MATHEMATICAL ARCHITECTURE FOR MODELS OF FLUID FLOW PHENOMENA. by Alexandr Labovschii

Transcription

1 MATHEMATICAL ARCHITECTURE FOR MODELS OF FLUID FLOW PHENOMENA by Alexandr Labovschii B.S. in Applied Mathematics and Computer Science, Moscow State University, 00 Submitted to the Graduate Faculty of the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 008

2 UNIVERSITY OF PITTSBURGH MATHEMATICS DEPARTMENT This dissertation was presented by Alexandr Labovschii It was defended on May 5th 008 and approved by Prof. William J. Layton, University of Pittsburgh Prof. Beatrice Riviere, University of Pittsburgh Prof. Ivan Yotov, University of Pittsburgh Prof. Giovanni Paolo Galdi, University of Pittsburgh Prof. Catalin Trenchea, University of Pittsburgh Dr. Myron Sussman, University of Pittsburgh Dissertation Director: Prof. William J. Layton, University of Pittsburgh ii

3 MATHEMATICAL ARCHITECTURE FOR MODELS OF FLUID FLOW PHENOMENA Alexandr Labovschii, PhD University of Pittsburgh, 008 This thesis is a study of several high accuracy numerical methods for fluid flow problems and turbulence modeling. First we consider a stabilized finite element method for the Navier-Stoes equations which has second order temporal accuracy. The method requires only the solution of one linear system (arising from an Oseen problem) per time step. We proceed by introducing a family of defect correction methods for the time dependent Navier-Stoes equations, aiming at higher Reynolds number. The method presented is unconditionally stable, computationally cheap and gives an accurate approximation to the quantities sought. Next, we present a defect correction method with increased time accuracy. The method is applied to the evolutionary transport problem, it is proven to be unconditionally stable, and the desired time accuracy is attained with no extra computational cost. We then turn to the turbulence modeling in coupled Navier-Stoes systems - namely, MagnetoHydroDynamics. Magnetically conducting fluids arise in important applications including plasma physics, geophysics and astronomy. In many of these, turbulent MHD (magnetohydrodynamic) flows are typical. The difficulties of accurately modeling and simulating turbulent flows are magnified many times over in the MHD case. We consider the mathematical properties of a model for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove existence, uniqueness and convergence of solutions for the simplest closed MHD model. Furthermore, iii

4 we show that the model preserves the properties of the 3D MHD equations. Lastly, we consider the family of approximate deconvolution models (ADM) for turbulent MHD flows. We prove existence, uniqueness and convergence of solutions, and derive a bound on the modeling error. We verify the physical properties of the models and provide the results of the computational tests. iv

5 TABLE OF CONTENTS PREFACE x.0 INTRODUCTION THE STABILIZED, EXTRAPOLATED TRAPEZOIDAL FINITE EL- EMENT METHOD FOR THE NAVIER-STOKES EQUATIONS Introduction Mathematical Preliminaries Stability and Convergence of the Stabilized Method Error estimates for time derivatives and pressure Physical Fidelity: Conservation of Integral Invariants Physical Fidelity: Predictions of the Turbulent Energy Cascade Kraichnan s Dynamic Analysis Applied to CNLEStab Computational Tests Conclusions A DEFECT CORRECTION METHOD FOR THE TIME-DEPENDENT NAVIER-STOKES EQUATIONS Introduction Previous results Mathematical Preliminaries and Notations Stability of the Velocity Error estimates Pressure Stability of the Pressure v

6 3.5. Error estimates for the pressure Computational Tests Comparison of the approaches A DEFECT CORRECTION METHOD FOR THE EVOLUTIONARY CONVECTION DIFFUSION PROBLEM WITH INCREASED TIME ACCURACY Introduction Notation and preliminaries Stability of the method Computational results NUMERICAL ANALYSIS OF A METHOD FOR HIGH PECLET NUMBER TRANSPORT IN POROUS MEDIA Introduction Notation and preliminaries Stability and error analysis of the Darcy problem The convection diffusion problem Stability of the method Error estimates for the method Conclusions LARGE EDDY SIMULATION FOR MHD FLOWS Introduction Existence and uniqueness for the MHD LES equations Notations and preliminaries Stability and existence for the model Regularity Accuracy of the model Limit consistency of the model Verifiability of the model Consistency error estimate Conservation laws vi

7 6.5 Alfvén waves APPROXIMATE DECONVOLUTION MODELS FOR MAGNETO- HYDRODYNAMICS Introduction Existence and uniqueness for the ADM MHD equations Accuracy of the model Limit consistency of the model Verifiability of the model Consistency error estimate Conservation laws Alfvén waves Computational results CONCLUSIONS AND FUTURE RESEARCH Conclusions Future Research BIBLIOGRAPHY vii

8 LIST OF TABLES Experimental convergence rates Number of CGS iterations for CNLE versus CNLEStab AV approximation. Re = Correction Step approximation. Re = AV approximation. Re = Correction Step approximation. Re = DCM. Re = 00000, T = /0, h = / ALTERNATIVE APPROACH. Re = 00000, T = /0, h = / Error estimates, ɛ = 0 5, T =, = h Error estimates, ɛ = 0 5, T =, = h Error estimates in (0, 0.75) (0, 0.75), ɛ = 0 5, = h Error estimates in (0, 0.75) (0, 0.75), ɛ = 0 5, = h Approximating the true solution, Re = 0 5, Re m = 0 5, Zeroth Order ADM Approximating the true solution, Re = 0 5, Re m = 0 5, First Order ADM Approximating the true solution, Re = 0 5, Re m = 0 5, Second Order ADM Approximating the average solution, Re = 0 5, Re m = 0 5, Zeroth Order ADM 69 7 Approximating the average solution, Re = 0 5, Re m = 0 5, First Order ADM 69 8 Approximating the average solution, Re = 0 5, Re m = 0 5, Second Order ADM 70 viii

9 LIST OF FIGURES Conservation of helicity, CNLE (α = 0) versus CNLEStab (α = ) Conservation of helicity for different mesh sizes in CNLESTAB (with α = ): as h gets smaller, helicity is conserved longer t = t = t = t = Computed solution u h, = h = / Computed solution u h, = h = / Computed solution u h, = h = / ADM Enstrophy vs. averaged MHD ADM Energy vs. averaged MHD ADM Energy vs. averaged MHD: zoom in ix

10 PREFACE I would lie to than everyone who has helped me during these five exciting years. I am (and will always be) very grateful to my advisor, Professor William Layton. His wise guidance, moral support and endless patience have helped me understand a lot about mathematics, creativity and life. I am also thanful to Professors Paolo Galdi, Beatrice Riviere, Ivan Yotov and Myron Sussman for their helpful discussions, their excellent teaching and willingness to help. This wor would not be possible without my collaborators. Chapter was written jointly with fellow colleagues Prof. Carolina Manica, Prof. Monia Neda and Prof. Leo Rebholz. Chapter 5 is a joint wor with Prof. Noel Heitmann. I have had many interesting and productive discussions with Professor Catalin Trenchea - our joint wor resulted in Sections 6, 7 and several ongoing projects. Than you - to my parents, who have supported and believed in me, throughout good times and bad. And last, but not least - I need to than my fiancee Anastasia. Her love and support are invaluable, and I appreciate everything she has done for me. x

11 .0 INTRODUCTION The accurate and reliable solution of fluid flow problems is important for many applications. In these one core problem is the Navier-Stoes equations, given by: u t + u u ν u + p = f, for x Ω, 0 < t T u = 0, x Ω, for 0 t T. In the numerical solution of higher Reynolds number flow problems some of the standard iterative methods fail. One common mode of failure is non-convergence of the iterative nonlinear and linear solvers used to compute the velocity and pressure at the new time levels. In Sections and 3 we introduce two unconditionally stable methods designed to overcome this type of failure. The method, introduced in Section, is the Cran-Nicolson Linear Extrapolation with Stabilization. The two main ingredients are the linear extrapolation of the velocity and the artificial viscosity stabilization. The method is unconditionally stable and second-order accurate. Most importantly, the method requires the solution of one linear system per time step, and this linear system is a discretized Oseen problem (with cell Reynolds number O()) plus an O(h) artificial viscosity operator. Thus, the standard iterative solvers and well-tested preconditioners can be used successfully, independent of how small the viscosity coefficient is. We also show that the stabilization in the method alters the numerical method s inetic energy rather than in its energy dissipation. We discuss the physical fidelity of the method and provide the results of numerical tests, that verify the accuracy and the convergence rates.

12 In Sections 3 and 4 we introduce the family of Defect Correction methods for time dependent fluid flow problems. There has been an extensive study and development of this approach for equilibrium flow problems, see e.g. Hemer[Hem8], Koren[K9], Heinrichs[Hei96], Layton, Lee, Peterson[LLP0], Ervin, Lee[EL06], and subsection 3.. for a review of this wor. Briefly, let a th order accurate discretization of the equilibrium Navier-Stoes equations (NSE) be written as NSE h (u h ) = f, (.0.) The DCM computes u h,..., u h as αh h u h + NSE h (u h ) = f, (.0.) αh h u h l + NSE h (u h l ) = f αh h u h l, for l =,...,, where the velocity approximations u h i are sought in the finite element space of piecewise polynomials of degree. It has been proven under quite general conditions (see, e.g., [LLP0]) that for the intermediate approximations of the equilibrium NSE u NSE u h l energy norm = O(h + h u NSE u h l energy norm ) = O(h + h l ), and thus, after l = steps, u u h energy norm = O(h ). Note that (.0.) requires solving an AV approximation times which is often cheaper and more reliable than solving (.0.) once. For many years, it has been widely believed that the method could be directly imported into implicit time discretizations of flow problems in the obvious quasistatic way. Unfortunately, this natural idea doesn t seem to be even stable (see Section 3.7). We give a critically important modification of the natural extension to time dependent problems, that we prove to be unconditionally stable (Theorem 3.) and convergent. Hence, we develop a method for which standard iterative solvers can be applied (for arbitrarily large Reynolds number); the method is unconditionally stable, computationally attractive and highly accurate: in order

13 to obtain an accuracy of O(h ), one needs to solve an artificial viscosity approximation times, which is often cheaper and more reliable (for high Reynolds number) than solving the NSE once. Section 4 presents a modification of this method, that allows to obtain extra time accuracy with almost no extra computational cost. In Chapter 5 we consider the coupling between the porous media problem ( p) = g u = p, and the convection diffusion problem φ t ɛ φ + u φ + cφ = f. This type of coupling is of great importance in a wide array of applications, including oil recovery and nuclear waste storage. The method introduced in this chapter is based on a consistent multiscale mixed method formulation, presented for the stationary convection diffusion problem by W. Layton [Layton0]. We couple the eddy viscosity discretization to the porous media problem, prove the stability of the method and trac the velocity error estimate from Darcy s problem to the convection diffusion to prove the near optimal error bound. In Sections 6 and 7 we consider turbulence modeling in MagnetoHydroDynamics. Even in the hydro-dynamic (flow governed by the Navier-Stoes equations) case the modern science does not yet have a good understanding of turbulent phenomena - due to the turbulence being diffusive, chaotic, irregular, highly dissipative. Another important characteristics of the turbulent flow is the continuum of scales (unsteady vortices can appear at different scales and interact with each other). From the critical length scale determined by Kolmogorov, the size of the smallest persistent eddy is O(Re 3/4 ), where the Reynolds number Re could be described as the ratio of advection coefficient to the diffusion coefficient. Hence, in order to accurately capture all physical properties of the three-dimensional flow, one needs to resolve the flow with O(Re 9/4 ) meshpoints. However, the Reynolds number for air flow around a car is of the order 0 6, around an airplane - 0 7, and it can achieve O(0 0 ) for some atmospheric 3

14 flows. Therefore, it is not computationally feasible to use direct numerical simulations for most of the turbulent flows. Hence - the modeling. Magnetically conducting fluids arise in important applications including climate change forecasting, plasma confinement, controlled thermonuclear fusion, liquid-metal cooling of nuclear reactors, electromagnetic casting of metals, MHD sea water propulsion. In many of these, turbulent MHD (magnetohydrodynamics [Alfv4]) flows are typical. The difficulties of accurately modeling and simulating turbulent flows are magnified many times over in the MHD case. They are evinced by the more complex dynamics of the flow due to the coupling of Navier-Stoes and Maxwell equations via the Lorentz force and Ohm s law. The flow of an electrically conducting fluid is affected by Lorentz forces, induced by the interaction of electric currents and magnetic fields in the fluid. The Lorentz forces can be used to control the flow and to attain specific engineering design goals such as flow stabilization, suppression or delay of flow separation, reduction of near-wall turbulence and sin friction, drag reduction and thrust generation. The mathematical description of the problem proceeds as follows. Assuming the fluid to be viscous and incompressible, the governing equations are the Navier- Stoes and pre- Maxwell equations, coupled via the Lorentz force and Ohm s law (see e.g. [Sher65]). Let Ω = (0, L) 3 be the flow domain, and u(t, x), p(t, x), B(t, x) be the velocity, pressure, and the magnetic field of the flow, driven by the velocity body force f and magnetic field force curl g. Then u, p, B satisfy the MHD equations: u t + (uu) Re u + S (B ) S (BB) + p = f, B t + Re m curl(curl B) + curl (B u) = curl g, u = 0, B = 0, (.0.3) in Q = (0, T ) Ω, with the initial data: u(0, x) = u 0 (x), B(0, x) = B 0 (x) in Ω, (.0.4) and with periodic boundary conditions (with zero mean): Φ(t, x + Le i ) = Φ(t, x), i =,, 3, Φ(t, x)dx = 0, (.0.5) Ω 4

15 for Φ = u, u 0, p, B, B 0, f, g. Here Re, Re m, and S are nondimensional constants that characterize the flow: the Reynolds number, the magnetic Reynolds number and the coupling number, respectively. Direct numerical simulation of a 3d turbulent flow is often not computationally economical or even feasible. On the other hand, the largest structures in the flow (containing most of the flow s energy) are responsible for much of the mixing and most of the flow s momentum transport. This led to various numerical regularizations; one of these is Large Eddy Simulation (LES) [S0], [J04], [BIL06]. It is based on the idea that the flow can be represented by a collection of scales with different sizes, and instead of trying to approximate all of them down to the smallest one, one defines a filter width δ > 0 and computes only the scales of size bigger than δ (large scales), while the effect of the small scales on the large scales is modeled. This reduces the number of degrees of freedom in a simulation and represents accurately the large structures in the flow. In Sections 6 and 7 we consider the problem of modeling the motion of large structures in a viscous, incompressible, electrically conducting, turbulent fluid. We introduce a family of approximate deconvolution models - referring to the family of models in [AS0]. Given the filtering widths δ and δ, the model computes w, q, W - the approximations to u δ,p δ,b δ. Here a δ, a δ denote two local, spacing averaging operators that commute with the differentiation. The ADM for the MHD reads w t + (GN w)(g N w)δ Re w S (G N W ) (G N W )δ + q = f δ, (.0.6a) W t + Re m curl(curl W ) + ((G N W )(G N w)δ ) ((G N w)(g N W )δ ) (.0.6b) = curl g δ, w = 0, W = 0, (.0.6c) subject to w(0, x) = u δ 0 (x), W (0, x) = B δ 0 (x) and periodic boundary conditions (with zero means). Here G N and G N 7.. are the deconvolution operators, that will be defined in Section We begin by proving the existence and uniqueness of solutions to the equations of the model, and that the solutions to the model equations converge to the solution of the MHD 5

16 equations in a wea sense as the averaging radii converge to zero. Then the physical fidelity of the models has to be established. For that we consider the conservation laws - and verify that the model s energy and helicities are also conserved, as they are for the MHD equations; the models also preserve the Alfvén waves - the unique property of the MHD flows. We perform the computational tests to verify the models verifiability, and we also conclude that in the situations when the direct numerical simulation is no longer available (flows with high Reynolds and magnetic Reynolds numbers), the solution can still be obtained by the ADM approach. 6

17 .0 THE STABILIZED, EXTRAPOLATED TRAPEZOIDAL FINITE ELEMENT METHOD FOR THE NAVIER-STOKES EQUATIONS. INTRODUCTION The accurate and reliable solution of fluid flow problems is important for many applications. In these one core problem is the Navier-Stoes equations, given by: find u : Ω [0, T ] R d (d =, 3), p : Ω (0, T ] R satisfying u t + u u ν u + p = f, for x Ω, 0 < t T u = 0, x Ω, for 0 t T, u = 0, on Ω, for 0 < t T, (..) u(x, 0) = u 0 (x), for x Ω, with the usual normalization condition that p(x, t) dx = 0 for 0 < t T when (..) Ω is discretized by accepted, accurate and stable methods, such as the finite element method in space and Cran-Nicolson in time, the approximation can still fail for many reasons. One common mode of failure is non-convergence of the iterative nonlinear and linear solvers used to compute the velocity and pressure at the new time levels. We consider herein a simple, second order accurate, and unconditionally stable method which addresses these failure modes. The method requires the solution of one linear system per time step. This linear system is a discretized Oseen problem plus an O(h) artificial viscosity operator - so the standard iterative solvers and well-tested preconditioners can be used successfully (the preconditioners are described, e.g., in chapter 8 of [ESW05]). Suppressing the spatial 7

18 discretization, the method can be written as (with time step = t and tuning parameter α = O()) u n+ = 0 and u n+ u n + U n+/ ( u n+ + u n ) ν ( u n+ + u n ) αh u n+ + ( p n+ + p n ) = f(t n+/ ) αh u n. (..) Here U n+/ := 3 u n u n is the linear extrapolation of the velocity to t n+/ from previous time levels. Thus, (..) is an extension of Baer s [B76] extrapolated Cran-Nicolson method. Artificial viscosity stabilization is introduced into the linear system for u n+ by adding αh u n+ to the LHS and correcting for it by αh u n (the previous time level) on the RHS. This is a nown idea in practical CFD, and liely has been used in practical computations with many different timestepping methods. To our nowledge however, it has only been proven unconditionally stable in combination with first order, bacward Euler time discretizations, e.g. E and Liu [EL0], Anitescu, Layton and Pahlevani [ALP04], Pahlevani [P06] for related stabilizations and also He [He03] for a two-level method based on Baer s extrapolated Cran-Nicolson method. The increase in accuracy from first order Bacward Euler with stabilization to second order in (..) (extrapolated CN with stabilization) is important. There is also a quite simple proof that (..) is unconditionally stable. We give the stability proof in Proposition.3 and then explore the effect the stabilization (and correction) in (..) have on the rates of convergence for various flow quantities. No discretization is perfect. However, simple and stable ones leading to easily solvable linear systems can be very useful. We therefore conclude with numerical tests which verify accuracy and decrease in complexity in the linear equation solver. William Layton first saw it used as a numerical regularization in 980 and it seems to have been nown well before that. It is related to the simple Kelvin-Voight model of viscoelasticity, Osolov [O80], Kalantarev and Titi [KT07]. 8

19 Defining the method precisely requires a small amount of notation. The spatial part of (..) is naturally formulated in X := H 0(Ω) d, Q := L 0(Ω). The finite element approximation begins by selecting conforming finite element spaces X h X, Q h Q satisfying the usual discrete inf-sup condition (defined in Section ). Denote the usual L norm and inner product by and (, ), and the space of discretely divergence free functions V h by: V h := {v h X h : (q h, v h ) = 0, q h Q h }. Define the explicitly sew-symmetrized trilinear form b (u, v, w) := (u v, w) (u w, v), (..3) and the extrapolation to t n+ := t n+t n+ by E[u h n, u h n ] := 3 uh n uh n, (..4) where u h j (x) is a nown approximation to u(x, t j ). The method studied is a -step method, so the initial condition and first step must be specified, but are not essential. We choose the Stoes Projection, defined in Section.. Algorithm. (Stabilized, extrapolated trapezoid rule). Let u h 0 be the Stoes Projection of u 0 (x) into V h. At the first time level (u h, p h ) (X h, Q h ) are sought, satisfying ( uh u h 0, v h ) + ν( ( uh + u h 0 ), v h ) + αh( u h, v h ) +b (u h 0, uh + u h 0, v h ) ( (ph + p h 0), v h ) = (f(t ), v h ) + αh( u h 0, v h ), v h X h, (..5) ( u h, q h ) = 0, q h Q h. 9

20 Given a time step > 0 and an O() constant α, the method computes u h, u h 3,, p h, p h 3, where t j = j and u h j (x) = u(x, t j ), p h j (x) = p(x, t j ). For n, given (u h n, p h n) (X h, Q h ) find (u h n+, p h n+) (X h, Q h ) satisfying ( uh n+ u h n, v h ) + ν( ( uh n+ + u h n ), v h ) + αh( u h n+, v h ) +b (E[u h n, u h n ], uh n+ + u h n, v h ) ( (ph n+ + p h n), v h ) = (f(t n+ ), v h ) + αh( u h n, v h ), v h X h, (..6) ( u h n+, q h ) = 0, q h Q h. We will refer to Algorithm. as CNLEStab (Cran-Nicolson with Linear Extrapolation Stabilized). If α = 0, i.e. if no stabilization is used, Algorithm. reduces to one studied by G. Baer in 976 [B76] and others, that we will refer to as CNLE. We shall show that Algorithm. (CNLEStab) is unconditionally stable and second order accurate, O( + h + spatial error). The extra stabilization terms added are O(h) because αh( (u h n+ u h n), v h ) = αh( ( uh n+ u h n ), v h ) h( u t ) = O(h). As stated above, each time step of the method requires the solution of only one linear Oseen problem at cell Reynolds number O(). Remar.. At the first time level, a nonlinear treatment of the trilinear term can be used instead of extrapolation: find (u h, p h ) (X h, Q h ), satisfying ( uh u h 0, v h ) + ν( ( uh + u h 0 ), v h ) + αh( u h, v h ) +b ( uh + u h 0, uh + u h 0, v h ) ( (ph + p h 0), v h ) = (f(t ), v h ) + αh( u h 0, v h ), v h X h, (..7) ( u h, q h ) = 0, q h Q h. We shall show that this modification affects neither the stability of the method nor the convergence rate of the velocity error approximation, but increases the convergence rate of pressure approximation. 0

21 The stabilization in the method alters the numerical method s inetic energy rather than in its energy dissipation. Proposition.4 and Section.5 show that Kinetic Energy in CNLEStab = L 3 [ uh n + αh u h n ], Energy Dissipation in CNLEStab = ν L 3 uh n. We shall show in Sections.5 and.6 that this has several interesting consequences. Section. collects some mathematical preliminaries for the analysis that follows. Sections.3 and.4 present a convergence analysis of the method (..). The modification of the method s inetic energy influences the norm in which convergence is proven. A basic convergence analysis is fundamental to a numerical method s usefulness but there are many important questions it does not answer. We try to address some of these in Section.5 and onward. In Section.5 we consider physical fidelity of a simulation produced by the method (..). One aspect of physical fidelity is conservation of important integral invariants of the Euler equations (ν = 0) and near conservation when ν is small. The conservation of the method s inetic energy when ν = 0 is clear from the stability proof in Section.3. The second important integral invariant of the Euler equations in 3d is helicity, [MT9],[DG0],[CCE03] and in d, enstrophy. Approximate conservation of these is explored in Section.5. Section.6 gives some insight into the predictions of (..) of flow statistics in turbulent flows. In Section.7 we present the results of the computational tests. These confirm the rates of convergence, predicted in Section.3.. MATHEMATICAL PRELIMINARIES Recall that (..) is naturally formulated in X := H 0(Ω) d, Q := L 0(Ω). The dual space of X is denoted by X (and its norm, by ), and V = {v X : (q, v) = 0, q Q} is the set of wealy divergence free functions in X. Norms in the Sobolev spaces H (Ω) d (or W (Ω) d ) are denoted by, and seminorms by.

22 Later analysis will require upper bounds on the nonlinear term, given in the following lemma. Lemma.. Let Ω R 3 or R. For all u, v, w X b (u, v, w) C(Ω) u v w, and If, in addition, v, v L (Ω), b (u, v, w) C(Ω) u u v w. b (u, v, w) C(Ω)( v L (Ω) + v L (Ω)) u w and b (u, v, w) C( u v L (Ω) + u v L (Ω)) w. Proof. See Girault and Raviart [GR86] for a proof of the first inequality. The second inequality follows from Hölder s inequality, the Sobolev embedding theorem and an interpolation inequality, e.g., [LT98]. The third bound follows from the definition of the sew-symmetric form and Hölder s inequality b (u, v, w) v L (Ω) u w + v L (Ω) u w, and Poincare s inequality, since w X. The proof of the last inequality can be found, e.g., in [LT98]. Throughout the chapter, we shall assume that the velocity-pressure finite element spaces X h X and Q h Q are conforming, have approximation properties typical of finite element spaces commonly in use, and satisfy the discrete inf-sup, or LBB h, condition inf q h Q h (q h, v h ) sup v h X v h h q h βh > 0, (..)

23 where β h is bounded away from zero uniformly in h. Examples of such spaces can be found in [GR79], [GR86], [G89]. In addition, we assume that an inverse inequality holds, i.e. there exists a constant C independent of h and, such that v Ch v, v X h. (..) We assume that (X h, Q h ) satisfy the following approximation properties typical of piecewise polynomials of degree (m, m ), [BS94]: inf u v v X h Ch m+ u m+, u H m+ (Ω), (..3) inf (u v) v X h Ch m u m+, u H m+ (Ω), (..4) inf p q Ch m p m, p H m (Ω). (..5) q Q h We will also use the following inequality, which holds under (..) and for all u V: inf v V h (u v) C(Ω) inf v X h (u v). (..6) The proof of (..6) can be found, e.g., in [GR79] (p.60, inequality (.)). Throughout the chapter we use the following Stoes Projection. Definition. (Stoes Projection). The Stoes projection operator P S : (X, Q) (X h, Q h ), P S (u, p) = (ũ, p), satisfies ν( (u ũ), v h ) (p p, v h ) = 0, ( (u ũ), q h ) = 0, (..7) for any v h X h, q h Q h. In (V h, Q h ) this formulation reads: given (u, p) (X, Q), find ũ V h satisfying ν( (u ũ), v h ) (p q h, v h ) = 0, (..8) for any v h V h, q h Q h. Under the discrete inf-sup condition (..), the Stoes projection is well defined. 3

24 Proposition. (Stability of the Stoes Projection). Let u, ũ satisfy (..8). The following bound holds ν ũ [ν u + dν inf p q h ], (..9) q h Q h where d is the dimension, d =, 3. Proof. Tae v h = ũ V h in (..8). This gives ν ũ = ν( u, ũ) (p q h, ũ). (..0) Using the Cauchy-Schwarz and Young inequalities, we obtain ν ũ ν u + ν 4 ũ (..) +dν inf p q h + ν q h Q h 4d ũ. Next, use the obvious inequality ũ d ũ. Combining the lie terms in (..) concludes the proof. In the error analysis we shall use the error estimate of the Stoes Projection (..8). Proposition. (Error estimate for the Stoes Projection). Suppose the discrete inf-sup condition (..) holds. Then the error in the Stoes Projection satisfies ν (u ũ) C[ν inf (u v h ) + ν inf p q h ], (..) v h X h q h Q h where C is a constant independent of h and ν. Proof. Decompose the projection error e = u ũ into e = u I(u) (ũ I(u)) = η φ, where η = u I(u), φ = ũ I(u), and I(u) approximates u in V h. Tae v h = φ V h in (..8). This gives ν φ = ν( η, φ) (p q h, φ). (..3) The Cauchy-Schwarz and Young inequalities lead to ν φ ν η + Cν inf p q h. (..4) q h Q h Since I(u) is an approximation of u in V h, we can tae infimum over V h. The proof is concluded by applying (..6) and the triangle inequality. 4

25 Remar.. Using the Aubin-Nitsche lift, one can obtain (see, e.g., [BDK8]) where C = C(ν, Ω). ( ) u ũ Ch inf (u v h ) + inf p q h, (..5) v h X h q h Q h The following variation on the discrete Gronwall Lemma is given in [HR90] as a remar to Lemma 5.. In this estimate, the first sum on the right hand side is only up to the next-to-last time step, which allows for an estimate with no smallness condition on. Lemma. (Discrete Gronwall). Let, B, a n, b n, c n, d n for integers n 0 be nonnegative numbers such that for N, if then for all > 0, a N + a N + n=0 N n=0 N b n d n a n + n=0 ( N N b n exp( d n ) n=0 N c n + B, n=0 ) N c n + B. The following results are readily obtained by Taylor series expansion. Lemma.3. Let = t n+ t n for all i and denote t n+/ = t n++t n. Let ψ(, t) be a function such that ψ t C 0 (0, T ; L (Ω)). Then there exists θ (0, ) such that and n=0 ψ(, t n+) ψ(, t n ) C ψ t (, t n+θ ). If ψ tt C 0 (0, T ; L (Ω)), then there exist θ, θ (0, ) such that ψ(, t n+) + ψ(, t n ) ψ(, t n+/ ) C ψ tt (, t n+θ ) 3 ψ(, t n) ψ(, t n ) ψ(, t n+/ ) C ψ tt (, t n+θ ). If ψ ttt C 0 (0, T ; L (Ω)), then there exists θ 3 (0, ) such that ψ(, t n+) ψ(, t n ) ψ t (, t n+/ ) C ψ ttt (, t n+θ3 ). 5

26 .3 STABILITY AND CONVERGENCE OF THE STABILIZED METHOD We start with the proof of unconditional stability, which is the mathematical ey to the good properties of the method, and motivates the more technical error analysis that follows. The unconditional stability of Algorithm. is proven in the following proposition. Proposition.3. [Stability of extrapolated trapezoidal method] Let f L (0, T ; H (Ω)). The stabilized, extrapolated trapezoid scheme (..5)-(..6) (and the scheme (..6)-(..7)) is unconditionally stable. For any h, > 0 and α 0, n 0 u h n+ + αh u h n+ + ν ( uh i+ + u h i ) u h 0 + αh u h 0 + ν f(t i+ ). Proof. Taing v h = uh +uh 0 V h in (..5) (and in (..7)) gives ( uh u h 0, uh + u h 0 ) + ν ( uh + u h 0 ) + αh( uh u h 0 Apply the Cauchy-Schwarz and Young inequalities. This gives, uh + u h 0 ) = (f(t ), uh + u h 0 ). (.3.) u h u h 0 + ν ( uh + u h 0 ) + αh uh u h 0 ν f(t ) + ν (uh + u h 0 ). (.3.) Thus, on the first time level we obtain the stability bound u h + ν ( uh + u h 0 ) + αh u h u h 0 + αh u h 0 + ν f(t ).(.3.3) Now consider (..6) for n ; let v h = uh n+ +uh n V h. This gives ( uh n+ u h n, uh n+ + u h n ) + ν ( uh n+ + u h n ) + αh( ( uh n+ u h n ), ( uh n+ + u h n )) = (f(t n+ ), uh n+ + u h n ). (.3.4) 6

27 Applying Cauchy-Schwarz and Young inequalities leads to u h n+ u h n + ν ( uh n+ + u h n ) + αh uh n+ u h n ν f(t n+ ) + ν (uh n+ + u h n ). (.3.5) Simplifying (.3.5) gives ( u h n+ u h n ) + ν ( uh n+ + u h n ) + αh( u h n+ u h n ) ν f(t n+ ). (.3.6) Summing (.3.6) over the time levels gives u h n+ + i= ν ( uh i+ + u h i u h + αh u h + ) + αh u h n+ i= ν f(t i+ ). (.3.7) Finally, using the bound on ( u h + αh u h ) from (.3.3), we obtain that for all n u h n+ + ν ( uh i+ + u h i u h 0 + αh u h 0 + ) + αh u h n+ ν f(t i+ ). (.3.8) This result, combined with Proposition., proves the Proposition. Hence the method is unconditionally stable. The question remains: how fast does u h converge to u? To evaluate the rates of convergence as h 0, we must mae a specific choice of X h, Q h. 7

28 Theorem.3. (Velocity Convergence Rates). Let the finite-element spaces (X h, Q h ) include continuous piecewise polynomials of degree m and m respectively (m ), and satisfy the discrete inf-sup condition (..) and approximation properties (..3)-(..5). Let C u L (0,T ;H m+ (Ω))h m 3 /, and u L (0, T ; H m+ (Ω)) L (0, T ; L (Ω)) C 0 (0, T ; H (Ω)), u L (0, T ; L (Ω)), u t L (0, T ; H m+ (Ω)) L (0, T ; L (Ω)), u tt L (0, T ; H (Ω)), p tt L (0, T ; L (Ω)). Then there is a C = C(ν, u, p, T ) < such that n {0,,..., N } the error in Algorithm. satisfies u(t n+ ) u h n+ + ( ) ν ( (u(t i+) u h i+) + (u(t i ) u h i ) ) +α h (u(tn+ ) u h n+) C(ν, u, p) ( h m + αh + ). The rest of this section will be devoted to proving this theorem. Proof. Consider the variational formulation corresponding to the Navier-Stoes equations (..), for any time t, in X h, (u t, v h ) + b (u, u, v h ) + ν( u, v h ) (p, v h ) = (f, v h ), v h X h. (.3.9) Then subtract (..6) from (.3.9), taen at t = t n+, to get (u t (t n+ ) uh n+ u h n, v h ) + ν( u(t n+ αh( (u h n+ u h n), v h ) + b (u(t n+ b (E[u h n, u h n ], uh n+ + u h n Let the velocity error be decomposed as ) ( uh n+ + u h n ), v h ) ), u(t n+ ), v h ), v h ) (p(t n+ ) p(t n+) + p(t n ), v h ) = 0.(.3.0) e n := u(t n ) u h n = (u(t n ) U n ) (u h n U n ) =: η n φ h n, (.3.) 8

29 where U n is the Stoes Projection of u n into V h (therefore φ h n V h, but η n ξ = e, φ h or η, define ξ n+ Add and subtract := ξ n+ +ξ n. / V h ). For ( u(t n+) u(t n ), v h ) + ν( ( u(t n+) u(t n ) ), v h ) +αh( (u(t n+ ) u(t n )), v h ) ( p(t n+) + p(t n ), v h ) +b (u(t n+ ) + E[u(t n ), u(t n )] + E[u h n, u h n ], u(t n+) + u(t n ), v h ) to (.3.0) to obtain the error equation (recall also that (q h, v h ) = 0, q h Q h ) where ( e n+ e n, v h ) + ν( e n+/, v h ) + αh( (e n+ e n ), v h ) = ( p(t n+) + p(t n ) q h, v h ) b (E[u h n, u h n ], e n+/, v h ) +b (E[e n, e n ], u(t n+) + u(t n ), v h ) + T (u, p; v h ), (.3.) T (u, p; v h ) = ( u(t n+) u(t n ) u t (t n+ ), v h ) + ν( ( u(t n+) + u(t n ) αh( (u(t n+ ) u(t n )), v h ) ( p(t n+) + p(t n ) +b (u(t n+ ), u(t n+) + u(t n ) u(t n+ ), v h ) ) u(t n+ ), v h ) p(t n+ ), v h ) b (E[u(t n ), u(t n )] u(t n+ ), u(t n+) + u(t n ), v h ). (.3.3) Using the error decomposition (.3.) and setting v h = φ h n+/ in (.3.) gives ( φh n+ φ h n ) + ν φ h n+/ + αh ( φh n+ φ h n ) = ( η n+ η n, φ h n+/) + ν( η n+/, φ h n+/) +αh( ( η n+ η n ), φ h n+/) ( p(t n+) + p(t n ) q h, φ h n+/) +b (E[u h n, u h n ], η n+/, φ h n+/) + b (E[η n, η n ], u(t n+) + u(t n ), φ h n+/) +b (E[φ h n, φ h n ], u(t n+) + u(t n ), φ h n+/) + T (u, p; φ h n+/), (.3.4) since b (E[u h n, u h n ], φ h n+/, φ h n+/) = 0. 9

30 Also it follows from the choice of the projection U n that ν( η n+/, φ h n+/) ( p(t n+) + p(t n ) q h, φ h n+/) = 0. Applying the Cauchy-Schwarz and Young s inequalities to the linear terms on the right hand side of (.3.4) gives ( φh n+ φ h n ) + 3ν 4 φh n+/ + αh ( φh n+ φ h n ) Cν η n+ η n + Cν αh ( η n+ η n ) + b (E[u h n, u h n ], η n+/, φ h n+/) + b (E[η n, η n ], u(t n+) + u(t n ), φ h n+/) + b (E[φ h n, φ h n ], u(t n+) + u(t n ), φ h n+/) + T (u, p; φ h n+/), (.3.5) For clarity, we analyze each of the remaining nonlinear terms on the RHS of (.3.5) individually. Here we use frequently Lemma. and the inverse estimate (..), together with Young s inequality. We start with the first nonlinear term in (.3.5). Adding and subtracting the quantity b (E[u(t n ), u(t n )], η n+/, φ h n+/), and using Lemma., followed by Young s inequality, we get b (E[u h n, u h n ], η n+/, φ h n+/) ν 6 φh n+/ + Cν E[u(t n ), u(t n )] η n+/ + Cν E[η n, η n ] η n+/ + C E[φ h n, φ h n ] / E[φ h n, φ h n ] / η n+/ φ h n+/. (.3.6) The first two terms involving the operator E[, ] can be bounded by using its definition (..4) and regularity assumptions on u, E[u(t n ), u(t n )] C and E[η n, η n ] 3 η n + η n.(.3.7) 0

31 For the third and fourth terms, we also need the inverse estimate (..), resulting in E[φ h n, φ h n ] E[φ h n, φ h n ] C( φ h n + φ h n ) ( φ h n + φ h n ), Ch ( φ h n + φ h n ), so that E[φ h n, φ h n ] / E[φ h n, φ h n ] / η n+/ φ n+/ Ch 3/ η n+/ ( φ h n + φ h n ) ( φ h n + φ h n+ ), (.3.8) Putting (.3.7) and (.3.8) bac into (.3.6), we have b (E[u h n, u h n ], η n+/, φ h n+/) ν 6 φh n+/ + Cν η n+/ +Cν ( η n + η n ) η n+/ +Ch 3/ η n+/ ( φ h n + φ h n + φ h n+ ). (.3.9) For the second trilinear term, use Lemma. and the assumption that u(t) is bounded for any t [0, T ]. Then we apply Young s inequality and (.3.7), resulting in b (E[η n, η n ], u(t n+) + u(t n ), φ h n+/) C E[η n, η n ] φ h n+/ Cν ( η n + η n ) + ν 6 φh n+/. (.3.0) The third trilinear term is bounded with the help of the third inequality in Lemma. and the regularity assumptions on u. As a result, b (E[φ h n, φ h n ], u(t n+) + u(t n ), φ h n+/) C E[φ h n, φ h n ] φ h n+/ Cν ( φ h n + φ h n ) + ν 6 φh n+/, (.3.) where the last step follows from Young s inequality.

32 Now, with (.3.9), (.3.0) and (.3.), the error equation (.3.5) can be rewritten as ( φh n+ φ h n ) + 9ν 6 φh n+/ + αh ( φh n+ φ h n ) Cν η n+ η n + Cν αh ( η n+ η n ) + Cν η n+/ +Cν ( η n + η n ) η n+/ +Cν ( η n + η n ) + Cν ( φ h n + φ h n ) +Ch 3/ η n+/ ( φ h n + φ h n + φ h n+ ) + T (u, p; φ h n+/), (.3.) and what is left is to bound T (u, p; φ h n+/). Each of its four linear terms can be bounded by the Cauchy-Schwarz and Young s inequalities, together with the estimates in Lemma.3. We tae care of one at a time below. ( u(t n+) u(t n ) u t (t n+/ ), φ h n+/) ν 80 φh n+/ +Cν 4 u ttt (t n+θ ), (.3.3) ν ( ( u(t n+) + u(t n ) u(t n+ )), φ h n+/) ν 80 φh n+/ +Cν 4 u tt (t n+θ ), (.3.4) αh ( ( u(t n+) u(t n ) ), φ h n+/) ν 80 φh n+/ +Cν α h u t (t n+θ3 ), (.3.5) ( p(t n+) + p(t n ) p(t n+/ ), φ h n+/) ν 80 φh n+/ +Cν 4 p tt (t n+θ4 ), (.3.6) for some θ, θ, θ 3, θ 4 (0, ).

33 For the two nonlinear terms in T (u, p; φ h n+/), use Lemma., Lemma.3 and Young s inequality, together with u(t) C, for any t [0, T ]. This gives b (E[u(t n ), u(t n )] u(t n+/ ), u(t n+) + u(t n ), φ h n+/) + b (u(t n+/ ), u(t n+) + u(t n ) u(t n+/ ), φ h n+/) C(Ω) ( 3 u(t n) u(t n ) u(t n+/ )) ( u(t n+) + u(t n ) ) φ h n+/ +C(Ω) ( u(t n+) + u(t n ) u(t n+/ )) u(t n+/ ) φ h n+/ Cν 4 u tt (t n+θ5 ) + ν 80 φh n+/, (.3.7) for some θ 5 (0, ). Combining (.3.3)-(.3.7), we have T (u, p; φ h n+/) ν 6 φh n+/ + Cν 4 ( u ttt (t n+θ ) + p tt (t n+θ4 ) ) +Cν 4 u tt (t n+θ5 ) + Cν α h u t (t n+θ3 ), (.3.8) so that error equation (.3.) gives ( φh n+ φ h n ) + ν φh n+/ + αh ( φh n+ φ h n ) Cν η n+ η n + Cν αh ( η n+ η n ) + Cν η n+/ +Cν ( η n + η n ) η n+/ +Cν ( η n + η n ) + Cν ( φ h n + φ h n ) +Ch 3/ η n+/ ( φ h n + φ h n + φ h n+ ) +Cν 4 ( u ttt (t n+θ ) + p tt (t n+θ4 ) ) +Cν 4 u tt (t n+θ5 ) + Cν α h u t (t n+θ3 ). (.3.9) 3

34 Multiply both sides of (.3.9) by and use (..),(..5) together with the approximation properties (..3)-(..5) of the spaces (X h, Q h ). Then sum over the time levels from to n, choosing U 0 = u h 0, which gives φ h 0 = 0, and φ h n+ + ν φ h i+/ + αh φ h n+ i= φ h + αh φ h + Cν h m+ u t L (0,T ;H m+ (Ω)) +Cν αh m+ u t L (0,T ;H m+ (Ω)) + Cν h m u L (0,T ;H m+ (Ω)) +Cν h 4m u L (0,T ;H m+ (Ω)) +Cν h m u L (0,T ;H m+ (Ω)) + Cν 4 ( u ttt L (0,T ;L (Ω)) + p tt L (0,T ;L (Ω)) ) +Cν 4 u tt L (0,T ;L (Ω)) + Cν α h u t L (0,T ;L (Ω)) +Cν ( φ h i + φ h i ) i= +Ch m 3/ u(t i+/ ) m+ ( φ h i + φ h i + φ h i+ ). (.3.30) i= Since u L (0, T ; H m+ (Ω)), the last two sums in (.3.30) can be combined as C u L (0,T ;H m+ (Ω))h m 3/ φ h n+ + C(h m 3/ + ν ) φ h i. i= Using the regularity of u and p, and the assumption that C u L (0,T ;H m+ (Ω))h m 3/ /, the error equation finally taes the form φh n+ + ν φ h i+/ + αh φ h n+ i= φ h + αh φ h + Cν ( + h + αh + h m )h m +Cν α h + C(ν + ν) 4 +C (ν + h m 3/ ) φ h i. (.3.3) i= To complete the proof bounds are needed for φ h in the above estimates. These bounds depend upon the way the first time step is taen, and there are two possibilities (..5) and (..7); we shall analyze both. Both lead to an optimal velocity error estimate. The more expensive method (..7) also leads to an optimal pressure error estimate (in Theorem.4.3 4

35 below). The error equation for φ h is the same as for φ h n except for the nonlinear terms, and is treated in the same way, except for the nonlinear term. Therefore, we go directly to the treatment of the nonlinear term in both cases (..5) and (..7). We start with formulation (..5). Adding and subtracting b (u h 0 u(t 0 ), u(t 0)+u(t ), v h ) to the nonlinear terms in (.3.0), we have b (u(t / ), u(t / ), v h ) b (u h 0, uh 0 + u h, v h ) = b (u(t / ), u(t / ), v h ) + b (u h 0, e /, v h ) b (e 0, u(t 0) + u(t ), v h ) + b (u(t 0 ), u(t 0) + u(t ), v h ). (.3.3) Taing v h = φ h /, the second and third terms in (.3.37) can be treated exactly as in (.3.6), (.3.0) and (.3.). The first and last are bounded as follows. Using Lemma.3 and the fact that there exists t θ (0, ) such that u(t / ) u(t 0 ) = u t (t θ ), we obtain b (u(t 0 ), u(t ) + u(t 0 ), φ h /) b (u(t / ), u(t / ), φ h /) = b (u(t 0 ), u(t / ) + C u tt (t θ ), φ h /) b (u(t / ), u(t / ), φ h /) b (u(t 0 ) u(t / ), u(t / ), φ h /) + C b (u(t 0 ), u tt (t θ ), φ h /) b (u t (t θ ), u(t / ), φ h /) + C b (u(t 0 ), u tt (t θ ), φ h /) b (u t (t θ ), u(t / ), φ h /) + ɛν φ h / + Cν 4. (.3.33) In order to bound the first term in (.3.33), we use integration by parts and Hölder s inequality to obtain b (u t (t θ ), u(t / ), φ h /) = (u t (t θ ) u(t / ), φ h /) + ( u t(t θ ), u(t / ) φ h /). (.3.34) Thus, b (u t (t θ ), u(t / ), φ h /) C( u t (t θ ) u(t / ) L (Ω) + u t (t θ ) u(t / ) L (Ω)) φ h / C φh /. (.3.35) 5

36 Now use the bounds (.3.33) and (.3.35) in the error analysis at the first time level (note that φ h / = φh, since φ h 0 = 0) to get φ h + ν φ h + αh φ h C[ν h m + ν h m + ν h m+ +ν α h 3 + ν αh m+ + ν h 4m +ν 5 + ν ]. (.3.36) If formulation (..7) is used, then, instead of (.3.37), we obtain, by adding and subtracting b ( uh +uh 0 u(t )+u(t 0 ) + u(t / ), u(t 0)+u(t ), v h ) to the nonlinear terms in first time level analog of (.3.0), the following b (u(t / ), u(t / ), v h ) b ( uh 0 + u h, uh 0 + u h, v h ) = b (u(t / ), u(t / ) u(t ) + u(t 0 ), v h ) +b ( uh 0 + u h, e /, v h ) b (e 0, u(t 0) + u(t ), v h ) +b ( u(t 0) + u(t ) u(t / ), u(t 0) + u(t ), v h ) (.3.37) Taing v h = φ h /, the second and third terms in (.3.37) can be treated exactly as in (.3.6), (.3.0) and (.3.). The first and last are similar, since, after application of Lemma. and regularity assumptions on u, both can be bounded as C (u(t / u(t 0) + u(t ) ) φ h / ɛν φ h / + Cν 4, with the help of Lemma.3 and Young s inequality. This leads to the upper bound φ h + ν φ h + αh φ h C[ν h m + ν h m + ν h m+ + ν α h 3 +ν αh m+ + ν h 4m + ν 5 + ν 5 ]. (.3.38) This bound is sharper than (.3.36), but it will not contribute to a higher order estimate. We thus insert the bound for φ h + αh φ h, obtained in (.3.36), into (.3.3), which gives φ h n+ + ν ( φh i+ + φ h i ) + αh φ h n+ C(ν + ν ) ( h m + α h + 4) + Cν ( 6 φ h i ). (.3.39)

37 Hence, it follows from the discrete Gronwall Lemma, that there exists C = C(ν, Ω, T, u, p) such that for any n 0 φ h n+ + ν ( φh i+ + φ h i ) + αh φ h n+ C ( h m + α h + 4). (.3.40) Finally, the statement of the theorem follows from the triangle inequality..4 ERROR ESTIMATES FOR TIME DERIVATIVES AND PRESSURE In order to prove pressure stability and convergence, we need to derive a bound on the time difference of the velocity error e n+ e n. Theorem.4.. Let the finite-element spaces (X h, Q h ) include continuous piecewise polynomials of degree m and m respectively (m ) and satisfy the discrete inf-sup condition. Let the assumptions of Theorem.3. be satisfied and u tt L (0, T ; L (Ω)), u tt L (0, T ; L (Ω)), u ttt L (0, T ; L (Ω)), p tt L (0, T ; L (Ω)). Then, if the finite element approximation u h n constant C = C(ν, u, p, T ) < such that is defined by (..5)-(..6), there exists a ν ( e n e n ) + ν ( e n + e n ) n + e i+ e i n + αh ( e i+ e i ) C(h m + α h + h ). (.4.) 7

38 If the finite element approximation u h n C = C(ν, u, p, T ) < such that is defined via (..7)-(..6), then there exists a ν ( e n e n ) + ν ( e n + e n ) n + e i+ e i n + αh ( e i+ e i ) C(h m + α h + h ). (.4.) Proof. Consider the error decomposition (.3.). Tae v h = φh n+ φh n V h in (.3.),(.3.3) to obtain φh n+ φ h n + ν φh n+ φ h n = ( η n+ η n, φh n+ φ h n ( p(t n+) + p(t n ) + αh ( φh n+ φ h n ) ) + ν( ( η n+ η n q h, φh n+ φ h n ) +b (E[u(t n ), u(t n )], e n+/, φh n+ φ h n ) b (E[η n, η n ], e n+/, φh n+ φ h n ) +b (E[φ n, φ n ], e n+/, φh n+ φ h n ) +b (E[e n, e n ], u(t n+) + u(t n ), φh n+ φ h n ) ), ( φh n+ φ h n )) +T (u, p; φh n+ φ h n ), (.4.3) 8

39 where, using Taylor expansion, T (u, p; φh n+ φ h n ) = ( u(t n+) u(t n ) +αh( ( η n+ η n u t (t n+/ ), φh n+ φ h n ) ), ( φh n+ φ h n )) +C ν( u tt (t n+θ ), ( φh n+ φ h n +C b (u tt (t n+θ ), u(t n+) + u(t n ) +C b (u(t n+/ ), u tt (t n+θ ), φh n+ φ h n ) +αh( ( uh n+ u h n ), ( φh n+ φ h n )) )) + C (u ttt (t n+θ ), φh n+ φ h n ), φh n+ φ h n ) +C (p tt (t n+θ ), ( φh n+ φ h n )), (.4.4) for some θ (0, ) and q h Q h. Also it follows from the definition of Stoes Projection that ν( ( η n+ η n ), ( φh n+ φ h n )) ( p(t n+) + p(t n ) q h, φh n+ φ h n ) = 0. (.4.5) We bound the four nonlinear terms on the right-hand side of (.4.3), using Lemma. and Cauchy-Schwarz and Young s inequalities. For the first term integrating by parts and applying Hölder s inequality gives b (E[u(t n ), u(t n )], e n+/, φh n+ φ h n ) = (E[u(t n ), u(t n )] e n+/, φh n+ φ h n ) + ( E[u(t n), u(t n )], e n+/ φh n+ φ h n ) = (E[u(t n ), u(t n )] e n+/, φh n+ φ h n ) C e n+/ φh n+ φ h n ɛ φh n+ φ h n + C ( e n+ + e n ). (.4.6) 9

40 Using the first bound from Lemma. and the inverse inequality (..), we obtain the bounds on the second and third nonlinear terms b (E[η n, η n ], e n+/, φh n+ φ h n ) Ch E[η n, η n ] e n+/ φh n+ φ h n ɛ φh n+ φ h n and, using also the intermediate result (.3.40) of Theorem.3., + Ch ( 3 η n η n ) ( e n+ + e n ) (.4.7), b (E[φ n, φ n ], e n+/, φh n+ φ h n ) Ch 3/ E[φ n, φ n ] ( e n+ + e n ɛ φh n+ φ h n ) φh n+ φ h n + Ch 3 (h m + α h + 4 ) ( e n+ + e n ). (.4.8) Finally, consider the fourth nonlinear term. Use the obvious identity 3 e n e n = e n+e n + (e n e n ) and the regularity of u. It follows from the last inequality of Lemma. that b (E[e n, e n ], u(t n+) + u(t n ) b ( e n + e n, φh n+ φ h n ), u(t n+) + u(t n ) + b (η n η n, u(t n+) + u(t n ) + b ( φh n φ h n ɛ φh n+ φ h n, u(t n+) + u(t n ) + C ( e n+ + e n ), φh n+ φ h n ), φh n+ φ h n ), φh n+ φ h n ) +C(h m + α h + 4 ) + C ( φh n φ h n ). (.4.9) Insert these bounds in (.4.3). The bound on T (u, p; φh n+ φh n ) is obtained as in the proof of Theorem.3.. Choosing ɛ = 4 gives φh n+ φ h n + ν φh n+ φ h n C ( e n+ + e n + αh (φh n+ φ h n ) ) + C ( φh n φ h n ) + C(h m + α h + 4 ) +Ch 3 (h m + α h + 4 ) ( e n+ + e n ). (.4.0) 30

41 At the first time level, tae v h = φh φh 0 ; taing U 0 = u h 0 in the initial error decomposition gives φ h 0 = 0. For the constant extrapolation (..5) we obtain φh φ h 0 + ν φh φ h 0 + αh (φh φ h 0 ) C(h m + α h + 4 ) + b (u t (t θ ), u /, φh φ h 0 ). (.4.) If we use (..7) instead of (..5) at the first time level, we have φh φ h 0 + ν φh φ h 0 + αh (φh φ h 0 ) C(h m + α h + 4 ) + b (u t (t θ ), u /, φh φ h 0 ). (.4.) Sum (.4.0) over the time levels n and add to (.4.) (or to (.4.) in the case of linear extrapolation). Multiply by to obtain φh i+ φ h i + ν φ h n+ + αh n C ( φh i+ φ h i ( φh i+ φ h i ) ) + C(h m + α h h 3 8 ) + +σ b (u t (t θ ), u(t / ), φh φ h 0 ), (.4.3) where σ = 0 for the constant extrapolation (..5) and σ = for the linear extrapolation (..7). For any n, add the inequalities (.4.3) at the time levels n + and n. Use the identity φ h n+ + φ h n = ( φh n+ φ h n 3 ) + ( φh n+ + φ h n ). (.4.4)

42 At any time level n we obtain ν ( φh n+ φ h n ) + ν ( φh n+ + φ h n ) + φh i+ φ h i + αh n Cν ν ( φh i+ φ h i ) +C(h m + α h h 3 8 ) ( φh i+ φ h i + +σ b (u t (t θ ), u(t / ), φh φ h 0 ). (.4.5) Next, decompose the last term in the right-hand side of (.4.5), using Lemma. and Young s inequality. This yields ) +σ b (u t (t θ ), u(t / ), φh φ h 0 Hence it follows from the discrete Gronwall Lemma that ) φh φ h 0 + C 3+σ. (.4.6) ν ( φh n+ φ h n ) + ν ( φh n+ + φ h n ) + φh i+ φ h i +αh ( φh i+ φ h i ) C(h m + α h σ + h 3 8 ).(.4.7) The proof of the theorem is now concluded by the triangle inequality. For the stability of pressure we will need the following á priori bounds Lemma.4. Let the assumptions of Theorem.4. hold. C = C(ν, u, p, T ) such that for any n Then there exists a constant ( uh n+ u h n uh i+ u h i ) ( e n+ e n e i+ e i + u(t i+) u(t i ) C, ) + ( u(t n+) u(t n ) ) C. Proof. Use the decomposition u h i = u(t i ) e i. The triangle inequality completes the proof. 3

43 Theorem.4. (Pressure Stability). Let (u h n, p h n) satisfy (..5)-(..6) (or (..7)-(..6)). Let f L (0, T ; H (Ω)) and let the assumptions of Theorem.4. be satisfied. Then, n ph i+ + p h i C(u h 0, f, β h ), where β h is the constant from the discrete LBB h condition (..). Proof. Consider (..6). Using the Cauchy-Schwarz inequality, the first bound from Lemma., the discrete LBB h condition (..) and the identity 3 uh n+ uh n = uh n+ +uh n + uh n+ uh n, we obtain β h ph n+ + p h n uh n+ u h n +αh ( uh n+ u h n + ν ( uh n+ + u h n ) ) + C ( uh n+ + u h n ) +C ( uh n+ u h n ) + f(t n+/ ). Sum over all time levels; the bounds of Lemma.4 complete the proof. We conclude this section by deriving the pressure error estimate. Theorem.4.3 (Pressure Convergence). Let (u h n, p h n) satisfy (..6) for n. Let (u h, p h ) satisfy the constant extrapolation (..5) or the linear extrapolation (..7). Then, under the assumptions of Theorem.4., n p(t i+/ ) p h i+/ C(ν, u, p, T )(h m + αh + h 3/ 4 + 3/+σ/ ), (.4.8) where σ = 0 for the constant extrapolation and σ = for the linear extrapolation. 33

44 Proof. Consider (.3.0), which holds true for any v h X h. Decompose the pressure approximation error into p(t n+ ) p h n+ = (p(t n+ ) I(p)) (p h n+ I(p)) = η n+ φ h n+, (.4.9) where φ h n+ Q h, I(p) is a projection of p(t n+ ) into Q h. Use the error decomposition (.4.9) in (.3.0) and apply the discrete LBB h condition to obtain for any n β h φ h n+ + φ h n e n+ e n +C ( e n+ + e n +ν ( e n+ + e n + C ( e n+ + e n ) ) + C ( e n e n ) + αh ( e n+ e n ) ) + C ( e n e n ) + η n+ + η n + Cν + C + Cαh. (.4.0) Hence from the triangle inequality we get β h (p(t n+) p h n+) + (p(t n ) p h n) e n+ e n +C ( e n+ + e n +C ( e n e n + C ( e n+ + e n ) ) + C ( e n e n ) ) + ν ( e n+ + e n ) +αh ( e n+ e n ) + Cν + C + Cαh + inf p(t n+) + (p(t n ) q h. (.4.) q h Q h On the first time level consider the constant extrapolation (..5). Using the discrete LBB h condition and (.3.36), we obtain the following bound (which can be improved in the case of linear extrapolation): β h (p(t ) p h ) + (p(t 0 ) p h 0) C( + h m + αh). (.4.) Add the inequalities (.4.) for all n, multiply by and add to (.4.). The proof is concluded by applying the result of Theorem.4. 34

45 .5 PHYSICAL FIDELITY: CONSERVATION OF INTEGRAL INVARIANTS We begin by proving that CNLEStab exactly conserves a modified inetic energy. Proposition.4. Let the boundary conditions be periodic; assume also f = ν = 0. Define Kinetic energy in (..6)= KE(t n ) := L 3 [ uh n + αh u h n ] The method exactly conserves inetic energy. Specifically, for all t n > 0 KE(t n ) = KE(0). Proof. Set v h = uh n+ +uh n and ν = f = 0 in (..5)-(..6). Exact conservation of helicity liely does not hold for CNLEStab. We thus consider approximate helicity conservation experimentally by considering and inviscid (ν = 0) fluid with no forcing term (f = 0). A comparison of the CNLE and CNLEStab under these conditions, in Figure, shows that, for a fixed mesh size, CNLEStab nearly conserves helicity, while CNLE does not. Both conserve inetic energies during these experiments. 35

46 Figure : Conservation of helicity, CNLE (α = 0) versus CNLEStab (α = ) A comparison of the CNLEStab performance for different mesh sizes is shown in Figure, confirming that as the mesh is refined, conservation of helicity improves..6 PHYSICAL FIDELITY: PREDICTIONS OF THE TURBULENT ENERGY CASCADE We consider the energy cascade predicted by (.) in the case of homogeneous, isotropic turbulence. Motivated by the consistency error argument, we consider the modified equation of the method (.). Since αh( u(t n+ ), v) αh( u(t n ), v) = αh( u t, v), we postulate a fluid with equations of motion given by: w : Ω [0, T ] R d, q : Ω (0, T ] R 36

47 Figure : Conservation of helicity for different mesh sizes in CNLESTAB (with α = ): as h gets smaller, helicity is conserved longer. satisfying: [w αh w] t + w w ν w + q = f, for x Ω, 0 < t T w = 0, x Ω, for 0 < t T, periodic boundary conditions on Ω, for 0 < t T, (.6.) w(x, 0) = u 0 (x), for x Ω, and the usual normalization condition in the periodic case that φ(x, t) dx = 0 on φ = Ω w, q, f, u 0 for 0 < t T. Thus we explore more subtle effects of the stabilization in Algorithm. through its modified equation (.6.). 37

48 We multiply (.6.) by w and integrate over the domain and time to obtain its precise energy balance given by T { w(t) + αh w(t) } + ν w(t) = 0 { w(0) + αh w(0) } + (f(t), w(t)). We can clearly identify three physical quantities of inetic energy, energy dissipation rate and power input. Let L denote the global length scale, e.g., L = vol(ω) /3 ; then these are given by Modified equations inetic energy: E model (w)(t) := L 3 { w(t) + αh w(t) },(.6.) Modified equations dissipation rate: ε model (w)(t) := ν L 3 w(t), (.6.3) Modified equations power input: P model (w)(t) := (f(t), w(t)). (.6.4) L3 The inetic energy has an extra term which reflects extraction of energy from resolved scales. The energy dissipation rate in the model (.6.3) is the same as for NSE equations. Equation (.6.) shares the common features of the Navier-Stoes equations which mae existence of an energy cascade liely, e.g. [F95], [P00]. First, (.6.) has the same nonlinearity as the Navier-Stoes equations, which pumps energy from larger to smaller scales. Next, the solution of (.6.) satisfies an energy equality in which its inetic energy and energy dissipation are readily discernible, and for ν = 0 the inetic energy is conserved through a large range of scales/wave-numbers. Since both conditions are satisfied we are to proceed to develop a quantitative theory of energy cascade of (.6.). 38

49 .6. Kraichnan s Dynamic Analysis Applied to CNLEStab The energy cascade will now be investigated more closely using the dynamical argument of Kraichnan, [K7]. Let Π model (κ) be defined as the total rate of energy transfer from all wave numbers < κ to all wave numbers > κ (not to confuse the wave number κ with the time step ). Following Kraichnan [K7] we assume that Π model (κ) is proportional to the total energy ( κe model (κ) ) in wave numbers of the order κ and to some effective rate of shear σ(κ) which acts to distort flow structures of scale /κ. That is: Π model (κ) σ(κ) κ E model (κ) (.6.5) Furthermore, we expect σ(κ) κ 0 p E model (p)dp (.6.6) The major contribution to (.6.6) is from p κ, in accord with Kolmogorov s localness assumption, [Kol4]. This is because all wave numbers κ should contribute to the effective mean-square shear acting on wave numbers of order κ, while the effects of all wave numbers κ can plausibly be expected to average out over the scales of order /κ and over times the order of the characteristic distortion time σ(κ). T 0 Let E(κ) := lim sup T E(κ, t) is the distribution of the time averaged inetic T energy by wave number. Here, we have E(κ, t) = L π =κ û(, t) where L - the reference length,, κ - the wave number vector and the wave number respectively, and û(, t) - the Fourier modes of the Navier-Stoes velocity. We shall say that there is an energy cascade if in some inertial range, Π model (κ) is independent of the wave number, i.e., Π model (κ) = ε model. Using the equations (.6.5) and (.6.6) we get E model (κ) ε /3 model κ 5/3 We have the relation E model (κ) ( + αhκ )E(κ). (.6.7) 39

50 Using (.6.7) we obtain: Model s cutoff lengthscale : αh, E(κ) ε /3 model κ 5/3, for κ (αh) /, E(κ) ε /3 model (αh) κ /3, for κ (αh) /. Therefore, (.6.) possesses an energy cascade with an enhanced inetic energy. The extra term in (.6.) triggers an accelerated energy decay of O(κ /3 ) beyond the cutoff length scale. Above the cutoff length scale (.6.) predicts the correct energy cascade of O(κ 5/3 )..7 COMPUTATIONAL TESTS We first test convergence rates for a problem with a nown exact solution. The example is one for which the true solution is nown, cos(π(z + t)) u = sin(π(z + t)), (.7.) sin(π(x + t)) and then the right-hand side f and initial condition u 0 are computed such that (.7.) satisfies (..). We selected this test problem because it is simple but already possesses complex rotational structures. For α =, ν = and final time T = 0.5, the calculated convergence rates in Table confirm what is predicted by Theorem.3. for (P, P ) discretization in space. Next we give a simple test of the positive effects of the stabilization on the methods complexity. The linear solver used in the simulations was (unpreconditioned) Conjugate Gradient Squared (CGS). On a h = /6 mesh in R 3, with ν = 500 and the same true solution (.7.), the number of CGS iterates needed for the first 8 solves of Cran-Nicolson with Linear Extrapolation (CNLE), i.e. α = 0, and CNLE with stabilization (CNLEStab, α > 0) are compared in Table. The linear system to be solved at each time step is also better conditioned when α > 0. 40

51 h u u h H (Ω) ratio rate / / / Table : Experimental convergence rates. time level CNLE CNLEStab Table : Number of CGS iterations for CNLE versus CNLEStab..8 CONCLUSIONS A simple second order time stepping algorithm for the Navier-Stoes equations was analyzed. It is a modification (by introduction of artificial viscosity stabilization and correction for the associated loss of accuracy) of the commonly used Cran-Nicolson scheme that requires the solution of only one linear system per time step. We not only proved that it is unconditionally stable and investigated how the rates of convergence for velocity and pressure behave, but we also went beyond error analysis. We showed that this scheme conserves inetic energy exactly, and provided experimental numerical evidence that it nearly conserves helicity, an 4

52 important integral invariant in three dimensional rotational flows. Dynamic analysis applied to their algorithm reveals the existence of an energy cascade with the correct statistics up to a cutoff length scale and with an accelerated energy decay above the cutoff length scale. Lastly, we presented more computational tests. The first confirms the velocity convergence rates obtained in the analysis in Section 3, and the second shows that even with a simple, unpreconditioned iterative method the linear system to be solved at each time step is better conditioned than the corresponding system without stabilization. 4

53 3.0 A DEFECT CORRECTION METHOD FOR THE TIME-DEPENDENT NAVIER-STOKES EQUATIONS 3. INTRODUCTION In the numerical solution of higher Reynolds number flow problems some of the standard iterative methods fail - see, e.g., [ES00] and remars in [S03] (p. 48, section..), [B96] (p.4), [C00] (project overview). Often failure means that the iterative method used to solve the linear and/or nonlinear system for the approximate solution at the new time level failed to converge within the time constraints of the problem or the resulting approximation had poor solution quality. The first type of failure can usually be overcome easily by using an upwind or artificial viscosity (AV) discretization at the expense of decreasing dramatically the accuracy of the method and possibly even altering the predictions of the simulation at the qualitative, O() level, therefore increasing the lielihood of the second type of failure. One interesting approach to attaining (by a convergent method) an approximate solution of desired accuracy is the defect correction method (DCM). Briefly, let a th order accurate discretization of the equilibrium Navier-Stoes equations (NSE) be written as NSE h (u h ) = f, (3..) The DCM computes u h,..., u h as αh h u h + NSE h (u h ) = f, (3..) αh h u h l + NSE h (u h l ) = f αh h u h l, for l =,...,, where the velocity approximations u h i polynomials of degree. are sought in the finite element space of piecewise 43

54 It has been proven under quite general conditions (see, e.g., [LLP0]) that for the intermediate approximations of the equilibrium NSE u NSE u h l energy norm = O(h + h u NSE u h l energy norm ) = O(h + h l ), and thus, after l = steps, u u h energy norm = O(h ). Note that (3..) requires solving an AV approximation times which is often cheaper and more reliable than solving (3..) once. In problems with high Reynolds number we may expect turbulence. In that case the DCM needs to be combined with appropriate turbulence models. These models tend to introduce extra nonlinearities (due to the closure of the model); it might be possible to incorporate them into the residual on the right-hand side, as was done in the quasistatic case by Ervin, Layton, Maubach [ELM00]. There has been an extensive study and development of this approach for equilibrium flow problems, see e.g. Hemer[Hem8], Koren[K9], Heinrichs[Hei96], Layton, Lee, Peterson[LLP0], Ervin, Lee[EL06], and subsection 3.. for a review of this wor. For many years, it has been widely believed that (3..) can be directly imported into implicit time discretizations of flow problems in the obvious way: discretize in time, given u h (t OLD ), the quasistatic flow problem for u h (t NEW ) is solved by applying (3..) directly, resulting in αh h u h (t NEW ) + B(u h (t NEW ), u h (t OLD )) + NSE h (u h (t NEW )) = f, (3..3) αh h u h l (t NEW ) + B(u h l (t NEW ), u h (t OLD )) + NSE h (u h l (t NEW )) = f αh h u h l (t NEW ), for l =,...,, where B is a time stepping operator (e.g., Bacward Euler), and is the degree of piecewise polynomials in the finite element space. Unfortunately, this natural idea doesn t seem to be even stable (see Section 3.7). On the other hand, there is a parallel development of DCM s, for initial value problems in which no spacial stabilization (such as αh h in (3..)) is used, but DCM is used to 44

55 increase the accuracy of the time discretization. This wor contains no reports of instabilities: see, e.g., Heywood, Rannacher[HR90], Hemer, Shishin[HSS], Lallemand, Koren[LK93], Minion[M04]. Yet, in spite of this parallel development and after 30+ years of studies of (3..), there has yet to be a successful extension of (3..) to time dependent flow problems. This chapter will present this extension of (3..) to the time dependent problem. We notice that the obvious extension, described above, is in fact unstable, see Section 3.7. We give a small but critically important modification of the above natural extension to time dependent problems, that we prove to be unconditionally stable (Theorem 3.) and convergent (Theorem 3.). We complement the stability proof of the modified DCM by a complete error analysis, which confirms the expected error in the resulting method: u(t n ) u h l (t n) energy norm = O( t a + h + h l ), l =,...,, where a is the order of accuracy of the (implicit) time stepping employed. The error analysis is necessarily technical. To eep the details under some control, we study the bacward Euler time discretization (It will be clear from our analysis that extension to more accurate time discretizations requires no new ideas and only more pages). In subsection 3.. we review important previous wor on DCM in space and DCM in time. Section 3. begins with (the inevitable) notation and preliminaries. Section 3.3, the heart of the chapter, gives the stability proof. The error analysis is given in Sections 3.4, 3.5. and Section 3.7 gives a numerical illustration. Consider the time dependent, incompressible Navier-Stoes equations u t Re u + u u + p = f, for x Ω, 0 < t T, (3..4) where Ω R d, d =, 3. u = 0, x Ω, 0 < t T, u(x, 0) = u 0 (x), x Ω, u Ω = 0, for 0 < t T, Before proceeding with the analysis we shall present carefully next the precise extension of (3..) to the time dependent NSE that we study. 45

56 Let X h X, Q h Q be finite-dimensional finite element spaces. Denote the finiteelement discretization of the Navier-Stoes operator by N h Re(u, p) u t Re h u + (u h )u + h p. Adding an artificial viscosity parameter to the inverse Reynolds number leads to the modified Navier-Stoes operator N h Re (u, p) u t (h + Re ) h u + (u h )u + h p. The method proceeds as follows: first we compute the AV approximation (u, p ) (X h, Q h ) via N h Re (u, p ) = f. The accuracy of the approximation is then increased by the correction step: (u, p ) (X h, Q h ), satisfying compute N h Re (u, p ) N h Re (u, p ) = f N h Re(u, p ). The Bacward Euler time discretization, combined with the two-step defect correction method in space leads to the following system of equations for (u h,n+, p h,n+ ), (u h,n+, p h,n+ ) (X h, Q h ), v h X h at t = t n+, n 0, with := t = t i+ t i u h,n, v h ) + (h + Re )( u h,n+, v h ) + b (u h,n+, u h,n+, v h ) (3..5) (p h,n+, v h ) = (f(t n+ ), v h ), u h,n, v h ) + (h + Re )( u h,n+, v h ) + b (u h,n+, u h,n+, v h ) (p h,n+, v h ) = (f(t n+ ), v h ) + h( u h,n+, v h ), ( uh,n+ ( uh,n+ where b (,, ) is the explicitely sew-symmetrized trilinear form, defined below. The initial value approximations are taen to be u h,0 = u h,0 = u s 0, where u s 0 is the modified Stoes projection of u 0 onto the space V h of discretely divergence-free functions (this projection and this space are defined in section 3.). The stability and error estimate for the modified Stoes projection are proven in the sections 3.3 and

57 3.. Previous results Many iterative methods can be written as a Defect Correction method, see e. g. Bohmer, Hemer, Stetter [BHS]. In the DCM we consider, no iterates occur; a small number of updates are calculated to increase the accuracy of the velocity and pressure approximations. Thus it is most similar to DCM s which are close to Richardson extrapolation (see, for example, Mathews, Fin [MF04]). In the late 970 s Hemer (Bohmer, Stetter, Heinrichs and others) discovered that DCM, properly interpreted, is good also for nearly singular problems. Examples for which this has been successful include equilibrium Euler equations (Koren, Lallemand [LK93]), high Reynolds number problems (Layton, Lee, Peterson [LLP0]), viscoelastic problems (Ervin, Lee [EL06]). There has also been interesting wor on Spectral Deferred Correction (SDC) for IVP s (e.g., Minion [M04], Bourlioux, Layton, Minion [BLM03], Kress, Gustafsson [KG0], Dutt, Greengard, Rohlin [DGR00]). With the exception of the SDC methods for time stepping, the majority of the results has been obtained for the equilibrium problems - an odd fact, since, e.g., for the Euler equations the time-dependent problem is natural. For example, it has not been nown apparently if the natural idea of time stepping combined with the DCM in space for the associated quasi-equilibrium problem is stable. 3. MATHEMATICAL PRELIMINARIES AND NOTATIONS Throughout this chapter the norm will denote the usual L (Ω)-norm of scalars, vectors and tensors, induced by the usual L inner-product, denoted by (, ). The space that velocity (at time t) belongs to, is X = H 0(Ω) d = {v L (Ω) d : v L (Ω) d d and v = 0 on Ω}. with the norm v X = v. The space dual to X, is equipped with the norm (f, v) f = sup v X v. 47

58 The pressure (at time t) is sought in the space Q = L 0(Ω) = {q : q L (Ω), q(x)dx = 0}. Ω Also introduce the space of wealy divergence-free functions X V = {v X : ( v, q) = 0, q Q}. For measurable v : [0, T ] X, we define and T v L p (0,T ;X) = ( v(t) p X dt) p, p <, 0 Define the trilinear form on X X X v L (0,T ;X) = ess sup v(t) X. 0 t T b(u, v, w) = Ω u v wdx. The following lemma is also necessary for the analysis Lemma 3.. There exist finite constants M = M(d) and N = N(d) s.t. M N and M = sup u,v,w X b(u, v, w) u v w <, N = sup u,v,w V b(u, v, w) u v w <. The proof can be found, for example, in [GR79]. The corresponding constants M h and N h are defined by replacing X by the finite element space X h X and V by V h X, which will be defined below. Note that M max(m h, N, N h ) and that as h 0, N h N and M h M (see [GR79]). Throughout the chapter, we shall assume that the velocity-pressure finite element spaces X h X and Q h Q are conforming, have typical approximation properties of finite element spaces commonly in use, and satisfy the discrete inf-sup, or LBB h, condition inf q h Q h (q h, v h ) sup v h X v h h q h βh > 0, (3..) where β h is bounded away from zero uniformly in h. Examples of such spaces can be found in [GR79]. We shall consider X h X, Q h Q to be spaces of continuous piecewise 48

59 polynomials of degree m and m, respectively, with m. The case of m = is not considered, because the optimal error estimate (of the order h) is obtained after the first step of the method - and therefore the DCM in this case is reduced to the artificial viscosity approach. The space of discretely divergence-free functions is defined as follows V h = {v h X h : (q h, v h ) = 0, q h Q h }. In the analysis we use the properties of the following Modified Stoes Projection Definition 3. (Modified Stoes Projection). Define the Stoes projection operator P S : (X, Q) (X h, Q h ), P S (u, p) = (ũ, p), satisfying for any v h V h, q h Q h. (h + Re )( (u ũ), v h ) (p p, v h ) = 0, (3..) ( (u ũ), q h ) = 0, In (V h, Q h ) this formulation reads: given (u, p) (X, Q), find ũ V h satisfying for any v h V h, q h Q h. (h + Re )( (u ũ), v h ) (p q h, v h ) = 0, (3..3) Define the explicitly sew-symmetrized trilinear form b (u, v, w) := (u v, w) (u w, v). The following estimate is easy to prove (see, e.g., [GR79]): there exists a constant C = C(Ω) such that b (u, v, w) C(Ω) u v w. (3..4) The proofs will require the sharper bound on the nonlinearity. improvable in R. This upper bound is Lemma 3. (The sharper bound on the nonlinear term). Let Ω R d, d =, 3. For all u, v, w X b (u, v, w) C(Ω) u u v w. 49

60 Proof. See [GR79]. We will also need the following inequalities: for any u V inf (u v) C(Ω) inf (u v), (3..5) v V h v X h The proof of (3..5) can be found, e.g., in Friedrich s inequality and (3..5). [HR90] inf u v C(Ω) inf (u v). (3..6) v V h v X h Define also the number of time steps N := T. [GR79], and (3..6) follows from the Poincare- We conclude the preliminaries by formulating the discrete Gronwall s lemma, see, e.g. Lemma 3.3. Let, B, and a µ, b µ, c µ, γ µ, for integers µ 0, be nonnegative numbers such that: a n + b µ µ=0 γ µ a µ + µ=0 c µ + B for n 0. Suppose that γ µ < for all µ, and set σ µ = ( γ µ ). Then a n + µ=0 µ=0 b µ e n µ=0 σµγµ [ c µ + B]. µ=0 3.3 STABILITY OF THE VELOCITY In this section we prove the unconditional stability of the discrete artificial viscosity approximation u h and use this result to prove stability of the higher order approximation u h. Over 0 t T < the approximations u h and u h are bounded uniformly in Re. Hence, the formulation (3..5) gives the unconditionally stable extension of the defect correction method to the time-dependent Navier-Stoes equations. We start by proving stability of the modified Stoes Projection, that we use as the approximation ũ 0 to the initial velocity u 0. 50

61 Proposition 3. (Stability of the Stoes projection). Let u, ũ satisfy (3..3). The following bound holds (h + Re ) ũ (h + Re ) u (3.3.) +d(h + Re ) inf p q h, q h Q h where d is the dimension, d =, 3. Proof. Tae v h = ũ V h in (3..3). This gives (h + Re ) ũ = (h + Re )( u, ũ) (3.3.) (p q h, ũ). Using the Cauchy-Schwarz and Young s inequalities, we obtain (h + Re ) ũ (h + Re ) u + h + Re ũ (3.3.3) 4 +d(h + Re ) inf p q h + h + Re ũ. q h Q h 4d Using the inequality ũ d ũ and combining the lie terms concludes the proof. Now we prove the main results of this section - stability of the AV approximation u h and the Correction Step approximation u h. Lemma 3.4 (Stability of the AV approximation). Let u h satisfy the first equation of (3..5). Let f L (0, T ; H (Ω)). Then for n = 0,..., N u h,n+ + Σ n+ i= (h + Re ) u h,i u s 0 + Σn+ h + Re i= f(t i). Also, if f L (0, T ; L (Ω)) and the time constraint T is finite, then there exists a constant C = C(T ) such that u h,n+ + Σ n+ i= (h + Re ) u h,i (3.3.4) C( u s 0 + Σ n+ i= f(t i) ). 5

62 Proof. Let v h = u h,n+ V h in the first equation of (3..5). Since b (u, v, v) = 0, we obtain u h,n+ (u h,n, u h,n+ ) + (h + Re ) u h,n+ (p h,n+, u h,n+ ) = (f(t n+ ), u h,n+ ). Since p h,n+ Q h and u h,n+ V h it follows that (p h,n+, u h,n+ ) = 0. Applying Cauchy- Schwartz and Young s inequalities gives u h,n+ u h,n + (h + Re ) u h,n+ (f(t n+ ), u h,n+ ). (3.3.5) The definition of the dual norm and the Young s inequality, applied to the inner-product on the right-hand side, lead to We obtain u h,n+ u h,n h + Re u h,n+ + (f n+, u h,n+ ) f n+ u h,n+ + h + Re u h,n+ (h + Re ) f(t n+). Summing (3.3.7) over all time levels and multiplying by gives (3.3.6) (h + Re ) f(t n+). (3.3.7) This proves the first part of Lemma. u h,n+ + (h + Re )Σ n+ i= uh,i u s 0 (3.3.8) + Σn+ h + Re i= f(t i). Consider (3.3.5). Apply the Cauchy-Schwarz and Young s inequalities to the right-hand side. Different choice of constants in the Young s inequality gives (f(t n+ ), u h,n+ ) f(t n+ ) u h,n+ uh,n+ + f(t n+) (3.3.9) and (f(t n+ ), u h,n+ ) f(t n+ ) u h,n+ 4 uh,n+ + f(t n+ ). (3.3.0) 5

63 Sum (3.3.5) over all time levels, using (3.3.9) at the time levels t 0, t,..., t n and (3.3.0) at t = t n+. We obtain u h,n+ u s 0 + Σ n+ i= (h + Re ) u h,i (3.3.) 4 uh,n+ + Σn i= u h,i + f(t n+ ) + Σn i= f(t i ). Multiply by 4 and simplify to obtain u h,n+ + 4Σ n+ i= (h + Re ) u h,i (3.3.) u s f(t n+ ) + Σ n i= f(t i ) + Σ n i= u h,i. For the finite time constraint T, the discrete Gronwall s lemma yields u h,n+ + 4Σ n+ i= (h + Re ) u h,i (3.3.3) e ( T ) ( u s 0 + Σ n+ i= f(t i) ). We use the result of Lemma 3.4 in the following Theorem 3. (Stability). Let u h, u h satisfy (3..5). Let f L (0, T ; H (Ω)). Then for n = 0,..., N : u h,n+, u h,n+ are bounded and u h,n+ h + (h + Re ) uh,n+ + Σ n+ i= (h + Re ) u h,i (3.3.4) h ( + (h + Re ) ) us 0 h +( + (h + Re ) ) Σn+ h + Re i= f(t i). Also, if f L (0, T ; L (Ω)) and the time constraint T is finite, then there exists a constant C = C(T ) such that u h,n+ h + (h + Re ) uh,n+ + Σ n+ i= (h + Re ) u h,i (3.3.5) C( u s 0 + Σ n+ i= f(t i) ). 53

64 It follows from (3.3.5) that both approximations u h and u h are bounded at any time level and for any viscosity, provided that the initial approximation and the forcing term are L -integrable. The rest of the section is devoted to the proof of Theorem 3.. Proof. Tae v h = u h,n+ V h in the second equation of (3..5). This gives ( uh,n+ u h,n ) + (h + Re ) u h,n+ (f(t n+ ), u h,n+ ) (3.3.6) The Cauchy-Schwarz and Young s inequalities give +h( u h,n+, u h,n+ ). ( uh,n+ u h,n ) + (h + Re ) u h,n+ (3.3.7) h + Re f(t n+) + h + Re u h,n+ 4 + h uh,n+ h + Re Multiply (3.3.7) by and simplify to obtain + h + Re 4 u h,n+. u h,n+ u h,n + (h + Re ) u h,n+ h + Re f(t n+) + Summing over all time levels leads to h uh,n+ h + Re. (3.3.8) u h,n+ + Σ n+ i= (h + Re ) u h,i (3.3.9) u s 0 + Σn+ h + Re i= f(t i) h + (h + Re ) Σn+ i= (h + Re ) u h,i. Inserting the bound on Σ n+ i= (h + Re ) u h,i from the stability result (3.3.8) in (3.3.9) gives u h,n+ + (h + Re )Σ n+ i= uh,i (3.3.0) u s 0 + h + (h + Re ) ( us 0 u h,n Σn+ h + Re i= f(t i) Σn+ h + Re i= f(t i) ).

65 Thus u h,n+ h + (h + Re ) uh,n+ + (h + Re )Σ n+ i= uh,i (3.3.) h ( + (h + Re ) ) us 0 h +( + (h + Re ) ) Σn+ h + Re i= f(t i). This proves the first statement of Theorem 3.. To conclude, consider (3.3.6); as in (3.3.9)-(3.3.0), use the Young s inequalities differently at different time levels to obtain and u h,n+ u h,n + (h + Re ) u h,n+ (3.3.) f(t n+ ) + 4 uh,n+ h + (h + Re ) uh,n+ + h + Re u h,n+, u h,i+ u h,i + (h + Re ) u h,i+ (3.3.3) h f(t i+) + uh,i+ + (h + Re ) uh,i+ + h + Re u h,i+, for i = 0,,.., n. Sum (3.3.3) over all time levels and add to (3.3.); multiply by 4 to obtain u h,n+ u s 0 + Σ n+ i= (h + Re ) u h,i (3.3.4) Σ n i= u h,i + 4 f(t n+ ) +Σ n i= f(t i ) h + (h + Re ) Σn+ i= (h + Re ) u h,i. Insert the bound on Σ n+ i= (h + Re ) u h,i from (3.3.3) into (3.3.4) and simplify. For the finite time constraint T, the discrete Gronwall s lemma yields u h,n+ h + (h + Re ) uh,n+ + Σ n+ i= (h + Re ) u h,i (3.3.5) (e ( T ) + 4e ( 4T ) h (h + Re ) )[ us 0 + Σ n i= f(t i ) ]. 55

66 The result of Theorem 3., combined with the result of Proposition 3., proves the unconditional stability of both u h,i and u h,i for any i ERROR ESTIMATES In this section we explore the error estimates in approximating the NSE velocity u by the Artificial Viscosity approximation u and the Correction Step approximation u. The results agree with the general theory of the defect correction methods: u u h energy norm C(h m + h), u u h energy norm C(h m +h ), where the velocity approximations u h and u h are sought in the finite-element space of piecewise polynomials of degree m. In the error analysis we shall use the error estimate of the Stoes projection (3..3). Proposition 3. (Error estimate for Stoes Projection). Suppose the discrete inf-sup condition (3..) holds. Then the error in the Stoes Projection satisfies (h + Re ) (u ũ) C[(h + Re ) inf v h V h (u v h ) (3.4.) where C is a constant independent of h and Re. +(h + Re ) inf p q h ], q h Q h Proof. Decompose the projection error e = u ũ into e = u I(u) (ũ I(u)) = η φ, where η = u I(u), φ = ũ I(u), and I(u) approximates u in V h. Tae v h = φ V h in (3..3). This gives (h + Re ) φ = (h + Re )( η, φ) (3.4.) (p q h, φ). Since Ω R d, we have φ d φ. Applying the Cauchy-Schwarz and Young s inequalities to (3.4.) gives (h + Re ) φ (h + Re ) η (3.4.3) +d(h + Re ) inf p q h. q h Q h 56

67 Since I(u) is an approximation of u in V h, we can tae infimum over V h. concluded by applying the triangle inequality. The proof is The following constants (depending upon Ω and u) are introduced in order to simplify the notation. Definition 3.. Let C u := u(x, t) L (0,T ;L (Ω)), C u := u(x, t) L (0,T ;L (Ω)), and introduce C, satisfying inf (u v) C inf (u v) C h m u H m+ Ch m. v V h v X h Also, using the constant C(Ω) from Lemma 3., we define C := 78C 4 (Ω). The main results of this section are presented in the following theorem: Theorem 3. (Error estimates). Let f L (0, T ; H (Ω)), let u h, u h satisfy (3..5), h + Re 4C u + (h + Re )C u + C C 4 (h + Re ) h 4m, u L (0, T ; H m+ (Ω)) L (0, T ; L (Ω)), u L (0, T ; L (Ω)), u t L (0, T ; H m+ (Ω)), u tt L (0, T ; L (Ω)), p L (0, T ; H m (Ω)). and Then there exists a constant C = C(Ω, T, u, p, f, h + Re ), such that max u(t i) u h,i + i N max u(t i) u h,i + i N ( ) n+ / (h + Re ) (u(t i ) u h,i ) C(h m + h + ), i= ( ) n+ / (h + Re ) (u(t i ) u h,i ) C(h m + h + h + ). i= 57

68 Hence, the second (Correction) step of the method gives an approximation of the true solution, that is improved by (roughly) an order of h compared to the first step (Artificial Viscosity) approximation. The goal of this section is to prove Theorem 3. - that is, that the method is of the first order in time and that the order of the approximation in space depends upon the step of defect correction procedure. Proof. By Taylor expansion, u(t n+ ) u(t n) = u t (t n+ ) ρ n+, where ρ n+ = u tt (t n+θ ), for some θ [0, ]. The variational formulation of the NSE, followed by the equations (3..5), gives for u X, p Q, u, u X h, p, p Q h, v V h ( u(t n+) u(t n ), v) + (h + Re )( u(t n+ ), v) + b (u(t n+ ), u(t n+ ), v) (3.4.4) (p(t n+ ), v) = (f(t n+ ), v) + h( u(t n+ ), v) (ρ n+, v), u h,n, v) + (h + Re )( u h,n+, v) + b (u h,n+, u h,n+, v) (3.4.5) (p h,n+, v) = (f(t n+ ), v), ( uh,n+ u h,n, v) + (h + Re )( u h,n+, v) + b (u h,n+, u h,n+, v) (3.4.6) (p h,n+, v) = (f(t n+ ), v) + h( u h,n+, v). ( uh,n+ Subtract (3.4.5) from (3.4.4). Introduce the error in the AV approximation e i := u(t i ) u h,i, i. This gives ( en+ e n, v) + (h + Re )( e n+, v) (3.4.7) +[b (u(t n+ ), u(t n+ ), v) b (u h,n+, u h,n+, v)] ((p(t n+ ) p h,n+ ), v) = h( u(t n+ ), v) (ρ n+, v). Adding and subtracting b (u h,n+, u(t n+ ), v) to the nonlinear terms in (3.4.7) gives b (u(t n+ ), u(t n+ ), v) b (u h,n+, u h,n+, v) (3.4.8) = b (e n+, u(t n+ ), v) + b (u h,n+, e n+, v). 58

69 Decompose the error e i = u(t i ) u h,i = u(t i ) ũ i + ũ i u h,i = η i φ h,i, (3.4.9) where ũ i V h is some projection of u(t i ) into V h, and η i = u(t i ) ũ i, φ h,i = u h,i ũ i, φ h,i V h, i. Tae v = φ h,n+ V h in (3.4.7) and use (3.4.8). Using also b (, φ h,n+, φ h,n+ ) = 0 and V h Q h, we obtain ( ηn+ η n, φ h,n+ ) ( φh,n+ φ h,n, φ h,n+ ) (3.4.0) +(h + Re )( η n+, φ h,n+ ) (h + Re ) φ h,n+ +b (η n+, u(t n+ ), φ h,n+ ) b (φ h,n+, u(t n+ ), φ h,n+ ) +b (u h,n+, η n+, φ h,n+ ) (p(t n+ ) q h,n+, φ h,n+ ) = h( u(t n+ ), φ h,n+ ) (ρ n+, φ h,n+ ). Apply the Cauchy-Schwarz and Young s inequalities to (3.4.0). d φ h,n+ for ɛ > 0 Since φ h,n+ φ h,n+ φ h,n + (h + Re ) φ h,n+ (3.4.) ɛ(h + Re ) φ h,n+ ηn+ η n + 4ɛ(h + Re ) +ɛ(h + Re ) φ h,n+ + (h + Re ) η n+ 4ɛ + b (η n+, u(t n+ ), φ h,n+ ) + b (φ h,n+, u(t n+ ), φ h,n+ ) +ɛ(h + Re ) φ h,n+ + d 4ɛ(h + Re ) + b (u h,n+, η n+, φ h,n+ ) inf p(t n+ ) q h,n+ q h Q h +ɛ(h + Re ) φ h,n+ h + 4ɛ(h + Re ) u(t n+) +ɛ(h + Re ) φ h,n ɛ(h + Re ) ρ n+.

70 We bound the nonlinear terms on the right-hand side of (3.4.), starting now with the first one. Use the bound (3..4), the regularity of u and Young s inequality to obtain b (η n+, u(t n+ ), φ h,n+ ) ɛ(h + Re ) φ h,n+ (3.4.) +C ηn+ h + Re. The second nonlinear term can be bounded, using the definition of b (,, ) and the regularity of u. This gives b (φ h,n+, u(t n+ ), φ h,n+ ) C u φh,n+ + C u ( φh,n+, φ h,n+ ) (3.4.3) C u φh,n+ + ɛ(h + Re ) φ h,n+ + C u 6ɛ(h + Re ) φh,n+. For the third nonlinear term of (3.4.), use the error decomposition to obtain b (u h,n+, η n+, φ h,n+ ) b (u(t n+ ), η n+, φ h,n+ ) (3.4.4) + b (η n+, η n+, φ h,n+ ) + b (φ h,n+, η n+, φ h,n+ ). Use the regularity of u and the inequality (3..4) to bound the first two terms on the righthand side of (3.4.4). Applying Lemma 3. to the third term gives b (φ h,n+, η n+, φ h,n+ ) C(Ω) φ h,n+ 3/ φ h,n+ / η n+. (3.4.5) We apply the Young s inequality to (3.4.5) with p = 4 3 (3.4.4) that and q = 4. Finally it follows from b (u h,n+, η n+, φ h,n+ ) ɛ(h + Re ) φ h,n+ (3.4.6) C + ( ηn+ h + Re + η n+ 4 ) 7C 4 (Ω) + 64ɛ 3 (h + Re ) 3 ηn+ 4 φ h,n+, where C(Ω) is the constant from Lemma

71 Tae ɛ = 6 in (3.4.). Using the bounds (3.4.)-(3.4.6), we obtain φ h,n+ φ h,n C + h + Re + h + Re C η ηn+ n h + Re +C(h + Re ) η n+ inf p(t n+ ) q h,n+ q h Q h C + h + Re h u(t n+ ) C + h + Re ρ n+ C + ( ηn+ h + Re + η n+ 4 ) +( C Cu u + h + Re + C (h + Re ) 3 ηn+ 4 ) φ h,n+. φ h,n+ (3.4.7) Sum (3.4.7) over all time levels and multiply by. assumptions of the theorem that It follows from the regularity ρ i+ C ρ i+ C. Therefore we obtain + φ h,n+ + (h + Re ) + C h + Re [ ηi+ φ h,i+ φ h,0 (3.4.8) η i + (h + Re ) η i + η i + η i 4 + inf q h Q h p(t i ) q h,i + h + ] (C u + C u h + Re + C (h + Re ) 3 ηi+ 4 ) φ h,i+. Tae ũ i in the error decomposition (3.4.9) to be the L -projection of u(t i ) into V h, for i. Tae ũ 0 to be u s 0. This gives φ h,0 = 0 and e 0 = η 0. Also it follows from Proposition 3. 6

72 that η 0 Ch m ; under the assumptions of the theorem the discrete Gronwall s lemma gives φ h,n+ + (h + Re ) C h + Re [ ηi+ η i + η i + η i 4 + inf q h Q h p(t i ) q h,i + h + ]. Using the error decomposition and the triangle inequality, we obtain φ h,i+ (3.4.9) e i+ (h + Re ) φ h,i+ + e n+ η n+ + φ h,n+, (3.4.0) e n+ η n+ + φ h,n+, η i+ + φ h,i+, (h + Re ) e i+ (h + Re ) η i+. Then it follows from (3.4.9),(3.4.0) that e n+ + C h + Re [ ηi+ (h + Re ) e i+ (3.4.) η i + η i + η i 4 + inf q h Q h p(t i ) q h,i + h + ]. Use the approximation properties of X h, Q h. Since the mesh nodes do not depend upon the time level, it follows from (3..5),(3..6) that η i C ηi+ η i Ch m, (3.4.) η i Ch m, ηi+ inf p(t i ) q h,i Ch m. q h Q h 6

73 Hence, we obtain from (3.4.),(3.4.) that u(t n+ ) u h,n+ + This proves the first statement of the theorem. (h + Re ) (u(t n+ ) u h,n+ ) (3.4.3) C h + Re [hm + h + ], where C = C(Ω, T, u, p, f). Now subtract (3.4.6) from (3.4.4). Introduce the error in the Correction Step approximation e i := u(t i ) u h,i, i. This gives ( en+ e n, v) + (h + Re )( e n+, v) (3.4.4) +[b (u(t n+ ), u(t n+ ), v) b (u h,n+, u h,n+, v)] ((p(t n+ ) p h,n+ ), v) = h( e n+, v) (ρ n+, v). Note that (3.4.4) differs from (3.4.7) only in the first term on the right-hand side. Using the Cauchy-Schwarz and Young s inequality, we obtain that for any ɛ > 0 Therefore, h( e n+, v) ɛ(h + Re ) v + 4ɛ(h + Re ) h e n+. (3.4.5) h( e n+, v) + 4ɛ(h + Re ) h ɛ(h + Re ) v (3.4.6) (h + Re ) e n+. Using the bound on n (h + Re ) e n+ from (3.4.3), we obtain Decompose the error h( e n+, v) ɛ(h + Re ) v (3.4.7) C + (h + Re ) 3 [hm+ + h 4 + h ]. e i = u(t i ) u h,i = u(t i ) ũ i + ũ i u h,i = η i φ h,i, (3.4.8) where η i = u(t i ) ũ i, φ h,i = u h,i ũ i, φ h,i V h, i. 63

74 To conclude, repeat the proof of the first statement of the theorem, replacing u h, e, φ h, η by u h, e, φ h, η, respectively, and using (3.4.7). Note that the term C h on the right- h+re hand side of (3.4.3), which was obtained from the bound on h( u(t n+ ), v), is now replaced by C [h m+ + h 4 + h ]. Hence, we obtain (h+re ) 3 u(t n+ ) u h,n+ + (h + Re ) (u(t n+ ) u h,n+ ) (3.4.9) C (h + Re ) 3 [hm + h 4 + h + ], where C = C(Ω, T, u, p, f). This completes the proof of Theorem 3.. Thus, we have derived the error estimates, that agree with the general theory of the defect correction methods. Namely, the Correction Step approximation u h is improved by an order of h, compared to the Artificial Viscosity approximation u h. Next we shall prove stability and derive the error estimates for the pressure. 3.5 PRESSURE This section gives the proof of stability and the convergence rates for pressure approximations p h and p h. For the pressure analysis we shall need the bounds on discrete time derivatives en+ e n and en+ e n. For pressure stability it is enough to bound these quantities by a constant, but a more subtle estimate is needed for proving the convergence rates. We start by proving this estimate as a theorem. Throughout this section we use the error decomposition e i j = u(t i ) u h,i j j =,, i =,..., n, introduced in (3.4.9),(3.4.8). = η i j φ h,i j, Also, taing ũ i = u s 0 on the initial time level gives φ h,0 = φ h,0 = 0 and e 0 = η 0, e 0 = η 0. It follows from Proposition 3. that η 0 Ch m and η 0 Ch m. 64

75 Theorem 3.3. Let the regularity assumptions of Theorem 3. be satisfied. Let p t L (0, T ; H m (Ω)), u ttt L (0, T ; L (Ω)). Also let min(h, (h + Re ) 3 ). Then for any time level n 0 and en+ e n + ( en+ e n + ( (h + Re ) ( ei+ e i ) ) / C(h m + h + ), i= (h + Re ) ( ei+ e i ) ) / C(h m + h + h + ). i= Proof. Start with the proof of the bound for φh,n+ n Tae v = φh,n+ φ h,n. Consider (3.4.7) with (3.4.8) for ( en+ e n, v) + (h + Re )( e n+, v) (3.5.) +b (e n+, u(t n+ ), v) + b (u h,n+, e n+, v) ((p(t n+ ) p h,n+ ), v) = h( u(t n+ ), v) (ρ n+, v), where ρ n+ = u t (t n+ ) u(t n+) u(t n ). φ h,n =: s h,n+ V h in (3.5.). Then consider (3.5.) at the previous time level and mae exactly the same choice v = s h,n+ V h. Subtract the equations, using the Taylor expansion to simplify the last term on the right-hand side. We obtain ( ηn+ η n + η n, s h,n+ ) (s h,n+ s h,n, s h,n+ ) (3.5.) +(h + Re )( ( ηn+ η n ), s h,n+ ) (h + Re ) s h,n+ +b (e n+, u(t n+ ), s h,n+ ) + b (u h,n+, e n+, s h,n+ ) b (e n, u(t n ), s h,n+ ) b (u h,n, e n, s h,n+ ) ( (p(t n+) p h,n+ ) (p(t n ) p h,n ), s h,n+ ) = h( ( u(t n+) u(t n+ ) ), s h,n+ ) C (ρ n+ t, s h,n+ ), where ρ n+ t = u ttt (t n+θ ) for some θ [0, ]. 65

76 Consider the nonlinear terms of (3.5.). Adding and subtracting b (e n, u(t n+ ), s h,n+ ) and b (u h,n+, e n, s h,n+ ) gives b (e n+, u(t n+ ), s h,n+ ) b (e n, u(t n ), s h,n+ ) (3.5.3) +b (u h,n+, e n+, s h,n+ ) b (u h,n, e n, s h,n+ ) = [b (e n+, u(t n+ ), s h,n+ ) b (e n, u(t n+ ), s h,n+ ) +b (e n, u(t n+ ), s h,n+ ) b (e n, u(t n ), s h,n+ )] +[b (u h,n+, e n+, s h,n+ ) b (u h,n+, e n, s h,n+ ) +b (u h,n+, e n, s h,n+ ) b (u h,n, e n, s h,n+ )]. Use the error decomposition (3.4.9). Since b (, s h,n+, s h,n+ ) = 0, it follows from (3.5.3) that b (e n+, u(t n+ ), s h,n+ ) b (e n, u(t n ), s h,n+ ) (3.5.4) +b (u h,n+, e n+, s h,n+ ) b (u h,n, e n, s h,n+ ) = b ( ηn+ η n, u(t n+ ), s h,n+ ) b (s h,n+, u(t n+ ), s h,n+ ) +b (e n+, u(t n+) u(t n ), s h,n+ ) + b (u h,n+, ηn+ η n, s h,n+ ) +b ( uh,n+ u h,n, e n, s h,n+ ). Use the regularity of u and the Cauchy-Schwarz and Young s inequalities to obtain the bounds on the terms in (3.5.4). It follows from (3..4) that for any ɛ > 0 ɛ(h + Re ) s h,n+ + b ( ηn+ η n C h + Re, u(t n+ ), s h,n+ ) (3.5.5) η (ηn+ n For the second term on the right-hand side of (3.5.4) use the regularity of u and the Cauchy- Schwarz and Young s inequalities to obtain ). b (s h,n+, u(t n+ ), s h,n+ ) ɛ(h + Re ) s h,n+ (3.5.6) C + h + Re C u s h,n+ + C u s h,n+. 66

77 The third nonlinear term on the right-hand side of (3.5.4) is bounded by b (e n+, u(t n+) u(t n ), s h,n+ ) (3.5.7) ɛ(h + Re ) s h,n+ C + en+ h + Re. For the fourth nonlinear term, add and subtract u(t n+ ) to the first term of the trilinear form. Using (3..4) and Lemma 3. leads to C + h + Re b (u h,n+, ηn+ η (ηn+ n η n, s h,n+ ) ɛ(h + Re ) s h,n+ (3.5.8) ) + C e n+ e n+ ( ηn+ η n ). For the fifth term add and subtract u(t n+ ) to the first term of the trilinear form to obtain u h,n b ( uh,n+, e n, s h,n+ ) b ( u(t n+) u(t n ), e n, s h,n+ ) (3.5.9) + b ( ηn+ η n, e n, s h,n+ ) + b (s h,n+, e n, s h,n+ ). Apply the result of Lemma 3. to the last trilinear form in (3.5.9) and use the Young s inequality with p = 4 3 and q = 4. This gives u h,n b ( uh,n+ 3ɛ(h + Re ) s h,n+ + C + h + Re en ( ηn+ η n ) + 67, e n, s h,n+ ) (3.5.0) C h + Re en C (h + Re ) 3 en 4 s h,n+.

78 Apply the Cauchy-Schwarz and Young s inequalities to (3.5.), using the bounds (3.5.4)- (3.5.0) for the nonlinear terms. This gives C + h + Re C + h + Re ηn+ s h,n+ s h,n + (h + Re ) s h,n+ (3.5.) 3ɛ(h + Re ) s h,n+ η n + η n + C(h + Re ) ( ηn+ η n ) C + inf p(t n+) p(t n ) qh,n+ q h,n h + Re q h Q h η [ (ηn+ n ) + e n + ( ηn+ η n ) e n ] +C e n e n + C ( ηn+ η n ) 4 C + h + Re h ( u(t n+) u(t n ) ) C + h + Re ρ n+ t Cu +C(C u + h + Re + (h + Re ) 3 en 4 ) s h,n+. Since u ttt L (0, T ; L (Ω)), we have ρ i+ t C ρ i+ t C. It follows from the assumption h and the result of Theorem 3. that max i e i C, max i e i C. 68

79 Tae ɛ = 6 all time levels n to obtain in (3.5.), simplify, multiply both sides of the inequality by and sum over +C i= s h,n+ + (h + Re ) s h,i+ s h, (3.5.) i= C + h + Re i= η i + η i [ ηi+ +(h + Re ) ( ηi+ η i ) + ( ηi+ η i ) +(h + Re ) ( ηi+ η i ) 4 + inf p(t i+) p(t i ) qh,i+ q h,i + h + ] q h Q h C + (h + Re ) (h + Re ) e i + C e i i= i= Cu (C u + h + Re + (h + Re ) 3 ei 4 ) s h,i+. Consider the error decomposition (3.4.9). Tae ũ i to be the L projection of u(t i ) into V h, for all i. Since the mesh nodes do not depend upon the time level, it follows from the approximation properties of X h, Q h and the regularity of u, p that i= η i + η i C ηi+ i= ηi+ η i + η i Ch m, (3.5.3) ( ηi+ η i ) Ch m, i= ( ηi+ η i ) 4 Ch 4m, i= i= inf p(t i+) p(t i ) q h Q h qh,i+ q h,i Ch m. Using (3.5.3) and (3.4.3), we derive from (3.5.) that +C i= s h,n+ + (h + Re ) s h,i+ s h, (3.5.4) +C[h m + h + ] Cu (C u + h + Re + (h + Re ) 3 ei 4 ) s h,i+. i= 69

80 Tae ũ 0 = u s 0 on the initial time level. This gives φ h,0 = 0 and e 0 = η 0 = u 0 u s 0. For the bound on s h, = φh, This gives φh,0, consider (3.5.) at n = 0 and tae v = φh, φh,0. ( e e 0, φh, φ h,0 ) + (h + Re )( e, ( φh, φ h,0 )) (3.5.5) +b (e, u(t ), φh, φ h,0 ) + b (u h,, e, φh, φ h,0 ) ((p(t ) p h, ), ( φh, φ h,0 )) = h( u(t ), ( φh, φ h,0 )) (ρ, φh, φ h,0 ). Rewrite the left-hand side of (3.5.5) so that we could use the properties of the modified Stoes projection (3..3) ( e e 0, φh, φ h,0 ) + (h + Re )( ( e e 0 ), ( φh, φ h,0 )) (3.5.6) +b (e, u(t ), φh, φ h,0 ) + b (u h,, e, φh, φ h,0 ) +(h + Re )( e 0, ( φh, φ h,0 )) ((p(t ) p h, ), ( φh, φ h,0 )) = h( u(t ), ( φh, φ h,0 )) (ρ, φh, φ h,0 ). Since φh, φh,0 V h and p h, Q h, it follows from the choice of initial approximation ũ 0 and from (3..3) that (h + Re )( e 0, ( φh, φ h,0 )) ((p(t ) p h, ), ( φh, φ h,0 )) = 0. (3.5.7) 70

81 Hence, using the Cauchy-Schwarz and Young s inequalities, we derive from (3.5.6) and (3.5.7) that for any ɛ, ɛ > 0 φh, φ h,0 + (h + Re ) ( φh, φ h,0 ) (3.5.8) ɛ φh, φ h,0 + C η η 0 +ɛ(h + Re ) φh, φ h,0 + C(h + Re ) ( η η 0 ) +ɛ φh, φ h,0 + C ρ + ɛ(h + Re ) ( φh, φ h,0 ) C + h + Re h ( u(t ) u 0 ) + ɛ φh, φ h,0 + Ch u 0 +b ( e e 0, u(t ), φh, φ h,0 ) + b (e 0, u(t ), φh, φ h,0 ) +b (u(t ), e, φh, φ h,0 ) + b (φ h,, e, φh, φ h,0 ) b (η, e, φh, φ h,0 ). Using the fact that φ h,0 = 0, we obtain b (, φ h,, φh, φh,0 ) = 0. The nonlinear terms in (3.5.8) are bounded by applying Cauchy-Schwarz and Young s inequalities. We obtain b ( e e 0, u(t ), φh, φ h,0 ) + b (e 0, u(t ), φh, φ h,0 ) (3.5.9) +b (u(t ), e, φh, φ h,0 ) + b (φ h,, e, φh, φ h,0 ) b (η, e, φh, φ h,0 ) C φ φh, h,0 + ɛ(h + Re ) ( φh, φ h,0 ) h + Re +ɛ(h + Re ) ( φh, φ h,0 ) C + h + Re (η η 0 ) +ɛ(h + Re ) ( φh, φ h,0 ) C + (h + Re ) 3 η 4 φh, φ h,0 +ɛ φh, φ h,0 + C η 0 + C η 0 +ɛ φh, φ h,0 + C η + C η +ɛ φh, φ h,0 + Ch η 4. 7

82 The inequalities (3.5.8)-(3.5.9) give φh, φ h,0 + (h + Re ) ( φh, φ h,0 ) (3.5.0) C (8ɛ + h + Re + C (h + Re ) 3 η 4 ) φh, φ 0 +5ɛ(h + Re ) ( φ φ 0 ) +C[ η η 0 + (h + Re ) ( η η 0 ) + ρ + h + Re h ( u(t ) u(t 0 ) ) + h u 0 + h + Re (η η 0 ) + η + η + h η 4 ]. It follows from the approximation properties of X h, Q h that φ h,0 + (h + Re ) ( φh, φ h,0 ) (3.5.) C[h m + h + ], φh, and the triangle inequality gives e e 0 + (h + Re ) ( e e 0 ) (3.5.) C[h m + h + ]. Insert the bound on φh, φh,0 apply discrete Gronwall s lemma. This leads to into (3.5.4). The restriction on the time step allows to φ h,n + (h + Re ) φh,n+ i= ( φh,i+ φ h,i ) (3.5.3) C[h m + h + ]. Using the triangle inequality we obtain en+ e n + (h + Re ) i= ( ei+ e i ) (3.5.4) C[h m + h + ]. This result proves the first statement of Theorem

83 For the bound on φh,n+ v = φh,n+ φ h,n φ h,n consider (3.4.4), n. Following the proof above, tae =: s h,n+, then consider (3.4.4) at the previous time level, mae the same choice v = s h,n+ and subtract the two equations. This leads to η n + η n, s h,n+ ) (s h,n+ s h,n, s h,n+ ) (3.5.5) ( ηn+ +(h + Re )( ( ηn+ η n ), s h,n+ ) (h + Re ) s h,n+ +b (e n+, u(t n+ ), s h,n+ ) + b (u h,n+, e n+, s h,n+ ) b (e n, u(t n ), s h,n+ ) b (u h,n, e n, s h,n+ ) ( (p(t n+) p h,n+ ) (p(t n ) p h,n ), s h,n+ ) = h( ( en+ e n ), s h,n+ ) C (ρ n+ t, s h,n+ ). The nonlinear terms in (3.5.5) are bounded in the same manner as those in (3.4.4), with s h, φ h and η replaced by s h, φ h and η. Using these bounds and the Cauchy-Schwarz and Young s inequalities, we obtain from (3.5.5) that C + h + Re C + h + Re ηn+ s h,n+ s h,n + (h + Re ) s h,n+ (3.5.6) 3ɛ(h + Re ) s h,n+ η n + η n + C(h + Re ) ( ηn+ η n ) C + inf p(t n+) p(t n ) qh,n+ q h,n h + Re q h Q h η [ (ηn+ n ) + e n + ( ηn+ η n ) e n ] +C e n e n + C ( ηn+ η n ) 4 C + h + Re h ( en+ e n ) C + h + Re ρ n+ t Cu +C(C u + h + Re + (h + Re ) 3 en 4 ) s h,n+. It follows from the assumption h and the result of Theorem 3. that max i e i C. 73

84 Tae ɛ = in (3.5.6), simplify, multiply both sides of (3.5.6) by and sum over all 6 time levels n. The bound on (h + Re ) n i= ( ei+ e i ) is obtained from (3.5.4). Using the approximation properties of X h, Q h and the triangle inequality, we obtain +C i= s h,n+ + (h + Re ) i= s h,i+ s h, (3.5.7) +C[h m + h 4 + h + ] Cu (C u + h + Re + (h + Re ) 3 ei 4 ) s h,i+. The bound on s h, = φh, φh,0 Consider (3.4.4) at n = 0 and tae v = s h, is obtained in the same manner as the bound on φh, = φh, φh,0. This leads to φh,0 φh, φ h,0 + (h + Re ) ( φh, φ h,0 ) (3.5.8) C (8ɛ + h + Re + C (h + Re ) 3 η 4 ) φh, φ h,0 +5ɛ(h + Re ) ( φh, φ h,0 ) +C[ η η 0 + (h + Re ) ( η η 0 ) + ρ + h + Re h ( e e 0 ) + h η 0 + h + Re (η η 0 ) + η + η + h η 4 ].. Use the bound on (h + Re ) ( e e0 ) from (3.5.). It follows from the approxima- tion properties of X h, Q h and the triangle inequality, that φ h,0 + (h + Re ) ( φh, φ h,0 ) (3.5.9) C[h m + h 4 + h + ] φh, and e e 0 + (h + Re ) ( e e 0 ) (3.5.30) C[h m + h 4 + h + ]. 74

85 Insert the bound on φh, φh,0 apply discrete Gronwall s lemma. This leads to φh,n+ into (3.5.7). The restriction on the time step allows to φ h,n + (h + Re ) i= ( φh,i+ φ h,i ) (3.5.3) C[h m + h 4 + h + ]. Using the triangle inequality we obtain en+ e n + (h + Re ) i= ( ei+ C[h m + h 4 + h + ]. e i ) (3.5.3) This completes the proof of Theorem Stability of the Pressure The stability of the pressure approximations p h and p h follows from the discrete inf-sup condition (3..). The required bound on the time derivative of velocity is obtained under the assumptions of Theorem 3.3. Theorem 3.4. Let f L (0, T ; H (Ω)). Let p h and p h satisfy the equations (3..5) and let the assumptions of Theorem 3.3 be satisfied. Then there exists a constant C = C(T, f, h + Re, u s 0) s.t. and p h,i+ C p h,i+ C Proof. Consider the first equation of (3..5). It holds true for v h X h. Apply the Cauchy- Schwarz inequality, divide both sides of the inequality by v h and regroup the terms, leaving only the pressure term on the left-hand side. Using Lemma 3. gives (p h,n+, v h ) v h u h,n uh,n+ + (h + Re ) u h,n+ (3.5.33) +M u h,n+ + f(t n+ ). 75

86 It follows from (3.5.33) and the discrete LBB condition (3..) that β h p h,n+ uh,n+ u h,n + (h + Re ) u h,n+ (3.5.34) +M u h,n+ + f(t n+ ). Decompose the first term on the right-hand side of (3.5.34), using the error decomposition and the triangle inequality. This gives u h,n uh,n+ u(t n+) u(t n ) + en+ e n. (3.5.35) Multiply both sides of (3.5.34) by and sum over the time levels. Using (3.5.35), we obtain β h +(h + Re ) p h,i+ u(t i+) u(t i ) + u h,i+ + u h,i+ + M The discrete Hölder s inequality gives ei+ e i (3.5.36) f(t i+ ). ( u h,i+ u h,i+ = ) ( ) = C( u h,i+ (3.5.37) u h,i+ ). Similarly, e i C( ei+ e i ei+ ) C( e i ). (3.5.38) ei+ The stability bound on n uh,i+ is obtained from Lemma 3.4. Using (3.5.38) and Theorem 3.3, it follows from (3.5.36) that β h C[ h + Re us 0 + (h + Re ) f(t i+ ) ]. (3.5.39) p h,i+ Hence, if the forcing term f is sufficiently smooth, the pressure approximation p h is stable. 76

87 Next, consider the second equation of (3..5). Apply the Cauchy-Schwarz inequality, divide both sides of the inequality by v h and regroup the terms. Following the outline of the proof above, we obtain β h p h,i+ u(t i+) u(t i ) + +(h + Re ) u h,i+ + M +h u h,i+ + e i (3.5.40) ei+ u h,i+ f(t i+ ). Use the discrete Hölder s inequality as in (3.5.37). It follows from Theorem 3.3 and Theorem 3. that β h C[ h + Re us 0 + (h + Re ) f(t i+ ) ]. (3.5.4) p h,i Error estimates for the pressure In this section we estimate the error in pressure approximations p(t i ) p h,i and p(t i ) p h,i. The results are obtained by using the inf-sup condition (3..) and the result of Theorem 3.3. The main result of the section is Theorem 3.5 (Pressure Convergence Rates). Let u, p, u h, p h, u h, p h satisfy the equations (3.4.4)-(3.4.6). Let the assumptions of Theorem 3.3 be satisfied. Then, for n 0 p(t i+ ) p h,i+ C[h m + h + ] (3.5.4) and p(t i+ ) p h,i+ C[h m + h + h + ]. (3.5.43) 77

88 Proof. Decompose the error in the pressure approximation p(t n+ ) p h,n+ = (p(t n+ ) q h,n+ ) (p h,n+ q h,n+ ) (3.5.44) =: γ n+ ψ h,n+, where q h,n+ is some projection of p(t n+ ) into Q h. Thus, ψ h,n+ Q h. Divide both sides of (3.5.) by v and regroup the terms. Use the result of Lemma 3. and the Cauchy-Schwarz inequality to obtain (ψ h,n+, v) v en+ e n (3.5.45) +(h + Re ) e n+ + M u(t n+ ) e n+ + M u h,n+ e n+ + inf q h Q h p(t n+ ) q h,n+ + h u(t n+ ) + ρ n+. Apply the discrete inf-sup condition. Multiply both sides of (3.5.45) by and sum over all time levels. Decomposing u h,n+ = u(t n+ ) e n+ gives +(h + Re ) + β h ψ h,i+ e i+ + M max 0 i n+ u(t i) +M e i+ + h inf q h Q h p(t i+ ) q h,i+ + e i (3.5.46) e i+ ei+ u(t i+ ) ρ i+. Applying the discrete Hölder s inequality and the triangle inequality and using Theorem 3.3 and Theorem 3. proves (3.5.4). Next, subtract (3.4.6) from (3.4.4). This gives for any v X h ( en+ e n, v) + (h + Re )( e n+, v) (3.5.47) +b (e n+, u(t n+ ), v) + b (u h,n+, e n+, v) ((p(t n+ ) p h,n+ ), v) = h( e n+, v) (ρ n+, v). 78

89 Following the proof above, we obtain +(h + Re ) + β h ψ h,n+ e i+ + M max 0 i n+ u(t i) +M e i+ + h inf p(t i+ ) q h,i+ + q h Q h e i (3.5.48) e i+ ei+ e i+ ρ i+. Applying the discrete Hölder s inequality and the triangle inequality and using Theorem 3.3 and Theorem 3. leads to (3.5.43). 3.6 COMPUTATIONAL TESTS We test the convergence rates for a two-dimensional problem with a nown exact solution. Consider the Chorin s vortex decay problem in the unit square Ω = (0, ). Tae u = cos(πx) sin(πy)exp( π t/re), (3.6.) sin(πx) cos(πy)exp( π t/re) p = 4 (cos(πx) + cos(πy))exp( 4π t/re), and then the right-hand side f and initial condition u 0 are computed such that (3.6.) satisfies (3..4). In order to reduce the influence of the time discretization error, the time step is taen to be very small: t = O(h 3 ). For Re =, Re = and final time T = /30, the calculated convergence rates in Tables 3-6 confirm what is predicted by Theorem 3. for (P, P ) discretization in space. The convergence rate of u u h L (0,T ;L (Ω)), predicted by Theorem 3., appears to be improvable in the case of moderate Reynolds number. However, for the flow with sufficiently large Reynolds number, the computed rates agree with those predicted by the theorem. 79

90 Table 3: AV approximation. Re =. h u u h L (0,T ;L (Ω)) rate u u h L (0,T ;H (Ω)) rate / / E /6.846E / Table 4: Correction Step approximation. Re =. h u u h L (0,T ;L (Ω)) rate u u h L (0,T ;H (Ω)) rate / / E / E / E Table 5: AV approximation. Re = h u u h L (0,T ;L (Ω)) rate u u h L (0,T ;H (Ω)) rate / / E / E /3.656E E

91 Table 6: Correction Step approximation. Re = h u u h L (0,T ;L (Ω)) rate u u h L (0,T ;H (Ω)) rate / / / E / E E COMPARISON OF THE APPROACHES For many years, it has been widely believed that (3..) can be directly imported into implicit time discretizations of flow problems in the obvious way: discretize in time, given u h (t OLD ), the quasistatic flow problem for u h (t NEW ) is solved by DCM of the form (3..). In this section we compare this approach and our method (3..5), applied to the same problem. Apply both methods to the one-dimensional singularly perturbed equation u t ɛu xx + u x = f Our method leads to the coupled pair of equations u n+,i u n,i (ɛ + h) un+ t u n+,i u n,i (ɛ + h) un+ t whereas the other method gives u n+,i u n,i t,i un+ h,i un+ h = f n+ i,i + u n+,i+,i + u n+ (ɛ + h) un+,i un+,i + u n+,i+ h u n+,i u n,i (ɛ + h) un+ t,i un+ h,i+ + un+,i+ un+,i h = f n+ i, + un+,i+ un+,i h h un+,i un+,i + u n+,i+, h = f n+ i 8 + un+,i+ un+,i h,i + u n+,i+ = f n+ i, + un+,i+ un+,i h h un+,i un+,i + u n+,i+. h

92 Consider 0 x, 0 t, u(0, t) = 0, u(, t) = 5, u(x, 0) = 0, f(x, t) = tx + 0x, ɛ = As one could have predicted, if we let the time interval be fixed and reasonably big ( t = 0.) and decrease the space-interval, both methods give almost the same results, since they mainly differ in treating the time-derivative. But if we fix h and monotonically decrease t, we immediately see the oscillations of the solution, obtained by the alternative method. Figures Fig.3-Fig.6 show the solution, obtained by our method (denoted by the solid line) and the solution, obtained by the alternative approach (dashed line on the graphs). The spacial mesh is fixed (with h = 0.0) and the time step t decreases to zero (see the captions). 5 h = 0.0, t = Figure 3: t = h = 0.0, t = Figure 4: t = 0.00 As we see, although this alternative approach uses the Correction Step approximation of the true solution on each time level (instead of the AV approximation), the results are 8

93 h = 0.0, t = Figure 5: t = h = 0.0, t = e Figure 6: t = worse even for a simple one-dimensional problem with the bounded domain and bounded right-hand side. We conclude the comparison of the methods by applying them to the Navier-Stoes equations in R. Consider the Chorin s vortex decay problem in the square Ω = (, ) with f(x, y, t) = π sin(πx)exp( 4π t/re) (3.7.) π sin(πy)exp( 4π t/re) and Re = 0 5. The final time is taen to be T = /0 and the mesh diameter is fixed at h = /4. As the time step t is decreased, the error estimates, obtained by the DCM (3..5), do not change - see the following table. At the same time, applying the alternative approach we obtain Hence, in the alternative approach the error increases as t tends to zero. 83

94 Table 7: DCM. Re = 00000, T = /0, h = /4 t u u h L (0,T ;L (Ω)) u u h L (0,T ;H (Ω)) Table 8: ALTERNATIVE APPROACH. Re = 00000, T = /0, h = /4 t u u h L (0,T ;L (Ω)) u u h L (0,T ;H (Ω)) We have seen from Figures Fig.3-Fig.6 that the alternative approach gives worse results than the DCM, when solving the convection diffusion equation. Comparing the Tables 7-8, we conclude that the Defect Correction Method (3..5) also behaves better, when applied to a more difficult Navier-Stoes problem. 84

95 4.0 A DEFECT CORRECTION METHOD FOR THE EVOLUTIONARY CONVECTION DIFFUSION PROBLEM WITH INCREASED TIME ACCURACY 4. INTRODUCTION Consider the evolutionary convection diffusion problem: find u : Ω [0, T ] R d (d =, 3) such that u t ɛ u + b u + gu = f, for x Ω, 0 < t T (4..) u = 0, on Ω, for 0 < t T, where u is the velocity field, ɛ is a diffusion coefficient, g is an absorbtion/reaction coefficient, and f is a forcing function. In the problems with high Peclet number (i.e. ɛ ) some iterative solvers fail to converge to a solution of (4..). We propose a certain Defect Correction Method (DCM), that is stable, computes a solution to (4..) for any ɛ with high space and time accuracy, and is computationally attractive. The general theory of Defect Correction Methods is presented, e.g., in Bohmer, Hemer, Stetter [BHS]. In the late 970 s Hemer (Bohmer, Stetter, Heinrichs and others) discovered that DCM, properly interpreted, is good also for nearly singular problems. Examples for which this has been successful include equilibrium Euler equations (Koren, Lallemand [LK93]), high Reynolds number problems (Layton, Lee, Peterson [LLP0]), viscoelastic problems (Ervin, Lee [EL06]). There has been an extensive study and development of the DC methods for equilibrium flow problems, see e.g. Hemer[Hem8], Koren[K9], Heinrichs[Hei96], Layton, Lee, 85

96 Peterson[LLP0], Ervin, Lee[EL06]. On the other hand, there is a parallel development of DCM s, for initial value problems in which no spacial stabilization is used, but DCM is used to increase the accuracy of the time discretization. This wor contains no reports of instabilities: see, e.g., Heywood, Rannacher[HR90], Hemer, Shishin[HSS], Minion[M04], Bourlioux, Layton, Minion [BLM03]. It was shown in [L07] that the natural idea of time stepping combined with the DCM in space for the associated quasi-equilibrium problem gives an oscillatory computed solution of poor quality. Another DC method was introduced for an evolutionary PDE, that was proven to be stable and accurate. The method presented in this chapter, is the modification (aiming at higher accuracy in time) of the DCM for the evolutionary PDEs, presented in [L07]. Compared to the method in [L07], the right hand side of the system is modified in the correction step, resulting in higher time accuracy with no extra computational cost. The method proceeds as follows: first we compute the AV approximation u h X h via L h ɛ+h(u h ) = f, where L h ɛ+h(u h ) = u h t (h + ɛ) u h + b u h + gu h. The accuracy of the approximation is then increased by the correction step: compute u h X h, satisfying L h ɛ+h(u h ) L h ɛ+h(u h ) = f L h ɛ (u h ) + B(u h ). Here B( ) is the time difference operator, that increases the accuracy of the discrete time difference for u t. 86

97 The Cran-Nicolson time discretization, combined with the two-step defect correction method in space leads to the following system of equations for u h,n+, u h,n+ X h, v h X h at t = t n+, n 0, with := t = t i+ t i u h,n, v h ) + (h + ɛ)( ( uh,n+ + u h,n ), v h ) + (b ( uh,n+ + u h,n ), v h ) (4..a) +g( uh,n+ + u h,n, v h ) = (f(t n+/ ), v h ), u h,n, v h ) + (h + ɛ)( ( uh,n+ + u h,n ), v h ) + (b ( uh,n+ + u h,n ), v h ) (4..b) + u h,n, v h ) = (f(t n+/ ), v h ) + h( ( uh,n+ + u h,n ), v h ) + B n (u h, v h ), ( uh,n+ ( uh,n+ +g( uh,n+ where B 0 (u, v) = ( ) u 4 5u 3 + 9u 7u + u 0, v (4..3a) 3 ( ) f(t3 ) 5f(t ) + 7f(t ) 3f(t 0 ) 6, v, B n (u, v) = ( ) u n+ 3u n+ + 3u n u n, v (4..3b) 3 + ( ) f(tn+ ) f(t n+ ) f(t n ) + f(t n ) 6, v, for n =,..., N, B N (u, v) = ( ) u N 7u N + 9u N 5u N 3 + u N 4, v (4..3c) 3 + ( ) 3f(tN ) 7f(t N ) + 5f(t N ) f(t N 3 ) 6, v. Depending on the current time level, we vary the templates - this demonstrates the resilience of the method. However, the condition N 4 needs to be satisfied, where N = T/ is the number of time levels. Note that the operator B is chosen so that for any n the true solution u satisfies ( u(t n+) u(t n ) +g( u(t n+) + u(t n ), v h ) + (h + ɛ)( ( u(t n+) + u(t n ) ), v h ) + (b ( u(t n+) + u(t n ) ), v h ), v h ) = (f(t n+/ ), v h ) + h( ( u(t n+) + u(t n ) ), v h ) + B n (u, v h ) + 4 (u (5) (t n+θ ), v h ), for some θ ]0, [. 87

98 Also, the only extra computational cost for the correction step is due to the storage of a few vectors u h,i. The Cran-Nicolson scheme is used for computing both u h and u h, but the time accuracy of the approximate solution u h is increased to be O( 4 ). The method is proven to be unconditionally stable over the finite time; it is also stable over all time under the assumption g b β > 0. In section 4. we briefly describe notation used and a few established results. Stability of the method is proven in Section 4.3. We conclude with the numerical results, proving the error estimates for the method - this is presented in Section NOTATION AND PRELIMINARIES We begin with a few definitions, assumptions, and forms used, and conclude the section with a statement of the method to be studied. The variational formulation of (4..) is naturally stated in X := H 0(Ω) d = {v : Ω R d, v L (Ω) d, v L (Ω) d, v = 0 on Ω}. We use the standard L norm,, and the usual norm on the Sobolev space H, namely. We mae several common assumptions. Remar 4.. There exists a constant β such that g b β > 0. The method is proven to be stable over a finite time even if the Assumption 4. doesn t hold. If it does, the method is stable over all time. We shall assume that the velocity finite element spaces X h X are conforming and have typical approximation properties of finite element spaces commonly in use. Namely, we tae X h to be spaces of continuous piecewise polynomials of degree, with. The interpolating properties of X h are given by the following assumption. 88

99 Remar 4.. For any function u X { } inf u χ + h (u χ) ch r+ u r+, r. χ X h We conclude the preliminaries by formulating the discrete Gronwall s lemma, see, e.g. [HR90] Lemma 4.. Let, B, and a µ, b µ, c µ, γ µ, for integers µ 0, be nonnegative numbers such that: a n + b µ γ µ a µ + c µ + B for n 0. µ=0 µ=0 µ=0 Suppose that γ µ < for all µ, and set σ µ = ( γ µ ). Then a n + µ=0 b µ e n µ=0 σµγµ [ c µ + B]. µ=0 4.3 STABILITY OF THE METHOD In this section we prove the unconditional stability of the discrete artificial viscosity approximation u h and use this result to prove stability of the higher order approximation u h. The approximations u h and u h are shown to be bounded uniformly in ɛ. Theorem 4.. Let f L (0, T ; L (Ω)). Let g b β >. If β < 0, let the length of the time step satisfy β < /4. Then the approximation u h, satisfying (4..a), is stable over the finite time T <. Specifically, there exist positive constants C, C such that for any n N u h,n+ + e C T ( (h + ɛ) ( uh,i+ + u h,i ) (4.3.) ) f(t i+/ ). u h,0 + C 89

100 If Assumption 4. is satisfied, then u h is stable over all time and u h,n+ + β + u h,i + uh,i+ (h + ɛ) ( uh,i+ + u h,i ) (4.3.) u h,0 + β f(t i+/ ). Proof. Tae v h = uh,n+ X h in (4..a). Apply Green s theorem to the last two terms +u h,n on the left hand side (with u h = 0 on Ω) and use the assumption g b β. The Cauchy-Schwartz and Young s inequalities, applied to the left hand side, yield u h,n+ u h,n + (h + ɛ) ( uh,n+ + u h,n ) (4.3.3) +β uh,n+ + u h,n (f(t n+/ ), uh,n+ + u h,n ). If Assumption 4. is satisfied, we bound the right hand side of (4.3.3) by (f(t n+/ ), uh,n+ + u h,n ) β f(t n+/) + β uh,n+ + u h,n. (4.3.4) Multiply (4.3.3) by and sum over the time levels i = 0,.., n +. Using (4.3.4), we obtain + β uh,i+ u h,n+ + + u h,i (h + ɛ) ( uh,i+ + u h,i ) (4.3.5) β f(t i+/), u h,0 + which proves stability over all time (provided that Assumption 4. is satisfied). If g b β with < β < 0, then we bound the right hand side of (4.3.3) by (f(t n+/ ), uh,n+ + u h,n ) 4 β f(t n+/) + β uh,n+ + u h,n. (4.3.6) It follows from (4.3.3),(4.3.6) and the triangle inequality that u h,n+ u h,n + (h + ɛ) ( uh,n+ + u h,n ) (4.3.7) 4 β f(t n+/) + β ( u h,n+ + u h,n ). 90

101 Multiply (4.3.7) by and sum over the time levels i = 0,.., n +. Under the condition β < /4 the discrete Gronwall s lemma yields u h,n+ + e ct ( (h + ɛ) ( uh,i+ + u h,i ) ) β f(t i+/), u h,0 + with c > 0. We now proceed to the proof of stability of u h. It follows from (4..3) that B i (u h, v h ) C v h for any v h X h, provided that the time difference uh,i+ u h,i u h,i. i = 0,..., N. Hence, we begin by establishing the bound for uh,i+ is bounded for any Lemma 4.. Let u h,0 H (Ω), f t L (0, T ; L (Ω)). Let g b β >. If β < 0, let the length of the time step satisfy β < /4. Then uh,n+ u h,n is bounded for the finite time T <. Specifically, there exist positive constants c, C = C(b, g, f, u h,0 ) such that u h,n u + (h + ɛ) (uh,i+ h,i ) i= ) e (C ct + β f(t i+/) f(t i / ). uh,n+ i= If Assumption 4. is satisfied, then uh,n+ C = C(b, g, f, u h,0 ) such that u h,n is bounded over all time: there exists + i= u h,n + uh,n+ β uh,i+ u h,i 4 i= C + u (h + ɛ) (uh,i+ h,i ) β f(t i+/) f(t i / ). i= 9

102 Proof. Consider (4..a) at any time level n. Tae v h = uh,n+ u h,n and mae the same choice for v h in (4..a) at the previous time level. Subtracting the resulting equations leads to ( uh,n+ u h,n + u h,n + (b (uh,n+, uh,n+ u h,n ) + u h,n (h + ɛ) (uh,n+ ) u h,n ), uh,n+ u h,n ) + g uh,n+ u h,n = ( f(t n+/) f(t n / ), uh,n+ u h,n ). Apply Green s theorem to the last two terms on the left hand side (with uh,n+ = 0 u h,n on Ω) and use the assumption g b β. Rewrite the first term on the left hand side, using the identity ( uh,n+ u h,n + u h,n This yields uh,n+ u h,n ) = uh,n+ u h,n uh,n u h,n., uh,n+ u h,n uh,n u h,n + u h,n (h + ɛ) (uh,n+ ) (4.3.8) + β uh,n+ u h,n ( f(t n+/) f(t n / ), uh,n+ u h,n ). Sum (4.3.8) over all time levels i =,.., n and consider the cases β > 0 and < β < 0 separately, as in the proof of Theorem 4.. If the Assumption 4. holds, this yields + uh,n+ u h,n u + (h + ɛ) (uh,i+ h,i ) (4.3.9) i= β uh,i+ u h,i uh, u h,0 + 4 β f(t i+/) f(t i / ). i= i= If < β < 0 and β < /4, it follows from the discrete Gronwall s lemma that uh,n+ u h,n u + (h + ɛ) (uh,i+ h,i ) (4.3.0) i= ( ) e ct uh, u h,0 + β f(t i+/) f(t i / ). i= 9

103 To complete the proof, we need a bound on uh, tae v h = uh, uh,0. This gives uh,0. Consider (4..a) at n = 0 and uh, u h,0 + (h + ɛ)( ( uh, + u h,0 ), ( uh, u h,0 )) (4.3.) +(b ( uh, + u h,0 ), uh, u h,0 ) + g( uh, + u h,0, uh, u h,0 ) = (f(t / ), uh, u h,0 ). Using the identity uh, +uh,0 = uh, uh,0 + u h,0, we can rewrite the last three terms in the left hand side of (4.3.). Applying Green s theorem as in the proofs above, yields u h,0 + (h + ɛ) ( uh, u h,0 ) + (h + ɛ)( u h,0, uh, u h,0 ) + β uh, u h,0 + (b u h,0, uh, u h,0 ) + g(u h,0, uh, u h,0 ) (f(t / ), uh, u h,0 ). uh, The Cauchy-Schwartz and Young s inequalities give uh, u h,0 + (h + ɛ) ( uh, u h,0 ) + β uh, u h,0 C, where C = b u h,0 + g u h,0 + (h + ɛ) u h,0 + f(t / ) <. Hence, if Assumption 4. is satisfied, or if < β < 0 and β < /4, we obtain the bound u h,0 + (h + ɛ) ( uh, u h,0 ) C <. (4.3.) uh, Inserting the bound on uh, uh,0 into (4.3.9) and (4.3.0) completes the proof. The unconditional stability of u h (uniform in ɛ) follows from Theorem 4. and Lemma 4.. The last term in the right hand side of (4..b) is bounded by means of Lemma

104 Theorem 4.. Let the assumptions of Theorem 4. and Lemma 4. be satisfied. Let g b β >. If β < 0, let the length of the time step satisfy β < /4. Then the approximation u h, satisfying (4..b), is stable over the finite time T <. Specifically, there exist positive constants c, C, C 3 such that for any n N e c T ( u h,n+ + (h + ɛ) ( uh,i+ + u h,i ) ) f(t i+/ ). C 3 + u h,0 + u h,0 + C If Assumption 4. is satisfied, then u h is stable over all time and u h,n+ + β + u h,i + uh,i+ (h + ɛ) ( uh,i+ + u h,i ) C + u h,0 + u h,0 + β f(t i+/ ), with C = C(b, g, f, u h,0 ). Proof. The proof resembles the proof of Theorem 4.. Tae v h = uh,n+ X h in (4..b). +u h,n Apply Green s theorem to the last two terms on the left hand side. This gives u h,n+ u h,n + (h + ɛ) ( uh,n+ + u h,n ) + β uh,n+ + u h,n (4.3.3) (f(t n+/ ), uh,n+ + u h,n ) + h( ( uh,n+ + u h,n ), ( uh,n+ + u h,n )) +B n (u h, uh,n+ + u h,n ). It is easy to verify that for any n where C = β B n (u h, uh,n+ + u h,n ) C + β uh,n+ + u h,n, (4.3.4) u h,i max + β max f(t i+) f(t i ). 0 i N 0 i N uh,i+ It also follows from Lemma 4. that this constant is finite, C <. 94

105 Using Cauchy-Schwartz and Young s inequalities gives h ( ( uh,n+ + u h,n ), ( uh,n+ + u h,n )) + u h,n (h + ɛ) (uh,n+ ) (4.3.5) h + u + (h + ɛ) (uh,n+ h,n ). (h + ɛ) Multiply (4.3.3) by, sum over the time levels and use (4.3.4),(4.3.5). Theorem 4. gives the bound on n uh,i+ (h + ɛ) ( ). The cases when Assumption 4. holds +u h,i and when < β < 0, β < /4 are treated as in the proof of Theorem 4.. Thus, the method is unconditionally stable for all time, provided that the Assumption 4. is satisfied. The approximate solutions u h and u h are bounded uniformly in h and ɛ. If the condition g b β is satisfied with < β < 0, then the assumption β < /4 is needed to conclude stability and uniform boundedness of the approximate solutions over a finite time. 4.4 COMPUTATIONAL RESULTS Based on the results of [L07] (and the general theory of Defect Correction Methods), we are expecting to obtain the following error estimates: u u h L (0,T ;L (Ω)) C(h + ), u u h L (0,T ;L (Ω)) C(h u u h L (0,T ;L (Ω)) + 4 ) C(h + h + 4 ). Consider the following transport problem in Ω = [0, ] [0, ]: find u satisfying (4..) with ɛ = 0 5, b = (, ) T, g = and f = [( + ɛπ ) sin(πx) sin(πy) + π sin(πx + πy)]e t. This problem has a solution u = sin(πx) sin(πy)e t. The results presented in the following tables are obtained by using the software F reef EM+ +. In order to draw conclusions about the convergence rate, we tae = h and = h. Note that the method needs the number of time steps N 4. 95

106 Table 9: Error estimates, ɛ = 0 5, T =, = h h u u h L (0,T ;L (Ω)) rate u u h L (0,T ;L (Ω)) rate / / / / Table 0: Error estimates, ɛ = 0 5, T =, = h h u u h L (0,T ;L (Ω)) rate u u h L (0,T ;L (Ω)) rate / / / The method doesn t resolve the problem of oscillations in the boundary layer, but the oscillations do not spread beyond the boundary layer. This is verified by the figure plots of the computed solution u h, as the mesh size and the time step are decreased. Figure 7: Computed solution u h, = h = /8 96

107 Figure 8: Computed solution u h, = h = /6 Figure 9: Computed solution u h, = h = /3 Finally, we compute the approximation error away from the boundary layer, namely in (0, 0.75) (0, 0.75). Table : Error estimates in (0, 0.75) (0, 0.75), ɛ = 0 5, = h h u u h L (0,T ;L (Ω)) rate u u h L (0,T ;L (Ω)) rate / / / Hence, the computational results verify the claimed accuracy of the method away from boundaries. Also, the oscillations of the computed solution do not spread outside of the boundary layer. 97