定常的磁流体力学方程的非线性Galerkin—Petrov最小二乘混合元法

本文提出了磁流体力学方程的一种非线性Galerkin-Petrov最小二乘混合元法,并导出了该方法解的存在性和误差估计.     

非定常的热传导-对流问题的Petrov最小二乘混合有限元法及其误差估计

文提出了非定常的热传导-对流方程的一种Petrov最小二乘混合有限元法.Petrov最小二乘混合有限元法可以回避Babuska-Brezzi条件的约束,使得有限元空间可以自由地选择并获得最优阶的误差估计.     

对流扩散反应方程的特征分裂最小二乘方法

1引言众所周知,最小二乘混合有限元方法具有两个显著的优点:一是不必满足经典混合元要求LBB条件,因此一般的有限元空间可供选择;二是算法系统是对称正定的,从而利于问题的求解.Pehlivanov等提出了一种最小二乘混合有限元算法求解椭圆型边值问题,并给出了H~1×H(div,·)模误差估计.之后,Cai等人把此方法推广应用到带有对流和反应项的二阶偏微分方程.近年来,最小二乘方法被应用到时间相关的问题.     

退化椭圆问题的最小二乘混合有限元方法

本文研究了电磁场中关于共振现象的一类退化的椭圆问题 ,提出了最小二乘混合有限元方法 .这一方法的好处是可以去掉传统混合元空间的LBB条件所得到的系数矩阵是对称正定的 ,使得法语解更加方便 .本文得到了最小二乘混合有限元方法的L2 和H1估计 .     

退化椭圆问题的最小二乘混合元逼近

１．引言我们将考虑具有退化系数的椭圆问题其中 Ω为 ＩＲ２中的一个凸多边形区域,定义为这里的ｇ＞０为分片线性连续函数．从物理背景来看,问题（１．１）来源于轴对称共振结构中的电磁场研究．Ｍａｒｉｎｉ,Ｐｉｅｔｒａ（１９９５）［４］研究了问题（１．１）的混合有限元逼近,并得到了最优误差估计． 此文,我们采用一种新的混合元,即最小二乘混合元方法［５］;对退化问题（１．１）进行逼近,利用插值投影证明了近似解具有最优阶精度的收敛性．比较起经典混合元方法来,最小二乘混合元方法有两个优越性：有限元空间不必满足Ｌ…     

定常的Navier-Stokes方程的非线性Galerkin/Petrov最小二乘混合元法

给出定常的Navier-Stokes方程的一种非线性Galerkin/Petrov最小二乘混合元法，该方法是将余量形式的Petrov最小二乘方法与非线性Galerkin混合元结合起来，使得速度和压力的混合元空间无需满足离散的Babuska-Brezzi稳定性条件，从而使得它们的有限元空间可以任意选择。并证明该方法的解的存在唯一性和收敛性。     

热传导对流问题的自适应最小二乘Galerkin/Petrov混合有限元法

对热传导对流问题提出了自适应Galerkin/Petrov最小二乘混合有限元法．该算法对任何速度和压力有限元空间的组合是相容和稳定的(不需要满足Babuka-Brezzi稳定性条件)．利用Verfürth的一般理论,得到了热传导对流问题的残量型的后验误差估计．最后通过几个数值算例验证了方法的有效性．     

最小二乘估计关于误差分布的稳健性

对于设计矩阵$X$是列降秩的线性统计模型, 本文讨论了最小二乘估计关于误差分布的稳健性, 给出了误差分布的最大类, 使得误差项的分布在此范围内变动时, 最小二乘估计在均方误差矩阵准则下是最优估计.     

二阶常微初值问题的时间间断最小二乘有限元法

采用时间间断最小二乘线性有限元方法求解二阶常微分方程初值问题.利用回收技巧及离散Gronwall引理证明了方法的稳定性.通过引入有限元空间上的范数,给出了方法在该范数意义下丰满的误差估计.数值实验验证了理论分析结果.     

Sobolev方程的最小二乘Galerkin有限元法

本文对于Sobolev方程提出并分析了两种新型数值方法：最小二乘Galerkin有限元法．这种方法的优越性在于不需要验证LBB条件,可以更好的选择有限元空间．误差估计表明在L~2(Ω))~2×L~2(Ω)范数意义下,这两种方法均具有最优收敛阶,并且关于时间分别具有一阶精确度和二阶精确度．     

Superconvergence of the split least‐squares method for second‐order hyperbolic equations

This article studies superconvergence phenomena of the split least‐squares mixed finite element method for second‐order hyperbolic equations. By selecting the least‐squares functional properly, the procedure can be split into two independent symmetric positive definite subprocedures, one of which is for the primitive unknown and the other is for the flux. Based on interpolation operators and an auxiliary projection, superconvergent H¹ error estimates for the primary variable u and L² error estimates for the introduced flux variable σ are obtained under the standard quasiuniform assumptions on finite element partition. A numerical example is given to show the performance of the introduced scheme. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 222‐238, 2014     

Reconstructed Discontinuous Approximation to Stokes Equation in a Sequential Least Squares Formulation

We propose a new least squares finite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by the patch reconstruction with one unknown per element. For the first step, we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace. By this space, we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure. In the second step, we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space. We derive error estimates for all unknowns under both $L^2$ norms and energy norms. Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method.     

A least‐squares finite element approximation for the compressible Stokes equations

This article studies a least‐squares finite element method for the numerical approximation of compressible Stokes equations. Optimal order error estimates for the velocity and pressure in the H¹ are established. The choice of finite element spaces for the velocity and pressure is not subject to the inf‐sup condition. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 62–70, 2000     

Normal and tangential continuous elements for least‐squares mixed finite element methods

We consider a finite element discretization of the primal first‐order least‐squares mixed formulation of the second‐order elliptic problem. The unknown variables are displacement and flux, which are approximated by equal‐order elements of the usual continuous element and the normal continuous element, respectively. We show that the error bounds for all variables are optimal. In addition, a field‐based least‐squares finite element method is proposed for the 3D‐magnetostatic problem, where both magnetic field and magnetic flux are taken as two independent variables which are approximated by the tangential continuous and the normal continuous elements, respectively. Coerciveness and optimal error bounds are obtained. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

An adaptive least squares mixed finite element method for the stress‐displacement formulation of linear elasticity

A least‐squares mixed finite element method for linear elasticity, based on a stress‐displacement formulation, is investigated in terms of computational efficiency. For the stress approximation quadratic Raviart‐Thomas elements are used and these are coupled with the quadratic nonconforming finite element spaces of Fortin and Soulie for approximating the displacement. The local evaluation of the least‐squares functional serves as an a posteriori error estimator to be used in an adaptive refinement algorithm. We present computational results for a benchmark test problem of planar elasticity including nearly incompressible material parameters in order to verify the effectiveness of our adaptive strategy. For comparison, conforming quadratic finite elements are also used for the displacement approximation showing convergence orders similar to the nonconforming case, which are, however, not independent of the Lamé parameters. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

电报方程H~1-Galerkin非协调混合有限元分析

主要研究一类电报方程的H~1-Galerkin非协调混合有限元方法,在任意四边形网格剖分下,其逼近空间分别取为类Wilson元与双线性Q_1元,在不需要满足LBB相容性条件及不采用传统的Ritz投影的情况下,得到了与常规有限元方法相同的L~2-模和H~1-模的误差估计,进一步拓展了H~1-Galerkin混合有限元和类Wilson元的应用范围.     

Mixed Finite Element Methods for Fourth Order Elliptic Optimal Control Problems

In this paper, a priori error estimates are derived for the mixed finite element discretization of optimal control problems governed by fourth order elliptic partial differential equations. The state and co-state are discretized by Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. The error estimates derived for the state variable as well as those for the control variable seem to be new. We illustrate with a numerical example to confirm our theoretical results.