二维发展型对流占优扩散方程的FD-SD法的后验误差估计

引言 对流占优扩散问题是流体力学中一个典型的模型问题,对其数值求解始终是众多学者相当关心的课题．［１１］中指出,即使对于线性问题,通常其解在外流边界附近也会产生剧烈变化．倘若在内流边界上所给出的边值函数存在不连续点时,则在沿过此不连续点的特征线（流线）附近会出现断层．因此在数值求解对流占优扩散问题时,尽管标准有限元法具有高阶精度,但常产生数值剧烈振荡Ｓ而古典人工粘性Ｇａｌｅｒｋｉｎ法虽具有较好的稳定性,但仅具有一阶精度．流线扩散法（Ｓｔｒｅａｍｌｉｎｅ　 Ｄｉｆｆｕｓｉｏｎ Ｍｅｔｈｏｄ,简称 ＳＤ…     

对流占优问题的无网格稳定化方法

应用标准的无网格方法求解对流占优问题时会出现数值伪振荡.针对此问题,给出了无网格方法中消除非稳定数值解的4种技术,即节点加密、增大节点影响半径、完全迎风无网格稳定化方法、自适应无网格稳定化方法.并将这4种技术应用于径向点插值方法求解一维或二维对流扩散方程.数值结果表明这4种技术均能有效地消除对流占优时的数值伪振荡现象,且自适应迎风无网格稳定化方法是4种技术中最有效的.     

非线性对流扩散问题的差分-流线扩散法

1．引言流线扩散法（简称SD方法）是由Huzhes和Brooks在1980年前后提出的一种数值求解对流占优扩散问题的新型有限元算法．随后，Johnson和N8vert将SD方法推广到发展型对流扩散问题（［1］，［2］，［3］）．熟知，对于对流扩散问题，标准有限元法虽具有高阶精度，但常产生数值振荡；古典人工粘性Galerkin法更具有较好的稳定性，但仅具有一阶精度．而（SD方法兼具良好的数值稳定性和高阶精度，因此得到了越来越多的重视，对于发展型对流扩散问题，传统的SD方法均采用时空有限元．这样做，虽然可使时间和空间方向上的精度很好的协调起…     

对流扩散方程的经济差分格式

１．引言 对流扩散方程是一类基本的运动方程,它可描述质量、热量的输运过程以及反应扩散过程等众多物理现象．寻找稳定、快速实用的数值方法,有着重要的理论和实际意义．标准的差分方法或有限元方法对它常常失效,根本原因在于“对流项”的存在．［１］提出了解对流扩散方程的特征线修正技术,这一方法考虑沿着特征线（流动方向）的离散,利用了对流扩散问题的物理力学性质,可以有效地克服数值振荡,保证数值解的稳定,尤其对“对流占优”的问题,这一方法有突出的优越性．这方面已有大量的理论和应用研究成果［２,３,７］．对大规模…     

二维非线性对流扩散方程的非振荡特征差分方法

１．引言 近十几年来,双曲守恒律问题的高分辨率格式已取得很大发展,具有局部自适应选取节点的非振荡插值算法（如 ＵＮＯ［１］, ＥＮＯ［２］等）在这些格式的构造中起着重要的作用．特征差分法是求解对流扩散问题的一种较为有效方法,但在求解具有陡峭前线问题时,也会产生非物理振荡阻（见４）．本文将把特征差分法与非振荡插值算法相结合构造对流扩散问题的高分辨率差分格式． ［１］中的 ＵＮＯ及［２］中的 ＥＮＯ插值都是一维的,有关讨论二维 ＵＮＯ及ＥＮＯ插值的文章还不多见,本文将构造二维基于六节点的二次非振荡插值以及…     

非线性对流扩散方程的UNO-MMOCAA差分方法

本文把MMOCAA差分方法与UNO插值相结合，提出了求解对流占优扩散问题的UN0—MMOCAA差分方法，它避免了基于高次(≥2)Lagrange插值的MMOCAA差分方法在方程解的陡峭前沿附近产生的振荡．本文通过引入辅助插值算于等方法，给出了非线性UNO—MMOCAA差分格式的误差分析．数值例子表明新格式无振荡。     

二维非线性对流扩散方程特征有限元的两重网络算法

针对二维非线性对流扩散方程,构造了特征有限元两重网格算法．该算法只需要在粗网格上进行非线性迭代运算,而在所需要求解的细网格上进行一次线性运算即可．对于非线性对流占优扩散方程,不仅可以消除因对流占优项引起的数值振荡现象,还可以加快收敛速度、提高计算效率．误差估计表明只要选取粗细网格步长满足一定的关系式,就可以使两重网格解与有限元解保持同样的计算精度．算例显示:两重网格算法比特征有限元算法的收敛速度明显加快．     

非定常线性对流扩散问题的算子分裂半显式有限元分析

在现代科学及工程领域中,存在着许多同时伴有物质传输和动力耗散两种过程的物理系统.在数学上,它们常归结为对流占优的对流扩散方程或以这种方程占主导的方程组.这类方程具有殆双曲性质,其解函数常呈现局部大梯度变化,使得传统的求解抛物问题的数值方法常常     

多孔介质驱动问题的混合元最小二乘特征有限元方法及其收敛分析

１．引言多孔介质二相驱动问题的数学模型是偶合的非线性偏微分方程组的初边值问题．该问题可转化为压力方程和浓度方程［１－４］．浓度方程一般是对流占优的对流扩散方程,它的对流速度依赖于比浓度方程的扩散系数大得多的Ｆａｒｃｙ速度．因此Ｄａｒｃｙ速度的求解精度直接影响着浓度的求解精度．为了提高速度的求解精度,７０年代Ｐ．Ａ．Ｒａｖｉａｔ和Ｊ．Ｍ．Ｔｈｏｍａｓ提出混合有限元方法［５］．Ｊ．ＤｏｕｇｌａｓＪｒ,Ｔ．Ｆ．Ｒｕｓｓｅｌｌ,Ｒ．Ｅ．Ｅｗｉｎｇ,Ｍ．Ｆ．Ｗｈｅｅｌｅｒ［１］－［４］,［９］,［１２］袁…     

一类非线性对流扩散问题的FDSD预测校正格式

1．引言由Hughes和Brooks门提出,并经Johnson等人［‘－‘1发展的流线扩散法（Streamline－DiffusionMetho人以下简称SD方法）是求解对流占优扩散问题（包括纯双曲问题）的一种有效的数值方法．由于良好的数值稳定性及其高阶收敛率,SD方法已广泛地应用于计算流体等诸多科学工程计算．然而,传统的sD方法利用时一空有限元求解发展型问题,导致对高维问题工作量过于庞大;其编程实现较复杂,对非线性问题也不便进行线性化处理．为使SD方法能够较简便地应用于高维和非线性问题,孙撒问提出了仅对空间域作有限元离散,而对时间域作差分…     

Convergence of Extrapolated BDF2 Finite Element Schemes for Unsteady Penetrative Convection Model

Extrapolated two-step backward difference (BDF2) in time and finite element in space discretization for the unsteady penetrative convection model is analyzed. Penetrative convection model employs a nonlinear equation of state making the problem more nonlinear. Optimal order error estimates are derived for the semi-discrete finite element spatial discretization. Two time discretization schemes based on linear extrapolation are proposed and analyzed, namely a coupled and a decoupled scheme. In particular, we show that although both schemes are unconditionally nonlinearly stable, the decoupled scheme converges unconditionally whereas coupled scheme requires that the time step be sufficiently small for convergence. These time discretization schemes can be implemented efficiently in practice, saving computational memory. Numerical computations and numerical convergence checks are presented to demonstrate the efficiency and the accuracy of the schemes.     

THE UPWIND FINITE ELEMENT SCHEME AND MAXIMUM PRINCIPLE FOR NONLINEAR CONVECTION-DIFFUSION PROBLEM

In this paper, a kind of partial upwind finite element scheme is studied for twodimensional nonlinear convection-diffusion problem. Nonlinear convection term approximated by partial upwind finite element method considered over a mesh dual to the triangular grid, whereas the nonlinear diffusion term approximated by Galerkin method. A linearized partial upwind finite element scheme and a higher order accuracy scheme are constructed respectively. It is shown that the numerical solutions of these schemes preserve discrete maximum principle. The convergence and error estimate are also given for both schemes under some assumptions. The numerical results show that these partial upwind finite element scheme are feasible and accurate.     

Energy-momentum conserving discretization of mixed shell elements for large deformation problems

In the present paper we consider structure-preserving integration methods in the context of mixed finite elements. The used low-order mixed finite elements typically exhibit improved coarse mesh accuracy. On the other hand energy-momentum (EM) consistent time-stepping schemes have been developed in the realm of nonlinear structural dynamics to enhance the numerical stability properties. EM schemes typically exhibit superior robustness and thus offer the possibility to use large time steps while still producing physically meaningful results. Accordingly, combining mixed finite element discretizations in space with EM consistent discretizations in time shows great promise for the design of numerical methods with superior coarse mesh accuracy in space and time. Starting with a general Hu-Washizu-type variational formulation we develop a second-order accurate structure-preserving integration scheme. The present approach is applicable to a large number of mixed finite element formulations. As sample application we deal with a specific mixed shell element. Numerical examples dealing with large deformations will show the improved coarse mesh accuracy in space and time of the advocated approach. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Alternating Direction Finite Volume Element Methods for Three-Dimensional Parabolic Equations

<正>This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes,one is analogous to Douglas finite difference scheme with second-order splitting error,the other two schemes have third-order splitting error,and the last one is an extended LOD scheme.The L~2 norm and H~1 semi-norm error estimates are obtained for the first scheme and second one,respectively.Finally,two numerical examples are provided to illustrate the efficiency and accuracy of the methods.     

A characteristics-mixed finite element method for Burgers’ equation

In this paper, we propose a new mixed finite element method, called the characteristics-mixed method, for approximating the solution to Burgers’ equation. This method is based upon a space-time variational form of Burgers’ equation. The hyperbolic part of the equation is approximated along the characteristics in time and the diffusion part is approximated by a mixed finite element method of lowest order. The scheme is locally conservative since fluid is transported along the approximate characteristics on the discrete level and the test function can be piecewise constant. Our analysis show the new method approximate the scalar unknown and the vector flux optimally and simultaneously. We also show this scheme has much smaller time-truncation errors than those of standard methods. Numerical example is presented to show that the new scheme is easily implemented, shocks and boundary layers are handled with almost no oscillations. One of the contributions of the paper is to show how the optimal error estimates inL ²(Ω) are obtained which are much more difficult than in the standard finite element methods. These results seem to be new in the literature of finite element methods.     

HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY

In this paper, efficient numerical schemes are proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave. By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator, finite difference method in time and spectral method in space are constructed for the considered model. The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term. The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time, and spectral accuracy in space. Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims. Finally, the decay rate of solutions is investigated.     

A second order characteristic finite element scheme for convection-diffusion problems

Summary. A new characteristic finite element scheme is presented for It is of second order accuracy in time increment, symmetric, and unconditionally stable. Optimal error estimates are proved in the framework of -theory. Numerical results are presented for two examples, which show the advantage of the scheme. Received November 22, 2000 / Revised version received July 11, 2001 / Published online October 17, 2001     

关于一个第二类变分不等式的有限元逼近

A new type of finite element scheme including the numerical integration modi-fication is presented for the second type variational inequality. Our methods really simplify the finite element analysis and practical calculation. The unique existence and stability of finite element solution are proved , and particularly the optimal order error estimates are derived under H^1 and L2 norms.     

Error estimates of H^1-Galerkin mixed finite element method for Schrodinger equation

An H1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.     

Second order modified method of characteristics mixed defect-correction finite element method for time dependent Navier–Stokes problems

In this paper, a second order modified method of characteristics defect-correction (SOMMOCDC) mixed finite element method for the time dependent Navier–Stokes problems is presented. In this method, the hyperbolic part (the temporal and advection term) are treated by a second order characteristics tracking scheme, and the non-linear term is linearized at the same time. Then, we solve the equations with an added artificial viscosity term and correct this solution by using the defect-correction technique. The error analysis shows that this method has a good convergence property. In order to show the efficiency of the SOMMOCDC mixed finite element method, we first present some numerical results of an analytical solution problem, which agrees very well with our theoretical results. Then, we give some numerical results of lid-driven cavity flow with the Reynolds number Re = 5,000, 7,500 and 10,000. From these numerical results, we can see that the schemes can result in good accuracy, which shows that this method is highly efficient.