Guaranteed computable bounds on quantities of interest in finite element computations

Abstract

We develop and compare a number of alternative approaches to obtain guaranteed and fully computable bounds on the error in quantities of interest of arbitrary order finite element approximations in the context of a linear second-order elliptic problem. In each case, the bounds are fully computable and do not involve any unknown multiplicative factors. Guaranteed computable bounds are also obtained for the case when the Dirichlet boundary conditions are non-homogeneous. This is achieved by taking account of the error incurred by the approximation of the Dirichlet data in the functional used to approximate the quantity of interest itself, which is found to generally give better results. Numerical examples are presented to show that the resulting estimators provide tight bounds with the effectivity index tending to unity from above

Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems

“…The key ingredient for goal-oriented error estimation is the formulation of an auxiliary problem, the dual problem to the primal problem, whose solution provides necessary information for reliable estimates of the error in the goal functional. Several strategies for goal-oriented error estimation have been proposed in the case of elliptic problems: goal-oriented error estimates based on energy norm of the errors in the primal and dual solutions were introduced in [48,45,46,49] and further developed by various authors, see for example [4,5], and references therein, error estimates using the dual-weighted residual method were proposed in [25,10,8]; functional a posteriori error estimates were developed in [43,50]; estimates based on the gradient-recovery method were considered in [39,38,47,44,40]; finally, goal-oriented estimates for discontinuous Galerkin methods in the case of second-order elliptic problems were derived in [32].…”

“…It is well known, see e.g. [5], that in order to obtain an efficient error estimator (effectivity indices remain close to unity), the dual problem must be approximated using a higher-order approximation than the one used in the finite element approximation of the primary problem. We propose in this paper, in order to approximate the dual solution, that a high-order discontinuous Galerkin method be used and applied on the same mesh as that used for the discretization of the primal problem.…”

“…If one prefers to keep the same mesh to solve the dual problem as that used for the primal finite element solution, while retaining an efficient error estimator, higher-order methods should be employed, see e.g. [5]. Furthermore, it is desirable that the discretization method for the dual problem allows for computationally inexpensive equilibrated-flux reconstruction.…”

A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method

“…The latter developed the so-called dual-weighted residual (DWR) method. The reliability of the DWR method was further investigated by Nochetto et al [27] and Ainsworth and Rankin [28]. Finally, a goal-oriented method aiming at adaptively controlling various models of a multiscale problem has been developed by Oden and co-worker [29].…”

“…This was later questioned by Nochetto et al [27] that showed that neglecting the higher order terms can cause a severe underestimation of the error and suggested safeguarded DWR method. Finally Ainsworth and Rankin [28] developed guaranteed and fully computable bounds on the error in quantities of interest. We close this introduction by mentioning that including the recent finding of Ainsworth and Rankin [28] in our multiscale DWR method is of great interest.…”