Dr. Caucao-Paillán, Sergio

We propose and analyze a new mixed finite element method for the coupling of the Stokes equations with a transport problem modelled by a scalar nonlinear convection–diffusion problem. Our approach is based on the introduction of the Cauchy fluid stress and two vector unknowns involving the gradient and the total flux of the concentration. The introduction of these further unknowns lead to a mixed formulation in a Banach space framework in both Stokes and transport equations, where the aforementioned stress tensor and vector unknowns, together with the velocity and the concentration, are the main unknowns of the system. 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 Babuška–Brezzi’s theory in Banach spaces, classical results on nonlinear monotone operators and certain regularity assumptions, are applied to prove the unique solvability of the continuous system. As for the associated Galerkin scheme, whose solvability is established similarly to the continuous case by using the Brouwer fixed-point theorem, we employ Raviart–Thomas approximations of order for the stress and total flux, and discontinuous piecewise polynomials of degree k for the velocity, concentration, and concentration gradient. With this choice of spaces, momentum is conserved in both Stokes and transport equations if the external forces belong to the piecewise constants and concentration discrete space, respectively, which constitutes one of the main features of our approach. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the good performance of the scheme and confirming the theoretical rates of convergence.

We propose and analyze a new mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Besides the velocity, our approach introduces the velocity gradient and a pseudostress tensor as further unknowns. As a consequence, we obtain a three-field Banach spaces-based mixed variational formulation, where the aforementioned variables are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation, and derive the corresponding stability bounds, employing classical results on nonlinear monotone operators. We then propose a semidiscrete continuous-in-time approximation on simplicial grids based on the Raviart–Thomas elements of degree k ≥ 0 for the pseudostress tensor and discontinuous piecewise polynomials of degree k for the velocity and the velocity gradient. In addition, by means of the backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove wellposedness and derive the stability bounds for both schemes, and under a quasi-uniformity assumption on the mesh, we establish the corresponding error estimates. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.

In this paper we consider a partially augmented fully-mixed variational formulation that has been recently proposed for the coupling of the stationary Brinkman–Forchheimer and double-diffusion equations, and develop an a posteriori error analysis for the 2D and 3D versions of the associated mixed finite element scheme. Indeed, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary (convex or non-convex) polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity and inf-sup condition of the forms involved, a suitable assumption on the data, stable Helmholtz decompositions in Hilbert and Banach frameworks, and the local approximation properties of the Clément and Raviart–Thomas operators. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical examples confirming the theoretical properties of the estimators and illustrating the performance of the associated adaptive algorithms, are reported. In particular, the case of flow through a 3D porous media with channel networks is considered.

In this work we present and analyze a fully-mixed formulation for the nonlinear model given by the coupling of the Navier–Stokes and Darcy–Forchheimer equations with the Beavers–Joseph–Saffman condition on the interface. Our approach yields non-Hilbertian normed spaces and a twofold saddle point structure for the corresponding operator equation. Furthermore, since the convective term in the Navier–Stokes equation forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with classical results on nonlinear monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. In particular, given an integer k ≥ 0, Raviart–Thomas spaces of order k, continuous piecewise polynomials of degree ≤k + 1 and piecewise polynomials of degree ≤k are employed in the fluid for approximating the pseudostress tensor, velocity and vorticity, respectively, whereas Raviart–Thomas spaces of order k and piecewise polynomials of degree ≤k for the velocity and pressure, constitute a feasible choice in the porous medium. A priori error estimates and associated rates of convergence are derived, and several numerical examples illustrating the good performance of the method are reported

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 Babuˇ ska–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 optimal a priori error estimates are derived. Several numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method.

We propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart–Thomas spaces of degree for the pseudostress tensor and discontinuous piecewise polynomial elements of degree for the velocity and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters.

We propose and analyze a new mixed finite element method for the problem of steady double-diffusive convection in a fluid-saturated porous medium. More precisely, the model is described by the coupling of the Brinkman–Forchheimer and double-diffusion equations, in which the originally sought variables are the velocity and pressure of the fluid, and the temperature and concentration of a solute. Our approach is based on the introduction of the further unknowns given by the fluid pseudostress tensor, and the pseudoheat and pseudodiffusive vectors, thus yielding a fully-mixed formulation. Furthermore, since the nonlinear term in the Brinkman–Forchheimer equation requires the velocity to live in a smaller space than usual, we partially augment the variational formulation with suitable Galerkin type terms, which forces both the temperature and concentration scalar fields to live in \(\mathrm {L}^4\). As a consequence, the aforementioned pseudoheat and pseudodiffusive vectors live in a suitable \(\mathrm {H}(\mathrm {div})\)-type Banach space. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram and Banach–Nečas–Babuška theorems, allow to prove the unique solvability of the continuous problem. As for the associated Galerkin scheme we utilize Raviart–Thomas spaces of order \(k\ge 0\) for approximating the pseudostress tensor, as well as the pseudoheat and pseudodiffusive vectors, whereas continuous piecewise polynomials of degree \(\le k + 1\) are employed for the velocity, and piecewise polynomials of degree \(\le k\) for the temperature and concentration fields. In turn, the existence and uniqueness of the discrete solution is established similarly to its continuous counterpart, applying in this case the Brouwer and Banach fixed-point theorems, respectively. Finally, we derive optimal a priori error estimates and provide several numerical results confirming the theoretical rates of convergence and illustrating the performance and flexibility of the method.

In this work we present and analyse a mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by the Navier–Stokes and the Darcy–Forchheimer equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. We consider the standard mixed formulation in the Navier–Stokes domain and the dual-mixed one in the Darcy–Forchheimer region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The well-posedness of the problem is achieved by combining a fixed-point strategy, classical results on nonlinear monotone operators and the well-known Schauder and Banach fixed-point theorems. As for the associated Galerkin scheme we employ Bernardi–Raugel and Raviart–Thomas elements for the velocities, and piecewise constant elements for the pressures and the Lagrange multiplier, whereas its existence and uniqueness of solution is established similarly to its continuous counterpart, using in this case the Brouwer and Banach fixed-point theorems, respectively. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, we report some numerical examples confirming the predicted rates of convergence, and illustrating the performance of the method.