Research Outputs
A three-field mixed finite element method for the convective Brinkman–Forchheimer problem with varying porosity
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.
A posteriori error analysis of a Banach spaces-based fully mixed FEM for double-diffusive convection in a fluid-saturated porous medium
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.
A posteriori error analysis of a mixed finite element method for the coupled Brinkman-Forchheimer and double-diffusion equations
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.
A fully-mixed formulation for the steady double-diffusive convection system based upon Brinkman-Forchheimer equations
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.
New mixed finite element methods for the coupled convective Brinkman-Forchheimer and double-diffusion equations
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
A Banach spaces-based fully-mixed finite element method for the stationary chemotaxis-Navier-Stokes problem
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.
Residual-based a posteriori error analysis for the coupling of the Navier–Stokes and Darcy–Forchheimer equations
In this paper we consider a mixed variational formulation that have been recently proposed for the coupling of the Navier–Stokes and Darcy–Forchheimer equations, and derive, though in a non-standard sense, a reliable and efficient residual-baseda posteriorierror estimator suitable for an adaptive mesh-refinement method. For the reliability estimate, which holds with respect to the square root of the error estimator, we make use of the inf-sup condition and the strict monotonicity of the operators involved, a suitable Helmholtz decomposition in non-standard Banach spaces in the porous medium, local approximation properties of the Clément interpolant and Raviart–Thomas operator, and a smallness assumption on the data. In turn, inverse inequalities, the localization technique based on triangle-bubble and edge-bubble functions in localLpspaces, are the main tools for developing the efficiency analysis, which is valid for the error estimator itself up to a suitable additional error term. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.
A Banach spaces-based mixed finite element method for the stationary convective Brinkman-Forchheimer problem
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.
A new non-augmented and momentum-conserving fully-mixed finite element method for a coupled flow-transport problem
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.
A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman-Forchheimer and double-diffusion equations
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.