Subdomain-based flux-free a posteriori error estimators

Abstract

A new residual-type flux-free error estimator is presented. It computes upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux-equilibration and resulting in a simple implementation that uses standard resources available in finite element codes. This is especially interesting for existing codes and 3D applications where the implementation of this technique is as simple as in 2D. Recall that on the contrary, the complexity of the flux-equilibration techniques increases drastically in the 3D case. Bounds for the energy norm of the error are used to produce upper and lower bounds of linear functional outputs, representing quantities of engineering interest. This new flux-free error estimator improves the effectivity of previous approaches (better accuracy in every test) and it can be used in the mechanical case for linear elements. The proposed approach demonstrates its efficiency in numerical tests producing sharp bounds of the reference error both for the energy and the quantities of interest.

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

“…As shown in [11] the bounds for the quantity of interest are not robust with respect to the advection parameter, since the effectivities of the bounds deteriorate as the advection term becomes dominant. In this work, sharper bounds which alleviate this behavior have been obtained combining stabilization techniques along with goal-oriented adaptivity.…”

“…The proposed strategies are an extension of the flux-free technique presented in [11] and the hybrid-flux technique presented in [10]. The flux-free estimates yield much sharper bounds than the hybrid-flux approach both for the stabilized and non-stabilized approaches.…”

“…This section is devoted to detail the computation of the piecewise polynomial dual estimatesq and r using the flux-free approach proposed in [11]. The strategy proposed in [1] can not be directly applied since the residuals are not orthogonal to V h .…”

“…In particular [1] presents a comparison of the performance of two of the main techniques to compute guaranteed bounds for quantities of interest in the context of the advection-reaction-diffusion equation: a standard residual type estimator (hybrid-flux) proposed in [10] and the new flux-free technique proposed in [11].…”

“…In the present paper we develop an extension of the techniques presented in [1] to compute guaranteed bounds for quantities of interest from stabilized approximations of the exact solution. Thus, strict bounds for quantities of interest are obtained using implicit residual error estimates, both using hybrid-flux techniques [13,14] and the flux-free technique first devised in [11].…”