigridalgorithmsforthebiharmonicequationdiscretizedbyMoneyelementonnonnestedmeshes.TOdefineamultigridalgorithm,certainintergridtransferoperatorhastobeconstructed.Throughtakingtheaveragesofthenodalvariables,weconstructanintergridtransferoperatorforMoneyelementonnonnestedmeshesthatsatisfiesacertainstableapproximationpropertywhichplaysakeyroleinmultigridmethodsfornonconformingplateelementsonnonnestedmeshes.Theso--calledregularity-approximaticnassurnptionisestablis…     

UNIFIED ANALYSIS OF TIME DOMAIN MIXED FINITE ELEMENT METHODS FOR MAXWELL'S EQUATIONS IN DISPERSIVE MEDIA

In this paper, we consider the time dependent Maxwell＇s equations when dispersive media are involved. The Crank-Nicolson mixed finite element methods are developed for three most popular dispersive medium models： the isotropic cold plasma, the one-pole Debye medium and the two-pole Lorentz medium. Optimal error estimates are proved for all three models solved by the Raviart-Thomas-Ndd@lec spaces. Extensions to multiple pole dispersive media are presented also.     

REVIEW ARTICLE MULTIGRID METHODS FOR OBSTACLE PROBLEMS

In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set.