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

Abstract

In this paper we propose a new mixed-primal formulation for heatdriven flows with temperature-dependent viscosity modeled by the stationary Boussinesq equations. We analyze the well-posedness of the governing equations in this mathematical structure, for which we employ the Banach fixedpoint theorem and the generalized theory of saddle-point problems. The motivation is to overcome a drawback in a recent work by the authors where, in the mixed formulation for the momentum equation, the reciprocal of the viscosity is a pre-factor to a tensor product of velocities; making the analysis quite restrictive, as one needs a given continuous injection that holds only in 2D. We show in this work that by adding both the pseudo-stress and the

A Dual-Mixed Approximation for a Huber Regularization of Generalized $p$-Stokes Viscoplastic Flow Problems

“…In consequence, we can state that problem (2) has a unique solution u ∈ V, which is clearly a unique solution for (1). Furthermore, by a direct application of the de Rahm's theorem, u is also a unique solution for (P).…”

“…Note that the main differences with the model in [16] are the assumption that the viscosity is a function of the infinitesimal stress tensor instead of the velocity gradient, and the presence of the regularized term. In order to obtain the dual-mixed formulation, we proceed as in [1] and set the strain rate tensor as an auxiliary unknown…”

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

“…On the other hand, in order to derive a fully-mixed formulation for (2.1)-(2.3), in which the essential boundary conditions become natural ones, we now proceed as in [22,Section 2] (see also [7,23,24]), and introduce the velocity gradient and the Bernoulli stress tensor as further unknowns t := ∇u and σ :…”

“…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.…”