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We present a fully computable a posteriori error estimator for piecewiselinear finite element approximations of reaction-diffusion problems with mixedboundary conditions and piecewise constant reaction coefficient formulated inarbitrary dimension. The estimator provides a guaranteed upper bound on theenergy norm of the error and it is robust for all values of the reactioncoefficient, including the singularly perturbed case. The approach is based onrobustly equilibrated boundary flux functions of Ainsworth and Oden (Wiley2000) and on subsequent robust and explicit flux reconstruction. This papersimplifies and extends the applicability of the previous result of Ainsworthand Vejchodsk\'y (Numer. Math. 119 (2011) 219-243) in three aspects: (i)arbitrary dimension, (ii) mixed boundary conditions, and (iii) non-constantreaction coefficient. It is the first robust upper bound on the error withthese properties. An auxiliary result that is of independent interest is thederivation of new explicit constants for two types of trace inequalities onsimplices.

computer methods in applied mechanics and engineering

Robust error bounds for finite element approximation of  reaction-diffusion problems with non-constant reaction coefficient in  arbitrary space dimension