Quasi-Optimal Error Estimates for the Incompressible Navier-Stokes ...

Quasi-Optimal Error Estimates for the Incompressible Navier-Stokes Problem Discretized by Finite Element Methods and Pressure-Correction Projection with Velocity Stabilization

arXiv:1609.00807v1 [math.NA] 3 Sep 2016

Daniel Arndt1 , Helene Dallmann1 and Gert Lube1 1 Institute for Numerical and Applied Mathematics, Georg-August University of G¨ottingen, D-37083, Germany d.arndt/h.dallmann/[email protected]

Abstract. We consider error estimates for the fully discretized instationary Navier-Stokes problem. For the spatial approximation we use conforming infsup stable finite element methods in conjunction with grad-div and local projection stabilization acting on the streamline derivative. For the temporal discretization a pressure-correction projection algorithm based on BDF2 is used. We can show quasi-optimal rates of convergence with respect to time and spatial discretization for all considered error measures. Some of the error estimates are quasi-robust with respect to the Reynolds number.

1. Introduction We consider the time-dependent Navier-Stokes equations ∂t u − ν∆u + (u · ∇)u + ∇p = f in (t0 , T ) × Ω, (1)

∇·u= 0

u=0

in (t0 , T ) × Ω,

in (t0 , T ) × ∂Ω,

u(0, ·) = u0 (·)

in Ω

in a bounded polyhedral domain Ω ⊂ Rd , d ∈ {2, 3}. Here u : (t0 , T ) × Ω → Rd and p : (t0 , T ) × Ω → R denote the unknown velocity and pressure fields for given viscosity ν > 0 and external forces f ∈ [L2 (t0 , T ; [L2 (Ω)]d ) ∩ C(t0 , T ; [L2(Ω)]d )]d . For the discretization with respect to time we use a splitting method called (standard) incremental pressure-correction projection method which is based on the backward differentiation formula of second order (BDF2). In the continuous problem u and p are coupled through the incompressibility constraint. The idea for e and solve pressure-correction projection methods is to define an auxiliary variable u e and p in two different steps such that the original velocity can be recovered for u from these two quantities. Such an approach was first considered by Chorin [1] and Temam [2]. An overview over different projection methods is given in [3]. Badia and Codina [4] analyzed the incremental pressure-correction algorithm with BDF1 time discretization. The incremental pressure-correction algorithm with BDF2 time discretization is discussed by Guermond in [5] for the unstabilized Navier-Stokes equations with ν = 1. Shen considered a different second order time discretization scheme in [6]. It turns out that this technique suffers from unphysical boundary conditions for the pressure that lead to reduced rates of convergence. To prevent this Timmermans proposed in [7] the rotational pressure-correction projection method that uses a divergence correction for the pressure. A thorough analysis for this has first been performed in [8] for the Stokes problem. For the spatial stabilization of the Navier-Stokes equations or related problems there exist many different approaches. Residual-based stabilization methods penalize the residual of the differential equation in the strong formulation and are hence consistent. See [9] for an overview. The bulk of non-symmetric form of the 1

2

stabilization terms and the occurrence of second order derivatives in the residual are drawbacks regarding the efficiency of this method. For the fully discretized case a local projection stabilization (LPS) PSPG-type stabilization for the discrete pressure is combined with LPS for the convective term in [10]. Stability and convergence is proven using an semi-implicit Euler scheme for the discretization in time. In [11] we considered the semi-discretized time-dependent incompressible NavierStokes problem where the spatial discretization had been performed with inf-sup stable finite element methods with grad-div stabilization and a stream line upwind local projection stabilization (LPS SU) of the convective term (see also Section 2 of the present paper). Inspired by the ideas in [12] for edge stabilized methods with equal order ansatz spaces, we were able to prove quasi-robust error estimates in case smooth solutions satisfy u ∈ [L∞ (t0 , T ; W 1,∞ (Ω))]d (which ensures uniqueness of the Navier-Stokes solution). The latter means that coefficients of the right hand side of the error estimate may depend on Sobolev norms of the solution (u, p) but not on critical physical parameters like 1/ν. In the present paper, we extend the analysis to the fully discrete incremental pressure-correction algorithm with BDF2 time discretization, see Section 3. The proposed approach is based on the paper [5] by Guermond (and preliminary considerations in [13]). It turns out that grad-div stabilization is again essential for the derivation of quasi-robust error estimates whereas the LPS gives additional control of dissipative terms, see Section 4 with the main result in Theorem 4.3. As the result is quasi-optimal in the spatial variables, it is not optimal in time. Therefore, in Section 5 we modify the analysis in [5] to improve the order of the temporal discretization. Although the resulting estimates are not quasi-robust, we nevertheless consider the dependence of the error estimates on ν and the choice of appropriate bounds for the stabilization parameters. In all the cases, it turns out that grad-div stabilization is essential while the LPS can be neglected for deriving quasi-robust error estimates. Therefore, in Section 7 numerical examples consider the confirmation of the analytical results with respect to rates of convergence as well as the the influence of the SU stabilization as subgrid model. A critical discussion of the results, can be found in Section 8. 2. Stabilized Finite Element Discretization for the Navier-Stokes Problem In this section, we describe the model problem and the spatial semi-discretization based on inf-sup stable interpolation of velocity and pressure together with local projection stabilization. 2.1. Time-Dependent Navier-Stokes Problem. In the following, we will consider the usual Sobolev spaces W m,p (Ω) with norm k · kW m,p (Ω) , m ∈ N0 , p ≥ 1. In particular, we have Lp (Ω) = W 0,p (Ω) and denote H m (Ω) := W m,2 (Ω). Moreover, the closed subspaces H01 (Ω) := W01,2 (Ω), consisting of functions in W 1,2 (Ω) with zero trace on ∂Ω, and L20 (Ω), consisting of L2 -functions with zero mean in Ω, will be used. The inner product in L2 (D) with D ⊆ Ω will be denoted by (·, ·)D . In case of D = Ω we omit the index. The variational formulation of problem (1) reads: Find U = (u, p) ∈ [L2 (t0 , T ; V ) ∩ L∞ (t0 , T ; [L2 (Ω)]d )] × L2 (t0 , T ; Q) where V × Q := [W01,2 (Ω)]d × L20 (Ω) such that (2)

(∂t u, v) + AG (u; U, V) = (f , v) ∀V = (v, q) ∈ V × Q

3

with the Galerkin form AG (w; U , V) := ν(∇u, ∇v) − (p, ∇ · v) + (q, ∇ · u) | {z } =:aG (U ,V)

1 + ((w · ∇)u, v) − ((w · ∇)v, u) . {z } |2

(3)

=c(w;u,v)

The skew-symmetric form of the convective term c is chosen for conservation purposes. In this paper, we will additionally assume that the velocity field u belongs to L∞ (t0 , T ; [W 1,∞ (Ω)]d ) which ensures uniqueness of the solution. 2.2. Finite Element Spaces. For a simplex T ∈ Th or a quadrilateral/hexahedron T in Rd , let Tˆ be the reference unit simplex or the unit cube (−1, 1)d. The bijective reference mapping FT : Tˆ → T is affine for simplices and multi-linear for quadrilatˆ l and Q ˆ l with l ∈ N0 be the set of polynomials of degree ≤ l erals/hexahedra. Let P and of polynomials of degree ≤ l in each variable separately. Moreover, we set Pl (Tˆ) on simplices Tˆ ˆ Rl (T ) := Ql (Tˆ) on quadrilaterals/hexahedra Tˆ. Define Yh,−l := {vh ∈ L2 (Ω) : v h |T ◦ FT ∈ Rl (Tˆ) ∀T ∈ Th }, Yh,l := Yh,−l ∩ W 1,2 (Ω).

For convenience, we write V h = Rku instead of V h = [Yh,ku ]d ∩V and Qh = R±kp instead of Qh = Yh,±kp ∩ Q. Assumption 2.1. Let V h and Qh be finite element spaces satisfying a discrete infsup-condition (4)

(∇ · v, q) ≥β>0 q∈Qh \{0} v∈V h \{0} k∇vk0 kqk0 inf

sup

with a constant β independent on h. 2.3. Stabilization. For a Galerkin approximation of problem (2)-(3) on an admissible partition Th of the polyhedral domain Ω, consider finite dimensional spaces V h × Qh ⊂ V × Q. Then, the semi-discretized problem reads: Find U h = (uh , ph ) : (t0 , T ) → V h × Qh such that for all V h = (v h , qh ) ∈ V h × Qh : (5)

(∂t uh , v h ) + AG (uh ; U h , V h ) = (f , v h ).

The semi-discrete Galerkin solution of problem (5) may suffer from spurious oscillations due to poor mass conservation or dominating advection. The idea of local projection stabilization (LPS) methods is to separate discrete function spaces into small and large scales and to add stabilization terms only on small scales. Let {Mh } be a family of shape-regular macro decompositions of Ω into dsimplices, quadrilaterals (d = 2) or hexahedra (d = 3). In the one-level LPSapproach, one has Mh = Th . In the two-level LPS-approach, the decomposition Th is derived from Mh by barycentric refinement of d-simplices or regular (dyadic) refinement of quadrilaterals and hexahedra. We denote by hT and h the diameter of cells T ∈ Th and M ∈ Mh . It holds hT ≤ h ≤ ChT for all T ⊂ M and M ∈ Mh . Assumption 2.2. Let the finite element space V h satisfy the local inverse inequality (6)

k∇vh k0,M ≤ Ch−1 kv h k0,M

∀v h ∈ V h , M ∈ Mh .

4

Assumption 2.3. There are quasi-interpolation operators ju : V → V h and jp : Q → Qh such that for all M ∈ Mh , for all w ∈ V ∩ [W l,2 (Ω)]d with 1 ≤ l ≤ ku + 1: (7)

kw − ju wk0,M + hk∇(w − ju w)k0,M ≤ Chl kwkW l,2 (ωM )

and for all q ∈ Q ∩ H l (M ) with 1 ≤ l ≤ kp + 1: (8)

kq − jp qk0,M + hk∇(q − jp q)k0,M ≤ Chl kqkW l,2 (ωM ) .

on a suitable patch ωM ⊃ M . Moreover, let kv − ju vkL∞ (M) ≤ Ch|v|W 1,∞ (M)

∀v ∈ [W 1,∞ (M )]d .

Let D M ⊂ [L∞ (M )]d denote a finite element space on M ∈ Mh for uh . For each M ∈ Mh , let πM : [L2 (M )]d → DM be the orthogonal L2 -projection. Moreover, we denote by κM := id − πM the so-called fluctuation operator. Assumption 2.4. The fluctuation operator κM = id − πM provides the approximation property (depending on DM and s ∈ {0, · · · , ku }): (9)

kκM wk0,M ≤ Chl kwkW l,2 (M) , ∀w ∈ W l,2 (M ), M ∈ Mh , l = 0, . . . , s.

A sufficient condition for Assumption 2.4 is [Ps−1 ]d ⊂ DM . In this work, we restrict ourselves to inf-sup stable element pairs. This means that the space of discretely divergence-free functions is non-empty: (10)

V hdiv := {v h ∈ V h : (∇ · v h , qh ) = 0

∀qh ∈ Qh } 6= ∅

Definition 2.5. For each macro element M ∈ Mh define the element-wise averaged streamline direction uM ∈ Rd by Z 1 u(x) dx. (11) uM := |M | M This choice gives the estimates (12)

|uM | ≤ CkukL∞ (M) ,

ku − uM kL∞ (M) ≤ Ch|u|W 1,∞ (M) .

Now, we can formulate the semi-discrete stabilized approximation: Find U h = (uh , ph ) : (t0 , T ) → V h ×Qh , such that for all V h = (v h , qh ) ∈ V h ×Qh : (13)

(∂t uh , v h ) + AG (uh ; U h , V h ) + sh (uh ; uh , v h ) + t(uh ; uh , v h ) = (f , v h )

with the streamline-upwind-type stabilization sh and the grad-div stabilization t according to X (14) τM (κM ((w M · ∇)u), κM ((y M · ∇)v))M sh (w h , u, y h , v) := M∈Mh

(15)

t(u, v) := γ(∇ · u, ∇ · v).

The set of non-negative stabilization parameters τM , γ has to be determined later on. For the grad-div stabilization at least γ > ν is assumed. Occasionally, we consider the error in a norm given by symmetrically testing the stabilized approximation without time derivatives: X kuk2LP S := νk∇uk20 + γk∇ · uk20 + τM kκ((uM · ∇)u)k20 M∈Mh

5

3. Pressure-Correction Projection Discretization For the discretization in time on the interval [t0 , T ] we consider N equidistant time steps of size ∆t = (T − t0 )/N yielding the set MT = {t0 , . . . , tN = T }. The scheme that we are using is a pressure-correction projection approach based on BDF2. In order to abbreviate the discrete time derivative we define the operator Dt by Dt un :=

(16)

3un − 4un−1 + un−2 . 2∆t

Defining Y h := V hdiv ⊕ ∇Qh ⊂ [L2 (Ω)]d the fully discretized and stabilized scheme reads: e nh ∈ V h , unh ∈ Y h and pnh ∈ Qh Find u n 3e uh − 4uhn−1 + uhn−2 e nh , v h ) unh , ∇v h ) + c(e unh , u , v h + ν(∇e 2∆t (17) e nh , ∇ · v h ) + sh (e e nh , u e nh , v h ) =(f n , v h ) + (phn−1 , ∇ · v h ) +γ(∇ · u unh , u

(18)

e nh |∂Ω =0 u

3unh − 3e unh n−1 n + ∇(ph − ph ), y h = 0 2∆t (∇ · unh , qh ) = 0 unh |∂Ω = 0

holds for all v h ∈ V h , y h ∈ Y h and qh ∈ Qh . From here on, we assume Qh ⊂ H 1 (Ω). We call (17) the convection-diffusion and (18) the projection step. Remark 3.1. By testing equation (18) with wh ∈ V hdiv we derive e nh , wh ) = 0 (unh − u

(19)

∀wh ∈ V hdiv .

e nh onto V hdiv and kunh k ≤ ke Hence, unh is the L2 (Ω) projection of u unh k.

Choosing a slightly bigger ansatz space Y 1h := V h + ∇Qh instead of Y h we can e nh and replace (17) by the equation eliminate the weakly solenoidal field u (20)

e nh , v h ) + ν(∇e e nh , v h ) unh , ∇v h ) + c(e unh ; u (Dt u

e nh , ∇ · v h ) e nh , v h ) + γ(∇ · u e nh , u + sh (e unh , u 7 n−1 5 n−2 1 n−3 = (f n , v h ) + ph − ph + ph , ∇ · v h 3 3 3 n e h |∂Ω = 0 u

and equation (18) by (21)

(∇(pnh − phn−1 ), ∇qh ) =

(n · ∇pnh )|∂Ω = 0.

e nh 3∇ · u , qh 2∆t

unh can then be recovered according to

e nh − ∇(pnh − phn−1 ). unh = u

This is the approach that is used in the implementation. The equivalence of the two formulations (17), (18) and (20), (21) has been considered by Guermond in [14] for a first order unstabilized projection scheme.

6

Remark 3.2. For the first time step we use a BDF1 instead of the BDF2 scheme. In particular, the convection-diffusion and the projection step in the fully discretized setting read: e nh ∈ V h , unh ∈ V hdiv and pnh ∈ Qh such that Find u ! e 1h − u0h u e 1h , v h ) u1h , ∇v h ) + c(e u1h ; u , v h + ν(∇e ∆t (22) e 1h , u e 1h , v h ) + γ(∇ · u e 1h , ∇ · v h ) = (f 1 , v h ) + (p0h , ∇ · v h ), +sh (e u1h , u

(23)

e 1h |∂Ω = 0 u !

e 1h u1h − u + ∇(p1h − p0h ), y h ∆t

= 0,

(∇ · u1h , qh ) = 0

u1h |∂Ω = 0

holds for all v h ∈ V h , y h ∈ Y h and qh ∈ Qh . e 0h = u0h = ju u(t0 ) and p0h = jp p(t0 ) The initial values are chosen according to u using the interpolation operators defined later on.

Definition 3.3. Consider sequences u = (u1 , . . . , uN ) ∈ AN of vector-valued and p = (p1 , . . . , pN ) ∈ B N of scalar-valued quantities, where A and B are normed spaces and 1 ≤ n ≤ N . The norms we want to bound the errors in are defined by kuk2l2 (t0 ,T ;A) := ∆t

N X

n=1

kun k2A ,

kukl∞ (t0 ,T ;A) := max kun kA , 1≤n≤N

kpk2l2 (t0 ,T ;B) := ∆t

N X

n=1

kpn k2B ,

kpkl∞ (t0 ,T ;B) := max kpn kB . 1≤n≤N

For quantities r that are continuous in time we identify r by its evaluation at the discrete points in time (r(t1 ), . . . , r(tN ))T . 4. Quasi-Robustness and Quasi-Optimal Spatial Error Estimates In the following, we follow the strategy described by Guermond in [5]. Compared to his work we consider all ν-dependencies as well as a grad-div and LPS SU. The idea is to first state quasi-optimal results for the initial time step with respect to spatial and temporal discretization. Afterwards we derive estimates for n ≥ 2 that are optimal with respect to the LPS norm but suboptimal for the energy error and hence for the pressure error as well. 4.1. The Interpolation Operator. For the interpolation into the discrete ansatz spaces, we consider (w h , rh ) as solution of the Stokes problem Find wnh ∈ V h and rhn ∈ Qh such that (24)

ν(∇w nh , ∇v h ) + γ(∇ · w nh , ∇ · v h ) − (rhn , ∇ · v h )

= ν(∇u(tn ), ∇v h ) + γ(∇ · u(tn ), ∇ · v h ) − (p(tn ), ∇ · v h )

(∇ · wnh , qh ) = (∇ · u(tn ), qh ) = 0

holds for all v h ∈ V h and qh ∈ Qh . We define errors according to (25)

ηnu = u(tn ) − wnh ,

enu = wnh − unh ,

ζ nu := u(tn ) − unh = η nu + enu ,

ηpn = p(tn ) − rhn ,

enp = rhn − pnh ,

ζpn := p(tn ) − pnh = ηpn + enp .

(26) (27)

e nh , e enu = wnh − u

n e e nh = η nu + e enu , ζ u := u(tn ) − u

7

According to [15, Theorem 1] the solution of the grad-div stabilized Stokes problem can be bounded by νk∇(u(tn ) − wh )k20 + γk∇ · (u(tn ) − wh )k20 νk∇η nu k20 ≤ C inf w h ∈V hdiv

+ ≤C

1 inf kp(tn ) − qh k20 γ qh ∈Qh

inf

w h ∈V hdiv

(ν + γ)k∇(u(tn ) −

wh )k20

! 1 2 + inf kp(tn ) − qh k0 . γ qh ∈Qh

This result can easily be extended to include the grad-div stabilization on the left-hand side and using inf-sup stability we arrive at νk∇η nu k20 + γk∇ · ηnu k20 + kenp k20 1 inf (ν + γ)k∇(u(tn ) − wh )k20 + inf kp(tn ) − qh k20 div qh ∈Qh γ wh ∈V h

≤C

!

.

Using interpolation results according to Assumption 2.3 and the local inverse inequality (Assumption 2.2) we get the bound (28)

νkη nu k21 + γk∇ · η nu k20 + kηpn k20 + h2 k∇ηpn k20

≤ C(ν + γ)h2ku ku(tn )k2W ku +1,2 + γ −1 h2kp +2 kp(tn )k2W kp +1,2

and using the Aubin-Nitsche trick (29)

kη nu k20 ≤ Ch2 (νkη nu k21 + γk∇ · η nu k20 )

≤ C((ν + γ)h2ku +2 ku(tn )k2W ku +1,2 + γ −1 h2kp +4 kp(tn )k2W kp +1,2 ).

This gives stability according to max (kwnh k20 + νkwnh k21 + γk∇ · wnh k20 + kpnh k20 )

1≤n≤N

(30)

≤ max (kη nu k20 + νk∇η nu k21 + γk∇ · η nu k20 + kηpn k20 1≤n≤N

+ ku(tn )k20 + νk∇u(tn )k21 + γk∇ · u(tn )k20 + kp(tn )k20 )

≤ C max ((ku(tn )k20 + νk∇u(tn )k21 + γk∇ · u(tn )k20 + kp(tn )k20 )) ≤ C. 1≤n≤N

4.2. Initial Error Estimates. Before deriving bounds for the case n ≥ 2 we consider the initialization step. Although both estimates follow the same approach, we nevertheless state both for the convenience of the reader. With the abbreviations δt un := un − un−1 we can derive the following result:

n n n−1 e t un := δt u = u − u D ∆t ∆t

Lemma 4.1. Provided the continuous solutions satisfy the regularity assumptions u ∈ W 2,∞ (t0 , T ; L2) ∩ L∞ (t0 , T ; W ku +1,2 )

p ∈ W 1,∞ (t0 , T ; H 1) ∩ L∞ (t0 , T ; W kp +1,2 ),

8

e1u = w1h − u e 1h and e1p = rh1 − p1h we obtain for the initial errors e1u = w1h − u1h , e ke1u k20 + ke e1u k20 + ∆tνk∇e e1u k20 + ∆tγk∇ · e e1u k20 + (∆t)2 k∇e1p k20 X 1 + ∆t τM kκM ((e u1M · ∇)e e1u )k20,M M∈Mh

(31)

e t η 1 k2 + ((∆t)2 )kD e t u(t1 ) − ∂t u(t1 )k2 + (∆t)2 kδt ∇r1 k2 . ((∆t)2 )kD u 0 0 h 0 + h−2z ∆tkη 1u k20 + 3∆th2z−2 |||η 1u |||2LP S + 3∆th2z−2 |||e e1u |||2LP S ! X 1 2 1 1 2 2 + ∆t max {τM |e uM | } kη u k1 + kκM (∇u(t1 ))k0 M∈Mh

M∈Mh

where z ∈ {0, 1} if

" 1 h2 K1 := + ∆t |u(t1 )|W 1,∞ + Ch2z |u(t1 )|2W 1,∞ + C |u(t1 )|2W 1,∞ 4 γ # h2z−2 1 2 2 2z−2 ku(t1 )k∞ + Ch ke uh k∞ < 1. +C γ

Proof. The proof is similar to the one for n ≥ 2. One difference is the different estimate for the pressure terms which simplifies to 1 (rh1 − p0h , ∇ · e e1u ) = −(∇(rh1 − rh0 ), e e1u ) ≤ ke e1 k2 + C∆tkδt ∇rh1 k20 . 8∆t u 0 and due to the linearity of the Stokes problem it holds the estimate (32)

k∇δt rh1 k20 = k∇δt (p(t1 ) − ηp1 )k20 ≤ k∇δt ηp1 k20 + k∇δt p(t1 )k20

≤ C(∆t)2 (kuk2W 1,∞ (t0 ,T ;W ku +1,2 ) + kpk2W 1,∞ (t0 ,T ;W kp +1,2 ) ).

The remaining parts of the proof with respect to the velocity errors are similar to the ones in Lemma 4.2. For an estimate on the gradient of the pressure error the projection equation (23) is utilized: ! e e1u − e1u 1 2 1 0 1 1 1 1 k∇ep k0 = (∇(ep − ep ), ∇ep ) = (δt rh , ∇ep ) + , ∇ep ∆t ≤ Ckδt rh1 k20 +

1 C C ke e1 k2 + k∇e1p k20 . Ckδt rh1 k20 + ke e1 k2 . (∆t)2 u 0 2 (∆t)2 u 0

e1u and hence Here we used that e1u is a L2 (Ω)-projection of e

ke e1u − e1u k20 ≤ C(ke e1u k20 + ke1u k20 ) ≤ Cke e1u k20 .

Therefore, (∆t)2 k∇e1p k20 can be bounded by the right-hand side in (31), too.

4.3. Error Estimates after Initialization. Next, we are interested in the discretization errors e enu , enu and enp for n ≥ 2. The approach is the same as for the initialization but does not yield quasi-optimal results with respect to the energy norm of the velocity. m m em Lemma 4.2. For all 1 ≤ m ≤ N the discretization error em u = u = w h − uh , e m m m e h and em wm p = rh − ph can be bounded by h −u

4 2 2 m m−1 2 ke em k0 + (∆t)2 k∇em p k0 u k0 + k2eu − eu 3 m X ∆tνk∇e enu k20 + ∆tγk∇ · e enu k20 + kδtt enu k20 + n=2

9

+2∆t

X

M∈Mh

(33)

n τM kκM ((e unM

·

∇)e enu )k20,M

!

. CG (∆t)2 (kuk2W 1,∞ (t0 ,T ;W ku +1,2 ) + kpk2W 1,∞ (t0 ,T ;W kp +1,2 ) ) +(ν + γ)h2ku +2−2z kuk2L∞ (t0 ,T ;W ku +1,2 )

+ γ −1 h2kp +4−2z kpk2L∞ (t0 ,T ;W kp +1,2 ) + (∆t)5 kukW 3,∞ (t0 ,T ;L2 ) ν + γ 2ku h2kp +2 + h kuk2L∞ (t0 ,T ;W ku +1,2 ) + kpk2L∞ (t0 ,T ;W kp +1,2 ) ν γν n 2 n 2s 2 uM | } . +h kukL∞ (t0 ,T ;W s+1,2 ) max max {τM |e 1≤n≤m M∈Mh

T where z ∈ {0, 1} and CG ∼ exp 1−K provided n " 1 h2 Kn := + C∆t |u(tn )|W 1,∞ + h2z |u(tn )|2W 1,∞ + |u(tn )|2W 1,∞ 4 γ #! h2z−2 0, we calculate: X enu k20,M + kηnu k0,M ke enu k0,M k∇u(tn )k∞,M ke T1 ≤ M∈Mh

≤ |u(tn )|W 1,∞ ke enu k20 +

(74)

X

M∈Mh

enu k0,M |u(tn )|W 1,∞ (M) kηnu k0,M ke

1 −2z n 2 enu k20 . h kηu k0 + |u(tn )|W 1,∞ + Ch2z |u(tn )|2W 1,∞ ke ≤ 4C For the term T2 , we have via integration by parts e nh )e T2 = (e unh · ∇η nu , e enu ) = −(e unh · ∇e enu , ηnu ) − ((∇ · u enu , η nu ) =: T21 + T22 .

Term T21 is the most critical one. We calculate using the local inverse inequality (Assumption 2.2) and Young’s inequality: X ke unh k∞,M k∇e enu k0,M kηnu k0,M T21 = −(e unh · ∇e enu , η nu ) ≤ (75)

≤C

X

M∈Mh

M∈Mh

n ke unh k∞,M ke enu k0,M h−1 M kη u k0,M

≤ Ch2z−2 ke unh k2∞ ke enu k20 + Ch−2z kη nu k20 .

Using (∇ · u(tn ), q) = 0 for all q ∈ L2 (Ω) and Young’s inequality with C > 0, we obtain e nh )η nu , e T22 = −((∇ · u enu ) = ((∇ · (η nu + e enu ))η nu , e enu ) X ≤ kη nu k∞,M k∇ · e enu k0,M + k∇ · η nu k0,M ke enu k0,M M∈Mh

(76)

≤

X ChM √ enu k0,M enu k0,M + k∇ · η nu k0,M ke √ |u(tn )|W 1,∞ (M) γ k∇ · e γ

M∈Mh

≤ Ch2−2z |||η nu |||2LP S + Ch2−2z |||e enu |||2LP S + C

h2z enu k20 . |u(tn )|2W 1,∞ ke γ

For T3 we have the splitting e nh )u(tn ), e e nh )η nu , e e nh )ju u(tn ), e enu ) = T22 + T32 enu ) + ((∇ · u enu ) = −((∇ · u T3 = ((∇ · u

35

and use the same estimate as in (76). Utilizing (∇ · u(tn ), q) = 0 for all q ∈ L2 (Ω) and Young’s inequality we obtain e nh , u · e |T32 | = |(∇ · u enu )| = |(∇ · (−ηnu − e enu + u(tn )), u(tn ) · e enu )| ≤ |(∇ · η nu , u(tn ) · e enu )| + |(∇ · e enu , u(tn ) · e enu )| X 1 √ ≤ ku(tn )k∞,M γk∇ · η nu k0,M √ ke en k0,M γ u M∈Mh

(77)

1 √ enu k0,M √ ke enu k0,M + ku(tn )k∞,M γk∇ · e γ

≤ Ch2−2z |||η nu |||2LP S + Ch2−2z |||e enu |||2LP S + C

h2z−2 ku(tn )k2∞ ke enu k20 . γ

Combining the above bounds (74)-(77) yields the claim.

Lemma B.3. The error with respect to the convective terms can be estimated as e nh , e c(u(tn ), u(tn ), e enu,h ) − c(e unh , u enu,h ) ≤

Cke enu,h k20

+

ke η nu,t k41 + ke η nu,h k41 ku(tn )k22 + 3 ν ν

!

+

ν n 2 ke e k 4 u,h 1

C (ke η nu,t k41 + ke η nu,h k41 + ke ηnu,h k20 ku(tn )k22 ) ν

enu,h , η e nu,h and η e nu,t as defined in (44), (47) and (48). with e

n e nu,t + e e nh we calculate for the convective term using Proof. Due to η ξ u,h = u(tn ) − u skew-symmetry and the estimates from Lemma B.1

(78) e nh , e c(u(tn ), u(tn ), e enu,h ) − c(e unh , u enu,h )

n n e nu,t , e enu,h ) enu,h ) + c(e unh , η enu,h ) + c(e η nu,t , u(tn ), e enu,h ) + c(e unh , e ξu,h , e = c(e ξ u,h , u(tn ), e n

n

e nu,t − e e nu,h , e = c(e ξ u,h , u(tn ), e enu,h ) + c(u(tn ) − η ξ u,h , η enu,h )

n e nu,t − e e nu,t , e + c(e η nu,t , u(tn ), e enu,h ) + c(u(tn ) − η ξ u,h , η enu,h )

e nu,h + η e nu,t , e e nu,t , u(tn ), e enu,h ) + c(u(tn ), η enu,h ) + c(e enu,h , u(tn ), e enu,h ) = c(e η nu,h + η e nu,h , η e nu,t + η e nu,h , e e nu,h + η e nu,t , e − c(e η nu,t + η enu,h ) + c(e enu,h , η enu,h ).

These terms can be estimated as

e nu,t , u(tn ), e e nu,t k0 ku(tn )k2 ke c(e η nu,h + η enu,h ) ≤ Cke η nu,h + η enu,h k1 C n ν n 2 e nu,t k20 ku(tn )k22 + ke ≤ ke η +η e k ν u,h 32 u,h 1 e nu,h + η e nu,t , e e nu,t k0 ku(tn )k2 ke c(u(tn ), η enu,h ) ≤ Cke η nu,h + η enu,h k1 ν n 2 C n e nu,t k20 ku(tn )k22 + ke η +η e k ≤ ke ν u,h 32 u,h 1 n n n n c(e eu,h , u(tn ), e eu,h ) ≤ Cke eu,h k0 ku(tn )k2 ke eu,h k1 C n 2 ν n 2 ≤ ke e k ku(tn )k22 + ke e k ν u,h 0 16 u,h 1 n n n n n n n n 2 e u,h , η e u,t + η e u,h , e e u,h k1 ke c(e η u,t + η eu,h ) ≤ Cke η u,t + η eu,h k1 C n ν n 2 e nu,h k41 + ke ≤ ke η u,t + η e k ν 16 u,h 1 1/2 3/2 e nu,h + η e nu,t , e e nu,t k1 ke c(e enu,h , η enu,h ) ≤ Cke enu,h k0 ke ηnu,h + η enu,h k1

36

C n 2 n ν n 2 e nu,t k41 + ke ke eu,h k0 ke ηu,h + η e k . 3 ν 16 u,h 1 For the last estimate we used the inequality √ √ a1/2 b3/2 = (ν −3/2 a)1/2 ( νb)3/2 ≤ C(((ν −3/2 a)1/2 )4 + (( νb)3/2 )4/3 ) ≤

≤ C(ν −3 a2 + νb2 ).

In combination, the error with respect to the convective terms is given by e nh , e c(u(tn ), u(tn ), e enu,h ) − c(e unh , u enu,h ) ≤

enu,h k20 Cke

+

ke η nu,t k41 + ke η nu,h k41 ku(tn )k22 + ν3 ν

!

+

ν n 2 e k ke 4 u,h 1

C (ke η nu,t k41 + ke ηnu,h k41 + ke η nu,h k20 ku(tn )k22 ). ν Appendix C. Discrete Gronwall Lemma

Lemma C.1. Let y n , hn , g n , f n be non-negative sequences satisfying for all 0 ≤ m ≤ [T /k] ym + k

m X

n=0 n

hn ≤ B + k

m X

[T /k]

(g n y n + f n ) with k

n=0

X

n=0 n −1

g n ≤ M.

Assume kg < 1 and let σ = max0≤n≤[T /k] (1 − kg ) . Then for all 0 ≤ m ≤ [T /k] it holds ! m m X X (79) ym + k hn ≤ exp(σM ) B + k fn . n=1

n=0

Proof. A proof of this result can be found in [21], for instance.

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