Robust a-posteriori estimator for advection-diffusion-reaction problems

Abstract

We propose an almost-robust residual-based a-posteriori estimatorfor the advection-diffusion-reaction model problem.The theory is developed in the one-dimensional setting. The numericalerror is measured with respect to a norm which was introduced bythe author in 2005 and somehow plays the role that the energy normhas with respect to symmetric and coercive differential operators.In particular, the mentioned norm possesses features that allow usto obtain a meaningful a-posteriori estimator, robust up to a factor,where is the global P{é}clet number of the problem. Various numericaltests are performed in one dimension, to confirm the theoreticalresults and show that the proposed estimator performs better thanthe usual one known in literature.We also consider a possible two-dimensional extension of our resultand only present a few basic numerical tests, indicating that theestimator seems to preserve the good features of the one-dimensionalsetting.