The versatility of an entropy inequality for the robust computation of convection dominated problems

We present a discrete inequality that exhibits versatile uses in convection dominated problems. Much like the thermodynamic entropy inequality, the sign of this so-called discrete entropy production allows us to determine unphysical regions in the numerical solution without any a-priori knowledge of the solution. Further, the sign of the discrete production also functions as an excellent indicator for mesh adaptation in convection-diffusion and other singular perturbation problems. We also show preliminary results for how the operator can be used to derive robust schemes for convection dominated problems. All the above applications i.e. (a) Detecting unphysical numerical behavior, (b) mesh adaptation and (c) stabilization, are robust in that they are achieved without any ad-hoc, user introduced, parameters. We show a range of numerical results that exhibit the efficacy of the operator.