Guaranteed and robusta posteriorierror estimates for singularly perturbed reaction–diffusion problems

Abstract

Abstract. We derive a posteriori error estimates for singularly perturbed reaction-diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a generic constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming method, like the piecewise linear finite element one. Our estimates are based on a H(div)-conforming reconstruction of the diffusive flux in the lowest-order RaviartThomas-Nédélec space linked with mesh dual to the original simplicial one, previously introduced by the last author in the pure diffusion case. They also rely on elaborated Poincaré, Friedrichs, and trace inequalities-based auxiliary estimates designed to cope optimally with the reaction dominance. In order to bring down the ratio of the estimated and actual overall energy error as close as possible to the optimal value of one, independently of the size of the reaction coefficient, we finally develop the ideas of local minimizations of the estimators by local modifications of the reconstructed diffusive flux. The numerical experiments presented confirm the guaranteed upper bound, robustness, and excellent efficiency of the derived estimates.Mathematics Subject Classification. 65N15, 65N30, 76S05.

Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods

“…This paper is a detailed description, completion, and extension of the results previously announced in [50] and [53]. Robust a posteriori error estimates for vertex-centered-like finite volume methods are then presented in [52] and [19]. For a complementary approach to a posteriori error estimation in locally conservative methods, evaluating the error in the velocity only (and alternatively in the velocity and the potential), we refer to [54].…”

Fully computable robust a posteriori error bounds for singularly perturbed reaction–diffusion problems

“…Moreover, numerical examples presented in [7] show that the resulting indicator fails to preserve the guaranteed upper bound property of the estimator presented in [2] and as such falls short of meeting the challenge. A computable upper bound for problem (1) discretised using a cell-centred finite volume method was presented by Cheddadi et al [5] based on the use of an associated dual mesh.…”

A unified framework for a posteriori error estimation for the Stokes problem

“…Its main ideas are very physical and can be traced back at least to the Prager-Synge equality [49]. Equilibrated flux estimates have recently been shown to be robust with respect to inhomogeneities, anisotropies, and reaction or convection dominance in [63,24,32] and with respect to the approximation polynomial degree in [13]. In a unifying spirit, similar to the present paper, they have been extended to the heat equation in [34].…”