Analysis of an augmented mixed‐primal formulation for the stationary Boussinesq problem

Abstract

In this article, we propose and analyze a new mixed variational formulation for the stationary Boussinesq problem. Our method, which uses a technique previously applied to the Navier-Stokes equations, is based first on the introduction of a modified pseudostress tensor depending nonlinearly on the velocity through the respective convective term. Next, the pressure is eliminated, and an augmented approach for the fluid flow, which incorporates Galerkin-type terms arising from the constitutive and equilibrium equations, and from the Dirichlet boundary condition, is coupled with a primal-mixed scheme for the main equation modeling the temperature. In this way, the only unknowns of the resulting formulation are given by the aforementioned nonlinear pseudostress, the velocity, the temperature, and the normal derivative of the latter on the boundary. An equivalent fixed-point setting is then introduced and the corresponding classical Banach Theorem, combined with the Lax-Milgram Theorem and the Babuška-Brezzi theory, are applied to prove the unique solvability of the continuous problem. In turn, the Brouwer and the Banach fixed-point theorems are used to establish existence and uniqueness of solution, respectively, of the associated Galerkin scheme. In particular, Raviart-Thomas spaces of order k for the pseudostress, continuous piecewise polynomials of degree ≤ k+1 for the velocity and the temperature, and piecewise polynomials of degree ≤ k for the boundary unknown become feasible choices. Finally, we derive optimal a priori error estimates, and provide several numerical results illustrating the good performance of the augmented mixed-primal finite element method and confirming the theoretical rates of convergence.

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

“…Proof. The results comes from a direct application of the Babuška-Brezzi theory (see [5,Lemma 3.6] or more precisely [20,Lemma 3.4]). In particular, the righthand side of (2.31) is obtained after bounding the functionals F w,φ (cf.…”

“…In particular, finite element methods approximating the solution of these equations have been developed by the mathematical community in the last two decades. For instance, the model with constant coefficients has been addressed via primal approaches in [11,13,19,23], whereas mixed-type schemes have been employed in [20,24,25], and in particular two different formulations based on a dual-mixed approach for the momentum equation, and a primal and mixed-primal one for the energy equation, are proposed in [21].…”

An augmented stress‐based mixed finite element method for the steady state Navier‐Stokes equations with nonlinear viscosity

“…Next, proceeding similarly as in (see also ), that is defining now the tensor using the incompressibility and the foregoing equation to eliminate the pressure, introducing the auxiliary unknowns which denote the strain and the vorticity, respectively, and observing from (2.5) that is now required to be symmetric, which improves the approach from , we arrive at the following system of equations with unknowns , and …”

An Augmented Mixed Finite Element Method for the Navier--Stokes Equations with Variable Viscosity

“…It is easy to see that they both satisfy (2.3) with (μ 1 , μ 2 ) = (2, 3) and (μ 1 , μ 2 ) = (α 0 , α 0 + α 1 ), respectively. Next, following [13] and [16], we observe that the first equation in (2.1) can be rewritten as the equilibrium equation…”