An augmented fully-mixed finite element method for a coupled flow-transport problem

In this paper we analyze the coupling of the Stokes equations with a transport problem modelled by a scalar nonlinear convection–diffusion problem, where the viscosity of the fluid and the diffusion coefficient depend on the solution to the transport problem and its gradient, respectively. An augmented mixed variational formulation for both the fluid flow and the transport model is proposed. As a consequence, no discrete inf-sup conditions are required for the stability of the associated Galerkin scheme, and therefore arbitrary finite element subspaces can be used, which constitutes one of the main advantages of the present approach. In particular, the resulting fully-mixed finite element method can employ Raviart–Thomas spaces of order k for the Cauchy stress, continuous piecewise polynomials of degree \(k + 1\) for the velocity and for the scalar field, and discontinuous piecewise polynomial approximations for the gradient of the concentration. In turn, the Lax–Milgram lemma, monotone operators theory, and the classical Schauder and Brouwer fixed point theorems are utilized to establish existence of solution of the continuous and discrete formulations. In addition, suitable estimates, arising from the combined use of a regularity assumption with the Sobolev embedding and Rellich–Kondrachov compactness theorems, are also required for the continuous analysis. Then, sufficiently small data allow us to prove uniqueness of solution and to derive optimal a priori error estimates. Finally, several numerical tests, illustrating the performance of our method and confirming the predicted rates of convergence, are reported.