Abstract
We propose and analyze a new mixed finite element method for the nonlinear problem given by the coupling of the steady Brinkman–Forchheimer and double-diffusion equations. Besides the velocity, temperature, and concentration, our approach introduces the velocity gradient, the pseudostress tensor, and a pair of vectors involving the temperature/concentration, its gradient and the velocity, as further unknowns. As a consequence, we obtain a fully mixed variational formulation presenting a Banach spaces framework in each set of equations. 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 formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and Babuška–Brezzi’s theory in Banach spaces, are applied to prove the unique solvability of the continuous and discrete systems. Appropriate finite element subspaces satisfying the required discrete inf-sup conditions are specified, and optimala priorierror estimates are derived. Several numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method.
Highlights
The phenomenon in which two scalar fields, such as heat and concentration of a solute, affect the density distribution in a fluid-saturated porous medium, referred to as double-diffusive convection, is a challenging multiphysics problem
Since the formulation for the double-diffusion equations is similar to the ones employed in [18,19], our present analysis certainly makes use of the corresponding results available there
Employing Raviart–Thomas spaces of order k ≥ 0 for approximating the pseudostress tensor and Bernoulli vectors, and discontinuous piecewise polynomials of degree k for the velocity, temperature, concentration and its corresponding gradients fields, we prove that the method is convergent with optimal rate
Summary
The phenomenon in which two scalar fields, such as heat and concentration of a solute, affect the density distribution in a fluid-saturated porous medium, referred to as double-diffusive convection, is a challenging multiphysics problem. According to the above bibliographic discussion, the goal of the present paper is to continue extending the applicability of the aforementioned Banach spaces framework by proposing a new fully-mixed formulation, without any augmentation procedure, for the coupled problem studied in [14, 28] To this end, we proceed as in [18] and introduce the velocity gradient and pseudostress tensors as auxiliary unknowns, and subsequently eliminate the pressure unknown using the incompressibility condition. To [17,18], we combine a fixed-point argument, classical results on nonlinear monotone operators, Babuska–Brezzi’s theory in Banach spaces, sufficiently small data assumptions, and the well known Banach fixed-point theorem, to establish existence and uniqueness of solution of both the continuous and discrete formulations In this regard, and since the formulation for the double-diffusion equations is similar to the ones employed in [18,19], our present analysis certainly makes use of the corresponding results available there. By ⟨·, ·⟩Γ we will denote the corresponding product of duality between H−1/2(Γ) and H1/2(Γ) (and between H−1/2(Γ) and H1/2(Γ))
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