Crank–Nicolson, Finite Element Approximation∗

Approximation of Time-Dependent, Viscoelastic Fluid Flow: Crank–Nicolson, Finite Element Approximation∗

Vincent J. Ervin† Norbert Heuer‡

Abstract. In this article we analyze a fully discrete approximation to the time dependent viscoelas- ´ ticity equations with an Oldroyd B constitutive equation in IRd, d´= 2, 3. We use a Crank–Nicolson discretization for the time derivatives. At each time level a linear system of equations is solved. To resolve the non-linearities we use a three step extrapolation for the prediction of the velocity and stress at the new time level. The approximation is stabilized by using a discontinuous Galerkin approximation for the constitutive equation. For the mesh parameter, h, and the temporal step size, ∆t, suﬃciently small and satisfying ∆t ≤ Chd/´ 4, existence of the approximate solution is proven. A priori error estimates for the approximation in terms of ∆t and h are also derived.

Key words. viscoelasticity, ﬁnite element method, fully discrete, discontinuous Galerkin

AMS Mathematics subject classiﬁcations. 65N30

1 Introduction

The accurate numerical simulations of time dependent viscoelastic flows are important in the ability to predict flow instabilities in non-Newtonian fluid mechanics. The underlying equations to be

∗Supported by FONDAP Program on Numerical Analysis and Fondecyt projects nos. 1010220/7010220 (both Chile) †Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634-0975, USA. email: [email protected]. Partially supported by the ERC Program of the National Science Foundation under Award Number ERC-9731680. This research was undertaken while a visitor in the Departamento de Ingenier´ıaMatem´atica, Universidad de Concepc´ıon. ‡GI2MA, Departamento de Ingenier´ıaMatem´atica,Universidad de Concepc´ıon,Casilla 160-C, Concepc´ıon,Chile. email: [email protected]

1 solved are the conservation of momentum and incompressibilty equations for ﬂuid ﬂow, coupled with a (hyperbolic) constitutive equation for the viscoelastic component of the stress. To avoid the introduction of spurious oscillations in the numerical approximation, some stabilization is needed in the discretization of the constitutive equation. This is commonly done via a discontinuous Galerkin (DG) approximation for the stress [2],[3],[14], or by using a Streamline Upwind Petrov Galerkin (SUPG) [7],[16] approximation for the constitutive equation.

In this paper we analyze a Crank-Nicolson, Finite Element Method (FEM) approximation scheme, and show that it is second order with respect to the time discretization (∆t). To date the only proofs of convergence for numerical approximations to time dependent problems in viscoelastic fluid flow, governed by a differential constitutive model, are given in [3], and [7]. This work extends the results obtained in [3], and [7]. In [3], Baranger and Wardi studied an implicit Euler time discretization, with a DG approximation for the stress, and showed that in IR2 the approximation of the velocity and the viscoelastic stress was first order in time, under the condition that ∆t ≤ Ch3/2. In [7], Ervin and Miles analyzed an implicit Euler time discretization with a SUPG discretization ´ of the constitutive equation and showed that, in IRd, the method was first order in time under the weaker condition of ∆t ≤ Chd/´ 2. No estimates for the approximation error for the pressure were given in [3],[7]. To obtain such an estimate one uses the discrete inf-sup condition together with the momentum equation. This requires a time differencing of the velocity approximation. For a first order temporal approximation for the velocity this would give an O(1) estimate for the error in the pressure. In this paper we are able to show that the Crank-Nicolson FEM approximation scheme generates a first order temporal approximation for the pressure.

Heywood and Rannacher in [10] studied a Crank–Nicolson approximation for the non-stationary Navier-Stokes equations. The algorithm they analyzed required the solution of a non-linear system at each time level. The authors oﬀered two suggestions to avoid having to solve a non-linear system while maintaining second order accuracy for the time discretization. These were: (i) linearize the non-linear system about the current approximation, and (ii) linearize the non-linear terms by using an extrapolation of the current and previous time level approximations (i.e. a two level scheme). This two level approach was implemented by Mu in [13] for the numerical simulation of the Ginzburg- Landau model of superconductivity. A comparison of the Crank-Nicolson method with other time stepping techniques for ﬂow problems is given in [17].

2 In forming a Crank–Nicolson approximation for viscoelasticity our goal was to have the approximation determined at each time level by the solution of a single linear system. To do so we use the extrapolation approach. Linearizing the non-linear system would still have involved the complica- tion of having the unknown velocity in the computation of the “edge jump contribution” arising from the DG discretization of the constitutive equation. We were not able to show second order accuracy in time using a two level discretization scheme. In the analysis the gradient of the velocity extrapolant is required to be bounded. We could not establish such a bound with a two level scheme. We therefore propose and analyze a three level scheme. However, the three level scheme analyzed can be considered a two level scheme for the time averaged variables. In deriving the error estimates we assume that the solution has the required regularity. For a discussion on the regularity issues associated with using the Crank-Nicolson discretization for the approximation of initial value problems we refer the reader to [10].

The paper is organized as follows. In section 2 we briefly describe the viscoelastic modeling equations. Herein we present the analysis for the Oldroyd B model, however the results can be readily extended to other differential models. Following the description of the model a variational formulation of the continuous problem is given. We then prove a perturbation result for the distance between the solution of the modeling equations and a nearby problem. The finite element approximation scheme is presented in section 3. The error analysis for the general scheme is then presented in section 4. Following in the appendix are several estimates used in the analysis of the general scheme, as well as an analysis of a suitable initialization procedure.

2 The Mathematical Model and the Approximating System

In this section we describe the modeling equations for viscoelastic fluid flow and the finite element approximation scheme.

2.1 The Mathematical Model

´ Consider a ﬂuid ﬂowing in a bounded, connected domain Ω ∈ IRd. The boundary of Ω, ∂Ω, is assumed to be Lipschitz. The vector n represents the outward unit normal to ∂Ω. The velocity

3 vector is denoted by u, pressure by p, total stress by T, and extra stress by τ. The deformation tensor, D(u), and the vorticity tensor, W (u), are given by

1 ³ ´ 1 ³ ´ D(u) = ∇u + (∇u)T ,W (u) = ∇u − (∇u)T . 2 2

The Oldroyd model can be described using an objective derivative [2],[11] denoted by ∂σ/∂tˆ , where

∂σˆ ∂σ := + u · ∇σ + g (σ, ∇u) , a ∈ [−1, 1] ∂t ∂t a and

ga(σ, ∇u) := σW (u) − W (u)σ − a(D(u)σ + σD(u)) 1 − a ³ ´ 1 + a ³ ´ = σ∇u + (∇u)T σ − (∇u)σ + σ(∇u)T . 2 2

Oldroyd’s model for stress employs a decomposition of the extra stress into two parts: a Newtonian

part and a viscoelastic part. So τ = τN + τV . The Newtonian part is given by τN = 2(1−α)D(u). The (1−α) represents that part of the total viscosity which is considered Newtonian. Hence α ∈ (0, 1) represents the proportion of the total viscosity that is considered to be viscoelastic in nature. For example, if a polymer is immersed within a Newtonian carrier ﬂuid, α is related to the percentage of polymer in the mix. The constitutive law is [2]

∂τˆ τ + λ V − 2αD(u) = 0 , (2.1) V ∂t where λ is the Weissenberg number, which is a dimensionless constant deﬁned as the product of the relaxation time and a characteristic strain rate [4]. For notational simplicity, the subscript, V , is dropped, and below τ will be used to denote the viscoelastic component of the extra stress.

The momentum balance for the ﬂuid is given by µ ¶ du Re = −∇p + ∇ · (2(1 − α)D(u) + τ) + f , (2.2) dt where Re is the Reynolds number, f the body forces acting on the ﬂuid, and du/dt := ∂u/∂t + u·∇u denotes the material derivative.

In addition to (2.1) and (2.2) we also have the incompressibility condition:

∇ · u = 0 in Ω.

4 To fully specify the problem, appropriate boundary conditions must also be given. A condition for the velocity is required on each of the boundaries, and the stress specified on the inflow boundary. For simplicity, we consider homogeneous Dirichlet condition for velocity. In this case, there is no inflow boundary, and, thus, no boundary condition is required for stress. Summarizing, the modeling equations are: µ ¶ ∂u Re + u · ∇u + ∇p − 2(1 − α)∇ · D(u) − ∇ · τ = f in Ω, (2.3) ∂t µ ¶ ∂τ τ + λ + u · ∇τ + g (τ, ∇u) − 2αD(u) = 0 in Ω, (2.4) ∂t a ∇ · u = 0 in Ω, (2.5)

u = 0 on ∂Ω, (2.6)

u(0, x) = u0(x) in Ω, (2.7)

τ(0, x) = τ0(x) in Ω. (2.8)

In [9], Guillope and Saut proved the following for the “slow-ﬂow” model of (2.3)-(2.8) (i.e. u · ∇u term in (2.3) is ignored):

1. local existence, in time, of a unique, regular solution, and

0 2. under a small data assumption on f, f , u0, τ0, the global existence (in time) of a unique solution for u and τ.

In contrast to the Navier–Stokes equations, well-posedness for general models in viscoelasticity is still not well understood. Results which are known fall into one of three types [15]:

1. for inital value problems, solutions have been shown to exist locally in time,

2. global existence (in time) of solutions if the initial conditions are small perturbations of the rest state, and

3. for steady-state problems, existence of solutions which are small perturbations of the analogous Newtonian case.

5 2.2 The Variational Formulation

In this section, we develop the variational formulation of (2.3)-(2.6). The following notation will be used. The L2(Ω) norm and inner product will be denoted by k·k and (·, ·), respectively. We use Hk k k to represent the Sobolev space W2 , and k·kk denotes the norm in H . When v(x, t) is deﬁned on the entire time interval (0,T ), we deﬁne Ã ! Z T 1/2 2 kvk∞,k := sup kv(·, t)kk , kvk0,k := kv(·, t)kk dt , kvk(t) := kv(·, t)k . 0

The following function spaces are used in the analysis: n o 1 1 Velocity Space : X := H0 (Ω) := u ∈ H (Ω) : u = 0 on ∂Ω , n o 2 Stress Space : S := τ = (τij): τij = τji; τij ∈ L (Ω); 1 ≤ i, j ≤ d´ n o 2 ∩ τ = (τij): u · ∇τ ∈ L (Ω), ∀u ∈ X , Z 2 2 Pressure Space : Q := L0(Ω) = {q ∈ L (Ω) : q dx = 0}, Z Ω Divergence − free Space : Z := {v ∈ X : q(∇ · v) dx = 0, ∀ q ∈ Q}. Ω The variational formulation of (2.3)-(2.6) proceeds in the usual manner. Taking the inner product of (2.3), (2.4), and (2.5) with a velocity test function, a stress test function, and a pressure test function respectively, we obtain µ ¶ ∂u Re + u · ∇u, v − (p, ∇ · v) + (2(1 − α)D(u) + τ , D(v)) = (f, v), ∀ v ∈ X, (2.9) ∂t µ µ ¶ ¶ ∂τ τ + λ + u · ∇τ + g (τ, ∇u) − 2αD(u) , σ = 0, ∀ σ ∈ S, (2.10) ∂t a (∇ · u, q) = 0, ∀ q ∈ Q. (2.11)

The space Z is the space of weakly divergence free functions. Note that the condition

(∇ · u, q) = 0, ∀ q ∈ Q, u ∈ X, is equivalent in a “distributional” sense to

(u, ∇q) = 0, ∀ q ∈ Q, u ∈ X, (2.12)

−1 1 where in (2.12), (·, ·) denotes the duality pairing between H and H0 functions. In addition, note that the velocity and pressure spaces, X and Q, satisfy the inf-sup condition (q, ∇ · v) inf sup ≥ β > 0. (2.13) q∈Q v∈X kqk kvk1

6 Since the inf-sup condition (2.13) holds, an equivalent variational formulation to (2.9)-(2.11) is: Find u ∈ Z, τ ∈ S satisfying µ ¶ ∂u Re + u · ∇u, v + (2(1 − α)D(u) + τ, D(v)) = (f, v), ∀ v ∈ Z, (2.14) ∂t µ µ ¶ ¶ ∂τ τ + λ + u · ∇τ + g (τ, ∇u) − 2αD(u), σ = 0, ∀ σ ∈ S. (2.15) ∂t a

We assume that the fluid flow satisfies the following properties:

kuk∞ , kτk∞ , k∇uk∞ , k∇τk∞ ≤ M, for all t ∈ [0,T ] . (2.16)

2.3 Perturbation Estimate

For the error analysis of the Crank-Nicolson time discretization of (2.14),(2.15), given below in (3.16)-(3.17), it is convenient to compare the approximation with the solution to a nearby problem. In this section we establish an error estimate between (u, τ, p) satisfying (2.3)–(2.8) and (w, η, r) the solution of a nearby problem — assuming both solutions exist.

Let (w, η, r) denote the solution of µ ¶ ∂w Re + w˜ · ∇w + ∇r − 2(1 − α)∇ · D(w) − ∇ · η = f in Ω, (2.17) ∂t µ ¶ ∂η η + λ + w˜ · ∇η + g (˜η, ∇w) − 2αD(w) = 0 in Ω, (2.18) ∂t a ∇ · w = 0 in Ω, (2.19)

w = 0 on ∂Ω, (2.20)

w(0, x) = u0(x) in Ω, (2.21)

η(0, x) = τ0(x) in Ω, (2.22) where w˜ = w˜ (w, x, t) ∈ X, andη ˜ =η ˜(η, x, t) ∈ S.

Analogous to (2.16) we assume that

kwk∞ , kηk∞ , k∇wk∞ , k∇ηk∞ ≤ M, for all t ∈ [0,T ] . (2.23)

Theorem 1 Assume that there exist (u, τ, p) ∈ (X, S, Q) satisfying (2.9)-(2.11), and (w, η, r) ∈ (X, S, Q) satisfying the analogous variational form of (2.17)-(2.22). Then, we have the following

7 error estimate:

Z T Z T ³ ´ ku − wk2 (T ) + kτ − ηk2 (T ) + kD(u − w)k2 dt ≤ C kw − w˜ k2 + kη − η˜k2 dt . (2.24) 0 0

Proof: Analogous to (2.14)-(2.15) we have that w, η satisfy µ ¶ ∂w Re + w˜ · ∇w, v + (2(1 − α)D(w) + η , D(v)) = (f, v), ∀ v ∈ Z, (2.25) ∂t µ µ ¶ ¶ ∂η η + λ + w˜ · ∇η + g (˜η, ∇w) − 2αD(w) , σ = 0, ∀ σ ∈ S. (2.26) ∂t a

Note that

u · ∇u − w˜ · ∇w = (u − w) · ∇u + w · ∇(u − w) + (w − w˜ ) · ∇w . (2.27)

Similarly,

u · ∇τ − w˜ · ∇η = (u − w) · ∇τ + w · ∇(τ − η) + (w − w˜ ) · ∇η , (2.28)

ga(τ, ∇u) − ga(˜η, ∇w) = ga(τ − η, ∇u) + ga(η, ∇(u − w)) + ga(η − η,˜ ∇w) . (2.29)

Letting ²u := u − w, ²τ := τ − η, subtracting (2.25)-(2.26) from (2.14)-(2.15) and using (2.27)-(2.29) we have µ ¶ ∂² Re u , v + Re (² · ∇u, v) + Re (w · ∇² , v) + (2(1 − α)D(² ),D(v)) + (² ,D(v)) ∂t u u u τ = (−(w − w˜ ) · ∇w, v), ∀ v ∈ Z, (2.30) µ ¶ ∂² (² , σ) + λ τ , σ + λ (² · ∇τ, σ) + λ (w · ∇² , σ) + λ (g (² , ∇u), σ) + λ (g (η, ∇² ), σ) τ ∂t u τ a τ a u

− (2αD(²u), σ) = −((w − w˜ ) · ∇η, σ) − λ (ga(η − η,˜ ∇w), σ) , ∀ σ ∈ S. (2.31)

Multiplying (2.30) by 2α and adding to (2.31) we obtain for the choice v = ²u, σ = ²τ

2 2 2 α Re k²ukt + 2α Re (²u · ∇u, ²u) + 2α Re (w · ∇²u, ²u) + 4α (1 − α) kD(²u)k + k²τ k 1 + λ k² k2 + λ (² · ∇τ, ² ) + λ (w · ∇² , ² ) + λ (g (² , ∇u), ² ) + λ (g (η, ∇² ), ² ) 2 τ t u τ τ τ a τ τ a u τ

= −2α((w − w˜ ) · ∇w, ²u) − ((w − w˜ ) · ∇η, ²τ ) − λ (ga(η − η,˜ ∇w), ²τ ) . (2.32)

Note that, using (2.19), we have

(w · ∇²u, ²u) = − (∇ · w ²u, ²u) − (w · ∇²u, ²u) = − (w · ∇²u, ²u) .

8 Thus,

(w · ∇²u, ²u) = 0 , (2.33) and similarly,

(w · ∇²τ , ²τ ) = 0 . (2.34)

Using (2.33),(2.34), equation (2.32) may be rewritten as

1 α Re k² k2 + λ k² k2 + 4α (1 − α) kD(² )k2 + k² k2 u t 2 τ t u τ

= −2α Re (²u · ∇u, ²u) − λ (²u · ∇τ, ²τ ) − λ (ga(²τ , ∇u), ²τ ) − λ (ga(η, ∇²u), ²τ )

−2α((w − w˜ ) · ∇w, ²u) − ((w − w˜ ) · ∇η, ²τ ) − λ (ga(η − η,˜ ∇w), ²τ ) . (2.35)

We now bound each of the terms on the right hand side of (2.35).

´ 2 −2α Re (²u · ∇u, ²u) ≤ 2α Re k²u · ∇uk k²uk ≤ 2α d Re k∇uk∞ k²uk . (2.36)

Similarly,

´ −λ (²u · ∇τ, ²τ ) ≤ λ k²u · ∇τk k²τ k ≤ λ d k∇τk∞ k²uk k²τ k λ d´ λ d´ ≤ k∇τk k² k2 + k∇τk k² k2 , (2.37) 2 ∞ u 2 ∞ τ

´ 2 −λ (ga(²τ , ∇u), ²τ ) ≤ λ kga(²τ , ∇u)k k²τ k ≤ 4λ d k∇uk∞ k²τ k , (2.38)

2 ´2 2 2 ´ 2 4λ d CK kηk∞ 2 −λ (ga(η, ∇²u), ²τ ) ≤ 4λ d kηk∞ k∇²uk k²τ k ≤ ²1 kD(²u)k + k²τ k , (2.39) ²1 (using Korn’s lemma)

´ 2 2 ´2 2 2 −2α((w − w˜ ) · ∇w, ²u) ≤ 2α d k∇wk∞ kw − w˜ k k²uk ≤ kw − w˜ k + α d k∇wk∞ k²uk , (2.40)

d´2 −((w − w˜ ) · ∇η, ² ) ≤ kw − w˜ k2 + k∇ηk2 k² k2 , (2.41) τ 4 ∞ τ

9 ´ 2 2 ´2 2 2 −λ (ga(η − η,˜ ∇w), ²τ ) ≤ λ 4d kη − η˜k k∇wk∞ k²τ k ≤ kη − η˜k + λ 4d k∇wk∞ k²τ k . (2.42)

Substituting (2.36)-(2.42) into (2.35) we obtain

2 1 2 2 2 α Re k²ukt + λ k²τ kt + (4α (1 − α) − ²1) kD(²u)k + k²τ k 2 Ã ! d´ ≤ k² k2 2α d´ Re k∇uk + λ k∇τk + α2 d´2 k∇wk2 u ∞ 2 ∞ ∞ Ã d´ + k² k2 4λ d´ k∇uk + λ k∇τk + 4λ2 d´2 k∇wk2 τ ∞ 2 ∞ ∞ ! ´2 4 2 2 ´2 2 d 2 + λ CK d kηk∞ + k∇ηk∞ ²1 4 + 2 kw − w˜ k2 + kη − η˜k2 . (2.43)

Applying Gronwall’s lemma, we obtain (2.24).

Of particular interest in what follows is the case corresponding to w˜ (x, t),η ˜(x, t) given by

∆t 1 3∆t 1 5∆t w˜ (·, t) := w(·, t − ) + w(·, t − ) − w(·, t − ) , (2.44) 2 2 2 2 2 ∆t 1 3∆t 1 5∆t andη ˜(·, t) := η(·, t − ) + η(·, t − ) − η(·, t − ) . (2.45) 2 2 2 2 2

Corollary 1 For w˜ and η˜ deﬁned in (2.44),(2.45) we have that (u, τ) and (w, η) given respectively by (2.14),(2.15), and (2.25),(2.26) satisfy

Z T Z T ³ ´ 2 2 2 4 2 2 ku − wk (T ) + kτ − ηk (T ) + kD(u − w)k dt ≤ C (∆t) kwttk + kηttk dt . 0 −5∆t/2 (2.46)

Proof: In view of (2.24), from (A.13) (in the appendix) we have Ã ! Z T ³ ´ Z T Z t ³ ´ 2 2 39 3 2 2 kw − w˜ k + kη − η˜k dt ≤ (∆t) kwttk + kηttk dµ dt 0 8 0 t−5∆t/2 Z T ³ ´ 39 3 5 2 2 ≤ (∆t) ∆t kwttk + kηttk dt . 8 2 −5∆t/2

10 3 Finite Element Approximation

In this section we formulate a fully discrete finite element method for solving the viscoelastic fluid flow equations, and prove the solvability of the approximation at each step (for sufficiently small ∆t, h). To avoid having a non-linear algebraic system for the Crank-Nicolson discretization, the approximation is a three–level scheme, involving computed approximations at the three previous time levels.

We begin by describing the ﬁnite element approximation framework and listing the approximating properties and inverse estimates used in the analysis. We assume throughout that the viscoelastic stress tensors, τ, η, are continuous. This assumption is consistent with that used in [3] of τ ∈ H2(Ω) for Ω ⊂ IR2.

d´ ´ Let Ω ⊂ IR (d = 2, 3) be a polygonal domain and let Th be a triangulation of Ω made of triangles (in IR2) or tetrahedrons (in IR3). Thus, the computational domain is deﬁned by

[ Ω = K; K ∈ Th.

We assume that there exist constants c1, c2 such that

c1h ≤ hK ≤ c2ρK , where hK is the diameter of triangle (tetrahedral) K, ρK is the diameter of the greatest ball (sphere) included in K, and h = maxK∈Th hK . Let Pk(A) denote the space of polynomials on A of degree no greater than k. Then we deﬁne the ﬁnite element spaces as follows. n o 2 Xh := v ∈ X ∩ C(Ω)¯ : v|K ∈ Pk(K), ∀K ∈ Th ,

Sh := {σ ∈ S : σ|K ∈ Pm(K), ∀K ∈ Th} , © ª Qh := p ∈ Q ∩ C(Ω)¯ : p|K ∈ Pq(K), ∀K ∈ Th ,

Zh := {v ∈ Xh :(q, ∇·v) = 0, ∀q ∈ Qh} .

Analogous to the continuous spaces, we assume that Xh and Qh satisfy the discrete inf-sup condition

(q, ∇ · v) inf sup ≥ β > 0 . (3.1) q∈Q h v∈Xh kqk kvk1

11 We summarize several properties of ﬁnite element spaces and Sobolev’s spaces which we will use in our subsequent analysis. For (w, r) ∈ Hk+1(Ω)d´ × Hq+1(Ω) we have (see [8]) that there exists

(U, P) ∈ Zh × Qh such that

k+1 kw − Uk + hk∇(w − U)k ≤ CI h kwkk+1 , (3.2)

q+1 kr − Pk ≤ CI h krkq+1 . (3.3)

m+1 d´×d´ Let T ∈ Sh be a Pm continuous interpolant of η. For η ∈ H (Ω) we have that

m+1 kη − T k + hk∇(η − T )k ≤ CI h kηkm+1 . (3.4)

Let ∆t denote the step size for t so that tn = n∆t, n = 0, 1, 2,...,N. For notational convenience, n we denote v := v(·, tn). Also, let f(t ) − f(t ) d f := n n−1 (3.5) t ∆t f n + f n−1 f¯n := (3.6) 2 1 1 f˜n := f n−1 + f n−2 − f n−3 . (3.7) 2 2

Note that for w, η given by (2.25),(2.26) and U, T by (3.2),(3.4), it follows from (2.23) and inverse estimates, [5], that n n n n kU k∞ , k∇U k∞ , kT k∞ , k∇T k∞ ≤ M˜ ≈ M. (3.8)

Below, for simplicity, we take M˜ = M.

The following norms are also used in the analysis:

|kvk| := max kvnk ∞,k 1≤n≤N k " # 1 XN 2 n 2 |kvk|0,k := kv kk ∆t . n=1

In order to describe the approximation of the constitutive equation by the method of discontinuous ﬁnite elements, following [2], we introduce ∂K−(u) := {x ∈ ∂K, u · n < 0}, where ∂K is the ± boundary of K and n is the outward unit normal and τ (u)(x) := lim²→0± τ(x + ²u).

We deﬁne X X Z ± ± ± ± (τ, σ)h := (τ, σ)K , hτ , σ ih,u := (τ (u): σ (u))|u · n| ds , (3.9) ∂K−(u) K∈Th K∈Th

12 and ± ± ± 1/2 hhτ iih,u := hτ , τ ih,u . (3.10)

The operator B on Xh × Sh × Sh is deﬁned by 1 B(u, τ, σ) := (u · ∇τ, σ) + (∇ · u τ, σ) + hτ + − τ −, σ+i . (3.11) h 2 h,u We have on applying Green’s Theorem to (3.11) that 1 B(u, τ, σ) := −(u · ∇σ, τ) − (∇ · u σ, τ) + hτ −, σ− − σ+i , (3.12) h 2 h,u which on combining with (3.11) yields some “coercivity” for B 1 B(u, τ, τ) = hhτ + − τ −ii2 . (3.13) 2 h,u

Also used in the analysis, for notation convenience, is the operator c, deﬁned on Xh × Xh × Xh, by

c(w, u, v) := (w · ∇u, v) , (3.14) and λˆ := 2α/λ.

As we are assuming “slow ﬂow”, i.e. Re ≡ O(1), we use a conforming ﬁnite element method to discretize the momentum equation.

Initialization of the Approximation Scheme The approximation scheme described, and analyzed below, is a three level scheme. To initialize n n the procedure suitable approximates are required for uh, and τh for n = 0, 1, 2. Here we state our assumptions on these initial approximates. (An initialization procedure is presented in the appendix.)

n 2 n 2 n 2 kuh − w(n∆t)k + kτh − η(n∆t)k + ∆t kD(uh − w(n∆t))k ³ ´ ≤ C (∆t)4 + C h2k + h2m + h2q+2 1 = G (∆t, h) , for n = 0, 1, 2 . (3.15) 3 I

Approximating System n n For n = 3, 4,...,N, ﬁnd uh ∈ Zh, τh ∈ Sh such that

n n n n n ¯n Re(dtuh, v) + Re c(u˜h, u¯h, v) + 2(1 − α)(D(u¯h),D(v)) + (¯τh ,D(v)) = (f , v) , v ∈ Zh (3.16) 1 (¯τ n, σ) + (d τ n, σ) + B(u˜n, τ¯n, σ) − λˆ(D(u¯n), σ) + (g (˜τ n, ∇u¯n), σ) = 0 , σ ∈ S . (3.17) λ h t h h h h a h h h

13 To ensure computability of the algorithm, we begin by showing that (3.16)-(3.17) is uniquely solvable

for uh and τh at each time step n. We use the following induction hypothesis. ° ° ° ° ° ° ° ° (IH1) °un−1° , °τ n−1° ≤ K. h ∞ h ∞

Lemma 1 Assume (IH1) is true. For a suﬃciently small step size ∆t, there exists a unique solution n n (uh, τh ) ∈ Zh × Sh satisfying (3.16)-(3.17).

Proof: For notational simplicity, in this proof we drop the subscript h from the variables. Choosing n n ˆ v = uh, σ = τh , multiplying (3.16) by λ and adding to (3.17) we obtain Re Re a(un, τ n; un, τ n) = λˆ(¯f n, un) + λˆ (un−1, un) − λˆ c(uñ, un−1, un) − λˆ(1 − α)(D(un−1),D(un)) ∆t 2 1 1 1 1 − λˆ(τ n−1,D(un)) − (τ n−1, τ n) + (τ n−1, τ n) − B(uñ, τ n−1, τ n) 2 2λ ∆t 2 1 1 + λˆ(D(un−1), τ n) − (g (˜τ n, ∇un−1), τ n) (3.18) 2 2 a where the bilinear form a(u, τ; v, σ) is defined as: Re Re 1 a(u, τ; v, σ) := λˆ (u, v) + λˆ c(uñ, u, v) + λˆ(1 − α)(D(u),D(v)) + (τ, σ) ∆t 2 2λ 1 1 1 + (τ, σ) + B(uñ, τ, σ) + (g (˜τ n, ∇u), σ) . (3.19) ∆t 2 2 a We now estimate the terms in a(un, τ n; un, τ n). We have

n n ´1/2 n |c(u˜ , u, u)| = |(u˜ · ∇u, u)| ≤ d ku˜ k∞ k∇uk kuk ´1/2 n ≤ d ku˜ k∞ Ck kD(u)k kuk , (using Korn’s lemma) ´ 2 2 2 dK Ck 2 ≤ ²1 kD(u)k + kuk , 4²1 n 1 + − 2 B(u˜ , τ, τ) = hhτ − τ ii n , 2 h,u˜ n n n n |(ga(˜τ , ∇u), τ )| ≤ 4 kτ˜ ∇uk kτ k ´1/2 n n ≤ 4d kτ˜ k∞ k∇uk kτ k ´1/2 n ≤ 4d Ck2K kτ k kD(u)k ´ 2 2 2 16dCk K n 2 ≤ ²2 kD(u)k + kτ k . ²2 Applying these inequalities to the bilinear form a(·, · ; ·, ·) yields Ã ! µ ¶ 1 dK´ 2C2 Re ² ² a(un, τ n; un, τ n) ≥ λˆ Re − k kunk2 + λˆ[(1 − α) − 1 ] − 2 kD(un)k2 ∆t 8²1 2 2

14 Ã ! ´ 2 2 1 1 8dCk K n 2 1 + − 2 + + − kτ k + hhτ − τ iih,u˜n . 2λ ∆t ²2 4 n o (1−α) λˆ(1−α) (1−α) 4 λˆ Choosing ²1 = 2Re , ²2 = 2 , and ∆t ≤ ´ 2 2 min Re , 16 , it follows that the bilinear form d Ck K a(·, · ; ·, ·) is positive. Hence, (3.18) has at most one solution. Since (3.18) is a ﬁnite dimensional linear system, the uniqueness of the solution implies the existence of the solution.

The discrete Gronwall’s lemma plays an important role in the following analysis.

Lemma 2 (Discrete Gronwall’s Lemma) [10] Let ∆t, H, and an, bn, cn, γn , (for integers n ≥ 0), be nonnegative numbers such that

Xl Xl Xl al + ∆t bn ≤ ∆t γn an + ∆t cn + H for l ≥ 0 . n=0 n=0 n=0

−1 Suppose that ∆t γn < 1, for all n, and set σn = (1 − ∆t γn) . Then, Ã !( ) Xl Xl Xl al + ∆t bn ≤ exp ∆t σn γn ∆t cn + H for l ≥ 0 . (3.20) n=0 n=0 n=0

4 A Priori Error Estimate

In this section we analyze the error between the ﬁnite element approximation given by (3.16)-(3.17) and the true solution. A priori error estimates for the approximation are given in Theorem 2.

d/´ 4 Theorem 2 There exists a constant c1 > 0 such that for ∆t < c1h , the ﬁnite element approximation (3.16)-(3.17) is convergent to the solution (u, τ) of (2.14)-(2.15) on the interval (0,T ) as

∆t, h → 0. In addition, the approximation (uh, τh) satisﬁes the following error estimates:

|kuh − uk|∞,0 + |kτh − τk|∞,0 ≤ F1(∆t, h) , (4.1) 1/2 |ku¯h − u¯k|0,1 + |kτ¯h − τ¯k|0,0 ≤ C (1 + T ) F1(∆t, h) + C F2(∆t) , (4.2) where ³ ´ k m q+1 F1(∆t, h) := C h |kwk|0,k+1 + h |kηk|0,m+1 + h |krk|0,q+1

15 ³ ´ k+1 m+1 + C h kwtk0,k+1 + h kηtk0,m+1 ³ ´ 2 + C (∆t) kwttk0,1 + kwtttk0,0 + kηttk0,1 + kηtttk0,0 + krttk0,0 Ã ! Z T ³ ´ 1/2 1/2 2 2 2 + C (GI (∆t, h)) + C (∆t) kwttk + kηttk dt , −5∆t/2 ³ 2 F2(∆t) := (∆t) k∇uk0,2 + k∇wk0,2 + k∇utk0,2 + k∇wtk0,2 + k∇uttk0,2 ´ + k∇wttk0,2 + k∇utttk0,2 + k∇wtttk0,2 + k∇uttttk0,2 + k∇wttttk0,2 , (4.3) and w, η, r satisfy (2.17)-(2.22),(2.44),(2.45), and GI (∆t, h), deﬁned in (3.15), represents the initialization error.

The structure of the proof of Theorem 2 is as follows.

n n n n n n Let U , T denote elements in Zh, Sh, satisfying (3.2) and (3.4), respectively, and deﬁne Λ , E , Γ , F ,

²w, ²η as

n n n n n n Λ = w − U , E = U − uh, (4.4)

n n n n n n Γ = η − T , F = T − τh , (4.5)

n n n n ²w = w − uh, ²η = η − τh . (4.6)

As introduced above, we use a bar to denote average between levels n and n − 1 and a tilde to denote extrapolation from levels n − 1, n − 2, and n − 3, i.e., 1 ³ ´ Λ¯ n = Λn + Λn−1 , 2 1 1 Λ˜ n = Λn−1 + Λn−2 − Λn−3 = 2Λ¯ n−1 − Λ¯ n−2 . 2 2

Step 1. We prove the following lemma.

Lemma 3 Under the induction hypothesis (IH) we have that for l = 3, 4,...,N,

l ° °2 ° °2 X ° ° ° l° ° l° ° n °2 °E ° + °F ° + ∆t D(E¯ ) ≤ G(∆t, h) + CGI (∆t, h) , (4.7) n=3 where ³ ´ 4 2 2 2 2 2 G(∆t, h) = C (∆t) kwttk0,1 + kwtttk0,0 + kηttk0,1 + kηtttk0,0 + krttk0,0 ³ ´ 2k 2 2m 2 2q+2 2 + C h |kwk|0,k+1 + h |kηk|0,m+1 + h |kkk|0,q+1 ³ ´ 2k+2 2 2m+2 2 + C h kwtk0,k+1 + h kηtk0,m+1 .

16 Step 2. We show that the induction hypothesis, (IH1), is true.

Step 3. We derive the error estimates in (4.1) and (4.2).

Step 1. Proof of Lemma 3: From (2.17)-(2.18), it is clear that the true solution (w, η, r) satisﬁes

n n n n n Re (dtw , v) + Re c(w˜ , w¯ , v) + 2(1 − α)(D(w¯ ),D(v)) + (¯η ,D(v)) ¯n n−1/2 = (f , v) + (r , ∇ · v) + R1(v) , ∀ v ∈ Zh, (4.8) 1 (¯ηn, σ) + (d ηn, σ) + B(w˜ n, η¯n, σ) − λˆ(D(w¯ n), σ) + (g (˜ηn, ∇w¯ n), σ) λ t a

= R2(σ) , ∀ σ ∈ Sh, (4.9) where

n n−1/2 n n n−1/2 R1(v) := Re (dtw − wt , v) + Re c(w˜ , w¯ − w , v) + 2(1 − α)(D(w¯ n − wn−1/2),D(v)) + (¯ηn − ηn−1/2,D(v)) , (4.10) and 1 R (σ) := (¯ηn − ηn−1/2, σ) + (d ηn − ηn−1/2, σ) + B(w˜ n, η¯n − ηn−1/2, σ) (4.11) 2 λ t t n n−1/2 n n n−1/2 − λˆ(D(w¯ − w ), σ) + (ga(˜η , ∇(w¯ − w ), σ) . (4.12)

Subtracting (3.16)-(3.17) from (4.8)-(4.9) we obtain the following equations for ²w and ²η:

n Re (dt²w, v) + Re c(u˜h, ²¯w, v) + 2(1 − α)(D(¯²w),D(v)) + (¯²η,D(v))

n−1/2 n n n = (r , ∇ · v) + R1(v) + Re c(u˜h − w˜ , w¯ , v) , ∀ v ∈ Zh, (4.13) 1 (¯² , σ) + (d ² , σ) + B(u˜n, ²¯ , σ) − λˆ(D(¯² ), σ) + (g (˜τ n, ∇²¯ ), σ) λ η t η h η w a h w n n n n n n = R2(σ) + B(u˜h − w˜ , η¯ , σ) + (ga(˜τh − η˜ , ∇w¯ ), σ) , ∀ σ ∈ Sh. (4.14)

n n n n n n Substituting ²w = E + Λ , ²η = F + Γ , v = E¯ , σ = F¯ into (4.13)-(4.14), we obtain

n ¯ n n ¯ n ¯ n ¯ n ¯ n ¯ n ¯ n ¯ n Re (dtE , E ) + Re c(u˜h, E , E ) + 2(1 − α)(D(E ),D(E )) + (F ,D(E )) = F1(E ) (4.15). 1 (F¯ n, F¯ n) + (d Fn, F¯ n) + B(u˜n, F¯ n, F¯ n) − λˆ(D(E¯ n), F¯ n) + (g (˜τ n, ∇E¯ n), F¯ n) = F (F¯ n) .(4.16) λ t h a h 2 where,

¯ n n−1/2 ¯ n ¯ n n n n ¯ n n ¯ n F1(E ) = (r , ∇ · E ) + R1(E ) + Re c(u˜h − w˜ , w¯ , E ) − Re (dtΛ , E )

17 n ¯ n ¯ n ¯ n ¯ n ¯n ¯ n −Re c(u˜h, Λ , E ) − 2(1 − α)(D(Λ ),D(E )) − (Γ ,D(E )) , 1 F (F¯ n) = R (F¯ n) + B(u˜n − w˜ n, η¯n, F¯ n) + (g (˜τ n − η˜n, ∇w¯ n), F¯ n) − (Γ¯ n, F¯ n) − (d Γn, F¯ n) 2 2 h a h λ t n ¯ n ¯ n ˆ ¯ n ¯ n n ¯ n ¯ n −B(u˜h, Γ , F ) + λ(D(Λ ), F ) − (ga(˜τh , ∇Λ ), F ).

Multiplying (4.15) by λˆ and adding to (4.16) yields the single equation 1 Reλˆ (d En, E¯ n) + Reλˆ c(u˜n, E¯ n, E¯ n) + 2λˆ(1 − α)(D(E¯ n),D(E¯ n)) + (F¯ n, F¯ n) + (d Fn, F¯ n) t h λ t n ¯ n ¯ n n ¯ n ¯ n ˆ ¯ n ¯ n + B(u˜h, F , F ) + (ga(˜τh , ∇E ), F ) = λF1(E ) + F2(F ) .

Note that Ã ! n n−1 n n−1 µ ° ° ¶ E − E E + E 1 ° °2 (d En, E¯ n) = , = kEnk2 − °En−1° , t ∆t 2 2∆t and similarly, µ ° ° ¶ 1 ° °2 (d Fn, F¯ n) = kFnk2 − °Fn−1° . t 2∆t Thus we have ˆ µ ° ° ¶ µ ° ° ¶ Reλ ° °2 1 ° °2 ° °2 1 ° °2 kEnk2 − °En−1° + kFnk2 − °Fn−1° + 2λˆ(1 − α) °D(E¯ n)° + °F¯ n° 2∆t 2∆t λ 1 ¯ n+ ¯ n− 2 ˆ n ¯ n ¯ n n ¯ n ¯ n ˆ ¯ n + hhF − F iih,u˜n = −Reλ c(u˜h, E , E ) − (ga(˜τh , ∇E ), F ) + λF1(E ) 2 h n + F2(F¯ ) . (4.17)

Multiplying (4.17) by 2∆t and summing from n = 3, . . . , l we have µ° ° ° ° ¶ µ° ° ° ° ¶ l l ° °2 ° °2 ° °2 ° °2 X ° °2 2 X ° °2 Reλˆ °El° − °E2° + °Fl° − °F2° + 4λˆ(1 − α) ∆t °D(E¯ n)° + ∆t °F¯ n° n=3 λ n=3 Xl h i Xl ˆ n ¯ n ¯ n n ¯ n ¯ n ˆ ¯ n ≤ 2∆t −Reλ c(u˜h, E , E ) − (ga(˜τh , ∇E ), F ) + 2λ∆t F1(E ) n=3 n=3 Xl n + 2∆t F2(F¯ ) . (4.18) n=3

We now estimate each term on the right hand side of (4.18):

n ¯ n ¯ n n ¯ n ¯ n |c(u˜h, E , E )| ≤ |(u˜h · ∇E , E )| ° ° ° ° ° n ¯ n° ° ¯ n° ≤ u˜h · ∇E E ° ° ° ° n ´1/2 ° ¯ n° ° ¯ n° ≤ ku˜hk∞ d ∇E E ° ° ° ° ° n ° ´1/2 ° n° ≤ Ck D(E¯ ) 2Kd E¯ ° ° 2 ´ 2 ° ° n 2 K dCk n 2 ≤ ²1 °D(E¯ )° + °E¯ ° . ²1

18 ° ° ° ° n ¯ n ¯ n ° n ¯ n ° °¯ n° |(ga(˜τh , ∇E ), F )| ≤ ga(˜τh , ∇E ) F ° ° ° ° n ´° ¯ n° °¯ n° ≤ 4 kτ˜h k∞ d ∇E F ° ° ° ° ° n ° ´° n° ≤ Ck D(E¯ ) 8Kd F¯ ° ° 2 ´2 2 ° ° n 2 16K d Ck n 2 ≤ ²2 °D(E¯ )° + °F¯ ° . ²2

Thus for the ﬁrst summation on the right hand side of (4.18) we have

Xl Xl ³ ´ ° ° ˆ n ¯ n ¯ n n ¯ n ¯ n ˆ ° ¯ n °2 2∆t [−Reλ c(u˜h, E , E ) − (ga(˜τh , ∇E ), F )] ≤ 2∆t Reλ²1 + ²2 D(E ) n=3 n=3 Xl ReλˆdK´ 2C2 ° ° Xl 16d´2K2C2 ° ° +2∆t k °E¯ n°2 + 2∆t k °F¯ n°2 .(4.19) n=3 ²1 n=3 ²2

n n n−1 Next we consider F1(E¯ ). For P and P elements in Qh satisfying (3.3),

|(rn−1/2, ∇·E¯ n)| = |(rn−1/2 − P¯n, ∇·E¯ n)|

≤ |(rn−1/2 − r¯n, ∇·E¯ n)| + |(¯rn − P¯n, ∇·E¯ n)| ° ° ° ° ° ° ° ° ° ° ≤ °rn−1/2 − r¯n° °∇·E¯ n° + °r¯n − P¯n° °∇·E¯ n° ° ° ° ° ° ° ° ° ° ° ≤ °rn−1/2 − r¯n° d´1/2 °∇E¯ n° + °r¯n − P¯n° d´1/2 °∇E¯ n° ° ° ° ° ° ° ° ° ° n ° ´1/2 ° n−1/2 n° ° n ° ´1/2 ° n n° ≤ Ck D(E¯ ) d °r − r¯ ° + Ck D(E¯ ) d r¯ − P¯

° ° 2 ´ ° °2 2 ´ ° ° n 2 Ck d ° n−1/2 n° Ck d n n 2 ≤ (²3 + ²4) °D(E¯ )° + °r − r¯ ° + °r¯ − P¯ ° . (4.20) 4²3 4²4

° ° n n n ¯ n n n n ° ¯ n° |c(u˜h − w˜ , w¯ , E )| ≤ k(u˜h − w˜ ) · ∇w¯ k E ° ° ´ n n n ° ¯ n° ≤ d k∇w¯ k∞ ku˜h − w˜ k E 1 1 ° ° ≤ dM´ ku˜n − w˜ nk2 + dM´ °E¯ n°2 2 h 2 ° ° ° ° 1 ° °2 ° °2 ° °2 ≤ dM´ °E¯ n° + dM´ °E˜ n° + dM´ °Λ˜ n° . (4.21) 2

n n ° n° n |(dtΛ , E¯ )| ≤ °E¯ ° kdtΛ k ° ° 1 ≤ °E¯ n°2 + kd Λnk2 . (4.22) 4 t

19 ° ° ° ° n ¯ n ¯ n ° ¯ n° ° n ¯ n° |c(u˜h, Λ , E )| ≤ E u˜h · ∇Λ ° ° ° ° ° ¯ n° n ´1/2 ° ¯ n° ≤ E ku˜hk∞ d ∇Λ ° ° ° ° ≤ °E¯ n°2 + K2d´°∇Λ¯ n°2 . (4.23)

° ° ° ° |(D(Λ¯ n),D(E¯ n))| ≤ °D(Λ¯ n)° °D(E¯ n)°

° n °2 1 ° n°2 ≤ ²5 °D(E¯ )° + °∇Λ¯ ° . (4.24) 4²5

° ° ° ° |(Γ¯ n,D(E¯ n))| ≤ °Γ¯ n° °D(E¯ n)°

° n °2 1 ° n°2 ≤ ²6 °D(E¯ )° + °Γ¯ ° . (4.25) 4²6

n For the R1(E¯ ) terms we have: ° ° ° °2 1 ° °2 |(d wn − wn−1/2, E¯ n)| ≤ °E¯ n° + °d wn − wn−1/2° . (4.26) t t 4 t t

|c(w˜ n, w¯ n − wn−1/2, E¯ n)| = |(w˜ n · ∇(w¯ n − wn−1/2), E¯ n)| ° ° ° ° ° ° ≤ °w˜ n · ∇(w¯ n − wn−1/2)° °E¯ n° ° ° ° ° n ´1/2 ° n n−1/2 ° ° ¯ n° ≤ kw˜ k∞ d °∇(w¯ − w )° E ° ° ° °2 ° °2 ≤ M °∇(w¯ n − wn−1/2)° + Md´°E¯ n° . (4.27)

° ° ° ° ° ° |(D(w¯ n − wn−1/2),D(E¯ n))| ≤ °D(w¯ n − wn−1/2)° °D(E¯ n)° ° °2 ° n °2 1 ° n n−1/2 ° ≤ ²7 °D(E¯ )° + °∇(w¯ − w )° . (4.28) 4²7

° °2 n n−1/2 n ° n °2 1 ° n n−1/2° |(¯η − η ,D(E¯ ))| ≤ ²8 °D(E¯ )° + °η¯ − η ° . (4.29) 4²8

20 n Combining (4.20)-(4.29) we have the following estimate for F1(E¯ ). ³ ´ n ° n °2 λˆF1(E¯ ) ≤ °D(E¯ )° λˆ(²3 + ²4 + ²6 + ²8) + λˆ2(1 − α)(²5 + ²7) µ ¶ ° ° ³ ´ ° °2 1 ° °2 + °E¯ n° 3λReˆ + λReˆ dM´ + λReˆ dM´ + °E˜ n° λReˆ dM´ 2 µ ¶ µ ¶ ° ° ³ ´ ° °2 1 Re ° °2 + °∇Λ¯ n° λReˆ dK´ 2 + λˆ2(1 − α) + kd Λnk2 λˆ + °Λ˜ n° λReˆ dM´ 4² t 4 µ 5 ¶ µ ¶ ° °2 ° °2 ° n n−1/2 ° ˆ ˆ 1 ° n n−1/2° ˆ 1 + °∇(w¯ − w )° λRe M + λ2(1 − α) + °dtw − wt ° λRe 4²7 4 µ ¶ ° ° µ ¶ ° °2 1 ° °2 1 + °Γ¯ n° λˆ + °η¯n − ηn−1/2° λˆ 4² 4² 6Ã ! 8Ã ! 2 ´ ° ° 2 ´ ° °2 C d ° °2 C d + °r¯n − P¯n° λˆ k + °rn−1/2 − r¯n° λˆ k . (4.30) 4²4 4²3

n Next we consider the terms in F2(F¯ ).

For the ﬁrst term in (4.31)

° ° n n n ¯ n n n n °¯ n° |((u˜h − w˜ ) · ∇η¯ , F )h| ≤ k(u˜h − w˜ ) · ∇η¯ k F ° ° ´3/2 n n n °¯ n° ≤ d k∇η¯ k∞ k(u˜h − w˜ )k F 1 1 ° ° ≤ d´3/2M k(u˜n − w˜ n)k2 + d´3/2M °F¯ n°2 2 h 2 ° ° ° ° ° °2 ° °2 1 ° °2 ≤ d´3/2M °E˜ n° + d´3/2M °Λ˜ n° + d´3/2M °F¯ n° . (4.32) 2

The second term is handled via

° ° n n n ¯ n n n n °¯ n° |(∇ · (u˜h − w˜ )η ¯ , F )| ≤ k∇ · (u˜h − w˜ )η ¯ k F ° ° ´ n n n °¯ n° ≤ d kη¯ k∞ k∇ · (u˜h − w˜ )k F ° ° ´3/2 n n °¯ n° ≤ d M k∇(u˜h − w˜ )k F ° ° ° ° ° ° ° ° ° ° ° ° ≤ d´3/2M °∇E˜ n° °F¯ n° + d´3/2M °∇Λ˜ n° °F¯ n° ° ° ° ° ° ° ° ° 3/2 ° n ° ° n° 3/2 ° n° ° n° ≤ d´ MCK °D(E˜ )° F¯ + d´ M °∇Λ˜ ° F¯ Ã ! ° °2 2 ° ° ° °2 ° n ° 1 3 2 CK n 2 ° n° ≤ ²9 °D(E˜ )° + d´ M 1 + °F¯ ° + °∇Λ˜ ° . (4.33) 4 ²9

21 For the third term n+ n− n+ |hη¯ − η¯ , F¯ i n n | = 0 , (4.34) h,(u˜h−w˜ ) by the continuity ofη ¯n.

° ° n n n ¯ n n n n °¯ n° |(ga(˜τh − η˜ , ∇w¯ ), F )| ≤ kga(˜τh − η˜ , ∇w¯ )k F ° ° ´ n n n °¯ n° ≤ 4d k∇w¯ k∞ kτ˜h − η˜ k F ° ° ´ n n 2 ´ °¯ n° ≤ 2dM kτ˜h − η˜ k + 2dM F ° ° ° ° ° ° ° °2 ° °2 ≤ 2dM´ °F¯ n° + 4dM´ °F˜ n° + 4dM´ °Γ˜ n° . (4.35)

° ° 1 ° ° |(Γ¯ n, F¯ n)| ≤ °F¯ n°2 + °Γ¯ n°2 . (4.36) 4

° ° 1 |(d Γn, F¯ n)| ≤ °F¯ n°2 + kd Γnk2 . (4.37) t 4 t

n n n n n n 1 n n n n+ n− n+ |B(u˜ , Γ¯ , F¯ )| ≤ |(u˜ · ∇Γ¯ , F¯ ) | + |(∇ · u˜ Γ¯ , F¯ )| + |hΓ¯ − Γ¯ , F¯ i n | . h h h 2 h h,u˜

Each of these terms may be bounded via:

° ° ° ° n ¯ n ¯ n ° n ¯ n° °¯ n° |(u˜h · ∇Γ , F )h| ≤ u˜h · ∇Γ F ° ° ° ° ´1/2 n ° ¯n° °¯ n° ≤ d ku˜hk∞ ∇Γ F ° ° ° ° ≤ °F¯ n°2 + dK´ 2 °∇Γ¯n°2 . (4.38)

1 1 ° ° ° ° |(∇ · uñ Γ¯ n, F¯ n)| ≤ °∇ · uñ Γ¯ n° °F¯ n° 2 h 2 h 1 ° ° ° ° ≤ d´k∇uñk °Γ¯ n° °F¯ n° 2 h ∞ 1 ° ° ° ° ≤ d´ C h−1 kuñk °Γ¯ n° °F¯ n° 2 i h ∞ ° ° 1 ° ° ≤ °F¯ n°2 + d´2 C2 K2 h−2 °Γ¯ n°2 . (4.39) 4 i

n+ n− n+ n |hΓ¯ − Γ¯ , F¯ ih,u˜n | = 0 , (by the continuity of Γ¯ ) . (4.40)

22 ° ° ° ° |(D(Λ¯ n), F¯ n)| ≤ °F¯ n° °D(Λ¯ n)° ° ° 1 ° ° ≤ °F¯ n°2 + °∇Λ¯ n°2 . (4.41) 4

° ° ° ° n ¯ n ¯ n ° n ¯ n ° °¯ n° |(ga(˜τh , ∇Λ ), F )| ≤ ga(˜τh , ∇Λ ) F ° ° ° ° ´ n ° ¯ n° °¯ n° ≤ 4d kτ˜h k∞ ∇Λ F ° ° ° ° ≤ °F¯ n°2 + 16d´2K2 °∇Λ¯ n°2 . (4.42)

n Now for the R2(F¯ ) terms. ° ° ° ° ° ° |(¯ηn − ηn−1/2, F¯ n)| ≤ °η¯n − ηn−1/2° °F¯ n° ° ° ° °2 1 ° °2 ≤ °F¯ n° + °η¯n − ηn−1/2° . (4.43) 4

° ° ° °2 1 ° °2 |(d ηn − ηn−1/2, F¯ n)| ≤ °F¯ n° + °d ηn − ηn−1/2° . (4.44) t t 4 t t

° ° ° °2 1 ° °2 |(D(w¯ n − wn−1/2), F¯ n)| ≤ °F¯ n° + °∇(w¯ n − wn−1/2)° . (4.45) 4

Next, ° ° n n n−1/2 n ° n n n−1/2 ° ° n° |(ga(˜η , ∇(w¯ − w ), F¯ )| ≤ °ga(˜η , ∇(w¯ − w ))° °F¯ ° ° ° ° ° ´ n ° n n−1/2 ° °¯ n° ≤ 4d kη˜ k∞ °∇(w¯ − w )° F ° ° ° °2 ° °2 ≤ °F¯ n° + 16d´2M 2 °∇(w¯ n − wn−1/2)° . (4.46)

For the ﬁrst term in (4.47) ° ° ° ° n n n−1/2 ¯ n n ´1/2 ° n n−1/2 ° °¯ n° |(w˜ · ∇(¯η − η ), F )h| ≤ kw˜ k∞ d °∇(¯η − η )° F ° ° ° °2 ° °2 ≤ °F¯ n° + dM´ 2 °∇(¯ηn − ηn−1/2)° . (4.48)

23 The second term in (4.47) is bounded via ° ° 1 1 ° ° ° ° |(∇ · w˜ n (¯ηn − ηn−1/2), F¯ n)| ≤ k∇ · w˜ nk °η¯n − ηn−1/2° °F¯ n° 2 2 ∞ ° ° 1 ° ° ° ° ≤ d´k∇w˜ nk °η¯n − ηn−1/2° °F¯ n° 2 ∞ ° ° ° °2 1 ° °2 ≤ °F¯ n° + d´2 M 2 °η¯n − ηn−1/2° . (4.49) 4

For the third term in (4.47) we have

n n−1/2 + n n−1/2 − n+ |h(¯η − η ) − (¯η − η ) , F¯ ih,w˜ n | = 0 , (4.50) by the continuity of η.

n Combining the estimates in (4.31)-(4.50) we obtain the following estimate for F2(F¯ ).

° °2 ³ ´ ° °2 ° °2 ³ ´ ° °2 n ° n° 3/2 ° n ° ° n° 3/2 ° n° |F2(F¯ )| ≤ °E˜ ° d´ M + °D(E˜ )° (²9) + °Λ˜ ° d´ M + °∇Λ˜ ° Ã ! Ã ! ˆ ° ° ˆ ° °2 λ ° °2 λ + °∇Λ¯ n° + 16d´2K2 + °∇(w¯ n − wn−1/2)° + 16d´2M 2 4 4 Ã Ã ! ! 2 ° ° ³ ´ ° °2 1 1 C 2 ° °2 + °F¯ n° d´3/2M + d´3M 2 1 + K + 8 + 2d´ M + 2λˆ + + °F˜ n° 4d´ M 2 4 ²9 λ µ ¶ ° ° ³ ´ ³ ´ ° °2 1 1 ° °2 ° °2 + °Γ¯ n° + d´2 C2 K2 h−2 + °Γ˜n° 4d´ M + °∇Γ¯ n° dK´ 2 4λ 4 i Ã ! µ ¶ ° ° 2 2 1 ° °2 1 d´ M + kd Γnk2 + °η¯n − ηn−1/2° + t 4 4λ 4 ° ° ³ ´ ° ° µ ¶ ° °2 ° °2 1 + °∇(¯ηn − ηn−1/2)° d´ M 2 + °d ηn − ηn−1/2° . (4.51) t t 4

Note that ° ° ° °2 1 1 ° °2 °F¯ n° ≤ kFnk2 + °Fn−1° , (4.52) 2 2 ° ° ° ° ° ° ° ° ° °2 3 ° °2 3 ° °2 3 ° °2 °F˜ n° ≤ °Fn−1° + °Fn−2° + °Fn−3° . (4.53) 2 2 2 with analogous estimates also holding for E¯ n.

With the following choices: ²1 = 3(1−α)/(18Re), ²2 = 3λˆ(1−α)/18, ²3 = ²4 = ²6 = ²8 = 3(1−α)/18,

²5 = ²7 = 3/36, ²9 = λˆ(1 − α)/60, substituting (4.19),(4.30),(4.51), and (4.52),(4.53) into (4.18) we obtain µ° ° ° ° ¶ µ° ° ° ° ¶ l l ° °2 ° °2 ° °2 ° °2 X ° °2 2 X ° °2 Reλˆ °El° − °E2° + °Fl° − °F2° + λˆ(1 − α) ∆t °D(E¯ n)° + ∆t °F¯ n° n=3 λ n=3

24 Xl Xl n 2 n 2 ≤ C1 ∆t kE k + C2 ∆t kF k n=3 n=3 Xl Xl Xl n 2 n 2 n 2 + C3 ∆t kΛ k + C4 ∆t k∇Λ k + C5 ∆t kdtΛ k n=0 n=0 n=3 ³ ´ Xl Xl Xl −2 n 2 n 2 n 2 + C6 + C7h ∆t kΓ k + C8 ∆t k∇Γ k + C9 ∆t kdtΓ k n=0 n=2 n=3 l l X ° °2 X ° °2 ° n n−1/2 ° ° n n−1/2° + C10 ∆t °∇(w¯ − w )° + C11 ∆t °dtw − wt ° n=3 n=3 l l X ° °2 X ° °2 ° n n−1/2° ° n n−1/2 ° + C12 ∆t °η¯ − η ° + C13 ∆t °∇(¯η − η )° n=3 n=3 l X ° °2 ° n n−1/2° + C14 ∆t °dtη − ηt ° n=3 l l X ° °2 X ° n n−1/2° n n 2 + C15 ∆t °r¯ − r ° + C16 ∆t kr − P k n=3 n=2 2 µ ¶ X ° °2 ° °2 ° °2 ° i° ° i° ° i ° + C17 ∆t °E ° + °F ° + °D(E¯ )° . (4.54) i=0

We now apply the interpolation properties of the approximating spaces to estimate the terms on the right hand side of (4.54). Using elements of order k for velocity, elements of order m for stress, and elements of order q for pressure, we have Ã ! Xl Xl Xl Xl n 2 n 2 2k n 2 2m n 2 ∆t k∇Λ k + ∆t k∇Γ k ≤ C h ∆t kw kk+1 + h ∆t kη km+1 n=0 n=2 ³ n=0 n=2´ 2k 2 2m 2 ≤ C h |kwk|0,k+1 + h |kηk|0,m+1 . (4.55)

Xl Xl Xl ∆t kΛnk2 + ∆t kΓnk2 + ∆t krn − Pnk2 n=0 n=0 n=3 Ã ! Xl Xl Xl 2k+2 n 2 2m+2 n 2 2q+2 n 2 ≤ C h ∆t kw kk+1 + h ∆t kη km+1 + h ∆t kr kq+1 ³ n=0 n=0 ´ n=3 2k+2 2 2m+2 2 2q+2 2 ≤ C h |kwk|0,k+1 + h |kηk|0,m+1 + h |krk|0,q+1 . (4.56)

° ° l l ° Z °2 X X ° 1 tn ∂Λ ° ∆t kd Λnk2 = ∆t ° 1 dt° t ° ° n=3 n=3 ∆t tn−1 ∂t Ã !Ã ! l µ ¶ Z Z Z µ ¶ X 1 2 tn tn ∂Λ 2 ≤ ∆t 1 dt dt dx n=3 ∆t Ω tn−1 tn−1 ∂t 2k+2 2 ≤ Ch kwtk0,k+1 , (4.57)

25 and similarly, Xl n 2 2m+2 2 ∆t kdtΓ k ≤ Ch kηtk0,m+1 . (4.58) n=3

Using (A.10),(A.11) and (A.12) we have

l X ° °2 ° n n−1/2 ° 4 2 ∆t °∇(w¯ − w )° ≤ C(∆t) kwttk0,1 , (4.59) n=3 l X ° °2 ° n n−1/2° 4 2 ∆t °dtw − wt ° ≤ C(∆t) kwtttk0,0 , (4.60) n=3 l X ° °2 ° n n−1/2° 4 2 ∆t °η¯ − η ° ≤ C(∆t) kηttk0,0 , (4.61) n=3 l X ° °2 ° n n−1/2 ° 4 2 ∆t °∇(¯η − η )° ≤ C(∆t) kηttk0,1 , (4.62) n=3 l X ° °2 ° n n−1/2° 4 2 ∆t °dtη − ηt ° ≤ C(∆t) kηtttk0,0 , (4.63) n=3 l X ° °2 ° n n−1/2° 4 2 ∆t °r¯ − r ° ≤ C(∆t) krttk0,0 . (4.64) n=3

Substituting (4.55)-(4.64) into (4.54) gives the estimate

µ° ° ° ° ¶ µ° ° ° ° ¶ l l ° °2 ° °2 ° °2 ° °2 X ° °2 2 X ° °2 Reλˆ °El° − °E2° + °Fl° − °F2° + λˆ(1 − α) ∆t °D(E¯ n)° + ∆t °F¯ n° n=3 λ n=3 Xl Xl ≤ C ∆t kEnk2 + C ∆t kFnk2 n=3 ³ n=3 ´ 4 2 2 2 2 2 + C (∆t) kwttk0,1 + kwtttk0,0 + kηttk0,1 + kηtttk0,0 + krttk0,0 ³ ´ 2k 2 2m 2 2q+2 2 + C h |kwk|0,k+1 + h |kηk|0,m+1 + h |krk|0,q+1 ³ ´ 2k+2 2 2m+2 2 + C h kwtk0,k+1 + h kηtk0,m+1 2 µ° ° ° ° ° ° ¶ X ° °2 ° °2 ° °2 + C ∆t °Ei° + °Fi° + °D(E¯ i)° , (4.65) i=0 where C denotes a constant independent of l, ∆t, h. Thus, combining (4.65) with (3.15) and, for ∆t suﬃciently small, applying Gronwall’s lemma to (4.65), estimate (4.7) follows.

Next we verify that the induction hypothesis (IH) holds.

26 Step 2. Veriﬁcation of (IH1) Assume that (IH) holds true for n = 1, 2, . . . , l − 1. By interpolation properties, inverse estimates and (4.7), we have that ° ° ° ° ° ° ° l ° ° l l° ° l° °uh° ≤ °uh − w ° + °w ° ∞ ° ° °∞ ° ∞ ° ° ° ° ≤ °El° + °Λl° + M ∞ ∞ ´ ° ° ´ ° ° − d ° l° − d ° l° ≤ C h 2 °E ° + C h 2 °Λ ° + M µ ¶ ´ 2 − d k−d/´ 2 m−d/´ 2 q+1−d/´ 2 ≤ C (∆t) h 2 + h + h + h + M. (4.66) µ ¶ ´ 2 − d k−d/´ 2 m−d/´ 2 q+1−d/´ 2 Note that the expression C (∆t) h 2 + h + h + h is independent of l. Hence, if we set k, m ≥ d/´ 2, q ≥ d/´ 2 − 1, and choose h, ∆t such that

d/´ 4 ´ ´ ´ 1 h hk−d/2, hm−d/2, hq+1−d/2 ≤ , ∆t ≤ , (4.67) C C1/2 then from (4.66) ° ° ° l ° °uh° ≤ M + 4 . ∞ ° ° ° ° Similarly it follows that °τ l ° ≤ M + 4. h ∞

Step 3. Proof of the Theorem 2. We have that

° °2 ° °2 ° °2 ° °2 ° °2 ° l l ° ° l l ° ° l l° ° l° ° l° °u − uh° + °τ − τh° ≤ 3 °u − w ° + 3 °E ° + 3 °Λ ° ° ° ° ° ° ° ° °2 ° °2 ° °2 + 3 °τ l − ηl° + 3 °Fl° + 3 °Γl° . (4.68)

Now, (4.1) follows from (4.68) using Corollary 1, Lemma 3, the approximation properties, and taking the maximum over l.

To establish (4.2), using (4.1)and (3.15), we have that

XN XN n n 2 n n 2 2 ∆t ku¯h − u¯ k ≤ ∆t kuh − u k ≤ T F1(∆t, h) , (4.69) n=1 n=1 PN n n 2 with the same estimate also valid for n=1 ∆t kτ¯h − τ¯ k .

PN n n 2 PN n n 2 To estimate n=1 ∆t k∇(u¯h − u¯ )k ≡ n=1 ∆t kD(u¯h − u¯ )k , note that XN XN XN ° ° XN ° ° n n 2 n n 2 ° ¯ n °2 ° ¯ n °2 ∆t kD(u¯h − u¯ )k ≤ 3 ∆t kD(u¯ − w¯ )k + 3 ∆t D(Λ ) + 3 ∆t D(E ) n=1 n=3 n=3 n=3

27 X2 n n 2 + ∆t kD(u¯h − u¯ )k . (4.70) n=1

The second, third, and fourth terms on the right hand side of (4.70) can be bounded using the interpolation properties, Lemma 3, and the initialization assumption (3.15), respectively. To bound the ﬁrst term we proceed as follows. For simplicity, we assume N is even.

N N µ ° ° ¶ X X ∆t ° °2 ∆t kD(u¯n − w¯ n)k2 ≤ kD(un − wn)k2 + °D(un−1 − wn−1)° n=3 n=3 2 (N−2)/2 µ° ° ° ° 3 X 2∆t ° °2 ° °2 ≤ °D(u2i − w2i)° + 4 °D(u2i+1 − w2i+1)° 2 6 i=1 ° ° ¶ ° °2 + °D(u2i+2 − w2i+2)°

using Simpson’s rule, µZ Z ¶ tN tN 3 2 7 4 2 ≤ kD(u − w)k dt + (∆t) |(kD(u − w)k )tttt| dt 2 t2 72 t2 using (2.46), Z Z T ³ ´ tN ³ 4 2 2 7 4 2 ≤ C (∆t) kwttk + kηttk dt + (∆t) k∇(u − w)k −5∆t/2 48 t 2 ´ 2 2 2 2 + 4 k∇(u − w)tk + 6 k∇(u − w)ttk + 4 k∇(u − w)tttk + k∇(u − w)ttttk dt Ã ! Z T ³ ´ 4 2 2 2 ≤ C (∆t) kwttk + kηttk dt + C F2(∆t) , (4.71) −5∆t/2 where F2(∆t) is deﬁned in (4.3).

The stated result, (4.2), now follows.

We are now in a position to consider the error estimate for the pressure. Similar to (2.9) the n approximation for the pressure ph satisﬁes the equation

n n n n n n Re(dtuh, v) + Re c(u˜h, u¯h, v) + 2(1 − α)(D(u¯h),D(v)) − (¯ph, ∇ · v) + (¯τh ,D(v)) ¯n = (f , v) , for all v ∈ Xh (4.72).

Corollary 2 With the hypotheses of Theorem 2, we have that Ã Ã ! N µ ¶ ¶ N 1/2 X ° °2 ° °2 1/2 X ° °2 ° n−1/2 n° ° n−1/2 n ° ° n−1/2 n° °u − u¯h° + °D(u − u¯h)° ∆t + °τ − τ¯h ° ∆t n=1 n=1 1/2 ≤ C (1 + T ) F1(∆t, h) + C F2(∆t)

28 ³ ´ 2 + C (∆t) kuttk0,0 + k∇uttk0,0 + kτttk0,0 , (4.73) Ã ! N 1/2 X ° °2 ° n−1/2 n° −1 1/2 1/2 °p − p¯h° ∆t ≤ C (∆t) T F1(∆t, h) + C (1 + T ) F1(∆t, h) + C F2(∆t) n=1 ³ ´ 2 + C (∆t) kuttk0,0 + k∇uttk0,0 + kτttk0,0 ³ ´ 2 +C (∆t) kutttk0,0 + kpttk0,0 + kfttk0,0 µ ° ° ¶ ° ° +C hq+1 |kpk| + hq+1(∆t)1/2 °p0° 0,q+1 q+1 Ã ! Z T 1/2 2 2 +C (∆t) kuttk dt . (4.74) −2∆t

Proof of Corollary 2: The estimate (4.73) follows from (4.2), the triangle inequality and (A.10),(A.12).

To estimate the error in the pressure, let P ∈ Qh be such that

q+1 kp − Pk ≤ CI h kpkq+1 . (4.75)

From the discrete inf-sup condition (3.1), and using (A.10), we have ° ° ¯n n ° ¯n n° 1 |(P − p¯h, ∇ · v)| P − p¯h ≤ sup β v∈Xh kvk1 1 |(pn−1/2 − P¯n, ∇ · v)| 1 |(pn−1/2 − p¯n, ∇ · v)| ≤ sup + sup h β v∈Xh kvk1 β v∈Xh kvk1 ³° ° ´ n−1/2 n 1 ° ° ° ° 1 |(p − p¯ , ∇ · v)| ≤ d´1/2 °pn−1/2 − p¯n° + °p¯n − P¯n° + sup h β β v∈X kvk  h 1  ÃZ !1/2 µ ¶  tn ° °  C ´1/2 3/2 2 q+1 n ° n−1° ≤ d (∆t) kpttk dt + h kp kq+1 + °p ° β  tn−1 q+1  1 |(pn−1/2 − p¯n, ∇ · v)| + sup h . (4.76) β v∈Xh kvk1

Subtracting (4.72) from (2.9)implies

n n−1/2 n−1/2 ¯n n−1/2 n n n (¯ph − p , ∇ · v) = (f − f , v) − Re (ut − dtu , v) − Re (dt(u − uh), v)

n−1/2 n n−1/2 n n n−1/2 n n−1/2 n − Re c(u − u˜ , u , v) − Re c(u˜ − u˜h, u , v) − Re c(u˜h, u − u¯h, v)

n−1/2 n n−1/2 n − 2(1 − α)(D(u − u¯h),D(v)) − (τ − τ¯h ,D(v)) .

Hence, n−1/2 n ° ° ° ° |(p − p¯h, ∇ · v)| ° n−1/2 ¯n° ° n−1/2 n° n n ≤ °f − f ° + Re °ut − dtu ° + Re kdt(u − uh)k kvk1

29 ° ° ´3/2 ° n−1/2 n° ´3/2 n n + Re d M °u − u˜ ° + Re d M ku˜ − u˜hk ³ ´ ° ° ° ° ´1/2 ° n−1/2 n ° ° n−1/2 n° + 2Kd Ck + 2(1 − α) °D(u − u¯h)° + °τ − τ¯h ° .(4.77)

All the terms on the right hand side of (4.77) may be bounded in a similar manner as:

ÃZ !1/2 ° ° tn ° n−1/2 n° 3/2 2 °f − ¯f ° ≤ C (∆t) kfttk dt , using (A.10), (4.78) tn−1 ÃZ !1/2 ° ° tn ° n−1/2 n° 3/2 2 °ut − dtu ° ≤ C (∆t) kutttk dt , using (A.11), (4.79) tn−1 Ã ! ° ° Z t 1/2 ° n−1/2 n° 3/2 n−1/2 2 °u − u˜ ° ≤ C (∆t) kuttk dt , using (A.13), (4.80) tn−3 ku˜n − u˜nk ≤ 2 F (∆t, h) , using (4.1), (4.81) h ° 1 ° ° n n n−1 n−1 ° °(u − uh) − (u − uh )° kd (un − un)k = ≤ 2 (∆t)−1 F (∆t, h) , using (4.1)(4.82). t h ∆t 1

As ° °2 ° °2 ° ° ° ° ° n−1/2 n° ° n−1/2 n° ° n ¯n°2 ° ¯n n°2 °p − p¯h° ≤ 3 °p − p¯ ° + 3 p¯ − P + 3 P − p¯h , combining the estimates from (4.76)-(4.82), and using (4.73), the stated result (4.74) now follows.

Note that the estimate for the pressure is only ﬁrst order with respect to the time discretization.

In concluding we again remark that the estimates in Theorem 2 and Corollary 2 are derived under the assumption that the solution to the continuous problem has the necessary regularity. For a discussion of the regularity assumption for the Navier-Stokes equations see [10].

A Appendix

On repeated integration by parts we have the following representations: Z tn n n−1/2 u = u(tn) = u + ut(·, t) dt tn−1/2 Z tn n−1/2 ∆t n−1/2 = u + ut + utt(·, t)(tn − t) dt (A.1) 2 tn−1/2 Z 2 tn n−1/2 ∆t n−1/2 1 (∆t) n−1/2 1 2 = u + ut + utt + uttt(·, t)(tn − t) dt . (A.2) 2 2 4 2 tn−1/2

30 Also Z tn n n−1 n−1 u = u + ∆t ut + utt(·, t)(tn − t) dt (A.3) tn−1

For un−1,un−2 and un−3 we have

Z t n−1 n−1/2 ∆t n−1/2 n−1/2 u = u(tn−1) = u − ut + utt(·, t)(t − tn−1) dt (A.4) 2 tn−1 2 Z t n−1/2 ∆t n−1/2 1 (∆t) n−1/2 1 n−1/2 2 = u − ut + utt − uttt(·, t)(t − tn−1) dt , (A.5) 2 2 4 2 tn−1

Z t n−2 n−1/2 3 ∆t n−1/2 n−1/2 u = u(tn−2) = u − ut + utt(·, t)(t − tn−2) dt (A.6) 2 tn−2 Z tn−1 n−2 n−1 n−1 u = u − ∆t ut + utt(·, t)(t − tn−2) dt , (A.7) tn−2

Z t n−3 n−1/2 5 ∆t n−1/2 n−1/2 u = u(tn−3) = u − ut + utt(·, t)(t − tn−3) dt (A.8) 2 tn−3 Z tn−1 n−3 n−1 n−1 u = u − 2∆t ut + utt(·, t)(t − tn−3) dt . (A.9) tn−3

Lemma 4 Z ° °2 tn ° n n−1/2° 1 3 2 °u¯ − u ° ≤ (∆t) kuttk dt . (A.10) 48 tn−1

Proof of Lemma 4:

° ° ° °2 ° °2 °1 ° °u¯n − un−1/2° = ° (un + un−1) − un−1/2° °2 ° Z "Z Z #2 1 tn tn−1/2 = utt(·, t)(tn − t) dt + utt(·, t)(t − tn−1) dt dx 4 Ω t tn−1 n−1/2  Z ÃZ !2 ÃZ !2 1 tn tn−1/2 ≤ 2  utt(·, t)(tn − t) dt + utt(·, t)(t − tn−1) dt  dx 4 Ω tn−1/2 tn−1 Z "Z Z tn tn 1 2 2 ≤ (utt(·, t)) dt (tn − t) dt 2 Ω t t n−1/2 n−1/2 # Z t Z t n−1/2 2 n−1/2 2 + (utt(·, t)) dt (t − tn−1) dt dx tn−1 tn−1 Z " µ ¶ Z µ ¶ Z # 3 tn 3 t 1 1 ∆t 2 1 ∆t n−1/2 2 = (utt(·, t)) dt + (utt(·, t)) dt dx 2 Ω 3 2 tn−1/2 3 2 tn−1 Z Z tn 1 3 2 = (∆t) (utt(·, t)) dt dx 48 Ω tn−1

31 Z tn 1 3 2 = (∆t) kuttk dt . 48 tn−1

Lemma 5 Z ° °2 tn ° n n−1/2° 1 3 2 °dtu − ut ° ≤ (∆t) kutttk dt . (A.11) 1280 tn−1

Proof of Lemma 5:

° ° ° °2 ° n−1/2°2 ° 1 n−1/2° °d un − u ° = ° (un − un−1) − u ° t t °∆t t ° µ ¶ Z "Z Z #2 2 tn t 1 2 n−1/2 2 = uttt(·, t)(tn − t) dt + uttt(·, t)(t − tn−1) dt dx 4 ∆t Ω t tn−1 n−1/2  µ ¶ Z ÃZ !2 ÃZ !2 2 tn t 1 2 n−1/2 2 ≤ 2  uttt(·, t)(tn − t) dt + uttt(·, t)(t − tn−1) dt  dx 4 ∆t Ω tn−1/2 tn−1 µ ¶ Z "Z Z 2 tn tn 1 2 4 ≤ 2 (uttt(·, t)) dt (tn − t) dt 4 ∆t Ω t t n−1/2 n−1/2 # Z t Z t n−1/2 2 n−1/2 4 + (uttt(·, t)) dt (t − tn−1) dt dx tn−1 tn−1 µ ¶ Z " µ ¶ Z µ ¶ Z # 2 5 tn 5 t 1 1 ∆t 2 1 ∆t n−1/2 2 = 2 (uttt(·, t)) dt + (uttt(·, t)) dt dx 4 ∆t Ω 5 2 tn−1/2 5 2 tn−1 Z Z tn 1 3 2 = (∆t) (uttt(·, t)) dt dx 1280 Ω tn−1 Z tn 1 3 2 = (∆t) kutttk dt . 1280 tn−1

For the vector u, u(i), i = 1,... d´, denotes the ith component of the vector.

Lemma 6 Z ° °2 3 tn ° n n−1/2 ° (∆t) 2 °∇(u¯ − u )° ≤ k∇uttk dt . (A.12) 48 tn−1

Proof of Lemma 6: Z (Z Z ) ° °2 tn tn−1/2 ° n n−1/2 ° 1 °∇(u¯ − u )° = ∇ utt(·, t)(tn − t) dt + utt(·, t)(t − tn−1) dt 4 Ω tn−1/2 tn−1 (Z Z ) tn tn−1/2 : ∇ utt(·, t)(tn − t) dt + utt(·, t)(t − tn−1) dt dx tn−1/2 tn−1

32 interchanging diﬀerentiation and integration Z (Z Z ) 1 tn tn−1/2 = ∇utt(·, t)(tn − t) dt + ∇utt(·, t)(t − tn−1) dt 4 Ω tn−1/2 tn−1 (Z Z ) tn tn−1/2 : ∇utt(·, t)(tn − t) dt + ∇utt(·, t)(t − tn−1) dt dx tn−1/2 tn−1 Ã ! d´ Z Z Z 2 X 1 tn tn−1/2 = ui (·, t)(t − t) dt + ui (·, t)(t − t ) dt dx 4 ttxj n ttxj n−1 i,j=1 Ω tn−1/2 tn−1 Ã ! Ã !  d´ Z Z 2 Z 2 X 1 tn tn−1/2 ≤ 2  ui (·, t)(t − t) dt + ui (·, t)(t − t ) dt  dx 4 ttxj n ttxj n−1 i,j=1 Ω tn−1/2 tn−1 " d´ Z Z Z X 1 tn ³ ´2 tn ≤ 2 ui (·, t) dt (t − t)2 dt 4 ttxj n i,j=1 Ω tn−1/2 tn−1/2 Z Z # tn−1/2 ³ ´2 tn−1/2 i 2 + uttxj (·, t) dt (t − tn−1) dt dx tn−1 tn−1 d´ Z µ ¶ Z X 1 1 ∆t 3 tn ³ ´2 = 2 ui (·, t) dt dx 4 3 2 ttxj i,j=1 Ω tn−1 Z 3 tn (∆t) 2 = k∇uttk dt . 48 tn−1

Lemma 7 Z ° °2 tn−1/2 ° n n−1/2° 39 3 2 °u˜ − u ° ≤ (∆t) kuttk dt . (A.13) 8 tn−3

Proof of Lemma 7: ° ° ° °2 ° °2 ° 1 1 ° °u˜n − un−1/2° = °(un−1 + un−2 − un−3) − un−1/2° ° 2 2 ° Z "Z Z tn−1/2 1 tn−1/2 = utt(·, t)(t − tn−1) dt + utt(·, t)(t − tn−2) dt Ω tn−1 2 tn−2 Z #2 1 tn−1/2 − utt(·, t)(t − tn−3) dt dx 2 tn−3  Z ÃZ !2 ÃZ !2 tn−1/2 1 tn−1/2 ≤ 3  utt(·, t)(t − tn−1) dt + utt(·, t)(t − tn−2) dt Ω tn−1 4 tn−2  ÃZ !2 1 tn−1/2 + utt(·, t)(t − tn−3) dt  dx 4 tn−3 " Z Z t Z t n−1/2 2 n−1/2 2 ≤ 3 (utt(·, t)) dt (t − tn−1) dt Ω tn−1 tn−1

33 Z t Z t 1 n−1/2 2 n−1/2 2 + (utt(·, t)) dt (t − tn−2) dt 4 t t n−2 n−2 # Z t Z t 1 n−1/2 2 n−1/2 2 + (utt(·, t)) dt (t − tn−3) dt dx 4 tn−3 tn−3 " Z µ ¶3 Z t µ ¶3 Z t 1 ∆t n−1/2 2 1 3∆t n−1/2 2 = 3 (utt(·, t)) dt + (utt(·, t)) dt Ω 3 2 tn−1 12 2 tn−2 # µ ¶3 Z t 1 5∆t n−1/2 2 + (utt(·, t)) dt dx 12 2 tn−3 Z Z t 39 3 n−1/2 2 ≤ (∆t) (utt(·, t)) dt dx 8 Ω tn−3 Z t 39 3 n−1/2 2 = (∆t) kuttk dt . 8 tn−3

Lemma 8 Z ° °2 tn−2 ° n n−1° 1 2 °u˜ − u ° ≤ ∆t kutk dt . (A.14) 4 tn−3

Proof of Lemma 8:

° ° °µ ¶ °2 ° °2 ° °2 ° 1 1 ° °1 ° °u˜n − un−1° = ° un−1 + un−2 − un−3 − un−1° = ° (un−2 − un−3)° ° 2 2 ° °2 ° Z "Z #2 1 tn−2 = 1 ut(·, t) dt dx 4 Ω tn−3 " # Z Z t Z t 1 n−2 n−2 2 ≤ 1 dt (ut(·, t)) dt dx 4 Ω tn−3 tn−3 Z t 1 n−2 2 = ∆t kut(·, t)k dt . 4 tn−3

B Appendix B

In this section we give an example of an initialization procedure for the Crank-Nicolson approximation described in (3.16),(3.17). There are three steps to the procedure: 2 2 2 2 Step 1. Apply a single step implicit Euler approximation to approximate uh, τh , (i.e. zh, θh). 2 2 Step 2. Apply a single step Crank-Nicolson approximation to compute uh, τh . 1 1 Step3. Interpolate to determine uh, τh .

34 We introduce the following notation: f(t ) − f(t ) d f := n n−2 (B.1) 2t 2∆t f n + f n−2 f˘n := (B.2) 2 + − + B2(u, τ, σ) := (u · ∇τ, σ)h + hτ − τ , σ ih,u . (B.3)

Similar to (3.13) for B(u, τ, τ), we have 1 1 B (u, τ, τ) = − (∇ · u τ, τ) + hhτ + − τ −ii2 . (B.4) 2 2 h 2 h,u ° ° ° ° 0 0 0 0 ° 0° ° 0° For the initial data we take uh = U , τh = T which implies that E = F = 0.

Initialization 0 0 0 0 2 2 Step 1. For zh = uh, θh = τh , ﬁnd zh ∈ Zh, θh ∈ Sh such that

2 0 2 2 2 2 Re(d2tzh, v) + Re c(uh, zh, v) + 2(1 − α)(D(zh),D(v)) + (θh,D(v)) = (f , v) , v ∈ Zh (B.5) 1 (θ2, σ) + (d θ2, σ) + B(u0 , θ2, σ) − λˆ(D(z2 ), σ) + (g (τ 0, ∇z2 ), σ) = 0 , σ ∈ S .(B.6) λ h 2t h h h h a h h h

1 2 0 1 2 0 2 2 Step 2. For zh = (zh + zh)/2, θh = (θh + θh)/2, ﬁnd uh ∈ Zh, τh ∈ Sh such that

2 1 2 2 2 1 Re(d2tuh, v) + Re c(zh, u˘h, v) + 2(1 − α)(D(u˘h),D(v)) + (˘τh ,D(v)) = (f , v) , v ∈ Zh (B.7) 1 (˘τ 2, σ) + (d τ 2, σ) + B (z1 , τ˘2, σ) − λˆ(D(u˘2 ), σ) + (g (θ1, ∇z2 ), σ) = 0 , σ ∈ S (B.8). λ h 2t h 2 h h h a h h h

1 2 0 1 2 0 Step 3. uh = (uh + uh)/2 , θh = (θh + θh)/2 .

For the error in Step 1 of the initialization we have:

2 2 Lemma 9 For zh, θh given by (B.5),(B.6)and w, η satisfying (2.25),(2.26) we have that

° °2 ° °2 ° 2 ° ° 2 ° 1 °zh − w(2∆t)° + °θh − η(2∆t)° ≤ E2 (∆t, h) , (B.9) where, µ ¶ ° °2 ° °2 ° °2 1 2k ° 0° 2m ° 0° 2q+2 ° 0° E2 (∆t, h) = C ∆t h °w ° + h °η ° + h °r ° k+1 m+1 q+1 µ ¶ ° °2 ° °2 ° °2 ° °2 ° °2 3 ° 0° ° 0° ° 0 ° ° 0° ° 0 ° + C (∆t) °wt ° + °∇wt ° + °wtt° + °ηt ° + °ηtt° µ ¶ ° °2 ° °2 2k+2 ° 0° 2m+2 ° 0° + C ∆t h °wt ° + h °ηt ° . (B.10) k+1 m+1

35 Proof: Proceeding in a analogous fashion to above (see also [7]) with

n n n n n n Ez := U − zh, Fz := T − θh , and noting that 0 0 0 0 0 0 0 zh = uh = U , θh = τh = T , and D(Ez) = 0 , we obtain ° ° ° ° ° ° ° ° ° °2 ° °2 ° °2 2 ° °2 Reλˆ °E2° + °F2° + ∆t λˆ(1 − α) °D(E2)° + ∆t °F2° z z z λ z ° °2 ° °2 ° 2° ° 2° ≤ C1∆t °Ez° + C2∆t °Fz° µ ¶ ° °2 ° °2 ° °2 ° °2 ° 0° ° 0° ° 2° ° 2° + C3∆t °Λ ° + °∇Λ ° + °∇Λ ° + C4∆t °d2tΛ ° µ ¶ ° °2 ° °2 ° °2 ° °2 ° 0° ° 2° ° 2° ° 2° + C5∆t °Γ ° + °Γ ° + °∇Γ ° + C6∆t °d2tΓ ° µ ¶ ° °2 ° °2 ° °2 ° 0 2° ° 0 2 ° ° 0 2° + C7 ∆t °w − w˜ ° + °∇(w − w˜ )° + °η − η˜ ° µ ¶ ° °2 ° °2 ° 2 2° ° 2 2° + C8 ∆t °d2tw − wt ° + °d2tη − ηt ° ° °2 ° 2 2° + C9 ∆t °r − P ° . (B.11)

2 2 2 2 2 2 Using zh − w(2∆t) = Ez + Λ , θh − η(2∆t) = Fz + Γ , and the approximation properties, the stated result follows.

2 2 Next we estimate the error in uh, τh generated in Step 2.

Note in this step a modiﬁed B operator, B2 is used. We have that ° ° 1 ° 1 ° |(∇ · zh τ, τ)| ≤ °∇ · zh τ° kτk ° ° ´ ° 1 ° 2 ≤ d °∇zh° kτk ∞° ° ´ −1 ° 1 ° 2 ≤ d CI h °zh° kτk ∞ −1 2 ≤ d´ CI K h kτk .

Therefore, from (B.4),

1 1 B (z1 , τ, τ) ≥ − d´ C K h−1 kτk2 + hhτ + − τ −ii2 . (B.12) 2 h 2 I 2 h,u

36 2 2 2 2 Lemma 10 For uh, τh given by (B.7),(B.8), w, η satisfying (2.25),(2.26), and E , F deﬁned in (4.4),(4.5), we have that

° °2 ° °2 ° °2 ° 2° ° 2° ° 2 ° °E ° + °F ° + ∆t °D(E )° ≤ E2(∆t, h) , (B.13) where, µ ¶ ° °2 ° °2 ° °2 2k ° 0° 2m ° 0° 2q+2 ° 0° E2(∆t, h) = C ∆t h °w ° + h °η ° + h °r ° k+1 m+1 q+1 µ ¶ ° °2 ° °2 2k+2 ° 0° 2m+2 ° 0° + C ∆t h °wt ° + h °ηt ° k+1 m+1 µ ¶ ° °2 ° °2 ° °2 ° °2 ° °2 4 ° 0° ° 0° ° 0 ° ° 0° ° 0 ° + C (∆t) °wt ° + °∇wt ° + °wtt° + °ηt ° + °ηtt° µ ¶ ° °2 ° °2 ° °2 ° °2 ° °2 5 ° 0 ° ° 0 ° ° 0 ° ° 0 ° ° 0 ° + C (∆t) °∇wtt° + °ηtt° + °∇ηtt° + °∇wttt° + °ηttt° . (B.14)

1 ˘ 2 ˘ 2 Proof: Proceeding in an analogous fashion to the general case with the B2(zh, F , F ) term on the left hand side of the equation bounded using (B.12), we obtain ° ° µ ¶ ° ° ° ° ° °2 4∆t 1 ° °2 ° °2 Re λˆ °E2° + 1 + − d´ C Kh−1∆t °F2° + λˆ(1 − α)∆t °D(E2)° λ 2 I ° °2 ° °2 ° 2° ° 2° ≤ C1∆t °E ° + C2∆t °F ° µ ¶ ° °2 ° °2 ° °2 ° °2 ° °2 ° 0° ° 0° ° 2° ° 2° ° 2° + C3∆t °Λ ° + °∇Λ ° + °Λ ° + °∇Λ ° + C4∆t °d2tΛ ° µ ¶ ° °2 ° °2 ° °2 ° °2 ° °2 ° 0° ° 0° ° 2° ° 2° ° 2° + C5∆t °Γ ° + °∇Γ ° + °Γ ° + °∇Γ ° + C6∆t °d2tΓ ° µ ¶ ° °2 ° °2 ° 1 1° ° 1 1° + C7 ∆t °zh − w˜ ° + °θh − η˜ ° µ ¶ ° °2 ° °2 ° °2 ° 2 1 ° ° 2 1° ° 2 1 ° + C8 ∆t °∇(w˘ − w )° + °η˘ − η ° + °∇(˘η − η )° µ ¶ ° °2 ° °2 ° 2 1° ° 2 1° + C9 ∆t °d2tw − wt ° + °d2tη − ηt ° ° °2 ° 1 1° + C10 ∆t °r − P ° . (B.15)

° ° ° ° ° 1 1°2 ° 1 1°2 To estimate zh − w˜ ( analogously, θh − η˜ ), we have ° ° ° °2 ° °2 ° °2 ° 1 ° °1 ° °z1 − w˜ 1° ≤ 2 °z1 − (w0 + w2)° + 2 ° (w0 + w2) − w˜ 1° . h ° h 2 ° °2 ° Now,

° °2 ° ° ° ° ° 1 ° ° °2 ° °2 2 °z1 − (w0 + w2)° ≤ °z0 − w0° + °z2 − w2° (B.16) ° h 2 ° h h ° °2 ° 0° 1 ≤ °Λ ° + E2 (∆t, h) . (B.17)

37 ° °2 ° 1 0 2 1° 1 For ° 2 (w + w ) − w˜ ° , using w˜ = w(·, ∆t/2) + w(·, −∆t/2)/2 − w(·, −3∆t/2)/2 and expanding about t = ∆t we have that ° ° Z ° °2 2∆t ° °2 °1 0 2 1° 3 2 4 ° 0 ° ° (w + w ) − w˜ ° ≤ C (∆t) kwttk dt ≤ C (∆t) °wtt° . (B.18) 2 −5∆t/2

° ° ° ° ° ° ° ° ° 2 1 °2 ° 2 1°2 ° 2 1 °2 ° 2 1°2 Estimates for ∇(w˘ − w ) , η˘ − η , ∇(˘η − η ) follow as in (A.10),(A.12), and d2tw − wt , ° ° ° 2 1°2 d2tη − ηt from (A.11). ° ° ° ° 2 2 2 2 The terms °d2tΛ ° , °d2tΓ ° are estimated as in (4.57).

On combining the above with the approximation properties, estimate (B.13) follows.

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