Recursos
Proyectos/Publicaciones
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.
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.