高次Galerkin有限元法的超收敛性

通过局部误差估计,对具有光滑边界的二阶常系数椭圆型方程,给出了高次Ｇａｌｅｒｋｉｎ 有限元法的超收敛性． 运用对称技巧和积分恒等式技巧,在局部对称矩形网格或三角形网格上,我们得到了改进的超收敛性（提高１－ ３ 阶）．     

The local superconvergence of the trilinear element for the three-dimensional Poisson problem

In this paper, we shall combine the finite element theory of Green?s function presented in this paper, the extrapolation technique and the local symmetric technique to investigate the local superconvergence of the trilinear element for the three-dimensional Poisson equation.     

导数小片插值恢复技术与超收敛性

１．引言 有限元超收敛的研究自七十年代起至今方兴未艾．现有的研究工作基本遵循两种途径：一是找出有限元插值逼近的超收敛点,然后再利用插值弱估计等手段导出有限元解本身所具有的超收敛性质［１,２］;二是利用各种后处理技术,如平均技术,投影技术,外插技术和插值有限元技术等[3,6],来导出经过后处理的有限元解的超收敛性．近年来一种新的超收敛后处理技术,即所谓的“Ｚ-Ｚ导数小片恢复技术”,得到众多的研究［７-１１］,并被 Ｂａｂｕｓｋａ等人认为是用于渐进准确的后验误差估计效果最好的技术之一［１２］．这种技术是利用…     

三维二次有限元梯度最大模的超逼近

作者证明了在一致四面体剖分下三维二次有限元的第一型弱估计,并给出了三维导数离散Green函数的估计,由此得到了四面体二次元梯度最大模的超逼近．通过这个超逼近还可以获得四面体二次元梯度最大模的超收敛．     

The Derivative Patch Interpolating Recovery Technique for Finite Element Approximations

A derivative patch interpolating recovery technique is analyzed for the finite element approximation to the second order elliptic boundary value problems in two dimensional case.It is shown that the convergence rate of the recovered gradient admits superc onvergence on the recovered subdomain, and is two order higher than the optimal global convergence rate (ultracovergence) at an internal node point when even order finite element spaces and local uniform meshes are used.     

Bivariate C 2 cubic spline quasi-interpolants on uniform Powell–Sabin triangulations of a rectangular domain

In this paper we construct discrete quasi-interpolants based on C ² cubic multi-box splines on uniform Powell–Sabin triangulations of a rectangular domain. The main problem consists in finding the coefficient functionals associated with boundary multi-box splines (i.e. multi-box splines whose supports overlap with the domain) involving data points inside or on the boundary of the domain and giving the optimal approximation order. They are obtained either by minimizing an upper bound for the infinity norm of the operator w.r.t. a finite number of free parameters, or by inducing the superconvergence of the gradient of the quasi-interpolant at some specific points of the domain. Finally, we give norm and error estimates and we provide some numerical examples illustrating the approximation properties of the proposed operators.     

A local superconvergence analysis of the finite element method for the Stokes equations by local projections

In this paper, we focus on a local superconvergence analysis of the finite element method for the Stokes equations by local projections. The local and global superconvergence results of finite element solutions are provided for the Stokes problem under some corresponding regularity assumptions. Conclusion can be drawn that the local superconvergence has advantages over the global superconvergence in two important aspects. On the one hand, it offsets theoretical limitation in practical applications. On the other hand, interior estimates are derived on the base of local properties of the domain without global smoothness for the exact solution and prior regularity of the problem globally over the whole domain.     

A local parallel superconvergence method for the incompressible flow by coarsening projection

In this article, we investigate a local parallel superconvergence method by coarsening projection for the incompressible Stokes flow. The method is a combination of the local superconvergence technique and the given framework of local parallel method. For the smooth subdomains, the local superconvergence method is applied in a higher order finite dimensional space corresponding to an appropriate coarse mesh on interior domain. Moreover, a useful and flexible local parallel method is designed to obtain the local parallel superconvergence results of presented method, which offset theoretical limitation of the model without the smoothness of the exact solution and a priori regularity of the underlying problem over the whole domain. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1209–1223, 2015     

特征值问题协调/非协调有限元后验误差分析

本文研究对称椭圆特征值问题的有限元后验误差估计,包括协调元和非协调元,具有下列特色:(1)对协调/非协调元建立了有限元特征函数uh的误差与相应的边值问题有限元解的误差在局部能量模意义下的恒等关系式,该边值问题的右端为有限元特征值λh与uh的乘积,有限元解恰好为uh.从而边值问题有限元解在能量模意义下的局部后验误差指示子,包括残差型和重构型后验误差指示子,成为有限元特征函数在能量模意义下的局部后验误差指示子.(2)讨论了协调有限元特征函数的基于插值后处理的梯度重构型后验误差估计,对有限元特征函数的导数得到了最大模意义下的渐近准确局部后验误差指示子.     

张量积二次长方体有限元梯度最大模的超逼近

对于某种三维椭圆边值问题,本文给出了长方体剖分下张量积二次长方体有限元的第一型弱估计以及离散导数Green函数的W^1,1半范估计,利用这两个估计本文获得了张量积二次长方体有限元梯度最大模的超逼近．进而,由超逼近也可以得到这种有限元梯度最大模的超收敛．     

Nitsche-mortaring for singularly perturbed convection–diffusion problems

In the present paper we analyse a finite element method for a singularly perturbed convection–diffusion problem with exponential boundary layers. Using a mortaring technique we combine an anisotropic triangulation of the layer region (into rectangles) with a shape regular one of the remainder of the domain. This results in a possibly non-matching (and hybrid), but layer adapted mesh of Shishkin type. We study the error of the method allowing different asymptotic behaviour of the triangulations and prove uniform convergence and a supercloseness property of the method. Numerical results supporting our analysis are presented.     

Some superconvergence results of high-degree finite element method for a second order elliptic equation with variable coefficients

In this paper, some superconvergence results of high-degree finite element method are obtained for solving a second order elliptic equation with variable coefficients on the inner locally symmetric mesh with respect to a point x ₀ for triangular meshes. By using of the weak estimates and local symmetric technique, we obtain improved discretization errors of O(h ^p+1 |ln h|²) and O(h ^p+2 |ln h|²) when p (≥ 3) is odd and p (≥ 4) is even, respectively. Meanwhile, the results show that the combination of the weak estimates and local symmetric technique is also effective for superconvergence analysis of the second order elliptic equation with variable coefficients.     

Maximum‐norm superapproximation of the gradient for the trilinear block finite element

For a model elliptic boundary value problem in three dimensions, we give the weak estimate of the first type for trilinear block elements and the estimate for W^1,1‐seminorm of the discrete derivative Green's function over rectangular partitions of the domain, from which we obtain maximum‐norm superapproximation of the gradient for the trilinear block finite element approximation. Furthermore, utilizing this superapproximation, we can also obtain maximum‐norm superconvergence of the gradient. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A finite element convergence analysis for 3D Stokes equations in case of variational crimes

We investigate a finite element discretization of the Stokes equations with nonstandard boundary conditions, defined in a bounded three-dimensional domain with a curved, piecewise smooth boundary. For tetrahedral triangulations of this domain we prove, under general assumptions on the discrete problem and without any additional regularity assumptions on the weak solution, that the discrete solutions converge to the weak solution. Examples of appropriate finite element spaces are given.     

SUPERCONVERGENCE ANALYSIS OF THE POLYNOMIAL PRESERVING RECOVERY FOR ELLIPTIC PROBLEMS WITH ROBIN BOUNDARY CONDITIONS

We analyze the superconvergence property of the linear finite element method based on the polynomial preserving recovery(PPR)for Robin boundary elliptic problems on triangulartions.First,we improve the convergence rate between the finite element solution and the linear interpolation under the H1-norm by introducing a class of meshes satisfying the Condition(α,σ,μ).Then we prove the superconvergence of the recovered gradients post-processed by PPR and define an asymptotically exact a posteriori error estimator.Finally,numerical tests are provided to verify the theoretical findings.     

A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimators relying on the solution of local subproblems and on a superconvergence result, respectively. Finally, we examine the relationship between the presented error estimators and compare their local components.

     

Optimal Interior and Local Error Estimates of a Recovered Gradient of Linear Elements on Nonuniform Triangulations

1.IntroductionConsideramodelellipticboundaryvalueproblemwhereflCRd,d=1,2,3,isaboundedpolyhedraldomainwithaLip8chitzboundaryfEL'(fl),aijareLipschitz-continuousfunctionsandthematrixA=(aij)issymmetricanduniformlypositivedefinitewithrespecttoxEn.Itisknownthatthefiniteelementmethodappliedto(1.1)mayproducesomesuperconvergencephenomenaeveniftheusedmeshesarenonuniform[5'8'9'1o'121.InarecentpaPer[61,aninteriorerrorestimatefortherecoveredgradientofGalerkinpiecewiselinearaPproximationshasbeenproposed…     

An a priori error analysis for the coupling of local discontinuous Galerkin and boundary element methods

In this paper we analyze the coupling of local discontinuous Galerkin (LDG) and boundary element methods as applied to linear exterior boundary value problems in the plane. As a model problem we consider a Poisson equation in an annular polygonal domain coupled with a Laplace equation in the surrounding unbounded exterior region. The technique resembles the usual coupling of finite elements and boundary elements, but the corresponding analysis becomes quite different. In particular, in order to deal with the weak continuity of the traces at the interface boundary, we need to define a mortar-type auxiliary unknown representing an interior approximation of the normal derivative. We prove the stability of the resulting discrete scheme with respect to a mesh-dependent norm and derive a Strang-type estimate for the associated error. Finally, we apply local and global approximation properties of the subspaces involved to obtain the a priori error estimate in the energy norm.

     

SUPERCONVERGENCE OF LEAST-SQUARES MIXED FINITE ELEMENTS FOR ELLIPTIC PROBLEMS ON TRIANGULATION

In this paper, we present the least-squares mixed finite element method and investigate superconvergence phenomena for the second order elliptic boundary-value problems over triangulations. On the basis of the L~2-projection and some mixed finite element projections, we obtain the superconvergence result of least-squares mixed finite     

Error estimates for the finite element approximation of linear elastic equations in an unbounded domain

In this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain. The finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error estimates show how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. A numerical example for Navier equations outside a circle in the plane is presented. Numerical results demonstrate the performance of our error estimates.