S.K. Gangaraj

Pollution error in the h-version of the finite element method and the local quality of the recovered derivatives

Abstract In this paper, we address the quality of the solution derivatives which are recovered from finite element solutions by local averaging schemes. As an example, we consider the Zienkiewicz-Zhu superconvergent patch-recovery scheme (the ZZ-SPR scheme), and we study its accuracy in the interior of the mesh for finite element approximations of solutions of Laplaces equation in polygonal domains. We will demonstrate the following: (1) In general, the accuracy of the derivatives recovered by the ZZ-SPR or any other local averaging scheme may not be higher than the accuracy of the derivatives computed directly from the finite element solution. (2) If the mesh is globally adaptive (i.e. it is nearly equilibrated in the energy-norm) then we can, practically always, gain in accuracy by employing the recovered derivatives instead of the derivatives computed directly from the finite element solution. (3) It is possible to guarantee that the recovered solution-derivatives have higher accuracy than the derivatives computed directly from the finite element solution, in any patch of elements of interest, by employing a mesh which is adaptive only with respect to the patch of interest (i.e. it is nearly equilibrated in a weighted energy-norm). (4) In practice, we are often interested in obtaining highly accurate derivatives (or heat-fluxes, stresses, etc.) only in a few critical regions which are identified by a preliminary analysis. A grid which is adaptive only with respect to the critical regions of interest may be much more economical for this purpose because it may achieve the desired accuracy by employing substantially fewer degrees of freedom than a globally adaptive grid which achieves comparable accuracy in the critical regions.

A posteriori estimation and adaptive control of the error in the quantity of interest. Part I: A posteriori estimation of the error in the von Mises stress and the stress intensity factor

In this paper we address the problem of a posteriori estimation of the error in an engineering quantity of interest which is computed from a finite element solution. As an example we consider the plane elasticity problem with the von Mises stress and the stress intensity factor, as the quantities of interest. The estimates of the error in the von Mises stress at a point are obtained by partitioning the error into two components with respect to the element which includes the point, the local and the pollution errors, and by constructing separate estimates for each component. The estimates of the error in the stress intensity factors are constructed by employing an extraction method. We demonstrate that our approach gives accurate estimates for rather coarse meshes and elements of various degrees. In Part II we will address the problem of the adaptive control of the error in the quantity of interest (the goal of the computation), and the construction of goal-adaptive meshes for one or multiple goals.

Guaranteed computable bounds for the exact error in the finite element solution Part I: One-dimensional model problem

Abstract This paper addresses the computation of guaranteed upper and lower bounds for the energy norm of the exact error in the finite element solution, and the exact error in any bounded linear functional. These bounds are constructed by employing approximate solutions of the element residual problems with equilibrated residual loads. The one-dimensional setting is used for the clarity of the ideas. All the arguments employed can be extended to the higher-dimensional case which will be discussed in Part II of this paper. The main result presented here is that the computed bounds are guaranteed for the exact error and not the error with respect to an enriched finite element solution, like the bounds proposed by other investigators and the bounds are guaranteed for any mesh, however coarse it may be. The quality of the bounds can be controlled by employing an inexpensive iterative scheme.

A model study of element residual estimators for linear elliptic problems: The quality of the estimators in the interior of meshes of triangles and quadrilaterals

In Babuska et al. (Comput. Meth. appl. Mech. Engng, 114, 307–378 (1994); Int. J. numer. Meth. Engng 37, 1073–1123 (1994)) we presented a model study of a posteriori error estimators in the interior of finite element meshes using a computer-based methodology. In this paper we investigate further the quality of element-residual error estimators. We analyzed several versions of the element-residual estimator and based on this study we propose recipes for robust estimators.

Guaranteed computable bounds for the exact error in the finite element solution?Part II: bounds for the energy norm of the error in two dimensions

This paper addresses the computation of guaranteed upper and lower bounds for the energy norm of the exact error in the finite element solution. These bounds are constructed in terms of the solutions of local residual problems with equilibrated residual loads and are rather sharp, even for coarse meshes. he sharpness of the bounds can be further improved by employing few iterations of a relatively inexpensive iterative scheme. he main result is that the bounds are guaranteed for the nergy norm of the exact error, unlike the bounds which ave been proposed in [13,14] which are guaranteed only for the nergy norm of the error with respect to an enriched (truth-esh) finite element solution. Copyright

Practical aspects of a-posteriori estimation for reliable finite element analysis

Abstract In this paper we study the problem of a-posteriori estimation of the error in the derivatives, strains and stresses in finite element solutions of the elasticity problem. Given any small patch of elements, we split the error in the patch into two components: the local (or near-field) error which is the response of the elastic continuum in the patch when the domain is loaded only by the near-field residuals, i.e. the residuals of the finite element solution in the elements in the neighborhood of the patch (the patch and a few, e.g. one, surrounding mesh-layers), and the pollution (or far-field) error which is the response in the patch when the domain is loaded by the residuals of the finite element solution in the elements outside the neighborhood of the patch (the far-field residuals). The local error can be estimated by employing element error indicators, which are determined using local computations in the neighborhood of the patch (element residual problems, or local averagings), while the estimation of the pollution error requires a global extraction. This extraction can be based on the ability of the code to compute energy inner products of error indicator functions corresponding to the finite element solution and finite element approximations of a few auxiliary functions which are determined (with negligible cost when a direct solver is employed) by solving the global finite element equations for a few additional auxiliary loads.

A posteriori estimation of the error in the recovered derivatives of the finite element solution

This work addresses the accuracy of the solution derivatives which are recovered by local averaging of the finite element solution. The main results of the study are: (1) The error in the locally averaged derivatives (e.g. the derivatives which are recovered by the Zienkiewicz-Zhu Superconvergent Patch Recovery (ZZ-SPR) or other similar local recoveries) can be more than the error in the derivatives computed directly from the finite element solution, especially in the case of unsmooth solutions and/or coarse meshes. (2) In order to determine which solution derivatives should be relied upon, the locally averaged ones or the ones computed directly from the finite element solution, one must be able to estimate their errors. It is shown that one can obtain indicators of the error in the derivatives recovered by the ZZ-SPR by employing an additional local averaging of the recovered derivatives (recycling of the ZZ-SPR) or by comparing the derivatives computed by the ZZ-SPR with the derivatives obtained using a different local averaging which takes into account the character of the exact solution (harmonic averaging).

Validation of recipes for the recovery of stresses and derivatives by a computer-based approach

In this paper, we present a methodology for checking the local quality of recipes for the recovery of stresses or derivatives from finite element solutions of linear elliptic problems. The methodology accounts precisely for the factors which affect the local quality of the recovered quantities, namely, the geometry of the grid, the polynomial degree and the type of the elements, the coefficients of the differential equation and the class of solutions of interest. We give examples of how the methodology can be used to obtain precise conclusions about the quality of a class of recipes, based on least-squares patch-recovery, in the interior of complex grids, like the ones employed in engineering computations. By using this approach, we were able to discover recipes which are much more robust than the ones which are currently in use in the various finite element codes.

e%-superconvergence in the interior of locally refined meshes of quadrilaterals: superconvergence of the gradient in finite element solutions of Laplace's and Poisson's equations

Abstract This paper is the third in a series in which we study the superconvergence of finite element solutions by a computer-based approach. In [1] we studied classical superconvergence and in [2] we introduced the new concept of η%-superconvergence and showed that it can be employed to determine regions of least error for the derivatives of the finite element solution in the interior of any grid of triangular elements. Here we use the same ideas to study the superconvergence of the derivatives of the finite element solution in the interior of complex grids of quadrilaterals of the type used in practical computations.

A posteriori estimation of the error in the error estimate

In this paper we address the problem of a posteriori estimation of the error in the error estimate. We consider the case of estimates for the error in the derivatives, the strains, or the stresses, which are constructed in terms of locally-computed element error indicators of the element residual, or the least-squares recovery type. The estimates of the error in the error estimate have the same structure as the original error estimates, and are determined by locally averaging (recycling) the original error indicators. The most accurate indicators of the error in the error indicators are obtained by employing a ‘harmonic’ basis in the recycling of the indicators, namely, a basis which locally satisfies the partial differential equation and the boundary conditions.