An augmented mixed-primal finite element method for a coupled flow-transport problem^∗

¹ Sección de Matemática, Sede Occidente, Universidad de Costa Rica, San Ramón de Alajuela, Costa Rica
mario.alvarez@ucr.ac.cr
² CI 2 MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
mguadamuz@ci2ma.udec.cl; ggatica@ci2ma.udec.cl
³ Institute of Earth Sciences, Géopolis UNIL-Mouline, University of Lausanne, 1015 Lausanne, Switzerland
ricardo.ruizbaier@unil.ch

Received: 27 August 2014
Revised: 16 February 2015

Abstract

In this paper we analyze the coupling of a scalar nonlinear convection-diffusion problem with the Stokes equations where the viscosity depends on the distribution of the solution to the transport problem. An augmented variational approach for the fluid flow coupled with a primal formulation for the transport model is proposed. The resulting Galerkin scheme yields an augmented mixed-primal finite element method employing Raviart−Thomas spaces of order k for the Cauchy stress, and continuous piecewise polynomials of degree ≤ k + 1 for the velocity and also for the scalar field. The classical Schauder and Brouwer fixed point theorems are utilized to establish existence of solution of the continuous and discrete formulations, respectively. In turn, suitable estimates arising from the connection between a regularity assumption and the Sobolev embedding and Rellich−Kondrachov compactness theorems, are also employed in the continuous analysis. Then, sufficiently small data allow us to prove uniqueness and to derive optimal a priori error estimates. Finally, we report a few numerical tests confirming the predicted rates of convergence, and illustrating the performance of a linearized method based on Newton−Raphson iterations; and we apply the proposed framework in the simulation of thermal convection and sedimentation-consolidation processes.