An posteriori error estimator for discontinuous Galerkin discretisations of convection-diffusion problems with application to Earth's mantle convection simulations

We present new a posteriori error estimates for the interior penalty discontinuous Galerkin method applied to non-stationary convection-diffusion equations. The focus is on strongly convection-dominated problems without zeroth-order reaction terms, which leads to the absence of positive L²-like components. An important specific example is the energy/temperature equation of the Boussinesq system arising from the modelling of mantle convection of the Earth. The key mathematical challenge of mitigating the effects of exponential factors with respect to the final time, arising from the use of Grönwall-type arguments, is addressed by an exponential fitting technique. The latter results to a new class of a posteriori error estimates for the stationary problem, which are valid in cases of convection and reaction coefficient combinations not covered by the existing literature. This new class of estimators is combined with an elliptic reconstruction technique to derive new respective estimates for the non-stationary problem, exhibiting reduced dependence on Grönwall-type exponents and, thus, offer more accurate estimation for longer time intervals. We showcase the superior performance of the new class of a posteriori error estimators in driving mesh adaptivity in Earth's mantle convection simulations, in a setting where the energy/temperature equation is discretised by the discontinuous Galerkin method, coupled with the Taylor-Hood finite element for the momentum and mass conservation equations. We exploit the community code ASPECT to present numerical examples showing the effectivity of the proposed approach.