Mario Álvarez, Gabriel N. Gatica, Ricardo Ruiz-Baier:
A posteriori error analysis for a sedimentation-consolidation system
In this paper we develop the a posteriori error 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 variable 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. The solvability of the continuous mixed--primal formulation along with a priori error estimates for a finite element scheme using Raviart-Thomas spaces of order k for the stress approximation, and continuous piecewise polynomials of degree \le k+1 for both velocity and concentration, have been recently established in [M. Alvarez et al., M3AS: Math. Models Methods Appl. Sci. 26 (5) (2016) 867-900]. In this work we derive two efficient and reliable residual-based a posteriori error estimators for that scheme. For the first estimator we make use of suitable ellipticiy and inf-sup conditions together with a Helmholtz decomposition and the local approximation properties of the Clement interpolant and Raviart-Thomas operator to show its reliability, whereas the efficiency follows from inverse inequalities and localization arguments based on triangle-bubble and edge-bubble functions. Next, we analyze 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.