对流扩散方程的时间间断时空有限体积元法

将时间间断的时空元思想与基于等距节点下三次Lagrange插值的超收敛有限体积元方法相结合,以三次Lagrange插值导数超收敛点为对偶剖分节点,引入插值投影算子,建立对流扩散方程的时间间断时空有限体积元格式.结合有限体积元分析与以Radau积分点为节点的Lagrange插值,证明了近似解的最优L∞(L2)-模误差估计.用单元正交分解法证明了格式在时间节点处的超收敛估计.最后给出数值算例验证了理论分析结果以及该方法的可行性和有效性.     

解Stokes方程的一种无散稳定二次有限体积法格式

在三角形网格上构造了一种求解Stokes方程的Lagrange二次有限体积法格式.取连续的二次有限元空间与间断的线性有限元空间分别作为Stokes方程的速度项与压力项的试探空间,从而保证了离散方程的速度解在宏元三角形单元上满足局部质量守恒性,且有限元空间对自然满足所谓的inf-sup条件.采用特殊的有限体积法映射与对偶剖分,求解Stokes方程的Lagrange二次有限体积法格式等价于相对应的有限元法格式,因此确保了有限体积法格式的无条件(无需约束三角形网格的几何形状)稳定性和关于速度项的最优阶H1范数的误差估计.最后,数值实验展示了理论结果的正确性以及有限体积法的数值模拟在计算流体力学中的有效性.     

解两点边值问题的一类修改的三次有限体积元法

构造了求解两点边值问题的一类修改的Lagrange型三次有限体积元法.试探函数空间取以四次Lobatto多项式的零点作为插值节点的Lagrange型三次有限元空间.将插值多项式的导数超收敛点(应力佳点)作为对偶单元的节点,检验函数空间取相应于对偶剖分的分片常数函数空间.证明了新方法具有最优的H1模和L2模收敛阶,讨论了在应力佳点导数的超收敛性,并通过数值实验验证了理论分析结果.     

二次有限体积法定价美式期权

本文考虑二次有限体积法定价美式期权.构造了隐式欧拉和Crank-Nicolson两种全离散二次有限体积格式,并得到相应的线性互补问题.采用基于超松弛迭代的模方法求解线性互补问题,并与投影超松弛迭代法作数值比较.数值实验结果表明Crank-Nicolson二次有限体积格式的求解效率高于隐式欧拉格式,模方法的求解速度较快,二次有限体积法的求解精度较高.     

热传导型半导体器件的二次元特征有限体积元方法及分析：H^1模估计

该文构造了热传导型半导体器件的全离散特征有限体积元格式,将特征线方法与有限体积元方法相结合,采用Lagrange型分片二次多项式空间和分片常数函数空间分别作为试探函数和检验函数空间,并进行误差分析,得到了最优阶 H1模误差估计结果.     

一维二阶椭圆型方程组的超收敛二次有限体积元方法

本文针对二阶椭圆型常微分方程组边值问题提出二次超收敛有限体积元方法,证明格式的H1和L2模误差估计,并给出应力佳点处的梯度超收敛估计.最后,编写计算格式的Fortran程序,用数值算例验证了理论分析的正确性和格式的有效性.     

热传导型半导体器件的二次元特征有限体积元方法及分析:H1模估计

该文构造了热传导型半导体器件的全离散特征有限体积元格式,将特征线方法与有限体积元方法相结合, 采用Lagrange型分片二次多项式空间和分片常数函数空间分别作为试探函数和检验函数空间,并进行误差分析,得到了最优阶H¹模误差估计结果.     

二维热传导型半导体瞬态问题的二次元特征有限体积元方法

将特征线方法与有限体积元方法相结合,采用Lagrange型分片二次多项式空间和分片常数函数空间分别作为试探函数和检验函数空间构造了二维热传导型半导体瞬态问题的全离散二次元特征有限体积元格式,并进行误差分析,得到了次优阶L^2模误差估计结果．     

SAV/重心插值配点法求解Allen-Cahn方程北大核心CSCD

采用标量辅助变量(scalar auxiliary variable, SAV)方法结合重心插值配点法求解二维Allen-Cahn方程.在时间方向上分别采用Crank-Nicolson格式、二阶向后差分格式离散,空间方向上采用重心Lagrange插值配点法离散,建立了两种无条件能量稳定SAV格式,并给出了重心插值配点格式的逼近性质.数值实验表明:两种SAV配点格式的时间收敛阶为二阶,并满足能量递减规律.与空间采用有限差分法离散对比,重心Lagrange配点格式具有指数收敛的特性.     

求解二维浅水方程的一种高分辨率有限体积法

本文将一种van Albada型可微的限制器函数引入到二维浅水方程的求解中，发展了一种求解二维浅水方程的有限体积法．数值实验结果表明，该方法不仅计算精度高，而且较其它求解二维浅水方程的高精度有限体积法，在数值解的收敛性能方面大有改善．     

Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems

Natural superconvergence of the least-squares finite element method is surveyed for the one-and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method. The second author was supported in part by the US National Science Foundation under Grant DMS-0612908.     

SUPERCONVERGENCE OF RITZ-VOLTERRA PROJECTION ON FINITE ELEMENT SPACES

The purpose of this paper is to study the superconvergence properties of Ritz-Volterra projection.Through construction a new type of Green function and making use of its properties and the principle of duality,the paper proves that the Ritz-Volterra projection defined on r-1 order finite element spaces of Lagrange type in one and two space variable cases possesses O(h2r~2)order and O(h4+1|Inh|)order nodal superconvergence,respectively,and the same type of superconver-gence results are demonstrated for the semidiscrete finite dement approximate solutions of Soboleve-quations.     

L^2-ERROR OF EXTRAPOLATION CASCADIC MULTIGRID （EXCMG）

Based on an asymptotic expansion of finite element, an extrapolation cascadic multigrid method （EXCMG） is proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid. In the case of multiple grids, both superconvergence error in H^1-norm and the optimal error in l2-norm are analyzed. The numerical experiment shows the advantage of EXCMG in comparison with CMG.     

非协调区域分解Lagrange乘子法的超收敛

本文讨论分析非协调区域分解Ｌａｇｒａｎｇｅ乘子法对二阶椭圆型方程Ｄｉｒｉｃｈｌｅｔ问题的有限元超收敛现象。文中通过利用积分恒等式，适宜地引进Ｌ２投影过渡以及高次插值后处理等技巧，经过一系列误差分析及估计，得到了高出半阶的超收敛结果，实现了非协调区域分解法与高精度算法的结合。     

Some recent advances on vertex centered finite volume element methods for elliptic equations

In this paper, we report our recent advances on vertex centered finite volume element methods (FVEMs) for second order partial differential equations (PDEs). We begin with a brief review on linear and quadratic finite volume schemes. Then we present our recent advances on finite volume schemes of arbitrary order. For each scheme, we first explain its construction and then perform its error analysis under both H ¹ and L ² norms along with study of superconvergence properties.     

Superconvergence of finite volume methods for the Stokes equations

A general superconvergence result of finite volume method for the Stokes equations is obtained by using a L² projection post‐processing technique. This superconvergence result can be applied to different finite volume methods and to general quasi‐uniform meshes.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

Convergence and superconvergence analysis of a new quadratic Hermite-type triangular element on anisotropic meshes

A new quadratic Hermite-type triangular finite element is conceived to solve a class of two-dimensional second-order elliptic boundary value problems. Its error estimates on anisotropic meshes are developed. Furthermore, we verify that some conditions set to the meshes contribute to the proof of its superconvergence properties, which can improve the approximation results. Numerical examples are given to confirm our theoretical analysis.     

FINITE ELEMENT METHODS FOR SOBOLEV EQUATIONS

A new high-order time-stepping finite element method based upon the high-order numerical integration formula is formulated for Sobolev equations, whose computations consist of an iteration procedure coupled with a system of two elliptic equations. The optimal and superconvergence error estimates for this new method are derived both in space and in time. Also, a class of new error estimates of convergence and superconvergence for the time-continuous finite element method is demonstrated in which there are no time derivatives of the exact solution involved, such that these estimates can be bounded by the norms of the known data. Moreover, some useful a-posteriori error estimators are given on the basis of the superconvergence estimates.     

Lagrange interpolation and finite element superconvergence

We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d‐dimensional Q_k‐type elements with d ≥ 1 and k ≥ 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H¹ norm. For d‐dimensional P_k‐type elements, we consider the standard Lagrange interpolation—the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d ≥ 2 and k ≥ d + 1 that such interpolation and the finite element solution are not superclose in both H¹ and L² norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33–59, 2004.