Grad-div stabilization for the time-dependent Boussinesq equations with inf-sup stable finite elements

Abstract

In this paper we consider inf-sup stable finite element discretizations of the evolutionary Boussinesq equations with a grad-div type stabilization. We prove error bounds for the method with constants independent on the Rayleigh numbers Keywords Boussinesq equations; inf-sup stable finite element methods; grad-div stabilization; High Rayleigh number flows * Instituto de Investigación en Matemáticas (IMUVA), Universidad de Valladolid, Spain. of heat. The functionp is given byp = p + ρ 0 gz /ρ, wherep is the pressure and −ρ 0 gz the hydrostatic pressure. For simplicity we only consider no slip boundary conditions for the velocityũ = 0, on ∂Ω, and for the temperature, both prescribed values and and no flux,whereñ denotes the exterior unit normal vector and ∂Ω = ∂Ω D ∪ ∂Ω N .In this paper we consider inf-sup stable mixed finite element approximations to the model (2) with grad-div stabilization. Our aim is to prove error bounds with constants independent on inverse powers of the viscosity ν and the thermal diffusivity κ > 0, or independent of Prandl and Rayleigh numbers (see (3) below). To this end we add to the Galerkin formulation a grad-div stabilization term. In [10] error bounds for the time-dependent Navier-Stokes equations with only grad-div stabilization and constants independent on inverse powers of the viscosity are obtained. The idea is to extend those bounds to the natural convection flow described by (2). The role of the grad-div stabilization term is explained in detail in the error analysis in the present paper (see Remark 1 below).There are several related works that should be mentioned. In [9] the authors prove error bounds for inf-sup stable mixed finite element approximations to the model (2) with both grad-div stabilization and local projection stabilization (LPS) of streamline-upwind (SUPG)-type for both the velocity and the temperature. The authors prove bounds for the velocity and the temperature with constants independent on inverse powers of ν and κ. The main drawback of [9] is that for proving the error bounds of the method the assumption ∇u h ∞ bounded (u h being the approximation to the velocity) is needed and no proof for this a priori bound is included. In [10] weaker assumptions (norms u h ∞ and ∇ · u h L 2d/(d−1) bounded) are needed to get the error bounds for the pressure once the bounds for the velocity are already obtained. However in [10] these a priori bounds are proved. Let us observe that bounds on the velocity are obtained in [10] whenever finite elements other than linears are used for the velocity in the case d = 3 so that the assumption ∇u h ∞ bounded for h → 0 appearing in [9] may not be valid for lower order finite elements. Error bounds for the pressure are missing in reference [9]. In this paper we get error bounds for the velocity, temperature and the pressure, without assuming a priori bounds for the velocity approximations and with only grad-div stabilization. The rate of convergence we prove is the same obtained in [9] for a scheme that apart from grad-div stabilizat...

Versatile Mixed Methods for Non-Isothermal Incompressible Flows

“…We note that, although the performance of this method is quite adequate from an accuracy standpoint, it is only weakly consistent. Most recently, de Frutos et al [32] derived an optimal set of stability and error estimates for grad-div stabilized, inf-sup stable mixed methods. These methods are effectively a subset of the methods constructed by Dallmann and Arndt in [18,4].…”