Ultraconvergence of the patch recovery technique

The ultraconvergence property of a derivative recovery technique recently proposed by Zienkiewicz and Zhu is analyzed for two-point boundary value problems. Under certain regularity assumptions on the exact solution, it is shown that the convergence rate of the recovered derivative at an internal nodal point is two orders higher than the optimal global convergence rate when even-order finite element spaces and local uniform meshes are used.

     

Ultraconvergence of ZZ patch recovery at mesh symmetry points

Summary. The ultraconvergence property of the Zienkiewicz-Zhu gradient patch recovery technique based on local discrete least-squares fitting is established for a large class of even-order finite elements. The result is valid at all rectangular mesh symmetry points. Different smoothing strategies are discussed and numerical examples are demonstrated. Mathematics Subject Classification (2000):65N30, 65N15, 65N12, 65D10, 74S05, 41A10, 41A25This research was partially supported by the National Science Foundation grants DMS-0074301, DMS-0079743, and INT-0196139     

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.     

Implicit a posteriori error estimation using patch recovery techniques

We develop implicit a posteriori error estimators for elliptic boundary value problems. Local problems are formulated for the error and the corresponding Neumann type boundary conditions are approximated using a new family of gradient averaging procedures. Convergence properties of the implicit error estimator are discussed independently of residual type error estimators, and this gives a freedom in the choice of boundary conditions. General assumptions are elaborated for the gradient averaging which define a family of implicit a posteriori error estimators. We will demonstrate the performance and the favor of the method through numerical experiments.     

两点边值问题有限元重构导数超强收敛性

基于两点边值问题 ,本文在改进的单元正交估计和连续性优化的基础上 ,研究了一种n次有限元单元块导数重构 ,该方法所获得的重构导数在单元块内部有n- 1个强超收敛点 .     

Ultraconvergence of the derivative of high-order finite element method for elliptic problems with constant coefficients

In this article, for second order elliptic problems with constant coefficients, the local ultraconvergence of the derivative of finite element method using piecewise polynomials of degrees k (k ≥ 2) is studied by the interpolation postprocessing technique. Under suitable regularity and mesh conditions, we prove that at an interior vertex, which is away from the boundary with a fixed distance, the gradient of the postprecessed finite element solution using piecewise polynomials of degrees k (k ≥ 2) converges to the gradient of the exact solution with order . Numerical experiments are used to illustrate our theoretical findings.     

变系数两点边值问题的有限元强校正格式

该文利用投影型插值,对于变系数两点边值问题,获得了一个高精度的有限元强校正格式,数值实验更验证了这一结果．     

Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements

Mathematical proofs are presented for the derivative superconvergence obtained by a class of patch recovery techniques for both linear and bilinear finite elements in the approximation of second‐order elliptic problems. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 151–167, 1999     

Finite element derivative interpolation recovery technique and superconvergence

A new finite element derivative recovery technique is proposed by using the polynomial interpolation method. We show that the recovered derivatives possess superconvergence on the recovery domain and ultraconvergence at the interior mesh points for finite element approximations to elliptic boundary problems. Compared with the well-known Z-Z patch recovery technique, the advantage of our method is that it gives an explicit recovery formula and possesses the ultraconvergence for the odd-order finite elements. Finally, some numerical examples are presented to illustrate the theoretical analysis.     

Ultraconvergence of finite element method by Richardson extrapolation for elliptic problems with inhomogeneous boundary conditions

In this article, Richardson extrapolation technique is employed to investigate the local ultraconvergence properties of Lagrange finite element method using piecewise polynomials of degrees () for the second order elliptic problem with inhomogeneous boundary. A sequence of special graded partition are proposed and a new interpolation operator is introduced to achieve order local ultraconvergence for the displacement and derivative.     

Application of Zienkiewicz–Zhu's error estimate with superconvergent patch recovery to hierarchical p-refinement

The Zienkiewicz–Zhu error estimate is slightly modified for the hierarchical p-refinement, and is then applied to three plane elastostatic problems to demonstrate its effectiveness. In each case, the error decreases rapidly with an increase in the number of degrees of freedom. Thus Zienkiewicz–Zhu's error estimate can be used in the hp-refinement of finite element meshes.     

Local recovery of a solenoidal vector field by an extension of the thin-plate spline technique

We show how one may interpolate a vector-valued function in two or three dimensions, whose value is (wholly or partly) known at a sufficient (but not large) number of points disposed in almost any configuration, under the condition that the interpolating function has zero divergence. The technique is based on the theory of thin-plate splines. One may use a similar scheme in the case where the data consist of flux integrals (or other linear functionals) of the unknown function.     

Diagram technique for the Hubbard model. II. Metal-insulator transition

Institute of Applied Physics, Moldavian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 85, No. 2, pp. 248–257, November, 1990.     

Optimization of a patch

We study optimal patterns of a patch made of an elastic anisotropic homogeneous material for covering a hole in a two-dimensional body possessing different physical characteristics. In addition to the optimization problem for inclusions in two-dimensional and three-dimensional elastic and piezoelectric bodies, we also consider similar problems for an arbitrary formally selfadjoint elliptic system of differential equations in multidimensional domains. A condition for the stationarity of the energy functional is obtained; for a free parameter the matrix of orthogonal transformations of the Euclidean space is taken; the result is based on an algebraic fact about small increments of orthogonal and unitary matrices. Bibliography: 23 titles. Illustrations: 1 figure.     

Symmetric marching technique for the Poisson equation. II. mixed boundary conditions

In paper I a symmetric marching technique for the discretized Poisson equation with Dirichlet boundary conditions was developed. In this paper, the symmetric marching technique is extended to cover mixed boundary value problems for Poisson equation. The results of some numerical experiments are also presented.