A mixed-primal finite element approximation of a sedimentation–consolidation system

Abstract

This paper is devoted to the mathematical and numerical analysis of a strongly coupled flow and transport system typically encountered in continuum-based models of sedimentation–consolidation processes. The model focuses on the steady-state regime of a solid–liquid suspension immersed in a viscous fluid within a permeable medium, and 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 variational formulation is based on an augmented mixed approach for the Brinkman problem and the usual primal weak form for the transport equation. Solvability of the coupled formulation is established by combining fixed point arguments, certain regularity assumptions, and some classical results concerning variational problems and Sobolev spaces. In turn, the resulting augmented mixed-primal Galerkin scheme employs Raviart–Thomas approximations of order [Formula: see text] for the stress and piecewise continuous polynomials of order [Formula: see text] for velocity and volume fraction, and its solvability is deduced by applying a fixed-point strategy as well. Then, suitable Strang-type inequalities are utilized to rigorously derive optimal error estimates in the natural norms. Finally, a few numerical tests illustrate the accuracy of the augmented mixed-primal finite element method, and the properties of the model.

Analysis and mixed-primal finite element discretisations for stress-assisted diffusion problems

“…In contrast, we will use a primal formulation for the diffusion equation. Then, following a similar approach to the one employed in [3] and [4], the existence and uniqueness of weak solutions to the coupled system will be established invoking the Lax-Milgram lemma, the Babuška-Brezzi theory, suitable regularity estimates, and fixed-point arguments permitting us to decouple the solid mechanics from the generalised Poisson problem. Nevertheless, while there are in fact certain similarities with [3] and [4], it is important to remark that the problems involved deal with very different models and that there are substantial differences between the respective analyses.…”

A mixed-primal finite element method for the coupling of Brinkman–Darcy flow and nonlinear transport

“…We stress that the local fluctuations of φ drive the flow patterns only through the external load in the momentum equations. In this sense, the coupling mechanisms considered here are somehow weaker than those studied in Alvarez et al (2015Alvarez et al ( , 2016a for transport flow in a single domain (where also viscosity was depending on φ).…”

“…In this section we proceed similarly as in Alvarez et al (2015) and Alvarez et al (2016a), to derive a suitable variational formulation of (2.1)-(2.4) and analyse its solvability by means of a fixed-point strategy.…”

“…The aim of this paper is to put together an extension of the results from Alvarez et al (2015Alvarez et al ( , 2016a and Alvarez et al (2016b) dealing with augmented and fully mixed finite element approximations of coupled flow and transport problems, and coupled Brinkman and Darcy flow, respectively. The , where, for any vector field v :…”

A new mixed finite element method for the <i>n</i>-dimensional Boussinesq problem with temperature-dependent viscosity

“…[5,Section 4.3]). Other problems and corresponding references where some of or all the above finite element subspaces have been employed include, among others, coupled flow-transport [8,9], natural convection with phase-change [7], Navier-Stokes [15,16], and Stokes-Darcy (see, e.g. [29], where the above restriction between the mesh sizes h and h was discussed).…”