Derivative superconvergent points in finite element solutions of Poisson's equation for the serendipity and intermediate families - a theoretical justification

Finite element derivative superconvergent points for the Poisson equation under local rectangular mesh (in the two dimensional case) and local brick mesh (in the three dimensional situation) are investigated. All superconvergent points for the finite element space of any order that is contained in the tensor-product space and contains the intermediate family can be predicted. In case of the serendipity family, the results are given for finite element spaces of order below 7. Any finite element space that contains the complete polynomial space will have at least all superconvergent points of the related serendipity family.

     

各向异性双二次元的自然超收敛性结果

The paper studies the convergence and the superconvergence of the biquadratic finite element for Poisson' problem on anisotropic meshes.By detailed analysis,it shows that the biquadratic finite element is anisotropically superconvergent at four Gauss points in the element.     

Analysis of a non-standard mixed finite element method with applications to superconvergence

We show that a non-standard mixed finite element method proposed by Barrios and Gatica in 2007, is a higher order perturbation of the least-squares mixed finite element method. Therefore, it is also superconvergent whenever the least-squares mixed finite element method is superconvergent. Superconvergence of the latter was earlier investigated by Brandts, Chen and Yang between 2004 and 2006. Since the new method leads to a non-symmetric system matrix, its application seems however more expensive than applying the least-squares mixed finite element method. Dedicated to Ivan Hlaváček on the occasion of his 75th birthday     

The streamline–diffusion method for a convection–diffusion problem with a point source

A linear singularly perturbed convection–diffusion problem with a point source is considered. The problem is solved using the streamline–diffusion finite element method on a class of Shishkin–type meshes. We prove that the method is almost optimal with uniform second order of convergence in the maximum norm. We also prove the existence of superconvergent points for the first derivative. Numerical experiments support these theoretical results.     

Superconvergent Cluster Recovery Method for the Crouzeix-Raviart Element

In this paper, we propose and numerically investigate a superconvergent cluster recovery (SCR) method for the Crouzeix-Raviart (CR) element. The proposed recovery method reconstructs a $C^0$ linear gradient. A linear polynomial approximation is obtained by a least square fitting to the CR element approximation at certain sample points, and then taken derivatives to obtain the recovered gradient. The SCR recovery operator is superconvergent on uniform mesh of four patterns. Numerical examples show that SCR can produce a superconvergent gradient approximation for the CR element, and provide an asymptotically exact error estimator in the adaptive CR finite element method.     

Mathematical analysis of Zienkiewicz—Zhu's derivative patch recovery technique

Zienkiewicz-Zhu's derivative patch recovery technique is analyzed for general quadrilateral finite elements. Under certain regular conditions on the meshes, the arithmetic mean of the absolute error of the recovered gradient at the nodal points is superconvergent for the second-order elliptic operators. For rectangular meshes and the Laplacian, the recovered gradient is superconvergent in the maximum norm at the nodal points. Furthermore, it is proved for a model two-point boundary-value problem that the recovery technique results in an “ultra-convergent” derivative recovery at the nodal points for quadratic finite elements when uniform meshes are used. © 1996 John Wiley & Sons, Inc.     

Pantograph-Catenary Dynamics: An Analysis of Models and Simulation Techniques

In this paper we analyze models and simulation techniques for the interaction of pantograph and catenary. Detailed models for catenary and pantograph and the propagation of waves are first investigated. Next, the semi discretization by the finite element method and the time integration are described. In this context numerical techniques like GGL-stabilization and superconvergent patch recovery are applied. The latter yields an error estimation for the finite element grid and shows the critical points of the system.     

Analysis of Sharp Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Parabolic Equations

In this paper, we study the superconvergence of the error for the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear parabolic equations when the alternating flux is used. We prove that if we apply piecewise $k$-th degree polynomials, the error between the LDG solution and the exact solution is ($k$+2)-th order superconvergent at the Radau points with suitable initial discretization. Moreover, we also prove the LDG solution is ($k$+2)-th order superconvergent for the error to a particular projection of the exact solution. Even though we only consider periodic boundary condition, this boundary condition is not essential, since we do not use Fourier analysis. Our analysis is valid for arbitrary regular meshes and for $P^k$ polynomials with arbitrary $k$ ≥ 1. We perform numerical experiments to demonstrate that the superconvergence rates proved in this paper are sharp.     

Cubic superconvergent finite volume element method for one-dimensional elliptic and parabolic equations

In this paper, a cubic superconvergent finite volume element method based on optimal stress points is presented for one-dimensional elliptic and parabolic equations. For elliptic problem, it is proved that the method has optimal third order accuracy with respect to H¹ norm and fourth order accuracy with respect to L² norm. We also obtain that the scheme has fourth order superconvergence for derivatives at optimal stress points. For parabolic problem, the scheme is given and error estimate is obtained with respect to L² norm. Finally, numerical examples are provided to show the effectiveness of the method.     

EFK方程一个新的低阶非协调混合有限元方法的高精度分析

对Extended Fisher-Kolmogorov(EFK)方程,利用EQ_1~(rot)元和零阶RaviartThomas(R-T)元建立了一个新的非协调混合元逼近格式.首先,证明了半离散格式逼近解的一个先验估计并证明了逼近解的存在唯一性.在半离散格式下,利用上述两种元的高精度分析结果以及这个先验估计,在不需要有限元解u_h属于L~∞的条件下,得到了原始变量u和中间变量v=-?u的H~1-模以及流量p=u的(L~2)~2-模意义下O(h~2)阶的超逼近性质.同时,借助插值后处理技术,证明了上述变量的具有O(h~2)阶的整体超收敛结果.其次,建立了一个新的线性化向后Euler全离散格式并证明了其逼近解的存在唯一性.另一方面,通过对相容误差和非线性项采取与传统误差分析不同的新的分裂技巧,分别导出了以往文献中尚未涉及的关于u和v在H~1-模以及p在(L~2)~2-模意义下具有O(h~2+τ)阶的超逼近性质,进一步地,借助插值后处理技术,得到了上述变量的整体超收敛结果.这里h和τ分别表示空间剖分参数和时间步长.最后,给出了一个数值算例,计算结果验证了理论分析的正确性.     

A superconvergent $L^{\infty}$-error estimate of RT1 mixed methods for elliptic control problems with an integral constraint

In this paper, we investigate the superconvergence property of mixed finite element methods for a linear elliptic control problem with an integral constraint. The state and co-state are approximated by the order $k=1$ Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. A superconvergent approximation of the control variable $u$ will be constructed by a projection of the discrete adjoint state. It is proved that this approximation have convergence order $h^{2}$ in $L^{\infty}$-norm. Finally, a numerical example is given to demonstrate the theoretical results.     

Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part 1: A smooth problem and globally quasi-uniform meshes

A class of a posteriori estimators is studied for the error in the maximum-norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems. The meshes are unstructured and, in particular, it is not assumed that there are any known superconvergent points. The estimators are based on averaging operators which are approximate gradients, ``recovered gradients', which are then compared to the actual gradient of the approximation on each element. Conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes. Asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large, in a controlled fashion to be detailed below, compared to the size of the elements.

     

Preconditioners for higher order edge finite element discretizations of Maxwell's equations

In this paper,we are concerned with the fast solvers for higher order edge finite element discretizations of Maxwell's equations.We present the preconditioners for the first family and second family of higher order N′ed′elec element equations,respectively.By combining the stable decompositions of two kinds of edge finite element spaces with the abstract theory of auxiliary space preconditioning,we prove that the corresponding condition numbers of our preconditioners are uniformly bounded on quasi-uniform grids.We also present some numerical experiments to demonstrate the theoretical results.     

Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension

Summary A semidiscrete Galerkin finite element method is defined and analyzed for nonlinear evolution equations of Sobolev type in a single space variable. Optimal orderL ^p error estimates are derived for 2p. And it is shown that the rates of convergence of the approximate solution and its derivative are one order better than the optimal order at certain spatial Jacobi and Gauss points, respectively. Also the standard nodal superconvergence results are established. Futher, it is considered that an a posteriori procedure provides superconvergent approximations at the knots for the spatial derivatives of the exact solution.     

Convergence theorems of shrinking projection methods for equilibrium problem, variational inequality problem and a finite family of relatively quasi-nonexpansive mappings

We introduce a W-mapping for a finite family of relatively quasi-nonexpansive mappings and construct an iterative scheme for finding a common element of the solution set of equilibrium problem, the solution set of the variational inequality problem for an inverse-strongly-monotone operator and set of common fixed points of a finite family of relatively quasi-nonexpansive mappings. Strong convergence theorems are presented in a 2-uniformly convex and uniformly smooth Banach space. Our results generalize and extend relative results.     

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.     

Finite volume element approximations of nonlocal reactive flows in porous media

In this article, we study finite volume element approximations for two‐dimensional parabolic integro‐differential equations, arising in the modeling of nonlocal reactive flows in porous media. These types of flows are also called NonFickian flows and exhibit mixing length growth. For simplicity, we consider only linear finite volume element methods, although higher‐order volume elements can be considered as well under this framework. It is proved that the finite volume element approximations derived are convergent with optimal order in H¹‐ and L²‐norm and are superconvergent in a discrete H¹‐norm. By examining the relationship between finite volume element and finite element approximations, we prove convergence in L^∞‐ and W^1,∞‐norms. These results are also new for finite volume element methods for elliptic and parabolic equations. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 285–311, 2000     

Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type

In this paper we study hybrid iterative scheme for finding a common element of set of solutions of generalized mixed equilibrium problem, set of common fixed points of finite family of weak relatively nonexpansive mapping and null spaces of finite family of γ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. Our results extend, improve and generalize the results of several authors which are announced recently. Application of our theorem to solution of equations of Hammerstein-type is of independent interest.     

Iterative algorithms with the regularization for the constrained convex minimization problem and maximal monotone operators

In this paper, we prove a strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the set of solutions of a finite family of variational inclusion problems in Hilbert space. A strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the solution sets of a finite family of zero points of the maximal monotone operator problem in Hilbert space is also obtained. Using our main result, we have some additional results for various types of non-linear problems in Hilbert space.     

Biquartic Finite Volume Element Metho d Based on Lobatto-Guass Structure

In this paper, a biquartic finite volume element method based on LobattoGuass structure is presented for variable coefficient elliptic equation on rectangular partition. Not only the optimal H1 and L2 error estimates but also some superconvergent properties are available and could be proved for this method. The numerical results obtained by this finite volume element scheme confirm the validity of the theoretical analysis and the effectiveness of this method.