Three-Dimensional Finite Element Superconvergent Gradient Recovery on Par6 Patterns

<正>In this paper,we present a theoretical analysis for linear finite element superconvergent gradient recovery on Par6 mesh,the dual of which is centroidal Voronoi tessellations with the lowest energy per unit volume and is the congruent cell predicted by the three-dimensional Gersho's conjecture.We show that the linear finite element solution u_h and the linear interpolation u_1 have superclose gradient on Par6 meshes. Consequently,the gradient recovered from the finite element solution by using the superconvergence patch recovery method is superconvergent to▽u.A numerical example is presented to verify the theoretical result.     

Polynomial preserving recovery for quadratic elements on anisotropic meshes

Polynomial preserving gradient recovery technique under anisotropic meshes is further studied for quadratic elements. The analysis is performed for highly anisotropic meshes where the aspect ratios of element sides are unbounded. When the mesh is adapted to the solution that has significant changes in one direction but very little, if any, in another direction, the recovered gradient can be superconvergent. The results further explain why recovery type error estimator is robust even under nonstandard and highly distorted meshes. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A superconvergent linear FE approximation for the solution of an elliptic system of PDEs

The aim of this work is to study a nonstandard piecewise linear finite element method for elliptic systems of partial differential equations. This nonstandard method was considered by the authors for scalar elliptic equations and for a planar elasticity problem. The method enables us to compute a superconvergent numerical approximation to the solution of the system of partial differential equations.     

Lower bounds for eigenvalues and postprocessing by an integral type nonconforming FEM

In this paper, we analyze some approximation properties of a nonconforming piecewise linear finite element with integral degrees of freedom. A nonconforming finite element method (FEM) is applied to second-order eigenvalue problems (EVPs).We prove that the eigenvalues computed by means of this element are smaller than the exact ones if the mesh size is small enough. The case when an EVP is defined on a nonconvex domain is considered. A superconvergent rate is established to a second-order elliptic problem by the introduction of nonstandard interpolated elements based on the integral type linear element. A simple postprocessing method applied to second-order EVPs is also proposed and analyzed. Finally, computational aspects are discussed and numerical examples are presented.     

Polynomial preserving recovery for meshes from Delaunay triangulation or with high aspect ratio

A newly developed polynomial preserving gradient recovery technique is further studied. The results are twofold. First, error bounds for the recovered gradient are established on the Delaunay type mesh when the major part of the triangulation is made of near parallelogram triangle pairs with ε‐perturbation. It is found that the recovered gradient improves the leading term of the error by a factor ε. Secondly, the analysis is performed for a highly anisotropic mesh where the aspect ratio of element sides is unbounded. When the mesh is adapted to the solution that has significant changes in one direction but very little, if any, in another direction, the recovered gradient can be superconvergent. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Superconvergent Recovery of the Gradient from Piecewise Linear Finite-element Approximations

We propose and justify the use of a simple scheme which recoversgradients from the piecewise linear finite-element approximationon triangular elements to the solution of a second-order ellipticproblem. The recovered gradient is a superconvergent estimateof the true gradient at the midpoints of element edges. A relatedscheme recovers the gradient at the element centroids.     

一类广义插值函数与广义有限元方法的后验估计

１．问题的提出具有快速振荡系数的微分方程大量出现在复合材料、多孔介质渗流等实际问题中．因为这类方程系数的激烈振荡性（受小尺度ε控制）,通常的有限元方法需花费巨大的计算工作量才能获得有意义的数值解（文［７］）,这对多维问题是无法承受的．８０年代发展起来的广义有限元法（文［１］［３］［５］）,为这类问题的解决提供了一条有效的新途径,它可在剖分步长ｈ＞＞　ε的情况下得到令人满意的数值结果．在广义有限元的理论分析中,因方程解的正则性估计通常与小尺度ε　有关,所以通常的有限元分析方法有一定的困难,目前尚未…     

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.

     

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.     

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.     

Adaptive operator splitting finite element method for Allen–Cahn equation

In this paper, a new numerical method is proposed and analyzed for the Allen–Cahn (AC) equation. We divide the AC equation into linear section and nonlinear section based on the idea of operator splitting. For the linear part, it is discretized by using the Crank–Nicolson scheme and solved by finite element method. The nonlinear part is solved accurately. In addition, a posteriori error estimator of AC equation is constructed in adaptive computation based on superconvergent cluster recovery. According to the proposed a posteriori error estimator, we design an adaptive algorithm for the AC equation. Numerical examples are also presented to illustrate the effectiveness of our adaptive procedure.     

A posteriori FE error control for p-Laplacian by gradient recovery in quasi-norm

A posteriori error estimators based on quasi-norm gradient recovery are established for the finite element approximation of the p-Laplacian on unstructured meshes. The new a posteriori error estimators provide both upper and lower bounds in the quasi-norm for the discretization error. The main tools for the proofs of reliability are approximation error estimates for a local approximation operator in the quasi-norm.

     

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.     

Recovery Type a Posteriori Error Estimates of Fully Discrete Finite Element Methods for General Convex Parabolic Optimal Control Problems

This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints. The time discretization is based on the backward Euler method. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. We derive the superconvergence properties of finite element solutions. By using the superconvergence results, we obtain recovery type a posteriori error estimates. Some numerical examples are presented to verify the theoretical results.     

双线性非协调有限元逼近的梯度恢复后验误差估计

给出了二阶椭圆方程的双线性非协调有限元逼近的梯度恢复后验误差估计.该误差估计是在Q_1非协调元上得到的,并给出了误差的上下界.进一步证明该误差估计在拟一致网格上是渐进精确地.证明依赖于clement插值和Helmholtz分解,数值结果验证了理论的正确性.     

Supercloseness analysis and polynomial preserving Recovery for a class of weak Galerkin Methods

In this article, we analyze convergence and supercloseness properties of a class of weak Galerkin (WG) finite element methods for solving second‐order elliptic problems. It is shown that the WG solution is superclose to the Lagrange interpolant using Lobatto points. This supercloseness behavior is obtained through some newly designed stabilization terms. A postprocessing technique using polynomial preserving recovery (PPR) is introduced for the WG approximation. Superconvergence analysis is performed for the PPR recovered gradient. Numerical examples are provided to illustrate our theoretical results.     

Gradient recovery type a posteriori error estimate for finite element approximation on non-uniform meshes

In this paper, we derive gradient recovery type a posteriori error estimate for the finite element approximation of elliptic equations. We show that a posteriori error estimate provide both upper and lower bounds for the discretization error on the non-uniform meshes. Moreover, it is proved that a posteriori error estimate is also asymptotically exact on the uniform meshes if the solution is smooth enough. The numerical results demonstrating the theoretical results are also presented in this paper.     

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.     

Polynomial Preserving Recovery for Anisotropic and Irregular Grids

Some properties of a newly developed polynomial preserving gradient recovery technique are discussed. Both practical and theoretical issues are addressed. Bounded-ness property is considered especially under anisotropic grids. For even-order finite element space, an ultra-convergence property is established under translation invariant meshes; for linear element, a superconvergence result is proven for unstructured grids generated by the Delaunay triangulation.     

Robust residual‐ and recovery‐based a posteriori error estimators for a multipoint flux mixed finite element discretization of interface problems

We consider a posteriori error estimation for a multipoint flux mixed finite element method for two‐dimensional elliptic interface problems. Within the class of modified quasi‐monotonically distributed coefficients, we derive a residual‐type a posteriori error estimator of the weighted sum of the scalar and flux errors which is robust with respect to the jumps of the coefficients. Moreover, we develop robust implicit and explicit recovery‐type estimators through gradient recovery in an H(curl)‐conforming finite element space. In particular, we apply a modified L² projection in the implicit recovery procedure so as to reduce the computational cost of the recovered gradient. Numerical experiments confirm the theoretical results.