Some recovery type error estimators for linear finite element method are analyzed under O(h ) (# > 0) regular grids. Superconvergence of order O(h ) (0 < # #) is established for recovered gradients by three different methods. As a consequence, a posteriori error estimators based on those recovery methods are asymptotically exact.

This is the first in a series of papers where a new gradient recovery method is introduced and analyzed. It is proved that the method is superconvergent for translation invariant finite element spaces of any order. The method maintains the simplicity, efficiency, and superconvergence properties of the Zienkiewicz-Zhu patch recovery method. In addition, under uniform triangular meshes, the method is superconvergent for the Chevron pattern, and ultraconvergent at element edge centers for the regular pattern. Applications of this new gradient recovery technique will be discussed in forthcoming papers.

Abstract. In this paper, we present a recovery type a posteriori error estimate and the superconvergence analysis for the finite element approximation of the distributed convex optimal control problems governed by integraldifferential equations. We provide the recovery type a posteriori error estimates for both the control and the state approximation, which is equivalent to the exact error generally. Under some strong conditions, it is not only equivalent, but also asymptotically exact. 1.

A polynomial preserving gradient recovery method is proposed and analyzed for bilinear element under quadrilateral meshes. It has been proven that the recovered gradient converges at a rate O(h 1+ρ) for ρ = min(α, 1), when the mesh is distorted O(h 1+α) (α > 0) from a regular one. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact.

Dedicated to Professor Zhong-ci Shi on the occasion of his 70th birthday 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.

Abstract. In this paper, the global superconvergence is analysed on two schemes (a mixed finite element scheme and a finite element scheme) for Maxwell’s equations in R3. Such a supercovergence analysis is achieved by means of the technique of integral identity (which has been used in the supercovergence analysis for many other equations and schemes) on a rectangular mesh, and then are generalized into more general domains and problems with the variable coefficients. Besides being more direct, our analysis generalizes the results of Monk. 1.

The convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a singularly perturbed model problem in one-dimensional setting are studied. By applying the DG method with appropriately chosen numerical traces, the existence and uniqueness of the DG solution, the optimal order L2 error bounds, and 2p + 1-order superconvergence of the numerical traces are established. The numerical results indicate that the DG method does not produce any oscillation even under the uniform mesh. Numerical experiments demonstrate that, under the uniform mesh, it seems impossible to obtain the uniform superconvergence of the numerical traces. Nevertheless, thanks to the implementation of the so-called Shishkin-type mesh, the uniform 2p + 1−order superconvergence is observed numerically.

Abstract. A numerical test case demonstrates that Lobatto and Gauss points are not natural superconvergent points for cubic and quartic finite elements under equilateral triangular mesh.

this article, we shall discuss a recovery technique that provides a superconvergence of order two on a whole domain or a sub-domain. A closely related work is an early study [6] by Lin, Yan, and Zhou who

Abstract. In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergence rate O(N −2 ln 2 N + ɛN −1.5 ln N)inadiscreteɛ-weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter ɛ. Numerical tests indicate that the rate O(N −2 ln 2 N) is sharp for the boundary layer terms. As a by-product, an ɛ-uniform convergence of the same order is obtained for the L 2-norm. Furthermore, under the same regularity assumption, an ɛ-uniform convergence of order N −3/2 ln 5/2 N + ɛN −1 ln 1/2 N in the L ∞ norm is proved for some mesh points in the boundary layer region. 1.