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.