A Stabilized Galerkin Scheme for the Convection-Diffusion-Reaction Equations

Transcription

1 Acta Appl Mat 14) 13: DOI 1.17/s A Stabilized Galerkin Sceme for te Convection-Diffusion-Reaction Equations Qingfang Liu Yanren Hou Lei Ding Qingcang Liu Received: 5 October 1 / Accepted: 5 July 13 / Publised online: 3 August 13 Springer Science+Business Media Dordrect 13 Abstract A fully discrete stabilized sceme is proposed for solving te time-dependent convection-diffusion-reaction equations. A time derivative term results in our stabilized algoritm. Te finite element metod for spatial discretization and te backward Euler or Crank-Nicolson sceme for time discretization are employed. Te long-time stability and convergence are establised in tis article. Finally, some numerical experiments are provided to confirm te teoretical analysis. Keywords Stabilized metod Finite element metod Convection-diffusion-reaction equations Stability Error estimates Subsidized by te Fundamental Researc Funds for te Central Universities Grant Nos and ), NSF of Cina Grant Nos and 11154) and te P.D. Programs Foundation of Ministry of Education of Cina Grant No ). Q. Liu Y. Hou B) Scool of Matematics and Statistics, Xi an Jiaotong University, Xi an 7149, P.R. Cina yrou@mail.xjtu.edu.cn Q. Liu qfliu1@mail.xjtu.edu.cn Q. Liu Y. Hou Center for Computational Geosciences, Xi an Jiaotong University, Xi an 7149, P.R. Cina L. Ding Scool of Electronic and Information Engineering, Xi an Jiaotong University, Xi an 7149, P.R. Cina dinglei@stu.xjtu.edu.cn Q. Liu Scool of Mecanics and Civil & Arcitecture, Nortwestern Polytecnical University, Xi an 7119, P.R. Cina qcliu8@163.com

2 116 Q. Liu et al. 1 Introduction We consider te time-dependent convection-diffusion-reaction equation u t u + b u + cu = f, x, t) Ω [,T], ux, ) = u, x Ω, ux, t) =, x, t) Ω [,T]. 1.1) Here Ω is a bounded domain in R m m = 1,, 3) wit a Lipscitz continuous boundary Ω, ux, t) is a scalar function representing some quality of a fluid flow suc as temperature or contaminant level, is a positive constant wic is called te diffusion coefficient, bx,t) L,T; L Ω)) m and cx,t) L,T; L Ω)) are given functions, f L,T; L Ω)) is a forcing function and T represents a finite time. During te last two decades, various numerical metods for solving te convectiondiffusion-reaction type equations ave been studied e.g., see te work of Codina [1], Huges et al. [ 4], Jon and Kaya [5, 6], Layton et al. [7 14], Guermond [15, 16], Heitmann [17], Santos and Almeida [18], Davis and Palevani [19]). In particular, a large number of works ave been devoted to te researc of stabilized metods see Codina [1] for a survey of some most popular metods). An alternative stabilized tecnique is te so-called artificial viscosity metods AV) wic add a suitable artificial viscosity term as a stabilized factor. For a given mes scale >, we define a finite element subspace V, ten te semi-discrete direct AV metod reads: find u t) V,forallt [,T] suc tat u t,v)+ + σ a ) u, v) + b u,v)+ cu,v)= f, v), v V, 1.) were σ a is a stabilization parameter and, ) denotes te usual L scalar product. However, 1.) is not identical to te system 1.1) because of te adding of σ a. Terefore, te most straigtforward artificial viscosity metod as been somewat less popular due to its poor performance. Recently, te progress in variational multiscale metods VMS) ave re-surged people s interests. Te VMS are originated by Huges et al. [ 4]. Tis metod acts only on te smallest refined mes scales modeled by extra bubble function degrees of freedom for finite element discretization. Te basic idea of VMS [ 7] is tat it introduces a classical Galerkin formulation involving an additional stabilization term in wic te coarse-scale residual is weigted. Tat is to find u t) V for all t [,T] suc tat for v V u t,v ) + u, v ) + b u,v ) + cu,v ) + LGLu,v ) = f, v) + LGf, v), 1.3) were Lu = u t u +b u +cu, G : V ˆV is a fine-scale Green function, V is te dual space of V and ˆV contains te unresolved fine scales. Huges and Sangalli [4]discussed some metodologies to determine te fine-scale Green function G and teir connection wit te streamline-upwind Petrov-Galerkin metod SUPG) and Residual-free Bubble RFB). Te VMS type metods ave become a ot researc topic recently. For instance, Layton et al. [7] applied VMS to te convection-diffusion-reaction equations and te more complex Navier-Stokes equations see te experiments in Jon and Kaya [5] and te analysis in Jon and Kaya [6]). As in te explanation of Kaya and Layton [8], some oter stabilized metods are essentially similar metods to te VMS, e.g., te subgrid eddy viscosity model inspired by Layton et al. [8], Guermond [15, 16] and Heitmann [17] and time relaxation tecnique introduced by Layton et al. [1, 11], tese metods sare te same idea as VMS tat adding an artificial viscosity term only for te iger frequency components.

3 A Stabilized Galerkin Sceme for te Convection-Diffusion-Reaction 117 In tis paper, we consider te stabilized sceme 3.1)and3.) for solving te convectiondominated convection-diffusion-reaction equations. Te stabilization tecnique used in tis paper is well-known. A similar stabilization can be found in recent book of Layton [14]. Moreover, Layton et al. [1] presented te same stabilization for te combination of forward and backward Euler sceme. Lately, Layton et al. [13] developed a second-order discretization for te Navier-Stokes equations plus applications to turbulence. A more accurate time discretization for te Navier-Stokes equations is investigated by Davis and Palevani [19]. HouandLiu[] proposed a stabilized semi-implicit Galerkin sceme for te evolutionary Navier-Stokes equations. Tis type of relaxation can weaken te fluctuations of iger frequency components, wic drives te iger frequency components decay rapidly witout destroying te order of accuracy of te lower frequency components. In fully discrete case, suc relaxation can ensure tat te fully discrete sceme eventually tends to te exact semidiscrete convection-diffusion-reaction equations. Te stabilization term or te so-called time relaxation term in tis article is actually te time derivative term rater tan te viscosity term like tat in VMS. Since te artificial viscosity term in te VMS metod is a nonzero term even wen te time step lengt tends to zero, its influence to te spatial accuracy of te approximate solution is unavoidable. Moreover, two more important points in regard to tis paper remain to be explained. One point is tat all te above mentioned examples only consider tis metod for solving te ODE system see Layton et al. [1]) and te Navier-Stokes equations see Layton et al. [13]). As far as we know, tere are no articles studying tis metod for solving te convection-dominant convection-diffusion-reaction equations. Te oter point is tat all te above mentioned metods only consider te local boundedness of te numerical solutions in te stability analysis. In tis paper, we are interested in te global boundedness of our stabilized metod, wic makes tis paper especially interesting to some extent. Tis paper is organized as follows: Sect. gives some notations frequently used in te rest of tis paper and recalls some classical regularity properties of te exact solution of te convection-diffusion-reaction equation 1.1). Section 3 presents our stabilized sceme and establises te stability results of tis stabilized metod. Te error estimates are investigated in Sect. 4. Some numerical tests are presented in Sect. 5 to support te previous analysis. Finally, we conclude in Sect. 6. Notations and Preliminaries We denote te functional spaces troug te following notations X = L Ω), V = H 1 Ω). For convenience of later analysis, we introduce te bilinear form on V V au,v) = Au, v V = u, v), u, v V were A is a linear continuous operator from V to V. Te variational formulation of 1.1) reads: find u V,for t [,T] suc tat u t,v)+ au,v) + b u, v) + cu, v) = f, v), v V, u) = u..1)

4 118 Q. Liu et al. Furtermore, we will assume tat tere exist positive constants M, κ 1, κ suc tat te exact solution of te convection-diffusion-reaction equation 1.1) satisfies te following inequalities ut) M, utt t) 1 κ 1, ut t) 1 κ..) Trougout tis paper, we use α to denote te H α Ω) norm for α R and always use κ to denote a generic positive constant depending only on te data, Ω, f, n, M, κ 1, κ ). Let >andk be a partition of Ω into triangles K or quadrilaterals K,assumedtobe uniformly regular as tending to. Te finite element subspace V is caracterized by tis partition. Let Π : V V denote te L ortogonal projection defined by Π v,v ) = v, v ), v V,v V. It is necessary to define a discrete analogue A = Π troug te condition tat u,v ) = u, v ), v V. Hereinafter, we will make te following common assumption. v Π v + v Π v 1 C v, v H Ω). We also assume tat tere is a constant μ > suc tat c 1 ) b x, t) μ >..3) 3 Stabilized Sceme and Its Stability Let u = Π u, we define te backward Euler time-stepping metod to find u all n = 1,...,J 1, J =[T/k], suc tat for v V V for dt u,v) + σ 1 a u u n,v) + a u,v) + b u,v) + c u,v) = f,v ). 3.1) were k is te time step lengt and d t u = 1 u u n k ), f = ft ), b = bt ), c = ct ). We also introduce te Crank-Nicolson sceme in time to find u V by setting dt u,v) + σ a u u n,v) + a ū,v) + b ū,v) + c ū,v) = f,v ) 3.) for v V were ū = 1 ) u + u n, f = 1 ft ) + ft n ) ),

5 A Stabilized Galerkin Sceme for te Convection-Diffusion-Reaction 119 t n = nk, σ 1 and σ are te stabilization parameter. For convenience, we use te unified notation σ to denote te stabilization parameter wic represents σ 1 or σ wen we discuss 3.1) or 3.) respectively. In fact, 3.1) will become te classical standard Galerkin sceme and 3.) is te classical Crank-Nicolson sceme wen σ =. It is easy to find tat for a given u n,v)will tend to zero wen te time step lengt k, wic confirms wat we said in te introduction. In tese stabilized scemes, an additional term involving an approximation to te Laplacian of te time derivative of te unknown is added to improve te stability of te resulting sceme. Te remaining part of tis paper is devoted to studying te stability and convergence of our stabilized algoritm 3.1) and3.). First of all, let us sow some discrete Gronwall lemmasusedin[1 3]. σ>, σau Lemma 3.1 Let d n be a positive sequence satisfying n, αd βd n γ, were α, β, γ are tree positive constants wit α β. Ten n, d n Lemma 3. Let d n, g n, p n be tree series satisfying ) β n d γ ) + γ α α β α β. d d n k g n d n + p n, n>. Ten, n >, ) N N N d n d exp k g i + k p i exp k g j ), n n + 1. i= i= Lemma 3.3 Let d n, g n, p n be tree series satisfying and wit kn = ξ. Ten d d n k j=i g n d n + p n, n n, N+i 1 N+i 1 N+i 1 k d n α, k g n β, k p n γ, i n, n=i n=i n=i d n β + γ ) exp α), n n + N. ξ Lemma 3.4 Let k, B and d n, g n, p n, q n be four series satisfying d n + k n n 1 g i k p i d i + k i= i= n q i + B, n. i=

6 1 Q. Liu et al. Ten ) n n 1 n d n + k g i exp k p i )k q i + B, n. i= i= i= Te following teorem gives te L -stability of te first-order time discretization sceme 3.1). Teorem 3.1 Let u X, ten tere exists a positive constant M suc tat te numerical solution u n of 3.1) satisfies u n + kσ u n M, n, provided tat kσ4μ + λ 1 ) were λ 1 is te smallest eigenvalue of operator A. Proof Taking v = ku we ave u in 3.1) and noticing te equality α β,α) = α + α β β, + u u n u n + kσ u + kσ u u n kσ u n + k u + k ) b u,u + k c u = k ) f,u.,u ) 3.3) Tanks to Young s inequality and.3), we obtain k ) ) b u,u + k c u,u kμu, k ) f,u k f 1 u k u + k f. 1 Combining te above estimates wit 3.3), we ave 1 + kμ ) u + kσ u + k u u n + kσ u n + k f ) Noticing tat we ave u λ 1 u, 1 + kμ + kλ ) 1 u σ kσ ) u 1 + kμ + kλ 1 u n + kσ u n + k f. 1

7 A Stabilized Galerkin Sceme for te Convection-Diffusion-Reaction 11 Te usage of te condition kσ4μ + λ 1 ) yields 1 + kμ + kλ ) 1 u + kσ u ) u n + kσ u n + k f ) Finally, using Lemma 3.1 to 3.5) gives te prescribed conclusion of tis teorem. Remark 3.1 From te above teorem, we know tat te stabilized parameter sould satisfy te following inequality kσ4μ + λ 1 ) 3.6) to enance te stability, wic gives a clue about ow to coose te stabilization parameter σ. It is wort remarking tat in order to obtain te global boundedness of te approximate solution in Teorem 3.1, te time step k and stabilization parameter σ sould satisfy 3.6). In fact, tis condition can be removed if we only want to obtain te local boundedness of te approximate solution. Inspection of te proof sows tat everyting after 3.4) is not necessary. Indeed from 3.4), it easily follows tat u + kσ u un + kσ u n + k f n 1 /. Tus te proof is obtained witout te bound on σ and even witout invoking te spectral estimate of A ). For a given positive constant ξ>andkn = ξ, wededucefrom3.4) tat k n +N 1 i=n u i M + ξ f = β 1 3,1, n ) Te following teorem presents te H 1 -stability of te first-order time discretization sceme 3.1). Teorem 3. Let u V be given, ten tere exists a constant M 1 suc tat te solution u n of 3.1) satisfies u n + kσ A u n M 1, n. provided k b L,T ;L Ω)) m + c L,T ;L Ω)) ) /. Proof We take v = ka u in 3.1), obtaining u + u u n u n + kσ A u + kσ A u A u n kσ A u n + k A u + k b u,a ) u + k c u,a ) ) u = k f,a u. 3.8) In view of Young s inequality and.3), tere olds k b u,a ) u + k c u,a ) u kμ u, k ) f,a u k f A u k A u + k f.

8 1 Q. Liu et al. Combining te above inequalities wit 3.8) admits k A u + kσ A u u n + kσ A u n + k f. For a given positive constant ξ>andkn = ξ, multiplying te above inequality by kσ 1 and summing for n = n 1,...,n + N, we obtain n +N 1 k i=n kσ A u i kσ n +N i=n 1 u i + kσ kσ A u n 1 + krσ f, n. Noticing 3.7) and te ypoteses of tis teorem, we obtain n +N 1 k i=n kσ A u i σ β 3,1 + κσ + krσ f = β 3,, n. Combining 3.7) wit te above inequality, tere olds k n +N 1 i=n u i + kσ A u i ) β3,1 + β 3, = β3, n. Let us give anoter estimate about te term kb u,a u ) + kc u, A u ) k b u,a ) u kb L u u n + ) u n A u k A u + k b L u n + u u n ), k c u,a ) u k c L u u n + ) u n A u k A u + k c L ). u n u n + u It follows from 3.8), te above estimates and te ypoteses in tis teorem tat u + kσ A u u n + kσ A u n + k b L + c L ) u n + k f. We set d n = u n + kσ A u n, g n = bn L + cn L ), p n = f n,

9 A Stabilized Galerkin Sceme for te Convection-Diffusion-Reaction 13 n +N 1 r bn L k g i = + cn L ) n +N 1 = β 1, k p i = ξ f n = β, n. i=n For n< + N, by using Lemma 3., tere exists a positive constant γ = u + kσ A u ) expβ 1) + β expβ 1 ) suc tat u n + kσ A u n γ, n< + N. For n + N, by using Lemma 3.3, tere exists a positive constant η = β + β 3 ξ ) expβ 1 ) suc tat u n + kσ A u n η, n + N. Coosing M 1 = maxγ, η), we will complete te proof of tis teorem. Te following teorem gives te stability of te second-order time discretization sceme 3.). Teorem 3.3 Tere exists a positive constant κ> suc tat te solution u n of 3.) is bounded u n + kσ n u n + k ū i κ, n J. i= Proof Taking v = kū in 3.), obtaining u u n + kσ u kσ u n + k ū + k b ū, ) ū + k c ū, ) ) ū = k f, ū. 3.9) Due to Young s inequality and.3), we ave k b ū, ) ū + k cū, ) ū kμ ū, k ) f, ū k f ū 1 k ū + k f. 1 Combining te above inequalities wit 3.9) implies u + kσ u + kμ ū + k ū i=n u n + kσ u n + k f. 1 Summing te above inequality, tere olds u + kσ u + kμ ūi + k ū i i=1 u + kσ u + k f i, n<j. 3.1) 1 Applying Lemma 3.4 to 3.1) yields te prescribed conclusion. i=1 i=1

10 14 Q. Liu et al. 4 Error Estimates For any given nonnegative integer n J, time step lengt k>, we set t n = nk, U n = Π ut n ), e n = U n un, en = U n un. We apply Π to.1) and rewrite it at t = t for all n<j, obtaining dt U,v ) + a U,v ) + b U,v ) + c U,v ) = f,v ) + r 1,v ) 4.1) or dt U,v ) + a Ū,v ) + b Ū,v ) + c Ū,v ) = f,v ) + r,v ) 4.) were t r 1 = 1 Π ut t) u t t ) ) dt, k t n r = 1 t Π ut t) ū t t ) ) dt = 1 k t n k t t n t t )t t n )Π u ttt t)dt, were ū t t ) = 1 u tt n ) + u t t )). Tanks to.), we ave r 1 κ k, 1 r 1 κ 1 k. Teorem 4.1 Under te conditions of Teorem 3., te following estimate olds ut n ) u n + kσ n e n + k were u n is te solution of 3.1). i= e i κ k + 4), n J, Proof Subtracting 3.1) from 4.1), we ave dt e,v) + σa e,v) + a e,v) + b e,v) + c e,v) = σa e n,v) + r 1,v ) + σa U U n,v), v V. Taking v = ke e in te above equality, we obtain + e e n e n + kσ e + kσ e e n kσ e n + k e + k ) ) b e,e + k c e,e = k ) ) r 1,e + kσa U U n,e. 4.3) By virtue of.),.3) and Young s inequality, we ave k b e,e ) + k c e,e ) kμ e,

11 A Stabilized Galerkin Sceme for te Convection-Diffusion-Reaction 15 k ) r 1,e k r 1 1 e k e + κ 1 k3, kσ ) a U U n,e kσ U U n e k e + σ κ k 3. Combining te above estimates wit 4.3), we ave 1 + kμ ) e + kσ e + k e e n + kσ e n + κ 1 + σ κ ) k 3. Summing te above inequality and noticing e = admits e + kσ e + kμ i=1 e i + k i=1 e i κk, n<j. Te combination of te above analysis and te relation tat I Π )ut n ) κm for all n J allows us to conclude te results of tis teorem. In te following analysis, we will devote to te error estimates of te Crank-Nicolson time discretization sceme 3.), wic will use te following lemma. Lemma 4.1 For te solution of 3.), te following estimate olds e n + kσ n e n + k i= provided tat σ k. Here ē i = 1 ei 1 + ei ). ē i κk4, n J, Proof Subtracting 3.) from 4.), we get dt e,v) + kσa e,v) + ka ē,v) + k b ē,v) + c ē,v) = kσa e n,v) + r,v ) + kσa U U n,v), v V. Taking v = kē in te above equality, tere olds e e n + kσ e kσ e n + k ē + ) ) c ē,e = k r, ē + kσa U U n, ē Tanks to.)and.3), te following estimates old k b ē, ē ) + c ē, ē k ) r, ē k r 1 ē k ) kμ ē, + k b ē ē, ē ). 4.4) + κ 1 k5, )

12 16 Q. Liu et al. kσ a U U n, ē ) kσ U U n ē k ē + κ σ k 3, Combining te above estimates wit 4.4) and noticing tat σ k,weave e + kσ e + kμ ē + k ē e n + kσ e n + κ 1 + κ ) k 5. Furtermore, summing above inequality, we find e + kσ e + kμ ēi + k ē i Tκ 1 + κ ) k 4, i=1 i=1 n<j. 4.5) Applying Gronwall Lemma 3.4 to 4.5), ten proof ends. Te combination of Lemma 4.1 wit I Π )ut n ) κm for all n J allows us to conclude te following error estimate. Teorem 4. For te solution of 3.), tere olds ut n ) u n + kσ n e n + k i= ē i κ k 4 + 4), n J. Remark 4.1 For te stability analysis in Teorem 3.1 and Teorem 3., we obtain te longtime stability of te approximate solution. For te convergence analysis in tis section, we don t prove te long-time convergence since Gronwall Lemma 3.4 is used wic is a tool to prove some local properties. For te long-time convergence, it s possible to be proved under some assumptions. Te interested readers can refer to te Heywood and Rannacer s work [4] to study te long-time convergence of tis sceme. Remark 4. Altoug for simplicity, we ave developed te stability and convergence for omogeneous boundary conditions. However, our analysis can be modified to cover te nonomogeneous boundary conditions wic is a similar argument. Te numerical tests in te next section ave no omogeneous boundary conditions, but te above results are also valid. 5 Numerical tests In tis section we present four numerical examples: one is a known analytical solution and te oters are tree model problems wit boundary layers. In te following simulations, we divide te domain Ω =[, 1] into triangles, wic are induced by te set of nodes i/m, j/m), i, j M, werem = Ω / is a positive integer. All te numerical experiments are carried out by using linear elements i.e., P 1 finite element). Te same metod can be applied to iger order finite element i.e., P I I>1) finite element). Moreover, we

13 A Stabilized Galerkin Sceme for te Convection-Diffusion-Reaction 17 Fig. 1 Mes1left)and Mesrigt) present te numerical results of te stabilized Crank-Nicolson sceme 3.). Te first-order backward Euler time discretization sceme 3.1) as te same numerical results. For simplicity, we consider te two-dimensional case and a similar idea can be applied in a straigtforward way to te tree-dimensional convection-diffusion-reaction problems. 5.1 Example 1 D Problem wit a Known Analytical Solution As te first example, we study te numerical performance in te simplest form. Te numerical studies are implemented wit te prescribed solution ux, t) = t cos ) x 1 x wit x = x 1,x ), b =, 1), c = 1, ux, ) = andt = 1. Te non-omogeneous Diriclet boundary conditions and te rigt and side f are cosen suc tat ux, t) fulfills 1.1). Moreover, since triangular elements are used, it is necessary to consider te dependence of te solution wit te mes direction. Figure 1 presents two meses wit different mes direction. In te following numerical experiments, we will compare te accuracy and te CPU time of our stabilized Crank-Nicolson metod CNStab) wit some oter numerical scemes, for example te classical Crank-Nicolson metod CN) wic is te case tat σ = in3.) and SUPG. Te SUPG wit Crank-Nicolson sceme for time discretization reads see [1]): find u V suc tat dt u,v ) + a ū,v ) + b ū,v ) + c ū,v ) + σ s dt u ū + b ū + c ū,b v ) = f,v ) + σ s f,b v ), v V were σ s is te stabilization parameter. Table 1 sows te results of te CN, CNStab and SUPG for various Peclet number Pe = b L,T ;L Ω)) m by using Mes 1 see Fig. 1). We say tat te metod is divergent wen te error value is greater tan 1. d.at means divergent at. We find tat CN loses stability wen Pe > 3.5. At tis moment, te stabilized metods, CNStab and SUPG, do stabilize te evolution of te iger frequency term. However, to our knowledge, te coice of stabilization parameter in SUPG is quite involved. Table sows te results of te CN, CNStab and SUPG for various Peclet number Pe byusingmesseefig.1). Table as te same results as te Table 1, wic verifies te independence of te solution wit te mes direction to some extent. Moreover, for Pe = 35 wic allows to verify

14 18 Q. Liu et al. Table 1 Results of CN, CNStab and SUPG wit different Peclet number Pe and stabilization parameter σ by using Mes 1 Metod Pe σ e L,T ;X) e L,T ;V) Time Pe σ e L,T ;X) e L,T ;V) Time CN s 35 d.at =.3 d.at = s CNStab s s SUPG s s CN 3.5 d.at =.48 d.at = s 35 d.at =.5 d.at = s CNStab s s SUPG s s Table Results of CN, CNStab and SUPG wit different Peclet number Pe and stabilization parameter σ by using Mes Metod Pe σ e L,T ;X) e L,T ;V) Time Pe σ e L,T ;X) e L,T ;V) Time CN s 35 d.at =.38 d.at = s CNStab s s SUPG s s CN 3.5 d.at =.46 d.at = s 35 d.at =.7 d.at = s CNStab s s SUPG s s Table 3 Convergence performance of CNStab and SUPG wit varying spacing but fixed time step k =.1 wen Peclet number Pe = 35 Metod σ e L,T ;X) Order e L,T ;V) Order CNStab 1/ SUPG CNStab 1/ SUPG CNStab 1/ SUPG CNStab 1/ SUPG tat = 1 4, te condition 3.6) kσ4μ + λ 1 ) defined in Teorem 3.1 will require σ 1. Te coices of σ in Table 1 are in good agreement wit our analysis in te previous section. Table 3 sows te relative errors and convergence rates of CNStab and SUPG scemes wit different space discretization size for Pe = 35 CN can t converge wen Pe = 35). To examine te convergence order wit respect to te spacing, we fix te time step k =.1 and varying spacing = 1/, 1/4, 1/8, 1/16 by a mid-point mes refinement. We coose te time step k small enoug to ensure tat te dominant error in te computa-

15 A Stabilized Galerkin Sceme for te Convection-Diffusion-Reaction 19 Table 4 Convergence performance of CNStab, SUPG and VMS wit varying time step k but fixed time step = 1/3 wen Peclet number Pe = 35 Metod k σ u,k u,k/ ρ k,l u,k u,k/ 1 ρ k,h 1 CNStab 1/ e SUPG e CNStab 1/ e e SUPG e e CNStab 1/ e e SUPG e e CNStab 1/ e e SUPG e e CNStab 1/ e e-7 SUPG e e-7 tions is te space discretization error. Table 3 presents te convergence order of CNStab and SUPG scemes and sows tat te CNStab sceme can get te same convergence rate as te SUPG. To obtain te convergence order wit respect to te time step k, we follow te metod in [5]. Because te approximate errors Ok γ ) + O μ ), we assume tat v,k x, t m ) vx,t m ) + C 1 x, t m )k γ + C x, t m ) μ. And define ρ k,h s = u,k u,k/ s u,k/ u,k/4 s 4γ γ γ 1 γ. In particular, ρ k,h s 4 for te corresponding convergence rate in time is of Ok ).Tocompute ρ k,h s, we fix te space discretization size = 1/3 and vary te time discretization size k = 1/8, 1/16, 1/3, 1/64, 1/18. Table 4 sows te relative errors and a set of ρ k,l and ρ k,h 1 using CNStab and SUPG scemes wit varying time step k wen Pe = 35. Tis table clearly suggests tat te concerned convergence orders of SUPG and CNStab in time are all of Ok ). 5. Example D Problem wit Internal and External Layers Te second issue to be considered ere is a two-dimensional convection-dominated problem wit te diffusion coefficient = 1 4, b = 1., 1.), c = 1andf =. Te initial condition is ux, ) = and boundary conditions are imposed tat u = atx =, x = 1, y = 1and y =, u = ifx.3 andu = 1if.3 x 1.. Tis example as been widely studied in some articles, for example, Santos and Almeida [18] presented a nonlinear subgrid metod for te advection-diffusion problems under tis conditions. It is well known tat tese conditions yield a solution wit two plateaus, an interior layer in te direction of te advection starting at.3, ) and an exponential external layer at x = 1.

16 13 Q. Liu et al. Fig. Example : Results left) and contours rigt) of te solutions using CN, CNStab and SUPG Te CN, CNStab and SUPG solutions and te corresponding contours of tese metods are exibited in Fig.. Te left parts are te solutions and te rigt parts are te corresponding contours of te solutions. We can see tat some spurious oscillations appear in te CN algoritm. Wile CNStab and SUPG can damp te spurious oscillations properly. Furtermore, as presented in [18], SUPG solution sows some local oscillations in te neigborood of te external layer. Our CNStab metod can preserve te same beavior as SUPG. Tis beavior is well sown in Fig..

17 A Stabilized Galerkin Sceme for te Convection-Diffusion-Reaction 131 Fig. 3 Example 3: Results left) and contours rigt) of te solutions using CN, CNStab and SUPG 5.3 Example 3 D Problem wit Source Term f = 1 Te tird issue to be considered ere is a two-dimensional convection-dominated problem wit te diffusion coefficient = 1 4, b =., ), c = 1 and a constant source term f = 1. We set te initial condition u = andu = on all te boundary so tat te exact solution is a45 slope, possessing parabolic layers at y =, y = 1 and exponential layer at x = 1. Te CN solution presents many oscillations as seen in Fig. 3. Te CNStab and te SUPG solutions present some local oscillations in te neigborood of te layers at y =, y = 1 and x = 1 wic can exibit te rigt beavior of te teoretical solution.

18 13 Q. Liu et al. Fig. 4 Example 4: Snapsots of CN left) andcnstabrigt) solutions at t =.1,.3, Example 4 D Problem wit Semicircular Internal Layers To verify te influence of te stabilization procedure along time, te last model is a two-dimensional convection-diffusion-reaction problem wit b = 1y,5 1x) and te Diriclet boundary conditions are imposed tat u = ify = and x<.15 or.375 <x.5, u = 1ify = and.15 x.375, u = ifx = ory = 1and

19 A Stabilized Galerkin Sceme for te Convection-Diffusion-Reaction x 1. Gunzburger et al. [6] as sown tat suc conditions will yield a solution wit two semicircular internal layers. Figures 4 sows a few snapsots of CN and CNStab solutions wit Pe = 35.Wesee tat CN solutions begin to diverge at t =.3 seefig.4c)). From tese figures, we see tat CNStab solutions can perform well on suppressing te spurious oscillations. 6 Conclusion A fully discrete stabilized sceme for te time-dependent convection-diffusion-reaction equations is studied in tis article. Tis is a well-known stabilization tecnique. Numerical analysis and some numerical examples in te specific context of te two-dimensional convection-diffusion-reaction equations are closely concerned in tis article. Concretely, we obtain te long-time stability results for any time T>. Error analysis is presented and numerical experiments are conducted to demonstrate te computational effectiveness of our stabilized metod. We give four numerical examples to compare te performance of our stabilized metod wit CN and SUPG. Te first consideration is to ceck tat te teoretical proven results in previous sections are also obtained numerically. Te numerical tests indicate tat our stabilized metod can get te teoretical convergence orders bot in space and in time, wic sare te same convergence rates as SUPG. Te second and tird issues considered are te bencmark problem wit internal, external, parabolic and exponential layers. Te numerical results can exibit te rigt beavior of te teoretical solution. Finally, we consider a solution wit yielding two semicircular internal layers to verify te influence of te stabilization procedure along time. Acknowledgements Te autors would like to tank te editor and te anonymous referees for teir elpful comments and suggestions, wic lead to substantial improvements of tis presentation. References 1. Codina, R.: Comparison of some finite element metods for solving te diffusion-convection-reaction equation. Comput. Metods Appl. Mec. Eng. 156, ). Huges, T.: Multiscale penomena: Green s functions, te Diriclet-to-Neumann formulation, subgrid scale models, bubbles and te origins of stabilized metods. Comput. Metods Appl. Mec. Eng. 17, ) 3. Huges, T., Mazzei, L., Jansen, K.: Large eddy simulation and te variational multiscale metod. Comput.Vis.Sci.3, ) 4. Huges, T., Sangalli, G.: Variational multiscale analysis: te fine-scale Green s function, projection, optimization, localization, and stabilized metods. SIAM J. Numer. Anal. 6, ) 5. Jon, V., Kaya, S.: A finite element variational multiscale metod for te Navier-Stokes equations. SIAM J. Sci. Comput. 6, ) 6. Jon, V., Kaya, S.: Finite element error analysis of a variational multiscale metod for te Navier-Stokes equations. Adv. Comput. Mat. 8, ) 7. Jon, V., Kaya, S., Layton, W.: A two-level variational multiscale metod for convection-dominated convection-diffusion equations. Comput. Metods Appl. Mec. Eng. 195, ) 8. Kaya, S., Layton, W.: Subgrid-scale eddy viscosity metods are variational multiscale metods. Tecnical report, University of Pittsburg, 3 9. Layton, W., Lee, H., Peterson, J.: A defect-correction metod for te incompressible Navier-Stokes equations. Appl. Mat. Comput. 19, 1 19 ) 1. Layton, W.: Superconvergence of finite element discretization of time relaxation models of advection. BIT Numer. Mat. 47, ) 11. Layton, W., Neda, M.: Truncation of scales by time relaxation. J. Mat. Anal. Appl. 35, )

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