Recursos
Proyectos/Publicaciones
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.
M. Álvarez, G.N. Gatica and R. Ruiz-Baier. An augmented mixed-primal finite element method for a coupled flow-transport problem. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 49, 5, pp. 1399-1427, (2015). DOI: http://dx.doi.org/10.
Proyecto de Investigación inscrito en la Universidad de Costa Rica: Análisis y Simulación Numérica de Fenómenos con Aplicaciones Biológicas y Biomecánicas. Actualmente se encuentra en desarrollo.
Proyecto de Investigación inscrito en la Universidad de Costa Rica: C0089: Métodos de Elementos Finitos Mixtos y Técnicas Afines para Modelos Matemáticos en Dinámica de Fluidos. Finalizado.
Proyecto de Investigación inscrito en la Universidad de Costa Rica: B7233 Métodos de Elementos Finitos Mixtos para Problemas Acoplados en Mecánica de Fluidos. Finalizado.
This paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi-augmented mixed-primal finite element method previously developed by us to numerically solve double-diffusive natural convection problem in porous media. The model combines Brinkman-Navier-Stokes equations for velocity and pressure coupled to a vector advection-diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo-stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart-Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual-based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart-Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance.
M. Álvarez, E. Colmenares, And F. A. Sequeira. A posteriori error analysis of a semi-augmented mixed finite element method for double-diffusive natural convection in porous media. Numerical Methods for Partial Differential Equations, Eq. 2024;e23090. DOI: http://dx.