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# Dr. Caucao-Paillán, Sergio

Nombre de publicación

Dr. Caucao-Paillán, Sergio

Nombre completo

Caucao Paillán, Sergio Andrés

Facultad

Email

scaucao@ucsc.cl

ORCID

20 results

## Research Outputs

Now showing 1 - 10 of 20

- PublicationA fully-mixed finite element method for the coupling of the Navier–Stokes and Darcy–Forchheimer equationsIn 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
- PublicationA Banach spaces-based analysis of a new mixed-primal finite element method for a coupled flow-transport problem(Computer Methods In Applied Mechanics And Engineering, 2020)
;Benavides, Gonzalo A.; ;Gatica, Gabriel N.Hopper, Alejandro A.In this paper we introduce and analyze a new finite element method for a strongly coupled flow and transport problem in , , whose governing equations are given by a scalar nonlinear convection–diffusion equation coupled with the Stokes equations. The variational formulation for this model is obtained by applying a suitable dual-mixed method for the Stokes system and the usual primal procedure for the transport equation. 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 solvability analysis, which constitutes the main advantage of the present approach. The resulting continuous and discrete schemes, which involve the Cauchy fluid stress, the velocity of the fluid, and the concentration as the only unknowns, are then equivalently reformulated as fixed point operator equations. Consequently, the well-known Schauder, Banach, and Brouwer theorems, combined with Babuška–Brezzi’s theory in Banach spaces, monotone operator theory, regularity assumptions, and Sobolev imbedding theorems, allow to establish the corresponding well-posedness of them. In particular, Raviart–Thomas approximations of order for the stress, discontinuous piecewise polynomials of degree for the velocity, and continuous piecewise polynomials of degree for the concentration, becomes a feasible choice for the Galerkin scheme. Next, suitable Strang-type lemmas are employed to derive optimal a priori error estimates. Finally, several numerical results illustrating the performance of the mixed-primal scheme and confirming the theoretical rates of convergence, are provided. - PublicationA five-field mixed formulation for stationary magnetohydrodynamic flows in porous media(Computer Methods in Applied Mechanics and Engineering, 2023)
;Angelo, Lady; 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. - PublicationA vorticity-based mixed formulation for the unsteady Brinkman-Forchheimer equations(Computer Methods in Applied Mechanics and Engineering, 2023)
;Anaya, Verónica ;Caraballo, Ruben; ; ;Ruiz-Baier, RicardoYotov , IvanWe 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. - PublicationA Banach spaces-based fully-mixed finite element method for the stationary chemotaxis-Navier-Stokes problem(Computers & Mathematics with Applications, 2023)
; ;Colmenares, Eligio ;Gatica, GabrielInzunza, CristianIn 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. - PublicationA posteriori error analysis of a Banach spaces-based fully mixed FEM for double-diffusive convection in a fluid-saturated porous mediumIn 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.
- PublicationA posteriori error analysis of a momentum and thermal energy conservative mixed FEM for the Boussinesq equationsIn this paper we complement the study of a new mixed finite element scheme, allowing conservation of momentum and thermal energy, for the Boussinesq model describing natural convection and derive a reliable and efficient residual-based a posteriori error estimator for the corresponding Galerkin scheme in two and three dimensions. More precisely, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces, such as local estimates, suitable Helmholtz decompositions and the local approximation properties of the Clément and Raviart–Thomas operators, we derive the aforementioned a posteriori error estimator on arbitrary (convex or non-convex) polygonal and polyhedral regions. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are employed to prove the local efficiency of the proposed a posteriori error estimator. Finally, to illustrate the performance of the adaptive algorithm based on the proposed a posteriori error indicator and to corroborate the theoretical results, we provide some numerical examples.
- PublicationAn augmented mixed FEM for the convective Brinkman-Forchheimer problem: A priori and a posteriori error analysis(Journal of Computational and Applied Mathematic, 2024)
; Esparza, JohannWe 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. - PublicationA Banach spaces-based mixed finite element method for the stationary convective Brinkman-Forchheimer problemWe 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.
- PublicationA new non-augmented and momentum-conserving fully-mixed finite element method for a coupled flow-transport problemWe 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.