The issue of reliability of computer predictions of physical events is examined as the goal of verification and validation processes. It is argued that verification, both solution and code verification, can be carried out with a high degree of confidence, even though much remains to be done to improve and advance verification procedures. It is validation of mathematical models that stands as the major bottleneck of reliable computer predictions. Uncertainty of input data represents a major feature of validation processes and must be quantified if models are to be judged valid or invalid. Examples are given from solid mechanics and heat transfer that demonstrate validation processes employing stochastic models and fuzzy set theories.

In this paper, we show that the piecewise linear finite element solution $u_h$ and the linear interpolation $u_I$ have superclose gradient for tetrahedral meshes, where most elements are obtained by dividing approximate parallelepiped into six tetrahedra. We then analyze a post-processing gradient recovery scheme, showing that the global $L^2$ projection of $\nabla u_h$ is a superconvergent gradient approximation to $\nabla u$.

In this paper, the global superconvergence analysis for the finite element approximation of the distributed optimal control governed by Stokes equations is discussed. For the control, a global superconvergence result is derived by applying patch recovery technique. For the state and the co-state, the global superconvergence results are derived by applying some postprocessing techniques for the bilinear-constant scheme over the uniform rectangular meshes. Based on the global superconvergence analysis, recovery type a posteriori error estimates are derived. It is shown that the recovery type a posteriori error estimators provided in this paper are asymptotically exact if the conditions for the superconvergence are satisfied.

In this paper we consider superconvergence and supercloseness in the least-squares mixed finite element method for elliptic problems. The supercloseness is with respect to the standard and mixed finite element approximations of the same elliptic problem, and does not depend on the properties of the mesh. As an application, we will derive more precise a priori bounds for the least squares mixed method. The superconvergence may be used to define a posteriori error estimators in the usual way. As a by-product of the analysis, a strengthened Cauchy-Buniakowskii-Schwarz inequality is used to prove the coercivity of the least-squares mixed bilinear form in a straight-forward manner. Using the same inequality, it can moreover be shown that the least-squares mixed finite element linear system of equations can basically be solved with one single iteration step of the Block Jacobi method.

In this paper we present a priori error analysis for mixed finite element approximation of quadratic optimal control problems. Optimal a priori error bounds are obtained. Furthermore, super-convergence of the approximation is studied.

For the nonconforming rotated $Q_1$ element over a mildly distorted quadrilateral mesh, we propose a superconvergence property at the element center, the vertices and the midpoints of four edges. Numerics are presented to confirm this observation.

The striking simplicity of averaging techniques in a posteriori error control of finite element methods as well as their amazing accuracy in many numerical examples over the last decade have made them an extremely popular tool in scientific computing. Given a discrete stress or flux $P_h$ and a post-processed approximation $A(p_h)$, the a posteriori error estimator reads $\eta_A := ||p_h - A(p_h)||$. There is not even a need for an equation to compute the estimator $\eta_A$ and hence averaging techniques are employed everywhere. The most prominent example is occasionally named after Zienkiewicz and Zhu, and also called gradient recovery but preferably called averaging technique in the literature.
The first mathematical justification of the error estimator $\eta_A$ as a computable approximation of the (unknown) error $||p - p_h||$ involved the concept of superconvergence points. For highly structured meshes and a very smooth exact solution $p$, the error $||p - A(p_h)||$ of the post-processed approximation $Ap_h$ may be (much) smaller than $||p - p_h||$ of the given $p_h$. Under the assumption that $||p - A(p_h)||$= h.o.t. is in relative terms sufficiently small, the triangle inequality immediately verifies reliability, i.e.,
                                   $|| p-p_h ||  \leq C_{rel} \eta_A + $h.o.t.,
and efficiency, i.e.,
                                    $\eta_A \leq C_{eff} || p-p_h || +$ h.o.t.,
of the averaging error estimator $\eta_A$. However, the required assumptions on the symmetry of the mesh and the smoothness of the solution essentially contradict the use of adaptive grid refining when $p$ is singular and the proper treatment of boundary conditions remains unclear.
This paper aims at an actual overview on the reliability and efficiency of averaging a posteriori error control for unstructured grids. New aspects are new proofs of the efficiency of all averaging techniques and for all problems.

A symmetric finite volume element scheme on quadrilateral grids is established for a class of elliptic problems. The asymptotic error expansion of finite volume element approximation is obtained under rectangle grids, which in turn yields the error estimates and superconvergence of the averaged derivatives. Numerical examples confirm our theoretical analysis.

In this paper, we study numerical approximations of eigenvalues when using projection method for spectral approximations of completely continuous operators. We improve the theory depending on the ascent of $T - \mu$ and provide a new approach for error estimate, which depends only on the ascent of $T_h - \mu_h$. Applying this estimator to the integral operator eigenvalue problems, we obtain asymptotically exact indicators. Numerical experiments are provided to support our theoretical conclusions.

Here we survey some previously published results and announce some that have been newly obtained. We first review some of the results in [3] on estimates for the finite element error at a point. These estimates and analogous ones in [4] and [7] have been applied to problems in a posteriori estimates [2], [8], superconvergence [5] and others [9], [10]. We then discuss the extension of these estimates to local estimates in $L_∞$ based negative norms. These estimates have been newly obtained and are applied to the problem of obtaining an asymptotically exact a posteriori estimator for the maximum norm of the solution error on each element.