New Mixed Finite Element Methods for Natural Convection with Phase-Change in Porous Media

Abstract

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.

Analysis of a semi-augmented mixed finite element method for double-diffusive natural convection in porous media

“…whereas, the symmetry of the pseudo-stress tensor is imposed in an ultra-weak sense (see e.g [8]) through the identity…”

A new mixed finite element method for the <i>n</i>-dimensional Boussinesq problem with temperature-dependent viscosity

“…[5,Section 4.3]). Other problems and corresponding references where some of or all the above finite element subspaces have been employed include, among others, coupled flow-transport [8,9], natural convection with phase-change [7], Navier-Stokes [15,16], and Stokes-Darcy (see, e.g. [29], where the above restriction between the mesh sizes h and h was discussed).…”

“…We highlight that (2.4) constitutes what we call an ultra-weak imposition of the symmetry of σ since η(v) : v ∈ H 1 (Ω) is a proper subspace of L 2 skew (Ω). This idea has also been applied in [7].…”

A fully-mixed finite element method for the steady state Oberbeck–Boussinesq system

“…Motivated by the vast applications and the challenging mathematical structure of such nonlinearly coupled system, the interest in analyzing it and in developing efficient numerical techniques to simulate related phenomena has significantly increased, see, e.g., [2,4,5,7,9,11,14,16,21,22,25,28,29,32,34,36,39,42,41,44,46,48] and the references therein. Those works include numerical algorithms based on finite volume approaches, standard finite element techniques, parallel and projection-based stabilization methods, spectral collocation, and mixed finite element methods; and they concentrate on heat-driven flows and double-diffusion convection, including cases in which the phenomena occur in porous enclosures, with either constant or variable physical parameters.…”

“…Some advantages of the proposed scheme include (a) the pressure is eliminated by the functional setting, and it can be recovered by postprocessing, (b) relaxed regularity requirements on temperature and concentration, yielding more flexibility in choosing approximation spaces, (c) the trace-free velocity gradient and the temperature and concentration gradients are now primary unknowns, Fully-mixed FEM for Oberbeck-Boussinesq equations (d) differently from the methods constructed in [11,16,24,41], the Dirichlet boundary conditions for the temperature and concentration are naturally introduced into the formulation, avoiding the use of either an extension or a boundary Lagrange multiplier, (e) this scheme does not involve any augmentation term (as done, e.g. in [6,7,9,10]), avoiding stabilization parameters for well-posedness of the continuous and discrete problems, as well as for the convergence of the method, (f) the analysis also applies to a more general model of cross-diffusion.…”