A Banach spaces-based mixed-primal finite element method for the coupling of Brinkman flow and nonlinear transport

Abstract

In this paper we consider a strongly coupled flow and nonlinear transport problem arising in sedimentation-consolidation processes in ${\textrm{R}}^n$, $n\in \big \{2,3\big \}$, and introduce and analyze a Banach spaces-based variational formulation yielding a new mixed-primal finite element method for its numerical solution. The governing equations are determined by the coupling of a Brinkman flow with a nonlinear advection–diffusion equation, in addition to Dirichlet boundary conditions for the fluid velocity and the concentration. The approach is based on the introduction of the Cauchy fluid stress and the gradient of its velocity as additional unknowns, thus yielding a mixed formulation in a Banach spaces framework for the Brinkman equations, whereas the usual Hilbertian primal formulation is employed for the transport equation. Differently from previous works on this and related problems, no augmented terms are incorporated, and hence, besides becoming fully equivalent to the original physical model, the resulting variational formulation is much simpler, which constitutes its main advantage, mainly from the computational point of view. The well-posedness of the continuous formulation is analyzed firstly by rewriting it as a fixed-point operator equation, and then by applying the Schauder and Banach theorems, along with the Babuška-Brezzi theory and the Lax-Milgram lemma. An analogue fixed-point strategy is employed for the analysis of the associated Galerkin scheme, using in this case the Brouwer theorem instead of the Schauder one. The resulting discrete scheme becomes momentum conservative for the fluid in an approximate sense. Next, a Strang-type lemma and suitable algebraic manipulations are utilized to derive the a priori error estimates, which, along with the approximation properties of the finite element subspaces, yield the corresponding rates of convergence. The paper is ended with several numerical results illustrating the performance of the mixed-primal scheme and confirming the theoretical decay of the error.