Modified characteristics projection finite element method for time-dependent conduction-convection problems

Transcription

1 Si and Wang Boundary Value Problems :151 DOI /s R E S E A R C H Open Access Modified caracteristics projection finite element metod for time-dependent conduction-convection problems Ziyong Si 1* and Yunxia Wang 2 * Correspondence: siziyong@pu.edu.cn 1 Scool of Matematics and Information Science, Henan Polytecnic University, Jiaozuo, 4543, P.R. Cina Full list of autor information is available at te end of te article Abstract In tis paper, we give a modified caracteristics projection finite element metod for te time-dependent conduction-convection problems, wic is gotten by combining te modified caracteristics finite element metod and te projection metod. Te stability and te error analysis sows tat our metod is stable and as optimal convergence order. In order to sow te effect of our metod, some numerical results are presented. From te numerical results, we can see tat te modified caracteristics projection finite element metod can simulate te fluid field, temperature field, and pressure field very well. MSC: 76M1; 65N12; 65N3; 35K61 Keywords: time-dependent conduction-convection problems; modified caracteristics metod; projection metod; stability analysis; error estimate 1 Introduction Te conduction-convection problem constitutes an important system of equations in atmosperic dynamics and dissipative nonlinear system of equations. Tere is a significant amount of literature as regards tis problem. An optimizing reduced Petrov-Galerkin least squares mixed finite element PLSMFE [1] metod for te non-stationary conductionconvection problems was given. An efficient sequential metod was developed to estimate te unknown boundary condition on te surface of a body from transient temperature measurements inside te solid [2]. An analysis of conduction natural convection conjugate eat transfer in te gap between concentric cylinders under solar irradiation [3]was carried out by Kim et al., Boland and Layton [4] gave an error analysis for finite element metods for steady natural convection problems. Newton type iterative metods [5 7] and defect-correction metods [8 1] for te conduction-convection equations were presented. Te projection metods, wic are efficient metods for solving te incompressible time-dependent fluid flow, were first introduced by Corin [11]and Temam[12]in te late 196s. Tis metod is based on a special time-discretization of te Navier-Stokes equations. In tis metod, te convection-diffusion and te incompressibility are dealt wit in two different sub-steps. Te velocity obtained in te convection-diffusion sub-step is projected in order to satisfy te weak incompressibility condition. Te projection metods 215 Si and Wang. Tis article is distributed under te terms of te Creative Commons Attribution 4. International License ttp://creativecommons.org/licenses/by/4./, wic permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to te original autors and te source, provide a link to te Creative Commons license, and indicate if canges were made.

2 Si and Wang Boundary Value Problems :151 Page 2 of 23 can be classified into tree families: te pressure-correction metod [13, 14], te velocitycorrection metod [15], andteconsistentsplittingsceme[16, 17], wic iscalleda gauge metod also [18]. Te convergence analysis of te semi-discrete projection metods can be found in Sen [19] and Guermond and Quartapelle [2]. In Guermond and Quartapelle [21], te projection metod was implemented by te finite element metod. It was used to solved te variable density Navier-Stokes equations in [22]. In [23], a gauge-uzawa projection metod was presented. Ten it was applied to te conduction-convection equations [24] and incompressible flows wit variable density [25]. As we know, te caracteristics metod is a igly effective metod for te advection dominated problems. Douglas and Russell [26] presented te modified metod of caracteristics first. It was extended to nonlinear coupled systems by Russell [27] intwoand tree spatial dimensions. A detailed analysis for te Navier-Stokes equations as been done by Dawson et al. [28] and numerical tests ave been presented by Buscagkia and Dari [29]. A second order MMOC mixed defect-correction finite element metod [3] for time-dependent Navier-Stokes problems was given. Notsu et al. gave a single-step caracteristics finite difference analysis for te convection-diffusion problems [31] and a single-step finite element metod for te incompressible Navier-Stokes equations [32]. El-Amrani and Seaid gave te error estimates of te modified metod of caracteristics finite element metods for te Navier-Stokes [33], natural, and mixed convection flows [34]. In [35], Acdou and Guermond gave te projection/lagrange-galerkin metod for te incompressible Navier-Stokes equations. In tis paper, we give te modified caracteristics projection finite element metod MCPFEM for te time-dependent conduction-convection problems, wic is gotten by combining te modified caracteristics finite element metod and te projection metod. We give stability and error analysis, wic sow tat our metod is stable and as optimal convergence order. In order to sow te efficiency of our metod, some numerical results are presented. At first, te numerical results of Bénard convection problems are given. Ten we give some numerical results of te termal driven cavity flow. From te numerical results, we can see tat MCPFEM can simulate te fluid field, temperature field, and pressure field very well. 2 Te modified caracteristics projection finite element metod for te time-dependent conduction-convection problems In tis paper, we consider te time-dependent conduction-convection problem in two dimensions wose coupled equations governing viscous incompressible flow and eat transfer for te incompressible fluid are Boussinesq approximations to te Navier-Stokes equations. For all t, t 1 ], find u, p, T X M W suc tat u t νu +u u + p = κν 2 gt + f, x, div u =, x, T t λνt + u T = b, x, ux,=u, Tx,=T, x, u =, T =, x, 1 were is a bounded domain in R 2 assumed to ave a Lipscitz continuous boundary. u =u 1 x, t, u 2 x, t T represents te velocity vector, px, t represents te pressure,

3 Si and Wang Boundary Value Problems :151 Page 3 of 23 Tx, t represents te temperature, κ represents te Grasoff number, λ = Pr 1, Pr is te Prandtl number, g represents te vector of gravitational acceleration, ν = Re 1, Re is te Reynolds number, and f and b are te forcing functions. In tis section, we aim to describe some notations and materials wic will be frequently used in tis paper. For te matematical setting of te conduction-convection problems 1, we introduce te Hilbert spaces X = H 1 2, W = H 1, } M = L 2 {ϕ = L 2 ; ϕ dx =. I is a quasi-uniform partition of into non-overlapping triangles, indexed by a parameter = max K I { K ; K = diamk}. We introduce te finite element subspace X X, M M, W W as follows: X = { v X C 2 ; v K P l K 2, K I }, M = { q M C ; q K P k K, K I }, W = { φ W C ; φ K P j K, K I }, were P l K is te space of piecewise polynomials of degree l on K, l 1, k 1, j 1 are tree integers. W = W H 1, and assume tat X, M satisfies te discrete LBB condition, tere exists β > suc tat dv, ϕ sup β ϕ, ϕ M, v X v were dv, ϕ = ϕ, v. Let V be te kernel of te discrete divergence operator, V = { v X ;q, v =, q M }. For eac positive integer N,let{J n :1 n N} be a partition of [, t 1 ]intosubintervals J n =t n 1, t n ], wit t n = n, = T 1 /N.Setu n = u, t n. Te caracteristic trace-back along te field u n 1 of a point x at time t n to t n 1 is approximately xx, t n 1 =x u n 1. Consequently, te yperbolic part in te first equation of 1attimet n is approximated by were u t + u n 1 u n un ū n 1, t ū n 1 = for any function w. { u n 1 x, x = x u n 1,, oterwise,

4 Si and Wang Boundary Value Problems :151 Page 4 of 23 Wit te previous notations, we get te projection FEM for te time-dependent conduction-convection problem 1, wic is a sligt transmutation of te projection FEM [13, 19] for te time-dependent Navier-Stokes equations. Algoritm 2.1 Projection FEM Start u as a solution of u, v =u, v andt, ψ = T, ψ, p =forallv V, ψ M. Step 1: Find û û X as te solution of u n, v + B u n, û, v + ν û, v = κν 2 gt n, v + f t, v, v X, were Bu, v, w = 1 2 u v, w 1 2 u w, v. Step 2: Find u V, p M as te solution of u û, v + d p, v =, v V, d q, u =, q M. 2 Step 3: Compute T W as te solution of te linear elliptic equation T T n, ψ + B u, T, ψ + λν T, ψ = bt, ψ, ψ W, 3 Bu, T, ψ = 1 2 u T, ψ 1 2 u ψ, T. Remark 2.1 Denote by P te ortogonal projector in L 2 2 onto V. We can readily ceck tat 2is equivalent to[19] u = P û. 4 Te MC time discretization, combined wit te projection finite element metod, leads to te following MC projection finite element metod. Algoritm 2.2 MC projection FEM Start wit u as a solution of u, v =u, v for all v V. Step 1: Find û were û X as te solution of u n, v + ν û, v = κν 2 gt n, v + f t, v, v V, 5 { u n = u n ẋ, ẋ = x un,, oterwise.

5 Si and Wang Boundary Value Problems :151 Page 5 of 23 Step 2: Find u V, p M as te solution of u û, v + b p, v =, v V, b q, u =, q M. 6 Step 3: Compute T W, te solution of te linear elliptic equation T Ṫ n, ψ + λν T, ψ = bt, ψ, ψ W, 7 were { Ṫ n = T nẋ, ẋ = x un,, oterwise. Remark 2.2 Define X x subset of W 1,, under te condition 1 2L n, L n = max 1 i n u n W t =x t tu n, t [t n 1, t ], 2 l N. SinceX is a 1, on te time step it is an easy matter to verify tat tis mapping as a positive Jacobian, since u n vanises on ; tis mapping is one-to-one and tis is a cange of variables from onto. Tis yields for any positive function φ on teestimatepleasesee[36] for details X φ t dx C φx dx. 3 Stability analysis Teorem 3.1 Stability If 1 2L n, L n = max 1 i n u i W 1,, te MC projection FEM is stable in te sense tat u N T N+1 2 N +2ν C u 2 + C T 2 + C 2ν u 2 N + λν T 2 N f t 2 N 1 +2C bt 1. Remark 3.1 We will prove te boundary of u n W 1, in te next section. Here, we use matematical induction metod. Proof Let v = u û u n, u Using 6, we deduce in 5, we obtain + ν û, u = κν 2 gt n, u û, v = u, v + dp, v, v X. Noting u =,weget + f t, u. û, u = u, u, 8

6 Si and Wang Boundary Value Problems :151 Page 6 of 23 û, u = u, u. 9 Ten we deduce u u n, u We arrive at + ν u u 2 u +2ν û u n 2 = κν2 gt n, u 2 2 u +2κν2 gt n, û + f t, u. +2 f t, û. 1 Now, we estimate te bound of u n 2 un 2. By te definition of X n x t n 1, we ave Hence, J X n x t n 1 = 1 x u n 1 1 yu n 1 1 det J X n x t n 1 =1+O. x u n yu n 1 2. Ten we get u n 2 un 2 = u dx u dx = u 1+O dx u dx. We ave u n 2 u n 2 C u n On te oter and, by Caucy-Scwarz inequality, we deduce 2κν 2 gt n, u 2Cκν 2 T n u Cκ 2 ν 4 T n 2 + C u Combining 1, 11, and 12, we get 2 u n 2 +2ν u 2 C u n 2 + f t 2 2ν 1 + 3ν u Cκ 2 ν 4 T + C u u Let ψ =2T in 7, we obtain 2 T Ṫ n, T +2λν T, T =2 bt, T. 14

7 Si and Wang Boundary Value Problems :151 Page 7 of 23 We deduce T 2 T n 2 + λν T 2 2 bt 1 + Ṫ n 2 T n 2. Similar to 11, we ave Ṫ n 2 T n 2 C u n 2. Ten we can get T 2 T + λν T 2 2 bt 1 + C u. 15 Adding 13and15, summing over all n from to N,wecanget u N T N+1 2 N +2ν u u 2 + T 2 N + C u + 2ν N f t Cκ2 ν 4 Using Gronwall lemma, we deduce 2 2 N + λν T 2 N T N +2 bt 1. u N T N+1 2 N +2ν C u 2 + C T 2 + C 2ν u 2 N + λν T 2 N f t 2 N 1 +2C bt 1. 4 Error analysis In order to get te error analysis, we give some lemmas first. Lemma 4.1 [37, 38] Let ex, n=[ un x ū n 1 x u t x, t n+u n x u n x] and let >be suc tat u C 4 [, T]; H 3 2. For t n >, we ave 1 d 2 gx n ex, n = 2 dt 2 were g n x t=ux t n tu n 1, t, u n x=ux, t n. Lemma 4.2 Let T n x Ṫ n 1 x ζ x, n= + u t ux, t n + O 2, 16 T t x, t u T, and let >be suc tat T C 4 [, T]; H 3. For t n >, we ave 1 d 2 γx n ζ x, n= 2 dt 2 + u t Tx, t n + O 2, were γ n x t=tx t n tu n 1, t, u n x=ux, t n.

8 Si and Wang Boundary Value Problems :151 Page 8 of 23 Lemma 4.3 Tere exists r : W W ; for all ψ W we ave ψ r ψ, φ =, φ W, 17 ψ r ψ dx =, r ψ ψ. 18 Wen ψ W r,q 1 q, we ave ψ r ψ s,q C r+s ψ r,q, 1 s m, r m Tere exists r : W W ; for all ψ W we ave ψ r ψ, φ =, φ W, r ψ ψ. 2 Wen ψ W r,q 1 q, we ave ψ r ψ s,q C r+s ψ r,q, 1 s m, r m Ten we define te Galerkin projection R, Q =R u, p, Q u, p : X, M X, M, suc tat νar u, v dq p, v +dϕ, R u=, u, p X, M, v, ϕ X, M. 22 Lemma 4.4 [39, 4] Te Galerkin projection R, Q satisfies R u + R u + Q p C k+1 ν u k+1 + p k, k =1, Error estimate for velocity and temperature Lemma 4.5 If 1 2L n, L n = max 1 i n u i W 1,, u, p, u t, and p t are sufficiently smoot, we ave u W 1, <+, ξ N+1 2 N + ξ n N 2 ν ˆξ + ξ N+1 + ˆξ 2 + ε N+1 N ε ξ n 2 N + λν ε 2 C 2 + 2k+1, 2 N + N ˆξ ε ξ n N 2 ν ξ n 2 N + λν ε ξ 2 + ε N C 2 + 2k+1,

9 Si and Wang Boundary Value Problems :151 Page 9 of 23 were ˆξ n = ûn Rn, ξ n = un Rn, ε = T r T, R n = R u n, p n, Cisapositiveconstant independent of and. Ṙ n Proof Subtracting R, v +ν R, v from bot sides of 5, we can get û R u n Ṙn = kν 2 g T n, v + f n R, v Defining η n = u n R n,wecanget, v + ν û R, v Ṙ n, v ν R, v. 24 ˆξ ξ n, v + ν ˆξ, v u ū n = ν u + p kν 2 gt f, v = η η n +, v + + ν u R u ū n u n ū n + ξ n + ξ n u n ū n, v + p ξ n, v + ξ n, v + kν 2 g T T n, v ν u + p kν 2 gt f, v, v, v u ū n = η η n +, v + + d Q p η η n, v +, v, v + ν u R, v + kν 2 g T T n, v ν u + p kν 2 gt f, v u n ū n ξ n, v + ξ n, v 25 + kν 2 g T T n, v. 26 Let v =2 ˆξ ˆξ i6, we can get 2 ξ + ˆξ ξ n 2 +2ν ˆξ u ū n = 2 +2 η η n, ˆξ +2 u n ū n, ˆξ + kν 2 g T T n, ˆξ 2 ν u + p kν 2 gt f, ˆξ +2 ξ n ξ n, ˆξ 5 A i, 27 i=1

10 Si and Wang Boundary Value Problems :151 Page 1 of 23 were u ū n A 1 = 2 A 2 =2 u n ū n, ˆξ, A 3 =2 η η n, ˆξ A 4 =2 u n ū n, ˆξ A 5 = kν 2 g T T n, ˆξ ν u + p kν 2 gt f f, ˆξ,, +2 ξ n ξ n, ˆξ,. Now, we estimate eac term A i, respectively. By Hölder inequality, we get A 1 C ē 2 + ν ˆξ By te definition of ẋ and x,wecanget ẋx, t n xx, t n = u n un. Using Taylor s formula, we obtain u n ū n = u n ẋ u n x u n u n un u n u n R n + R n u n. Terefore, we ave u n ū n u n u n R n + R n un C k+1 + ξ n. Ten we deduce A 2 C u n ū n ˆξ C 2k+1 + ξ n 2 ν + 8 ˆξ Now, we estimate te boundedness of A 3.Weave η η n = η η n dx = t n t η t n t x, θ dθ dx η t t η t x, θ dθ dx L 2 [t n,t ]:L 2. 3

11 Si and Wang Boundary Value Problems :151 Page 11 of 23 By te definition of ˆ Hence, J Xˆ x t n = det J ˆ Ten we get Xx Xx t n, we can get 1 x u n 1 1 yu n 1 1 x u n yu n 1 2 t n =1+O. η n η n 1 = sup v 1 η n ˆη n, v Let Gx=x ˆ v V. [ = sup v 1 η n xvx dx v V sup v 1 v V Xx vx v ˆ Xx + sup v V C 2 v 1 t n 1,ten Gx C,and t n 1 2 η n 1 x vx v ˆ X n x t n 1 dx t t n C 2 v 2. η n 1 zv Xˆ x n t n 1 dz. d dt v Xˆ x t 1 2 dt dx η n zv Xˆ x n t n 1 1+O t 2 ] dz Similarly, we ave v ˆ Xx Ten we deduce t n 1 v C. η n η n 1 C η n. 31 By 3and31, we ave η η n 1 C k+1 u n + C k Terefore, we get A 3 η η n 1 ˆξ C 2 2k+1 + C 2k+1 + ν ˆξ Similarly, we obtain A 4 C ξ + ν ˆξ 2 16.

12 Si and Wang Boundary Value Problems :151 Page 12 of 23 For te term A 5, by Taylor s formula, we can get T T n C,ten A 5 = kν 2 g T T n, ˆξ = kν 2 g T T n, ˆξ + kν 2 g T n T n, ˆξ Ck 2 ν Ck 2 ν 4 T n T n 2 + ν ξ Combining 27, 28, 2, and 34, we arrive at ˆξ 2 ξ n 2 + ˆξ ξ n ν ˆξ 2 + ν s 2 s n C 2 + 2k+1 + C ξ + Ck2 ν 4 T n T n Subtracting 1 r T r Ṫ n, ψ +λν r T, ψ from bot sides of 7, we can get T r T Ṫ n r Ṫ n r T r Ṫ n = T r T Ṫ n r Ṫ n =, ψ + λν T r T, ψ, ψ λν r T, ψ + bt, ψ 2, ψ + λν T r T, ψ + ζ t, ψ. 36 Letting ε = Ṫ r Ṫ, θ = T r T, θ n = Ṫ n r Ṫ n,andψ =2ε in 36, we can get 2 ε ε n, ε +2λν ε, ε =2 θ θ n, ε Similarly to 32, we get +2 ζ t, ε ε n ε n C ε n, θ θ n 1 C k+1 u n + C k ε n ε n, ε. Ten we deduce Namely, ε 2 ε + ε ε n C r+1 + C 3 + λν ε ε 2 ε n 2 + ε ε n +2λν ε 2 + C ε n 2 + λν ε C 2r+1 + C 3 + C ε n

13 Si and Wang Boundary Value Problems :151 Page 13 of 23 Adding 35to37, weget ξ 2 ξ + ξ ξ n ν ˆξ 2 + ε 2 ε + ε ε n + λν ε C 2r+1 + C 3 + C ξ + C ε. Summing over n from to N gives ˆξ N+1 2 N + ˆξ ξ n 2 N N 2 ν ˆξ 2 + ε N+1 + N ε ε n C 2 + 2k+1 + C By Gronwall lemma, we obtain ˆξ N N + N ˆξ ε Using 6, we get ρ N + λν ε 2 N ε + ξ. ξ n N 2 ν ˆξ ξ n 2 N + λν ε ξ ˆξ, v b p, v =. Let v =2ξ,wecanget ξ 2 ˆξ 2 + ξ ˆξ 2 =. Ten we ave ξ N N + ˆξ ξ n N 2 ν ˆξ ε N C 2 + 2k ε N+1 N ε ξ n 2 N + λν ε 2 C 2 + 2k+1. Using te inequality see [41], Remark 1.6 and [19], 2 P u C u

14 Si and Wang Boundary Value Problems :151 Page 14 of 23 we ave ξ N+1 2 N + + ˆξ ξ n N 2 ν ξ 2 + ε N+1 N ε ξ n 2 N + λν ε 2 C 2 + 2k Using te triangle inequality, we deduce u W 1, u R W 1, + R W 1,. Via te inverse inequality, v W 1, C 1 v see [36], we can get u W 1, C 1 u R + R. 2 We tus finis te proof. Teorem 4.6 Error estimates for te velocity and temperature If 1 2L n, u, p, u t, and p t are sufficiently smoot, we ave N u N+1 u N+1 2 C 2 + 2k+1, 39 1 N 2 ν u u 2 C 2 + 2k, 4 N T N+1 T N+1 2 C 2 + 2k+1, 41 1 N 2 ν T T 2 C 2 + 2k. 42 Proof Using triangle inequality, 23, and Lemma 4.3, we can get tis teorem. 4.2 Error estimates for te pressure Te following teorem on te pressure is a consequence of te previous teorem on te velocity. Teorem 4.7 Error estimate for pressure If 1 2L n, u, p, u t, and p t are sufficiently smoot, we ave for all 1 n N, p p C + k. Proof By 6, we deduce p p, v u ū n ξ = ν u + p kν 2 gt f ξ n, v, v

15 Si and Wang Boundary Value Problems :151 Page 15 of 23 ν ˆξ η η n, v +, v, v, v + kν 2 g T T n, v. u n ū n ξ n +, v + ξ n + ν u R By te LBB condition and Caucy-Scwarz inequality, we get p p u ū n ν u + p kν 2 gt f + C 1 ξ ξ n + ν ˆξ + C u n ū n + C η η n + C ξ n ξ n ν u R + kν 2 g T T n. Using 27, 28, 2, and 34, we arrive at p p C + k. Tus,wefinisteproof. 5 Numerical experiments In order to sow te effect of our metod, we give some numerical results in tis section. 5.1 Bénard convection problem Te first experiments is Bénard convection problem in te domain =[,5] [, 1] wit te forcing f =andb =.Figure1 displays te initial and boundary conditions for velocity u and temperature T. It means tat te boundary conditions for te velocity are te no-slip boundary condition u =on, termal insulation υ T = on te lateral boundaries, and a fixed temperature difference between top and bottom boundaries. Here, we coose = 1/16, =.1, and te finite element space is a Taylor-Hood finite element space. Here, we use te software package FreeFEM++ [42] for our program. First, we set κ =1 4, λ =1.,ν =1..Figure2 gives te numerical temperature at t =.5,.1,.15, and 1.. Figure 3 gives te numerical pressure at t =.5,.1,.15, and 1.. Figure 4 gives te numerical streamline at t =.5,.1,.15, and1.. Figure 1 Pysics model of Bénard convection problem.

16 Si and Wang Boundary Value Problems :151 Page 16 of 23 Figure 2 Numerical temperatures of Bénard convection problem wit κ =1 1 4 at different times. Figure 3 Numerical pressures of Bénard convection problem wit κ =1 1 4 at different times.

17 Si and Wang Boundary Value Problems :151 Page 17 of 23 Figure 4 Numerical streamlines of Bénard convection problem wit κ =1 1 4 at different times. Ten we set κ =1 5, λ =1.,ν =1..Figure5 gives te numerical temperature at t =.5,.1,.15, and 1.. Figure 6 gives te numerical pressure at t =.5,.1,.15, and 1. Figure 7 gives te numerical streamline at t =.5,.1,.15, and 1. Fromte numerical results, we can see tat MCPFEM can simulate te fluid field, temperature field and pressure field very well, and it works well for a ig Grasoff number κ. 5.2 Termal driven cavity flow problem Here, we consider te termal driven flow in an enclosed square =[,1] 2 wit te forcing f =andb =, and te initial and boundary conditions are given by Figure 8.Itmeans tat te boundary conditions for velocity is no-slip boundary condition u =on,and termal insulation υ T = on te top and bottom boundaries, and a fixed temperature difference between left and rigt boundaries. Here, we coose = 1/32, =1 4,andte finite element space is a Taylor-Hood finite element space. We coose λ =1,ν =1,κ =1 5 and 1 6 respectively. Figures 9 and 1 give te numerical results for κ =1 5 and 1 6, respectively. From te numerical results, we can see tat MCPFEM can simulate te fluid field, temperature field, and pressure field very well. Te numerical experiments confirm our teoretical analysis and demonstrate te efficiency of our metod.

18 Si and Wang Boundary Value Problems :151 Page 18 of 23 Figure 5 Numerical temperatures of Bénard convection problem wit κ =1 1 5 at different times. Figure 6 Numerical pressures of Bénard convection problem wit κ =1 1 5 at different times.

19 Si and Wang Boundary Value Problems :151 Page 19 of 23 Figure 7 Numerical streamline of Bénard convection problem wit κ =1 1 5 at different times. Figure 8 Pysics model of termal driven cavity flow.

20 Si and Wang Boundary Value Problems :151 Page 2 of 23 Figure 9 Numerical results of termal driven cavity flow wit κ =1 1 5 at different time, left panels te numerical streamlines, middle panels te numerical pressures, and rigt panels te numerical temperatures.

21 Si and Wang Boundary Value Problems :151 Page 21 of 23 Figure 1 Numerical results of termal driven cavity flow wit κ =1 1 6 at different times, left panels te numerical streamlines, middle panels te numerical pressures, and rigt panels te numerical temperatures.

22 Si and Wang Boundary Value Problems :151 Page 22 of 23 Competing interests Te autors declare tat tey ave no competing interests. Autors contributions Te autors contributed equally in tis article. Tey read and approved te final manuscript. Autor details 1 Scool of Matematics and Information Science, Henan Polytecnic University, Jiaozuo, 4543, P.R. Cina. 2 Scool of Materials Science and Engineering, Henan Polytecnic University, Jiaozuo, 4543, P.R. Cina. Acknowledgements Te autors would like to tank te anonymous referees for teir valuable suggestions and comments, wic elped to improve te quality of te paper. Tis work of te autors was supported in part by Cinese NSF Grant Nos and Received: 4 Marc 215 Accepted: 19 August 215 References 1. Luo, Z, Cen, J, Navon, IM, Zu, J: An optimizing reduced PLSMFE formulation for non-stationary conduction-convection problems. Int. J. Numer. Metods Fluids 6, Garcia, J, Cabeza, J, Rodriguez, A: Two-dimensional non-linear inverse eat conduction problem based on te singular value decomposition. Int. J. Term. Sci. 48, Kim, D, Coi, Y: Analysis of conduction-natural convection conjugate eat transfer in te gap between concentric cylinders under solar irradiation. Int. J. Term. Sci. 48, Boland, J, Layton, W: Error analysis for finite element metod for natural convection problems. Numer. Funct. Anal. Optim. 11, Si, ZY, He, YN: A coupled Newton iterative mixed finite element metod for stationary conduction-convection problems. Computing 89, Si, ZY, Sang, YQ, Zang, T: New one- and two-level Newton iterative mixed finite element metods for stationary conduction-convection problem. Finite Elem. Anal. Des. 47, Si, ZY, Zang, T, Wang, K: A Newton iterative sceme mixed finite element metod for stationary conduction-convection problems. Int. J. Comput. Fluid Dyn. 24, Si, ZY, He, YN: A defect-correction mixed finite element metod for stationary conduction-convection problems. Mat.Probl.Eng.211, Article ID doi:1.1155/211/ Si, ZY, He, YN, Wang, K: A defect-correction metod for unsteady conduction convection problems I: spatial discretization. Sci. Cina Mat. 54, Si, ZY, He, YN, Zang, T: A defect-correction metod for unsteady conduction-convection problems II: time discretization. J. Comput. Appl. Mat. 236, Corin, AJ: Numerical solution of te Navier-Stokes equations. Mat. Comput. 22, Temam, R: Sur l approximation de la solution des équations de Navier-Stokes par la métode des pas fractionnaires II. Arc. Ration. Mec. Anal. 33, Guermond, JL, Sen, J: On te error estimates of rotational pressure-correction projection metods. Mat. Comput. 73, Sun, HY, He, YN, Feng, XL: On error estimates of te pressure-correction projection metods for te time-dependent Navier-Stokes equations. Int. J. Numer. Anal. Model. 8, Guermond, JL, Sen, J: Velocity-correction projection metods for incompressible flows. SIAM J. Numer. Anal. 41, Guermond, JL, Sen, J: A new class of truly consistent splitting scemes for incompressible flows. J. Comput. Pys. 192, Sen, J, Yang, X: Error estimates for finite element approximations of consistent splitting scemes for incompressible flows. Discrete Contin. Dyn. Syst., Ser. B 8, E, W, Liu, JG: Gauge metod for viscous incompressible flows. Commun. Mat. Sci. 1, Sen, J: On error estimates of te projection metods for te Navier-Stokes equations: first-order scemes. SIAM J. Numer. Anal. 29, Guermond, JL, Quartapelle, L: On te approximation of te unsteady Navier-Stokes equations by finite element projection metods. Numer. Mat. 8, Guermond, JL, Quartapelle, L: Calculation of incompressible viscous flows by an unconditionally stable projection FEM. J. Comput. Pys. 132, Guermond, JL, Salgado, A: A splitting metod for incompressible flows wit variable density based on a pressure Poisson equation. J. Comput. Pys. 228, Nocetto, R, Pyo, J: Te gauge-uzawa finite element metod. Part I: te Navier-Stokes equations. SIAM J. Numer. Anal. 43, Nocetto, R, Pyo, JH: Te finite element gauge-uzawa metod. Part II: te Boussinesq equations. Mat. Models Metods Appl. Sci. 16, Pyo, J, Sen, J: Gauge-Uzawa metods for incompressible flows wit variable density. J. Comput. Pys. 221, Douglas, J Jr., Russell, TF: Numerical metods for convection-dominated diffusion problems based on combing te metod of caracteristics wit finite element or finite difference procedures. SIAM J. Numer. Anal. 19, Russell, TF: Time stepping along caracteristics wit incomplete iteration for a Galerkin approximation of miscible displacement in porous media. SIAM J. Numer. Anal. 22, Dawson, CN, Russell, TF, Weeler, MF: Some improved error estimates for te modified metod of caracteristics. SIAM J. Numer. Anal. 26,

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