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Mathematical Modelling and Analysis Publisher: Taylor&Francis and VGTU Volume 20 Number 5, September 2015, 658{680 http://www.tandfonline.com/TMMA http://dx.doi.org/10.3846/13926292.2015.1091394 ISSN: 1392-6292 c Vilnius Gediminas Technical University, 2015 eISSN: 1648-3510 Modified Method of Characteristics Variational Multiscale Finite Element Method for Time Dependent Navier-Stokes Problems∗ Zhiyong Sia, Yunxia Wangb and Xinlong Fengc aSchool of Mathematics and Information Science, Henan Polytechnic University Jiaozuo 454003, P.R. China bSchool of Materials Science and Engineering, Henan Polytechnic University Jiaozuo 454003, P.R. China cCollege of Mathematics and Systems Science, Xinjiang University Urumqi 830046, P.R. China E-mail(corresp.): [email protected] E-mail: [email protected] E-mail: [email protected] Received March 6, 2015; revised August 26, 2015; published online September 15, 2015 Abstract. In this paper, a modified method of characteristics variational multiscale (MMOCVMS) finite element method is presented for the time dependent Navier- Stokes problems, which is leaded by combining the characteristics time discretization with the variational multiscale (VMS) finite element method in space. The theoretical analysis shows that this method has a good convergence property. In order to show the efficiency of the MMOCVMS finite element method, some numerical results of analytical solution problems are presented. First, we give some numerical results of lid-driven cavity flow with Re = 5000 and 7500 as the time is sufficient long. From the numerical results, we can see that the steady state numerical solutions of the time-dependent Navier-Stokes equations are obtained. Then, we choose Re = 10000, and we find that the steady state numerical solution is not stable from t = 200 to 300. Moreover, we also investigate numerically the flow around a cylinder problems. The numerical results show that our method is highly efficient. Keywords: modified method of characteristics, VMS finite element method, time dependent Navier-Stokes problems, characteristics-based method, numerical tests. AMS Subject Classification: 76D05; 76M10; 65M60; 65M12. ∗ This work was supported in part by the NSF of China (Grant No. 11301156, 11401177 and 11271313). MMOC-VMSFEM for Time Dependent N-S Problems 659 1 Introduction In this paper, we consider the time-dependent Navier-Stokes problems 8 > ut − ν4 u + (u · r)u + rp = f; x 2 Ω × [0;T ]; <> r · u = 0; x 2 Ω × [0;T ]; (1.1) > u(x; 0) = u0(x); x 2 Ω; :> u(x; t) = 0; x 2 @Ω × [0;T ]; where Ω is a bounded domain in R2 assumed to have a Lipschitz continuous T boundary @Ω. u = (u1(x; t); u2(x; t)) represents the velocity vector, p(x; t) represents the pressure, f(x; t) is the body force, ν = 1=Re the viscosity num- ber, Re is the Reynolds number. Developing efficient finite element methods for the Stokes and Navier-Stokes equations is a key component in the incompressible flow simulation. There are some iterative methods for solving the stationary Navier-Stokes equations un- der some strong uniqueness conditions were presented by some authors [15, 18, 19]. The VMS method was first introduced by Hughes and his coworkers in [22,23]. The basic idea is splitting the solution into resolved and unresolved scale, representing the unresolved scales in terms of the resolved scales, and using this representation in the variational equation for the resolved scales. And there are many works devoted to this method, e.g. VMS method for the Navier-Stokes equations [26]; a two-level VMS method for convection- dominated convection-diffusion problems [29]; VMS methods for turbulent flow [9,30,31]; large-eddy simulation (LES) [24,25,32]; subgrid-scale models for the incompressible flow [22,44]. In [27,28], John et al. presented the error analysis of the two different kinds of VMS for the Navier-Stokes equations. The main difference is the definition of the large scales projection (L2-projection in [28] and elliptic projection in [27]). There is another class of VMS method which rely on a three-scale decomposition of the flow field into large, resolved small and unresolved scales [8]. In the characteristics method, which is a highly effective method for ad- vection dominated problems, the hyperbolic part (the temporal and advection term) is treated by a characteristic tracking scheme. In 1982, Douglas and Rus- sell [11] presented the modified method of characteristics finite element method (MMOCFEM) firstly. Russell [36] extended it to nonlinear coupled systems in two and three spatial dimensions. The second order in time method for lin- ear convection diffusion problems had been given by Ewing and Russell [13]. A characteristics mixed finite element method for advection-dominated trans- port problems was presented by Arbogast [2] and for Navier-stokes equations by Buscagkia and Dari [7], respectively. In [34], a detail theory analysis for Navier-Stokes equations have been done by Pironneau, who obtained subop- timal convergence rates of the form O(hm + 4t + hm+1=4t) and improved by Dawson et al [10]. In [4], Boukir et al. presented a second-order time scheme based on the characteristics method and spatial discretization of finite element type for the incompressible Navier-Stokes equations. An optimal error estimate for the Lagrange-Galerkin mixed finite element approximation of the Navier-Stokes equations was given by S¨uliin [43]. In [38, 39], one order and Math. Model. Anal., 20(5):658{680, 2015. 660 Z. Si, Y. Wang and X. Feng second order MMOC mixed defect-correction finite element methods for time- dependent Navier-Stokes problems were given. In [41], second order in time MMOC variational multiscale finite element method for the time-dependent Navier-Stokes equation was shown. In [40], we presented modified character- istics gauge-Uzawa finite element method for the conduction-convection equa- tion. In [42], the modified characteristics finite element method for the Navier- Stokes/Darcy problem was shown. In this paper, we present a modified method of characteristics VMS finite element method based on the L2 projection for the time dependent Navier- Stokes equations. In our method, the hyperbolic part (the temporal and ad- vection term) is treated by a characteristic tracking scheme. The VMS based on projection finite element method is used for spatial discretization. The error analysis shows that this method has a good convergence property. In order to show the efficiency of the MMOCVMS finite element method, we firstly present some numerical results of analytical solution problems. The numerical results show that the convergence rates are O(h3) in the L2-norm for u, O(h2) in the semi H1-norm for u and O(h2) in the L2-norm for p by using the Taylor-Hood element, which agrees very well with our theoretical results. Then, some nu- merical results of the lid-driven cavity flow with Re = 5000 and 7500 are given. We present the numerical results as the time are sufficient long enough, then the solution of the Navier-Stokes should approximate the solution of the steady state cases. From the numerical results, we can see that a steady state numeri- cal solutions of the time-dependent Navier-Stokes equations are obtained. And the numerical solutions are in good agreement with that of the steady Navier- Stokes equations shown by Ghia et al. [14] and Erturk et al. [12]. Finally, we present some numerical results for Re = 10000. It shows that the solution is mainly periodic with small variations in the amplitude of the time evolution at the monitoring points as the time is sufficient long. And the phase portraits of the monitoring points show that the variations in amplitude yield a solution which is quasi-periodic. Furthermore, we present numerical simulations for the two-dimensional fluid flow around a cylinder. It is observed from these numerical results that the scheme can result in good accuracy, which shows that this method is highly efficient. The rest of this paper is organized as follows. In Section 2, the func- tional settings of the Navier-Stokes equations are presented. In Section 3 the MMOCVMS finite element methods is proposed. The following Section 4 we derive optimal error estimate for the new algorithm. Numerical experiments will be carried out in Section 5 and some concluding remarks will be given in the final section. 2 Functional settings of the Navier-Stokes equations In this section, we aim to describe some notations and results which will be frequently used in this paper. Firstly we introduce the standard Sobolev spaces Z T i 2 n X @ v H (0;T ; Φ) = v 2 Φ; dt < +1 ; @ti 0≤i≤n 0 Φ MMOC-VMSFEM for Time Dependent N-S Problems 661 equipped with the norm 2 1 Z T i 2 X @ v kvkHn(Φ) = dt : @ti 0≤i≤n 0 Φ Here Φ denotes a Hilbert space with the norm k · kΦ. Especially, when n = 0 we note 1 Z T 2 2 kvkL2(Φ) = kvkΦdt : 0 And, we define 1 L (0;T ; Φ) = v 2 Φ; ess sup kvkΦ < +1 0≤t≤T with the norm kvkL1(Φ) = ess sup kvkΦ: 0≤t≤T For the mathematical setting of the Navier-Stokes problems (1.1), we intro- duce the following Hilbert spaces Z 1 2 2 2 X := H0 (Ω) ;M := L0(Ω) = ' 2 L (Ω); 'dx = 0 ; Ω V := fv 2 X;(r · v; q) = 0 8q 2 Mg : The following assumptions and results are recalled (see [18]). (A1) There exists a positive constant C0 which only depends on Ω, such that kuk0 ≤ C0kruk0 8u 2 X: (A2) Assume that Ω is smooth, hence the unique solution (v; q) 2 (X; M) of the steady Stokes problem −4v + rq = g; r · v = 0; in Ω; vj@Ω = 0 for any prescribed g 2 L2(Ω)2 exits and satisfies kvk2 + kqk1 ≤ ckgk0; where c > 0 is a generic constant depending on Ω, which may stand for different 2 values at its different occurrences.
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