Computer‐based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's, and the elasticity equations

Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (II)

In this paper,we prove superconvergence results for the vector
variable when lowest order triangular mixed finite elements of
Raviart-Thomas type [17] on uniform triangulations are used,
i.e., that the 
$H(\mbox{div;} \Omega)$
 -distance between the
approximate solution and a suitable projection of the real solution
is of higher order than the 
$H(\mbox{div;} \Omega)$
 -error. We
prove
results for both Dirichlet and Neumann boundary conditions. Recently,
Duran [9] proved similar results for rectangular mixed finite
elements, and superconvergence along the Gauss-lines for rectangular
mixed finite elements was considered by Douglas, Ewing, Lazarov and
Wang in [11], [8] and [18]. The triangular case
however needs some extra effort. Using the superconvergence results,
a simple postprocessing of the approximate solution will give an
asymptotically exact a posteriori error estimator for the
 
$L^2(\Omega)$
 
-error in the approximation of the vector variable.

/pdf/superconvergence-and-a-posteriori-error-estimation-in-ieqlsgs44s.pdf

Superconvergence and a posteriori error estimation in triangular mixed finite elements

This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of elliptic problems. We analyze two estimators based on recovery operators for the gradient of the approximate solution. By using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution.

One of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation. This estimator is equivalent to the error in the energy norm under rather general conditions. However, we show that for the asymptotic exactness, the regularity assumption on the mesh is not merely technical. While doing this, we analyze the relation between superconvergence and asymptotic exactness for some particular examples.

On the asymptotic exactness of error estimators for linear triangular finite elements

Superconvergence results on mildly structured triangulations

Piecewise linear finite element approximations to two-dimensional Poisson problems are treated. For simplicity, consideration is restricted to problems having Dirichlet boundary conditions and defined on rectangular domains Ω which are partitioned by a uniform triangular mesh. It is also required that the solutions u ∈ H3 (Ω). A method is proposed for recovering the gradients of the finite element approximations to a root mean square accuracy of O(h2), both at element edge mid-points and element vertices, using simple averaging schemes over adjacent elements. Piecewise linear interpolants (respectively discontinuous and continuous) are then fitted to these recovered gradients, and are shown to be O(h2) estimates for ∇u in the L2-norm, and thus superconvergent. A discussion is given of the extension of the results to problems with more general region and mesh geometries, boundary conditions and with solutions of lower regularity, and also to other second-order elliptic boundary value problems, e.g. the problem of planar linear elasticity.

A unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations

This paper is concerned with the phenomenon of gradient superconvergence of finite element approximations to the solutions of two-dimensional second order elliptic boundary value problems. The original concept of gradient superconvergence is that there exist certain points in each element at which the gradient of the finite element approximation has a higher rate of convergence to the gradient of the true solution in a discrete analogue of a Lebesgue norm than that found globally in the Lebesgue norm itself.

G. Goodsell

Papers

A unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations

Some Gradient Superconvergence Results in the Finite Element Method