The Crank-Nicolson Extrapolation Stabilized Finite Element Method for Natural Convection Problem

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 393494, 22 pages http://dx.doi.org/10.1155/2014/393494 Research Article The Crank-Nicolson Extrapolation Stabilized Finite Element Method for Natural Convection Problem Yunzhang Zhang1,2 and Yanren Hou3 1 School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China Department of Mathematics, Nanjing University, Nanjing 210093, China 3 School of Mathematics and Statistics and Center for Computational Geosciences, Xi’an Jiaotong University, Xi’an 710049, China 2 Correspondence should be addressed to Yunzhang Zhang; Received 16 March 2014; Revised 20 April 2014; Accepted 20 April 2014; Published 25 May 2014 Academic Editor: Ming Li Copyright © 2014 Y. Zhang and Y. Hou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper studies a fully discrete Crank-Nicolson linear extrapolation stabilized finite element method for the natural convection problem, which is unconditionally stable and has second order temporal accuracy of 𝑂(Δ𝑡2 + ℎΔ𝑡 + ℎ𝑚 ). A simple artificial viscosity stabilized of the linear system for the approximation of the new time level connected to antidiffusion of its effects at the old time level is used. An unconditionally stability and an a priori error estimate are derived for the fully discrete scheme. A series of numerical results are presented that validate our theoretical findings. 1. Introduction 𝑇𝑡 − ∇ ⋅ (𝑘∇𝑇) + (𝑢 ⋅ ∇) 𝑇 = 𝛾 Natural convection flow has many thermal engineering applications such as in double-glazed windows, solar collectors, cooling devices for electronic instruments, gas-filled cavities around nuclear reactor cores, and building insulation. Typically, fluid flow and heat transfer are governed by the partial differential equation system of momentum, mass, and energy conservation, but in the case of natural convection, the so-called Boussinesq approximation is generally used. The natural convection problem which we consider is for bounded, polyhedral domains Ω𝑒 ⊂ Ω in R𝑑 (𝑑 = 2, 3) with dist(𝜕Ω𝑒 , 𝜕Ω) > 0, the simulation time 𝑡∗ , and the force field 𝛾 : Ω × (0, 𝑡∗ ] → R; find the velocity 𝑢 : Ω × (0, 𝑡∗ ] → R𝑑 , the pressure 𝑝 : Ω × (0, 𝑡∗ ] → R, and the temperature 𝑇 : Ω × (0, 𝑡∗ ] → R satisfying [1] 𝑇=0 𝑢𝑡 − PrΔ𝑢 + (𝑢 ⋅ ∇) 𝑢 + ∇𝑝 = Pr Ra 𝜁𝑇, 𝑢=0 on 𝜕Ω𝑒 , 𝑢|𝑡=0 = 𝑢0 , 𝑔 𝜁 = 󵄨󵄨 󵄨󵄨 , 󵄨󵄨𝑔󵄨󵄨 𝑢 ≡ 0 in Ω − Ω𝑒 = Ω𝑠 , ∇ ⋅ 𝑢 (𝑥, 𝑡) = 0 in Ω𝑒 , on Γ𝑇 , 𝜕𝑇 =0 𝜕𝑛 𝑇|𝑡=0 = 𝑇0 , in Ω, on Γ𝐵 , in Ω, (1) where 𝜁 is a unit vector in the direction of gravity, 𝑛 is the outward unit normal to Ω, and Γ𝑇 = 𝜕Ω \ Γ𝐵 where Γ𝐵 is a regular open subset of 𝜕Ω, Pr is Prandtl number, Ra is Rayleigh number, and 𝑘 > 0 is thermal conductivity parameter. Moreover, 𝑘 = 𝑘𝑒 in Ω𝑒 and 𝑘 = 𝑘𝑠 in Ω𝑠 , where 𝑘𝑒 and 𝑘𝑠 are positive constants. A global-in-time existence result for a more general natural convection problem (NavierStokes/Fourier model) is given in [2]. Many authors have worked hard to study for a great variety of efficient numerical schemes for the natural convection problem [3–17] and relevant research [18, 19]. We mention only a few papers here. [3, 4] are the early papers by using mixed finite element (FE) method. Çıbık and Kaya [5] have formulated a projection-based stabilization FE technique for solving the steady-state natural convection problems. The global stabilizations are added for both velocity 2 Mathematical Problems in Engineering and temperature variables and these effects are subtracted from the large scales. Galvin et al. [7] consider the problem of poor mass conservation in mixed FE algorithms for flow problems with large rotation-free forcing in the momentum equation. Zhang et al. [8] have presented a subgrid stabilized defect-correction method for steady-state natural convection problem. Shi and Ren [11] have proposed a least squares Galerkin-Petrov nonconforming mixed FE method for stationary conduction-convection problems. Luo et al. [12] have given an optimizing reduced Petrov-Galerkin least squares mixed FE for the nonstationary conductionconvection problem. Boland and Layton [1] have derived stability properties and error estimates for the mixed FE spatial discretization case when used to approximate heat flow in a fluid enclosed by a solid medium. Benı́tez and Bermúdez have presented a second order Lagrange-Galerkin method for natural convection problems in [17]. In [20, 21], a stability analysis of thermal natural convection in superposed fluid and porous layers is carried out. Our goal in this paper is to solve time-dependent natural convection problem efficiently and accurately. Usually fully implicit schemes are (almost) unconditionally stable, but one has to solve a system of nonlinear equations at each time step. Although an explicit scheme is much easier in computation, it suffers a restricted time step size from the stability requirement. A popular approach is based on an implicit scheme for the linear term and a semi-implicit scheme or an explicit scheme for the nonlinear term. There are numerous works on the Crank-Nicolson and relevant high order scheme for the Navier-Stokes (NS) equations [22–30]. The Crank-Nicolson linear extrapolation (CNLE) scheme for NS equations was first studied by Baker in [23]. The second and third order CNLE methods are introduced and analysed in [24]. A stabilized extrapolated trapezoidal FE method is given in [25] for the NS equations. A variational multiscale method based on the CNLE scheme for the NS equations is proposed in [26]. He et al. [27–29] have studied the NS equations based on the Crank-Nicolson extrapolation (Crank-Nicolson/Adams-Bashforth, or two level methods) schemes. In [31], we have studied fully implicit CrankNicolson scheme for natural convection problem. We consider herein a simple, second order accurate, and unconditionally stable fully discrete Crank-Nicolson linear extrapolation stabilized (CNSLE) FE method for natural convection problem which requires the solution of one linear system per time step. Suppressing the spatial discretization, the method is 𝑢𝑛+1 + 𝑢𝑛 𝑢𝑛+1 − 𝑢𝑛 − PrΔ ( ) Δ𝑡 2 + (𝑈𝑛+1/2 ⋅ ∇) ( 𝑝𝑛+1 + 𝑝𝑛 𝑢𝑛+1 + 𝑢𝑛 )+( ) 2 2 − 𝜇ℎΔ𝑢𝑛+1 = Pr Ra 𝜁 ( 𝑇𝑛+1 + 𝑇𝑛 ) − 𝜇ℎΔ𝑢𝑛 , 2 ∇ ⋅ 𝑢𝑛+1 = 0, 𝑇𝑛+1 + 𝑇𝑛 𝑇𝑛+1 − 𝑇𝑛 − ∇ ⋅ (𝑘∇ ( )) Δ𝑡 2 + (𝑈𝑛+1/2 ⋅ ∇) ( 𝑇𝑛+1 + 𝑇𝑛 ) − 𝜇ℎΔ𝑇𝑛+1 2 = 𝛾𝑛+1/2 − 𝜇ℎΔ𝑇𝑛 , (2) where the time step Δ𝑡 > 0, the constant 𝜇 = 𝑂(1), and 𝑈𝑛+1/2 = (3/2)𝑢𝑛 − (1/2)𝑢𝑛−1 is the linear extrapolation of the velocity to 𝑡𝑛+1/2 from previous time levels. It is a three time levels scheme. Artificial viscosity stabilizations are introduced into the linear systems for 𝑢𝑛+1 and 𝑇𝑛+1 by adding −𝜇ℎΔ𝑢𝑛+1 and −𝜇ℎΔ𝑇𝑛+1 to the left-hand sides (LHS) and correcting them by −𝜇ℎΔ𝑢𝑛 and −𝜇ℎΔ𝑇𝑛 on the righthand sides (RHS), respectively. To the best of th (...truncated)