Mixed-primal and fully-mixed formulations for the convection-diffusion-reaction system based upon Brinkman–Forchheimer equations

Abstract

We introduce and analyze new mixed formulations, within Banach space-based frameworks, for numerically solving the model given by the coupling of the Brinkman–Forchheimer equations with a convection-diffusion-reaction phenomenon. Specifically, for the former, we consider a pseudostress-velocity mixed formulation, whereas for the latter we analyze both primal and mixed approaches. In particular, for the mixed one the convection-diffusion-reaction part is reformulated by introducing the diffusion vector as an additional unknown, thus leading to a fully-mixed formulation of the coupling. On the other hand, in the mixed-primal setting, the Dirichlet boundary condition for the concentration is enforced through a suitable Lagrange multiplier, whereas this requirement is avoided in the fully mixed approach. We establish the well-posedness of both formulations using a fixed-point strategy and prove the well-posedness of the uncoupled problems by relying on recently established solvability results for perturbed saddle-point problems in Banach spaces, together with the Banach–Nečas–Babuška theorem and the Babuška–Brezzi theory. Additionally, we provide a discrete analysis for both approaches under specific hypotheses on arbitrary finite element spaces. For instance, for each integer k ≥ 0, we consider tensor and vector Raviart–Thomas subspaces of order k for the pseudostress and the diffusion, respectively, along with piecewise polynomial subspaces of degree  ≤ k for the velocity and concentration. This choice yields stable Galerkin schemes for the fully-mixed approach, for which optimal theoretical convergence rates are achieved. Finally, we illustrate the theoretical results through several numerical examples, comparing both approaches and discussing the advantages of each.