Aposteriori error estimation for an augmented mixed-primal method applied to sedimentation–consolidation systems

Abstract

In this paper we develop the aposteriorierror analysis of an augmented mixed-primal finite element method for the 2D and 3D versions of a stationary flow and transport coupled system, typically encountered in sedimentation–consolidation processes. The governing equations consist in the Brinkman problem with concentration-dependent viscosity, written in terms of Cauchy pseudo-stresses and bulk velocity of the mixture; coupled with a nonlinear advection– nonlinear diffusion equation describing the transport of the solids volume fraction. We derive two efficient and reliable residual-based aposteriorierror estimators for a finite element scheme using Raviart–Thomas spaces of order kfor the stress approximation, and continuous piecewise polynomials of degree ≤k +1for both velocity and concentration. For the first estimator we make use of suitable ellipticity and inf–sup conditions together with a Helmholtz decomposition and the local approximation properties of the Clément interpolant and Raviart–Thomas operator to show its reliability, whereas the efficiency follows from inverse inequalities and localisation arguments based on triangle-bubble and edge-bubble functions. Next, we analyse an alternative error estimator, whose reliability can be proved without resorting to Helmholtz decompositions. Finally, we provide some numerical results confirming the reliability and efficiency of the estimators and illustrating the good performance of the associated adaptive algorithm for the augmented mixed-primal finite element method.