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Exact Bounds for Linear Outputs of the Advection-Diffusion-Reaction Equation Using Flux-Free Error Estimates

The paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. The proposed approach is an alternative to the standard residual type estimators (hybrid-flux), circumventing the need of flux-equilibration following a flux-free error estimation strategy. The presented estimator provides sharper estimates than the ones provided by both the standard hybrid-flux techniques and other flux-free techniques.

Accurate upper and lower error bounds by solving flux-free local problems in “stars”

Two implicit residual type estimators yielding upper bounds of the error are presented which do not require flux equilibration. One of them is based on the ideas introduced in [MAC 00, CAR 99, MOR 03, PRU 02]. The new approach introduced here is based on using the estimated error function rather than the estimated error norms. Once the upper bounds are computed, also lower bounds for the error are obtained with little supplementary effort.

Simultaneous upper and lower bound error assessment solving local problems on stars

Publisher Summary The goal of the application of computational mechanics to engineering practice is often the evaluation of particular quantities of interest. These quantities are also denoted as engineering outputs of the problem. The goal of an adaptive procedure is to control efficiently the accuracy of these outputs, which represent the relevant engineering quantities. It is worth noting that goal oriented error estimates have the attractive feature of being directly computable from energy estimates of the error. To bind the error in the quantity of interest, simultaneous lower and upper bounds of the energy error are needed. Standard implicit residual type error estimators provide upper bounds for the energy norm of the error. These estimators require the computation of equilibrated fluxes and use the flux jumps along the interelement boundaries. The computational cost of these fluxes is very high, especially in 3D. This chapter presents new implicit residual type estimators yielding upper bounds of the error that do not require flux equilibration. Once the upper bounds are computed, also lower bounds for the error are obtained with little supplementary effort.