Dr. Caucao-Paillán, Sergio

We propose and analyse an augmented mixed finite element method for the pseudo stress–velocity formulation of the stationary convective Brinkman–Forchheimer problem inRd, d∈ {2,3}. Since the convective and Forchheimer terms 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 written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram theorem, allow to prove the unique solvability of the continuous problem. The finite element discretization involves Raviart–Thomas spaces of order k≥0 for the pseudostress tensor and continuous piecewise polynomials of degree ≤k+1 for the velocity. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. In addition, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the form involved, a suitable assumption on the data, a stable Helmholtz decomposition, 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, some numerical examples illustrating the performance of the mixed finite element method, confirming the theoretical rate of convergence and the properties of the estimators, and showing the behaviour of the associated adaptive algorithms, are reported. In particular, the case of flow through a 2D porous media with fracture networks is considered.

We introduce and analyze a partially augmented fully mixed formulation and a mixed finite element method for the coupled problem arising in the interaction between a free fluid and a poroelastic medium. The flows in the free fluid and poroelastic regions are governed by the Navier–Stokes and Biot equations, respectively, and the transmission conditions are given by mass conservation, balance of fluid force, conservation of momentum and the Beavers–Joseph–Saffman condition. We apply dual-mixed formulations in both domains, where the symmetry of the Navier–Stokes and poroelastic stress tensors is imposed in an ultra-weak and weak sense. In turn, since the transmission conditions are essential in the fully mixed formulation, they are imposed weakly by introducing the traces of the structure velocity and the poroelastic medium pressure on the interface as the associated Lagrange multipliers. Furthermore, since the fluid convective term requires the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin-type terms. Existence and uniqueness of a solution are established for the continuous weak formulation, as well as a semidiscrete continuous-in-time formulation with nonmatching grids, together with the corresponding stability bounds and error analysis with rates of convergence. Several numerical experiments are presented to verify the theoretical results and illustrate the performance of the method for applications to arterial flow and flow through a filter.

In this paper we present and analyze a new mixed finite element method for the nonlinear problem given by the stationary convective Brinkman–Forchheimer equations with varying porosity. Our approach is based on the introduction of the pseudostress and the gradient of the porosity times the velocity, as further unknowns. As a consequence, we obtain a mixed variational formulation within a Banach spaces framework, with the velocity and the aforementioned tensors as the only unknowns. The pressure, the velocity gradient, the vorticity, and the shear stress can be computed afterwards via postprocessing formulae. A fixed-point strategy, along with monotone operators theory and the classical Banach theorem, are employed to prove the well-posedness of the continuous and discrete systems. Specific finite element subspaces satisfying the required discrete stability condition are defined, and optimal a priori error estimates are derived. Finally, several numerical examples illustrating the performance and flexibility of the method and confirming the theoretical rates of convergence, are reported.

In this paper we introduce and analyze new Banach spaces-based mixed finite element methods for the stationary nonlinear problem arising from the coupling of the convective Brinkman-Forchheimer equations with a double diffusion phenomenon. Besides the velocity and pressure variables, the symmetric stress and the skew-symmetric vorticity tensors are introduced as auxiliary unknowns of the fluid. Thus, the incompressibility condition allows to eliminate the pressure, which, along with the velocity gradient and the shear stress, can be computed afterwards via postprocessing formulae depending on the velocity and the aforementioned new tensors. Regarding the diffusive part of the coupled model, and additionally to the temperature and concentration of the solute, their gradients and pseudoheat/pseudodiffusion vectors are incorporated as further unknowns as well. The resulting mixed variational formulation, settled within a Banach spaces framework, consists of a nonlinear perturbation of, in turn, a nonlinearly perturbed saddle-point scheme, coupled with a usual saddle-point system. A fixed-point strategy, combined with classical and recent solvability results for suitable linearizations of the decoupled problems, including in particular, the Banach-Nečas-Babuška theorem and the Babuška-Brezzi theory, are employed to prove, jointly with the Banach fixed-point theorem, the well-posedness of the continuous and discrete formulations. Both PEERS and AFW elements of order l>0 for the fluid variables, and piecewise polynomials of degree

We propose and analyze a new mixed finite element method for the nonlinear problem given by the stationary convective Brinkman–Forchheimer equations. In addition to the original fluid variables, the pseudostress is introduced as an auxiliary unknown, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the aforementioned tensor and the velocity. As a consequence, we obtain a mixed variational formulation consisting of a nonlinear perturbation of, in turn, a perturbed saddle point problem in a Banach spaces framework. In this way, and differently from the techniques previously developed for this model, no augmentation procedure needs to be incorporated into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that recently established solvability results for perturbed saddle-point problems in Banach spaces, along with the well-known Banach–Nečas–Babuška and Banach theorems, are applied to prove the well-posedness of the continuous and discrete systems. The finite element discretization involves Raviart–Thomas elements of order for the pseudostress tensor and discontinuous piecewise polynomial elements of degree for the velocity. Stability, convergence, and optimal a priori error estimates for the associated Galerkin scheme are obtained. Numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method. In particular, the case of flow through a 2D porous media with fracture networks is considered.

This paper develops the a priori analysis of a mixed finite element method for the filtration of an incompressible fluid through a non-deformable saturated porous medium with heterogeneous permeability. Flows are governed by the Brinkman–Forchheimer and Darcy equations in the more and less permeable regions, respectively, and the corresponding transmission conditions are given by mass conservation and continuity of momentum. We consider the standard mixed formulation in the Brinkman–Forchheimer domain and the dual-mixed one in the Darcy region, and we impose the continuity of the normal velocities by introducing suitable Lagrange multiplier. The finite element discretization involves Bernardi–Raugel and Raviart–Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for the Lagrange multiplier. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. Numerical tests illustrate the theoretical results.

We propose and analyze an augmented mixed formulation for the time-dependent Brinkman–Forchheimer equations written in terms of vorticity, velocity and pressure. The weak formulation is based on the introduction of suitable least squares terms arising from the incompressibility condition and the constitutive equation relating the vorticity and velocity. 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 based on stable Stokes elements for the velocity and pressure, and continuous or discontinuous piecewise polynomial spaces for the vorticity. In addition, by means of the backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove well-posedness and derive the stability bounds for both schemes, and 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 introduce and analyze a Banach spaces-based approach yielding a fully-mixed finite element method for numerically solving the stationary chemotaxis-Navier-Stokes problem. This is a nonlinear coupled model representing the biological process given by the cell movement, driven by either an internal or an external chemical signal, within an incompressible fluid. In addition to the velocity and pressure of the fluid, the velocity gradient and the Bernouilli-type stress tensor are introduced as further unknowns, which allows to eliminate the pressure from the equations and compute it afterwards via a postprocessing formula. In turn, besides the cell density and the chemical signal concentration, the pseudostress associated with the former and the gradient of the latter are introduced as auxiliary unknowns as well. The resulting continuous formulation, posed in suitable Banach spaces, consists of a coupled system of three saddle point-type problems, each one of them perturbed with trilinear forms that depend on data and the unknowns of the other two. The well-posedness of it is analyzed by means of a fixed-point strategy, so that the classical Banach theorem, along with the Babuška-Brezzi theory in Banach spaces, allow to conclude, under a smallness assumption on the data, the existence of a unique solution. Adopting an analogue approach for the associated Galerkin scheme, and under suitable hypotheses on arbitrary finite element subspaces employed, we apply the Brouwer and Banach theorems to show existence and then uniqueness of the discrete solution. General a priori error estimates, including those for the postprocessed pressure, are also derived. Next, a specific set of finite element subspaces satisfying the required stability conditions, and yielding approximate local conservation of momentum, is introduced, which, given an integer , is defined in terms of Raviart-Thomas spaces of order k and piecewise polynomials of degree ≤k only. The respective rates of convergence of the resulting Galerkin method are then provided. Finally, several numerical experiments confirming the latter and illustrating the good performance of the method, are reported.

We introduce and analyze a new mixed variational formulation for a stationary magnetohydrodynamic flows in porous media problem, whose governing equations are given by the steady Brinkman–Forchheimer equations coupled with the Maxwell equations. Besides the velocity, magnetic field and a Lagrange multiplier associated to the divergence-free condition of the magnetic field, a convenient translation of the velocity gradient and the pseudostress tensor are introduced as further unknowns. As a consequence, we obtain a five-field Banach spaces based mixed variational formulation, where the aforementioned variables are the main unknowns of the system. The resulting mixed 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 a sufficiently small data assumption, are applied to prove the unique solvability of the continuous and discrete systems. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. The finite element discretization involves Raviart–Thomas elements of order for the pseudostress tensor, discontinuous piecewise polynomial elements of degree for the velocity and the translation of the velocity gradient, Nédélec elements of degree for the magnetic field and Lagrange elements of degree for the associated Lagrange multiplier. Stability, convergence, and optimal a priori error estimates for the associated Galerkin scheme are obtained. Numerical tests illustrate the theoretical results.

In this paper we consider a Banach spaces-based fully-mixed variational formulation that has been recently proposed for the coupling of the stationary Brinkman–Forchheimer and double-diffusion equations, and develop the first reliable and efficient residual-based a posteriori error estimator for the 2D and 3D versions of the associated mixed finite element scheme. For the reliability analysis, and due to the nonlinear nature of the problem, we employ the strong monotonicity of the operator involving the Forchheimer term, in addition to inf-sup conditions of some of the resulting bilinear forms, along with a stable Helmholtz decomposition in nonstandard Banach spaces, which, in turn, having been recently derived, constitutes another distinctive feature of the paper, and local approximation properties of the Raviart–Thomas and Clément interpolants. On the other hand, inverse inequalities, the localization technique through 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 estimator and illustrating the performance of the associated adaptive algorithms, are reported. In particular, the case of flow through a 2D porous media with an irregular channel networks is considered.