Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients

Abstract

General error estimates are proved for a class of finite element schemes for nonstationary thermal convection problems with temperaturedependent coefficients. These variable coefficients turn the diffusion and the buoyancy terms to be nonlinear, which increases the nonlinearity of the problems. An argument based on the energy method leads to optimal error estimates for the velocity and the temperature without any stability conditions. Error estimates are also provided for schemes modified by approximate coefficients, which are used conveniently in practical computations.

Stability of a second‐order method for phase change in porous media flow

“…Apart from proposing a conforming Galerkin method for the space discretisation of the governing equations (momentum, mass, energy, and enthalpy), our contribution focuses also on deriving stability bounds for the discrete solutions. This result generalises other studies, so far focused on the natural or thermal convection of fluids without phase change, see for instance [1,[4][5][6]8]. In particular, this note summarises the recent results reported in [2] and presents three new numerical tests.…”

One‐level and two‐level finite element methods for the Boussinesq problem

“…Applying the above inequalities with (16) and summing it about n from 0 to J, applying Lemmas 3.2 and 3.3, we get…”

“…These assumptions are known to cause nonlocal compatibility conditions on the given data as discussed in Heywood and Rannacher in the case of the Navier‐Stokes equations. As that have been stated in theorem 1 of Tabata and Tagami, we do not touch on the behavior of the solutions near the initial time but confine ourselves to an ideal case. Hereafter, C 1 , C 2 ,…, denote some positive constants depending only on Ω.…”

Well-Posedness of the Infinite Prandtl Number Model for Convection With Temperature-Dependent Viscosity

“…An upper bound on the Nusselt number for the model under investiagtion which scales like Ra 1 2 was derived in [6]. We also need to consider accurate and effective numerical schemes for the system, especially long time behaviors (see [1] for the case of constant viscosity and long time behavior, [7] for finite time approximation).…”