Abstract
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 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 a posteriori error estimators for a finite element scheme using Raviart–Thomas spaces of order k for the stress approximation, and continuous piecewise polynomials of degree ≤k+1 for 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.
Highlights
The phenomenon of gravitational sedimentation of relatively small particles within viscous fluids is of considerable importance in a number of diverse applications related for instance to wastewater treatment, mineral processing, volcanology, or hemodynamics
We derive in [3], two efficient and reliable residual-based a posteriori error estimators for an augmented mixed-primal finite element approximation of a stationary viscous flow and transport problem, which serves as a prototype model for sedimentation–consolidation processes and other phenomena where the transport of species concentration within a viscous fluid is of interest
The following model describes the steady state of the sedimentation–consolidation process consisting on the transport and suspension of a solid phase into an immiscible fluid contained in a vessel
Summary
The phenomenon of gravitational sedimentation of relatively small particles within viscous fluids is of considerable importance in a number of diverse applications related for instance to wastewater treatment, mineral processing, volcanology, or hemodynamics. The purpose of this work is to provide reliable and efficient residual-based a posteriori error estimators for the steady sedimentation–consolidation system studied in [2]. Estimators of this kind are frequently employed to guide adaptive mesh refinement in order to guarantee an adequate convergence behaviour of the Galerkin approximations, even under the eventual presence of singularities. We derive in [3], two efficient and reliable residual-based a posteriori error estimators for an augmented mixed-primal finite element approximation of a stationary viscous flow and transport problem, which serves as a prototype model for sedimentation–consolidation processes and other phenomena where the transport of species concentration within a viscous fluid is of interest.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
