Error analysis for incompressible viscous flow based on a sequential regularization formulation

ERROR ANALYSIS FOR INCOMPRESSIBLE VISCOUS FLOW BASED ON A SEQUENTIAL REGULARIZATION FORMULATION LU XILIANG NATIONAL UNIVERSITY OF SINGAPORE 2006 ERROR ANALYSIS FOR INCOMPRESSIBLE VISCOUS FLOW BASED ON A SEQUENTIAL REGULARIZATION FORMULATION LU XILIANG (M.Sc. National University of Singapore) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2006 Acknowledgements Prof. Lin Ping has served as my advisor since a meeting in July 2000, in my more than five years in the Department of Mathematics at National University of Singapore, Prof. Lin committed significant time and effort in support of my completion of the research. I am most grateful to have Prof. Lin as my supervisor, and I hope our working relationship may continue in the future. At the same time, I would like to thank Prof. Liu Jian-Guo for his kind help, I would like to thank Prof. Olivier Pironneau, Prof. Fr´ed´eric Hecht and Prof. Antoine Le Hyaric, who made the free software FreeFEM++ which saves me a lot of time on writing code. In addition, I would like to express my thanks to my colleagues and friends, Dr. Tang Hongyan, Dr. Zhang Ying, Dr. Liang Kewei, Pan Suqi and Jia Shuo who gave me a lot of help and encouragement in my research and my life. I would like to thank National University of Singapore for awarding me the Research Scholarship and providing me with a helpful research environment. Finally I would like to dedicate this dissertation to my parents. i Contents Summary i Introduction 1.1 Navier-Stokes Equations and Numerical Methods . . . . . . . . . . . 1.2 Sequential Regularization Method . . . . . . . . . . . . . . . . . . . . Preliminary Some Estimations to SRM Equations 12 3.1 Various Estimations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Convergence of the SRM Solution . . . . . . . . . . . . . . . . . . . . 24 Existence and Uniqueness 28 4.1 A Priori Estimations for Semi-Discrete Equations . . . . . . . . . . . 28 4.2 Existence and Uniqueness of the Solution . . . . . . . . . . . . . . . . 33 Error Estimations to the Time Discrete Scheme 41 5.1 First Order Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.2 Crank-Nicolson Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Error Analysis to Finite Element Method 54 6.1 Error Estimation to a Spatially Discrete Scheme . . . . . . . . . . . . 55 6.2 Error Estimation to Full Discrete Scheme . . . . . . . . . . . . . . . . 63 ii iii In the case of α1 =0 70 7.1 Energy Estimations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.2 Finite Element Estimation . . . . . . . . . . . . . . . . . . . . . . . . 74 Numerical Simulations 79 8.1 Flow with Exact Expression . . . . . . . . . . . . . . . . . . . . . . . 81 8.2 Cavity Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.3 Flow past a circular cylinder. . . . . . . . . . . . . . . . . . . . . . . 95 Conclusion, Final Remark and Future Work 103 Bibliography 107 Summary In this dissertation we will discuss error analysis of a sequential regularization method (SRM) to solve time dependent incompressible Navier-Stokes equations. From both theoretical and numerical point of view, the most difficult part to solve Navier-Stokes equations is how to deal with the divergence free condition. This sequential regularization formulation can treat this difficulty efficiently. We review a few existing numerical methods for Navier-Stokes equations in chapter 1, especially projection method and penalty method. We then introduce the sequential regularization method, including the derivation and various formulations of this method. Chapter is a preliminary part. We collect a number of well-known inequalities which will be used in our analysis in later chapters. In chapter 3, we prove a few lemmas for equations related to sequential regularization formulation. Chapters to will include our main results. The existence and uniqueness of the sequential regularization solution are proved in chapter 4. In chapter 5, we consider semidiscretization in time. We obtain the optimal error estimation for each scheme. In chapter 6, the error estimation of semi-discretization in spatial variables and fully discrete scheme are obtained. We focus on a special case of SRM (α1 = 0) in chapter and give similar error estimation for semi-discretization in spatial variables. We include three numerical examples in chapter 8. The first example has exact i ii solution and can be used to verify the convergence. The second and third examples are both well-known problems in computational fluid dynamics. We can compare our results with benchmark solutions. In the last chapter, we conclude this thesis and point out a few directions which we will work in future. Chapter Introduction 1.1 Navier-Stokes Equations and Numerical Methods To describe the motion of an incompressible flow in a two dimensional or three dimensional domain, Navier-Stokes equations are always the governing equations. A derivation will be given in the appendix A. Assume Ω is a bounded connected domain in Rn (n = 2, 3) with smooth boundary. Let u describe the velocity field of the fluid, and p describe the pressure, then (u, p) satisfies dimensionless NavierStokes equations: ut − ν u + (u · ∇)u + ∇p = f , (1.1) divu = 0, (1.2) u|∂Ω = g, where ν = , Re u(0) = u0 , (1.3) and Re is Reynolds number. Although the Navier-Stokes equations were derived in the 19th century, our understanding of them is still limited, such as that the regularity of the solutions in a general three dimensional domain is still open, which is one of ”the Millennium Prob1 Chapter Introduction lems”. Numerical simulation of the Navier-Stokes equations also has a long history. A lot of methods have been proposed. The standard discretization which include finite difference method, finite volume method, conformal finite element method on the divergence free space, non-conformal finite element method and spectral method are discussed in [30][11][15][26][17][8]. The main difficulty to solve the Navier-Stokes equations numerically is that the velocity field u and the pressure p are coupled by incompressible constraint divu = 0. we need to find a proper way to dispose this difficulty when we try to use direct discretization. There are some approaches to overcome this difficulty, such as projection method and penalty method. In the following, we will briefly introduce these two methods, and then present sequential regularization method. The projection method was firstly introduced by Chorin and Temam [10][38]. It is one kind of fractional-step methods. We solve an intermediate velocity un+ regardless of the divergence free condition, then project it to the divergence free space by the Helmohltz-Hodge decomposition. Hence it decouples the velocity and the pressure. The semi-discrete scheme of the projection method can be written as follows. Intermediate Velocity Solve un+ from    un+ −un t − ν un+ + (un · ∇)un = f (tn+1 ),   un+ 21 |∂Ω = 0. (1.4) Projection Step Find (un+1 , pn+1 ) be solution of   un+1 −un+  + ∇pn+1 = 0,  t   divun+1 = 0,      un+1 · n|∂Ω = 0. (1.5) To solve the equations at the projection step, we take the divergence operator for Chapter Introduction both sides of un+1 −un+ t + ∇pn+1 = 0, and obtain a Poisson equation for pn+1 with homogeneous Neumann boundary condition. We can then solve pn+1 and update un+1 by substituting pn+1 into the first equation. The scheme (1.4) - (1.5) and its variations have been widely used to find numerical solution of Navier-Stokes equations. The convergence has been analyzed as well, see [31][32][34]. We will give a few remarks on the projection method, especially for scheme (1.4) - (1.5). Remark 1.1 1. In the equations (1.4), to calculate the intermediate velocity, there are different ways to approximate the nonlinear convection term. One simple mod1 ification is (un · ∇)un+ . Hence the equations are still second order elliptic, the artificial stabilizer is not required, but the resulting stiffness matrix is not symmetric. 2. To obtain un+1 , we project the intermediate velocity un+ to the divergence free space by solving equation (1.5). But the solution un+1 is only in L2 , consequently, it is not necessary to satisfy the homogeneous Dirichlet boundary condition (we use un+1 · n|∂Ω = instead of un+1 |∂Ω = 0). 3. In the equations (1.5), the pressure pn+1 satisfies homogeneous Neumann boundary condition, but we not have any explicit boundary condition for pressure in Navier-Stokes equations, which also prevents us from obtaining accurate approximation for the pressure near the boundary. Besides the projection method, reformulation methods are also widely used to overcome the difficulty of divergence free condition. The idea is to reformulate the original system by adding a term involving a small parameter . When tends to 0, the solution of the reformulated system will approach the exact solution. The penalty method, artificial incompressibility method and the sequential regularization method belong in this category. Penalty Method Chapter Numerical Simulations 101 Figure 8.51: Re=100, T=30 1.5 0.5 −0.5 −1 −1.5 10 10 10 Figure 8.52: Re=100, T=35 1.5 0.5 −0.5 −1 −1.5 Figure 8.53: Re=100, T=40 1.5 0.5 −0.5 −1 −1.5 Chapter Numerical Simulations 102 Figure 8.54: Re=100, T=45 1.5 0.5 −0.5 −1 −1.5 10 10 10 Figure 8.55: Re=100, T=50 1.5 0.5 −0.5 −1 −1.5 Figure 8.56: Re=100, T=55 1.5 0.5 −0.5 −1 −1.5 Chapter Conclusion, Final Remark and Future Work We have already discussed a lot about the sequential regularization formulation applied to the Navier-Stokes equations. We proved existence, uniqueness of the sequential regularization formulation, obtained error estimation of the discrete system, and tested some numerical examples. There are still a lot of issues remained. Firstly we consider Navier-Stokes equations in a two dimensional domain and a finite time interval. Can we extend our results to a long time or to a three dimensional domain? The answer seems to be positive. To work out the long time case, we need a lemma which similar to lemma 3.1, but the choice of and constant C not depend on time interval T . For 3D case, we know the solution of the Navier-Stokes equations may blow up at finite time. We can apply the technique in analysis of Navier-Stokes equations to obtain reformulation solution in a short time. Secondly, let us check the convergence results. Recall when α1 = 1, we obtain the optimal convergence result i.e. theorem 3.5. A natural question is: when α1 = 0, may we obtain the similar property? Before answering this question, we should better look at the sequential regularization formulation from another point of view. 103 Chapter Conclusion, Final Remark and Future Work 104 For simplification, we only discuss Stokes equations. Consider (u, p) be the solution of the steady state Stokes equations with homogenous Dirichlet boundary condition: − u + ∇p = f , (9.1) divu = 0. (9.2) The penalty method when applied to Stokes equations can be formulated as, (for a given small positive constant ) − u + ∇p = f , (9.3) divu + p = 0. (9.4) When we introduce the iteration of the penalty method, with a given initial guess of pressure p0 , we have a sequence (us , ps ) satisfy the following equations, − us + ∇ps = f , (9.5) divus + (ps − ps−1 ) = (9.6) for s = 1, 2, . . Then we can easily derive error estimation us − u + ∇(ps − p) ≤ (C )s . The key part in the proof is the estimation of the solution of the non-homogenous Stokes equation (see the proof of lemma 3.1). But things will be different for time dependent case. Consider non-homogenous time depend Stokes equation: ut − u + ∇p = f , (9.7) divu = g, (9.8) u|t=0 = 0. (9.9) u|∂Ω = 0; We only have T ( u 2 + ut T + ∇p )dt ≤ C( ( f + ∇g + gt (H ) )dt), (9.10) Chapter Conclusion, Final Remark and Future Work 105 where (H ) is dual space of H . From this point of view, perhaps we can not expect similar results for time dependent (Navier-)Stokes equations when we consider an iterative penalty solution. Fortunately the sequential regularization formulation discussed in this thesis can overcome this difficulty. Consider the non-homogenous time dependent Stokes equations: ut − u + ∇p = f , (9.11) divut + divu = g, (9.12) u|∂Ω = 0; Once the basic compatibility holds ( T ( u 2 + ut u|t=0 = 0. ∂Ω (9.13) g = 0), we have T + ∇p )dt ≤ C( ( f + ∇g )dt), (9.14) where the constant C only depends on the domain Ω and time T. There is no gt at right hand side. From this result, we can obtain convergence. Finally, let us move to the error analysis of the finite element method. We obtained uh − u ≤ h √ C for P1 approximation in chapter 6, and indicated that this result can be extended to higher order polynomial approximation without any difficulty. But from our numerical experiments, we observed that the numerical result with P2 element is significantly better than that with P1 element. It seems that the error does not change with in the test. This may imply that the optimal error estimation for P2 element should have no in the denominator. This result is reasonable since we have an extra property, i.e. inf-sup condition for (P2 , P1 ) finite element space, which is not used in our analysis. It is well-known that when (Vh , Qh ) satisfies inf-sup condition, the finite element estimation to the Navier-Stokes equations can be obtained. The SRM solution us can be viewed as us = u + s v, where u is the solution of the Navier-Stokes equations, and v is a function whose norm is independent of the choice of . If our finite element space satisfies inf-sup Chapter Conclusion, Final Remark and Future Work 106 condition implicitly (we only have finite element space for u, p is recovered by divu), the optimal error estimation should be independent of . It is one of our future work to obtain this estimation. Bibliography [1] R. A. Adams, Sobolev Spaces, Academic Press, 1975. [2] P. Andreas, Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations, B. G. Teubner Stuttgart, 1997. [3] U. Ascher, P. Lin, Sequential regularization methods for higher index DAEs with constraint singularities: Linear index-2 case, SIAM J. Numer. Anal., 33, (1996), pp. 1921-1940. [4] J. Baumgarte, Stabilization of constraints and integrals of motion in dynamics systems, Comput. Methods Appl. Mech. Engrg., (1972), pp. 1-16. [5] B. Brefort, J. M. Ghidaglia, R. Temam, Attractors for the penalized NavierStokes equations, SIAM J. Math. Anal., 19, (1988), pp. 1-21. [6] K. Brenan, S. Campbell, L. Petzold, Numerical Solutions of Initial-Value Problems in Differential-Algebraic Equations, Elsevier, New York, 1989. [7] Susanne C. Brenner, Li-Yeng Sung, Linear Finite Element Methods For Planar Linear Elasticity, Math. of Comp., Vol. 59, (1992), pp. 321-338. [8] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spetral Methods in Fluid Dynamics, Springer-Verlag, Berlin, New York, 1987. [9] J. Chorin, E. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd Edition, Springer-Verlag, 1993. 107 Bibliography 108 [10] J. Chorin, Numerical Solution of the Navier-Stokes equations, Math. Comp., 22, (1968), pp. 745-762. [11] P. G. Ciarlet, J. L. Lions,(editor) Handbook of numerical analysis, Elsevier Science Pub, 1990-1997. [12] L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998. [13] D. Gartling, A Test Problem for Outflow Boundary Conditions: Flow Over a Backward-facing Step, Int. J. Numer. Methods Fluids, Vol. 11, (1990), pp. 953-967. [14] U. Ghia, K. N. Ghia, C. T. Shin, High-Re Solutions for Incompressivle Flow Using the Navier-Stokes Equations and a Mutigrid Methods, J. Comp. Phys., 48, (1982), pp. 387-411. [15] M. D. Gunzburger, Finite Element Methods for Viscous Incompressible Flows, a guide to theory, practice and algorithms, Academic Press, Inc. 1989. [16] S. Hermann, The Navier-Stokes Equations, an elementary functional analytic approach, Birkhauser Verlag, 2001. [17] J. G. Heywood, The Navier-Stokes equations: On the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29, (1980), pp. 639-681. [18] J. G. Heywood, R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem, I: Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19, (1982), pp. 275-311. Bibliography 109 [19] J. G. Heywood, R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem, psrt II: Stability of solutions and erroe estimates uniform in time, SIAM J. Numer. Anal., 23, (1986), pp. 750-777. [20] J. G. Heywood, R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem, part III: Smoothing peoperty and higher order error estimates for spatial discretization, SIAM J. Numer. Anal., 25, (1988), pp. 489-512. [21] J. G. Heywood, R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem, part IV: Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27, (1990), pp. 353-384. [22] V. Girault, P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag Berlin Heidelberg, 1986. [23] P. Lin, A Sequential Regularization Methods for Time-Dependent Incompressibel Navier-Stokes Equations, SIAM J. Numer. Anal., Vol. 34, No. 3, (1997), pp. 1051-1071. [24] P. Lin, Regularization Methods for Differential Equations and Their Numerical Solutions, Ph.D. thesis, Department of Mathematics, University of British Columbia, 1995. [25] P. Lin, X. Chen, M. Ong, Finite element methods based on a new formulation for the non-stationary incompressible Navie-Stokes equations, Int. J. Numer. Meth. Fluids, 46, (2004), pp. 1169-1180. [26] J. G. Liu, J. Liu, R. L. Pego, Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate, to appeare. Bibliography 110 [27] Wade S. Martinson, Paul I. Barton, A differentiation index for partial differential-algebraic equations, SIAM J. Sci. Comp., Vol. 21, No. 6, pp. 22952315. [28] V. A. Patel, Symmetry of the flow around a circular cylinder, J. of Comp. Phy., Vol. 71, No. 1, July 1987, pp. 65-99. [29] R. Peyret, (editor) Handbook of computational fluid mechanics, London Academic Press, 1996. [30] L. Quartapelle, Numerical Solution of the Incompressible Navie-Stokes Equations, Birkhauser, 1993. [31] J. Shen, On error estimates of the projection methods for the Navier-Stokes equations: first-order schemes, SIAM J. Numer. Anal., 29, (1992), pp. 55-77. [32] J. Shen, On error estimates of some higher-order projection and penaltyprojection methods for the Navier-Stokes equations, Numer. Math., 62, (1992), pp. 49-73. [33] J. Shen, On error estimates of the penalty method for the unsteady NavierStokes equations, SIAM J. Numer. Anal., 32, (1995), pp. 386-403. [34] J. Shen, On error estimates of the projection methods for the Navier-Stokes equations: second-order schemes, Math. Comp., 65, (1996), pp. 1039–1065. [35] G. Strang, G.J. Fix, An analysis of the finite element method, Princtice-Hall, 1973. [36] R. Temam, Navier-Stokes Equations, North-Holland Amsterdam, 1977. [37] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edition, CBMS-NSF Reginal Conference Series in Applied Mathematics, 1995. Bibliography 111 [38] R. Temam, Sure L’approximation de la solution des equations de Navier-Stokes par la m´ethods des pas fractionnaires II, Arch. Rational Mech. Anal., 33, (1969), pp. 377-385. [39] T. E. Tezduyar, J. Liou, D. K. Ganjoo, Incompressible flow computations based on the vorticity-stream function and velocity-pressure formulations, Computers and Structures, Volume 35, Issue 4, 1990, Pages 445-472. [40] C. Williamson, Vortex dynamics in the cylinder wake, Annual Review of Fluid Mechanics, 1996, 28, pp. 477C539. [41] L. A. Ying, Finite Element Methods, Lecture Notes, 1998. Appendix A Derivation of Navier-Stokes Equations We will derive the governing equations, Navier-Stokes equations here, which is based on the assumption that the fluid is incompressible, the conservation laws of mass, momentum and energy. In [9], more details can be found. Suppose Ω is an open bounded domain in Rn , where n = 2, 3, the functions u and p are defined as velocity field and pressure, which are functions depending on both spatial variables x and time variable t. We define the total derivative operator D Dt = ∂ ∂t + u · ∇, then we have Reynolds transport theorem as D Dt f dV = V (t) ( V (t) ∂f + div(f u))dV ∂t (9.15) where V is an arbitrary volume, f = f (x, t) describes an arbitrary intensive property of the fluid. A1: Conservation of Mass The fluid density ρ is an intensive property of the fluid, we can then apply Reynolds transport theorem to obtain ( V ∂ρ + div(ρu))dV = 0. ∂t 112 (9.16) Bibliography 113 since V is arbitrary volume, (9.16) is equivalent to ∂ρ + div(ρu) = 0, ∂t (9.17) Dρ + ρ∇ · u = 0. Dt (9.18) or For incompressible fluid, the incompressible condition divu = will yield to Dρ Dt = 0. A2: Balance of Momentum The momentum is also an intensive property of the fluid, hence we can apply Reynolds transport theorem again. Define the momentum function as m, then we have D Dt mdV = V ( V ∂m + ∇ · (mu))dV, ∂t (9.19) where ∇ · (mu) represents the divergence of the convective momentum flux tensor. The Newton’s second law states that the rate of change of momentum of a portion of the fluid equals the force applied to it, therefore we have ∂m + ∇ · (mu) − ρf − ∇ · σ = 0, ∂t (9.20) where ρf is the body force and σ is stress tensor. From the definition of momentum, we can replace m as ρu to obtain ∂ρu + ∇ · (ρuu) − ρf − ∇ · σ = 0. ∂t (9.21) Simplify the above equation and apply compressible condition, we have ρ ∂u + ρ(u · ∇)u = ρf + ∇ · σ. ∂t (9.22) Since we only consider Newtonian fluids which the shear stress is proportional to the velocity gradient in this article, the stress tensor σ can be presented as σ = −pI + τ , where τ = λ(∇ · u)I + µ(∇u + (∇u)T ). If we define D = (∇u + (∇u)T ), we Bibliography 114 have physical meanings of these functions and parameters. τ is the deviatoric stress tensor, D is the strain tensor, µ is the first coefficient of viscosity and λ is the second coefficient of viscosity. Substituting stress tensor into the balance of momentum equation to obtain ∂u + (u · ∇)u = − ∇p + ν u + f , ∂t ρ where ν = µ ρ (9.23) is called the coefficient of kinematic viscosity. Under the assumption of ρ be constant, we have the Navier-Stokes equations: ut − ν u + (u · ∇)u + ∇p = f , ρ (9.24) divu = 0. (9.25) A3: Transformation to Dimensionless Form In this section we will discuss the scaling properties of the Navier-Stokes equations with the aim of introducing the Reynolds number which measures the effect of viscosity of the flow. For a given problem, let characteristic length and velocity be L and U , the time scale T defined by T = L , U then we define the following dimensionless variables, x = x , L u = u , U Define Reynolds number Re = LU , ν t = t , T p = p , ρU f = Tf . U substituting these new variables into Navier- Stokes equations and drop the primes to have dimensionless form of the NavierStokes equations: ut − u + (u · ∇)u + ∇p = f , Re divu = 0. (9.26) (9.27) Bibliography 115 A4: Vorticity and Streamline Functions For a given flow, if the velocity field is u, the vorticity field can be defined as ω = ∇ × u. Particularly if the flow is two dimensional, u = (u, v), then vorticity is ω = ∂x v − ∂y u. We still focus on the two dimensional flow of which the continuity equation is ∂x u+∂y = 0. There exists a scalar function ψ (streamline function), unique up to an additive constant, satisfies ∂x ψ = −v and ∂y ψ = u. Let ψ, ω be unknown functions, we can rewrite the Navier-Stokes equation as the equivalent vorticity-streamline form, ψ = −ω, (9.28) ∂ω ∂ψ ∂ω ∂ψ ∂ω ∂f2 ∂f1 − ω+ − = − . ∂t Re ∂y ∂x ∂x ∂y ∂x ∂y (9.29) A5: Boundary Conditions The Navier-Stokes equations are supplemented by boundary conditions. We use homogenous Dirichlet boundary condition in this thesis, which is reasonable in both experiment and mathematics. For instance, we consider a solid wall around the domain where the velocity is on the boundary, which is the so-called no-slip condition. From mathematical point of view, to make the solution unique, we need Dirichlet boundary condition for the term u. The Navier-Stokes equations are time depen- dent, so after adding initial condition for u, we can obtain the full version of our governing equation as following. ut − u + (u · ∇)u + ∇p = f , Re divu = 0, u|∂Ω = 0, (9.30) (9.31) u(0) = u0 . (9.32) Name: Lu Xiliang Degree: Ph.D Department: Mathematics Thesis Title: Error Analysis for Incompressible Viscous Flow Based on a Sequential Regularization Formulation Abstract We consider the sequential regularization method (SRM) for the unsteady incompressible Navier-Stokes equations. The thesis presents the formulation, existence and uniqueness of SRM solution, error analysis of discrete scheme of sequential regularization formulation. Based on sequential regularization formulation, we could solve Navier-Stokes equations numerically with strongly enforced divergence free constraint and with easily constructed finite element spaces (without consideration of inf-sup condition). Artificial boundary condition for pressure is not needed as well. We provide some numerical tests in the end of thesis. Keywords: Sequential Regularization method (SRM), Navier-Stokes Equations, Finite Element Analysis, Penalty Method, Computational Fluid Dynamics, Incompressible Viscous Flow. [...]... error analysis 1.2 Sequential Regularization Method In [3], an iterative reformulation method is introduced to solve the differentialalgebraic equations (DAE) It is so-called sequential regularization method (SRM) The DAE is an ordinary differential equation coupled with an algebraic equation Partial differential algebraic equations (PDAE) can be viewed as an extension of DAE The Navier-Stokes equations... equation (2.1), we immediately have that the lemma 2.1 and 2.2 are ¯ also true for trilinear form ¯ For simplicity, we focus on B and ¯ in the thesis, but b b all results we obtain later are valid for B and b as well Chapter 3 Some Estimations to SRM Equations we recall that the sequential regularization reformulation of Navier-Stokes equations reads: given initial pressure p0 , for s = 1, 2, 3 · · · , solve... Baumgart’s stabilization, that is, reformulate the incompressible condition as (divut +αdivu)+ p = 0 This is the idea of sequential regularization method (SRM) Chapter 1 Introduction 6 developed in [23] The SRM is an iterative penalty method for the Baumgart stabilized formulation Let two constants α1 ≥ 0 and α2 > 0 We replace the incompressible condition by α1 divut + α2 divu = 0 and modify it as α1 divut... mathematically equivalent to divu = 0 If we use divut = 0 instead of divu = 0, it is equivalent to take divergence of equation (1.1) We thus obtain a socalled pressure-Poisson equation But there is a weak instability associated with this formulation, i.e given small perturbation for the second equation (divut = δ), then divu = δt will increase when time goes on The so-called Baumgart’s stabilization... is a constant which only depends on • Gronwall inequality in differential form: Let y(t) be a nonnegative, absolutely continuous function in [0, t] and satisfy for almost every t, the differential inequality: y (t) ≤ a( t)y(t) + b(t), where a( t) and b(t) are nonnegative, summable functions in [0, t] Then we have: y(t) ≤ e t 0 a( s)ds t y(0) + b(s)ds 0 • Discrete Gronwall inequality: Let y n , an , bn and... small to be used for long time computation The penalty method also causes an initial layer for the pressure To see this, consider divu + p = 0 at initial time t = 0 We obtain p(0) = 0 since divu(0) = 0, but the exact pressure for Navier-Stokes equations may be nonzero at the initial time This layer will disappear after an O( ) time To avoid initial layer, we can combine the penalty method with Baumgart’s... equations are important examples of PDAE In [23], how to apply SRM to the Navier-Stokes equations was considered This thesis is a continuation of the work in that paper When we consider a DAE (or PDAE) of form xt + Ax + By + q = 0, (1.9) Cx + r = 0 Where A, B, C can be matrixes (DAE) or differential operators (PDAE) Since equation (1.9) is an index 2 DAE, it is ill-posed in a certain sense, and direct... Navier-Stokes equations As we mentioned before, reformulation methods can be applied to Navier-Stokes equations For instance, if we modify the equation divu = 0 as divu + p = 0, then we have the penalized formulation The penalty method is well-posed, but in practice, we need to choose very small to reduce the reformulation error That could make the system stiff When we use explicit time discretization, the time... discretization does not work well Regularization methods such as penalty method reformulate the second equation by adding a small perturbation term which involves y, to reduce the index of system Then we can apply straightforward discretization to the perturbed system We will focus our attention on the Navier-Stokes equations (1.1) - (1.3) in the next section Chapter 1 Introduction 5 The Navier-Stokes equations... chapter In this chapter, we assume existence of the solution and establish some energy estimations for the equations (3.4) - (3.5) This chapter will be organized as follows In the first section we will obtain a few estimations for (3.4) - (3.5) Most of them will be used in later chapters, to obtain the error estimation for the discrete system At second part of this chapter, we will prove the reformulation . ERROR ANALYSIS FOR INCOMPRESSIBLE VISCOUS FLOW BASED ON A SEQUENTIAL REGULARIZATION FORMULATION LU XILIANG NATIONAL UNIVERSITY OF SINGAPORE 2006 ERROR ANALYSIS FOR INCOMPRESSIBLE VISCOUS FLOW. so-called sequential regularization method (SRM). The DAE is an ordinary differential equation coupled with an algebraic equation. Partial differential algebraic equations (PDAE) can be viewed as an. but all results we obtain later are valid for B and b as well. Chapter 3 Some Estimations to SRM Equations we recall that the sequential regularization reformulation of Navier-Stokes equations reads: