A posteriori error analysis for a viscous flow-transport problem^∗

¹ Sección de Matemática, Sede de Occidente, Universidad de Costa Rica, San Ramón de Alajuela, Costa Rica. .
mario.alvarezguadamuz@ucr.ac.cr
^∗∗ Present address: CI 2 MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
mguadamuz@ci2ma.udec.cl
² CI 2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
ggatica@ci2ma.udec.cl
³ Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, OX2 6GG Oxford, UK.
ruizbaier@maths.ox.ac.uk

Received: 26 September 2015
Revised: 8 January 2016
Accepted: 22 January 2016

Abstract

In this paper we develop an a posteriori error analysis for an augmented mixed-primal finite element approximation of a stationary viscous flow and transport problem. The governing system corresponds to a scalar, nonlinear convection-diffusion equation coupled with a Stokes problem with variable viscosity, and it 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 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 ≤ k + 1 for both velocity and concentration, have been recently established in [M. Alvarez et al., ESAIM: M2AN 49 (2015) 1399–1427]. Here we derive two efficient and reliable residual-based a posteriori error estimators for that scheme: for the first estimator, and under suitable assumptions on the domain, we apply a Helmholtz decomposition and exploit local approximation properties of the Clément interpolant and Raviart−Thomas operator to show its reliability. On the other hand, its efficiency follows from inverse inequalities and the localization arguments based on triangle-bubble and edge-bubble functions. Secondly, an alternative error estimator is proposed, whose reliability can be proved without resorting to Helmholtz decompositions. Our theoretical results are then illustrated via some numerical examples, highlighting also the performance of the scheme and properties of the proposed error indicators.