Recursos
Proyectos/Publicaciones
In this paper we study a stationary double-diffusive natural convection problem in porous media given by a Navier-Stokes/Brinkman type system, for describing the velocity and the pressure, coupled to a vector advection-diffusion equation relate to the heat and substance concentration, of a viscous fluid in a porous media with physical boundary conditions. The model problem is rewritten in terms of a first-order system, without the pressure, based on the introduction of the strain tensor and a nonlinear pseudo-stress tensor in the fluid equations. After a variational approach, the resulting weak model is then augmented using appropriate redundant penalization terms for the fluid equations along with a standard primal formulation for the heat and substance concentration. Then, it is rewritten as an equivalent fixed-point problem. Well-posedness results for both the continuous and the discrete schemes are stated, as well as the respective convergence result under certain regularity assumptions combined with the Lax-Milgram theorem, and the Banach and Brouwer fixed-point theorems. In particular, Raviart-Thomas elements of order k are used for approximating the pseudo-stress tensor, piecewise polynomials of degree ≤k and ≤k+1 are utilized for approximating the strain tensor and the velocity, respectively, and the heat and substance concentration are approximated by means of Lagrange finite elements of order ≤k+1. Optimal a priori error estimates are derived and confirmed through some numerical examples that illustrate the performance of the proposed semi-augmented mixed-primal scheme.
M. Álvarez, E. Colmenares, and F. A. Sequeira. Analysis of a semi-augmented mixed finite element method for double-diffusive natural convection in porous media. Computers and Mathematics with Applications, vol. 114, pp. 112- 131, (2022). DOI: http://dx.doi.org/10.1016/j.
This paper is devoted to the mathematical and numerical analysis of a model describing the interfacial flow-transport interaction in a porous-fluidic domain. The medium consists of a highly permeable material, where the flow of an incompressible viscous fluid is governed by Brinkman equations (written in terms of vorticity, velocity and pressure), and a porous medium where Darcy’s law describes fluid motion using filtration velocity and pressure. Gravity and the local fluctuations of a scalar field (representing for instance, the solids volume fraction or the concentration of a contaminant) are the main drivers of the fluid patterns on the whole domain, and the Brinkman-Darcy equations are coupled to a nonlinear transport equation accounting for mass balance of the scalar concentration. We introduce a mixed-primal variational formulation of the problem and establish existence and uniqueness of solution using fixed-point arguments and small-data assumptions. A family of Galerkin discretizations that produce divergence-free discrete velocities is also presented and analysed using similar tools to those employed in the continuous problem. Convergence of the resulting mixed-primal finite element method is proven, and some numerical examples confirming the theoretical error bounds and illustrating the performance of the proposed discrete scheme are reported.
M. Álvarez, G.N. Gatica, and R. Ruiz-Baier. A mixed-primal finite element method for the coupling of Brinkman-Darcy flow and nonlinear transport. IMA Journal of Numerical Analysis, vol. 41, 1, pp. 381-411, (2021). DOI: http://dx.doi.org/10.
This article is concerned with the mathematical and numerical analysis of a steady phase change problem for non-isothermal incompressible viscous flow. The system is formulated in terms of pseudostress, strain rate and velocity for the Navier–Stokes–Brinkman equation, whereas temperature, normal heat flux on the boundary, and an auxiliary unknown are introduced for the energy conservation equation. In addition, and as one of the novelties of our approach, the symmetry of the pseudostress is imposed in an ultra-weak sense, thanks to which the usual introduction of the vorticity as an additional unknown is no longer needed. Then, for the mathematical analysis two variational formulations are proposed, namely mixed-primal and fully-mixed approaches, and the solvability of the resulting coupled formulations is established by combining fixed-point arguments, Sobolev embedding theorems and certain regularity assumptions. We then construct corresponding Galerkin discretizations based on adequate finite element spaces, and derive optimal a priori error estimates. Finally, numerical experiments in 2D and 3D illustrate the interest of this scheme and validate the theory.
M. Álvarez, G. Gatica, B. Gómez-Vargas, and R. Ruiz-Baier. New mixed finite element methods for natural convection with phase-change in porous media. Journal of Scientific Computing, vol. 80, pp. 141-174, (2019). DOI: https://link.springer.
In this paper we study a phase change problem for non-isothermal incompressible viscous flows. The underlying continuum is modelled as a viscous Newtonian fluid where the change of phase is either encoded in the viscosity itself, or in the Brinkman–Boussinesq approximation where the solidification process influences the drag directly. We address these and other modelling assumptions and their consequences in the simulation of differentially heated cavity flows of diverse type. A second order finite element method for the primal formulation of the problem in terms of velocity, temperature, and pressure is constructed, and we provide conditions for its stability. We finally present several numerical tests in 2D and 3D, corroborating the accuracy of the numerical scheme as well as illustrating key properties of the model.
J. Woodfield, M. Álvarez, B. Gómez-Vargas, and R. Ruiz-Baier. Stability and finite element approximation of phase change models for natural convection in porous media. Journal of Computational and Applied Mathematics, vol. 360, pp.117-137, (2019).DOI: http://dx.doi.org/10.1016/j.
In this paper we develop the a posteriori error analysis of an augmented mixed-primal finite element method for the 2D and 3D versions of a stationary flow and transport coupled system, typically encountered in sedimentation–consolidation processes. The governing equations consist in the Brinkman problem with concentration-dependent viscosity, written in terms of Cauchy pseudo-stresses and bulk velocity of the mixture; coupled with a nonlinear advection – nonlinear diffusion equation describing the transport of the solids volume fraction. We derive two efficient and reliable residual-based a posteriori error estimators for a finite element scheme using Raviart–Thomas spaces of order k for the stress approximation, and continuous piecewise polynomials of degree ≤k+1 for both velocity and concentration. For the first estimator we make use of suitable ellipticity and inf–sup conditions together with a Helmholtz decomposition and the local approximation properties of the Clément interpolant and Raviart–Thomas operator to show its reliability, whereas the efficiency follows from inverse inequalities and localisation arguments based on triangle-bubble and edge-bubble functions. Next, we analyse an alternative error estimator, whose reliability can be proved without resorting to Helmholtz decompositions. Finally, we provide some numerical results confirming the reliability and efficiency of the estimators and illustrating the good performance of the associated adaptive algorithm for the augmented mixed-primal finite element method.
M. Álvarez, G.N. Gatica and R. Ruiz-Baier. A posteriori error estimation for an augmented mixed-primal method applied to sedimentation-consolidation systems. Journal of Computational Physics, vol. 367, pp. 322- 346, (2018). DOI: http://dx.doi.org/10.
We analyse the stability of a second-order finite element scheme for the primal formulation of a Brinkman-Boussinesq model where the solidification process influences the drag and the viscosity. The problem is written in terms of velocity, temperature, and pressure, and we produce numerical approximations to the flow observed in heated cavities and near ice sheets.
M. Álvarez, B. Gómez-Vargas, R. Ruiz-Baier and J. Woodfield. Stability of a second-order method for phase change in porous media flow. Proceedings in Applied Mathematics and Mechanics, 25 April, (2018). DOI: http://dx.doi.org/10.
We develop the a posteriori error analysis for a mixed finite element method applied to the coupling of Brinkman and Darcy equations in 3D, modelling the interaction of viscous and non-viscous flow effects across a given interface. The system is formulated in terms of velocity and pressure within the Darcy subdomain, together with vorticity, velocity and pressure of the fluid in the Brinkman region, and a Lagrange multiplier enforcing pressure continuity across the interface. The solvability of a fully-mixed formulation along with a priori error bounds for a finite element method have been recently established in Álvarez et al. ( Comput Methods Appl Mech Eng 307:68–95, 2016). Here we derive a residual-based a posteriori error estimator for such a scheme, and prove its reliability exploiting a global inf-sup condition in combination with suitable Helmholtz decompositions, and interpolation properties of Clément and Raviart–Thomas operators. The estimator is also shown to be efficient, following a localisation strategy and appropriate inverse inequalities. We present numerical tests to confirm the features of the estimator and to illustrate the performance of the method in academic and application-oriented problems.
M. Álvarez, G.N. Gatica and R. Ruiz-Baier. A posteriori error analysis of a fully-mixed formulation for the Brinkman-Darcy problem. Calcolo, vol.54, 4, pp. 1491- 1519, (2017). DOI: http://dx.doi.org/10.
We propose and analyze a fully-mixed finite element method to numerically approximate the flow patterns of a viscous fluid within a highly permeable medium (an array of low concentration fixed particles), described by Brinkman equations, and its interaction with non-viscous flow within classical porous media governed by Darcy’s law. The system is formulated in terms of velocity and pressure in the porous medium, together with vorticity, velocity and pressure of the viscous fluid. In addition, and for sake of the analysis, the tangential component of the vorticity is supposed to vanish on the whole boundary of the Brinkman domain, whereas null normal components of both velocities are assumed on the respective boundaries, except on the interface where suitable transmission conditions are considered. In this way, the derivation of the corresponding mixed variational formulation leads to a Lagrange multiplier enforcing the pressure continuity across the interface, whereas mass balance results from essential boundary conditions on each domain. As a consequence, a typical saddle-point operator equation is obtained, and hence the classical Babuška–Brezzi theory is applied to establish the well-posedness of the continuous and discrete schemes. In particular, we remark that the continuous and discrete inf–sup conditions of the main bilinear form are proved by using suitably chosen injective operators to get lower bounds of the corresponding suprema, which constitutes a previously known technique, recently denominated TT-coercivity. In turn, and consistently with the above, the stability of the Galerkin scheme requires that the curl of the finite element subspace approximating the vorticity be contained in the space where the discrete velocity of the fluid lives, which yields Raviart–Thomas and Nédélec finite element subspaces as feasible choices. Then we show that the aforementioned constraint can be avoided by augmenting the mixed formulation with a residual arising from the Brinkman momentum equation. Finally, several numerical examples illustrating the satisfactory performance of the methods and confirming the theoretical rates of convergence are reported.
M. Álvarez, G.N. Gatica and R. Ruiz-Baier. A posteriori error analysis for a viscous flow-transport problem. ESAIM: Mathematical Modeling and Numerical Analysis, vol. 50, 6, pp. 1789-1816, (2016). DOI: http://dx.doi.org/10.
This paper is devoted to the mathematical and numerical analysis of a strongly coupled flow and transport system typically encountered in continuum-based models of sedimentation–consolidation processes. The model focuses on the steady-state regime of a solid–liquid suspension immersed in a viscous fluid within a permeable medium, and the governing equations consist in the Brinkman problem with variable viscosity, written in terms of Cauchy pseudo-stresses and bulk velocity of the mixture; coupled with a nonlinear advection — nonlinear diffusion equation describing the transport of the solids volume fraction. The variational formulation is based on an augmented mixed approach for the Brinkman problem and the usual primal weak form for the transport equation. Solvability of the coupled formulation is established by combining fixed point arguments, certain regularity assumptions, and some classical results concerning variational problems and Sobolev spaces. In turn, the resulting augmented mixed-primal Galerkin scheme employs Raviart–Thomas approximations of order kk for the stress and piecewise continuous polynomials of order k+1k+1 for velocity and volume fraction, and its solvability is deduced by applying a fixed-point strategy as well. Then, suitable Strang-type inequalities are utilized to rigorously derive optimal error estimates in the natural norms. Finally, a few numerical tests illustrate the accuracy of the augmented mixed-primal finite element method, and the properties of the model.
M. Álvarez, G.N. Gatica and R. Ruiz-Baier. A mixed-primal finite element approximation of a sedimentation-consolidation system. M3AS: Mathematical Models and Methods in Applied Sciences, vol. 26, 5, pp. 867-900, (2016). DOI: http://dx.doi.org/10.