A Banach spaces-based analysis of a new mixed-primal finite element method for a coupled flow-transport problem

Abstract

In this paper we introduce and analyze a new finite element method for a strongly coupled flow and transport problem in Rn, n∈{2,3}, whose governing equations are given by a scalar nonlinear convection–diffusion equation coupled with the Stokes equations. The variational formulation for this model is obtained by applying a suitable dual-mixed method for the Stokes system and the usual primal procedure for the transport equation. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the solvability analysis, which constitutes the main advantage of the present approach. The resulting continuous and discrete schemes, which involve the Cauchy fluid stress, the velocity of the fluid, and the concentration as the only unknowns, are then equivalently reformulated as fixed point operator equations. Consequently, the well-known Schauder, Banach, and Brouwer theorems, combined with Babuška–Brezzi’s theory in Banach spaces, monotone operator theory, regularity assumptions, and Sobolev imbedding theorems, allow to establish the corresponding well-posedness of them. In particular, Raviart–Thomas approximations of order k≥0 for the stress, discontinuous piecewise polynomials of degree ≤k for the velocity, and continuous piecewise polynomials of degree ≤k+1 for the concentration, becomes a feasible choice for the Galerkin scheme. Next, suitable Strang-type lemmas are employed to derive optimal a priori error estimates. Finally, several numerical results illustrating the performance of the mixed-primal scheme and confirming the theoretical rates of convergence, are provided.