Pressure Estimate

In order to derive an upper bound for the pressure error, we proceed in a similar manner as in Corollary3.1.3where stability of ph is established.

Theorem 3.3.1 (Pressure estimate).

Let (u, p, θ) : [0, T]→V^div×Q×Θ,(u_h, p_h, θ_h) : [0, T]→V^div_h ×Q_h×Θ_h be solutions of (2.9)-(2.10) and (3.1)-(3.2) satisfying

u∈L^∞(0, T; [W^1,∞(Ω)]^d), u_h∈L^∞(0, T; [L^∞(Ω)]^d).

Then we obtain the a priori estimate for the semi-discrete pressure errorξ_p,h =p−p_h for 0≤t≤T

kξ_p,hk²_L2(0,t;L²(Ω)) ≤ C β_h²

(

k∂_tξ_u,hk²_L2(0,t;L²(Ω))+β²kgk²_L∞(0,t;L^∞(Ω))kξ_θ,hk²_L2(0,t;L²(Ω))

+ kuk²_L2(0,t;L^∞(Ω))+ku_hk²_L2(0,t;L^∞(Ω))

kξ_u,hk²_L∞(0,t;L²(Ω))

+ Z t

0

ν+C max

M∈M_h{γ_M⁻¹ku_hk²_∞,M}+ max

M∈M_h{τ_M^u|u_M|²}+ max

M∈M_h{γ_Md}|||ξ_u,h|||²_{LP S}dτ +

Z t 0

max

M∈M_h{τ_M^u|u_M|²} ^X

M∈M_h

τ_M^u|u_M|²kκ^u_M(∇u)k²_0,Mdτ )

with a constant C >0 independent of the problem parameters, hM, hL and the solutions.

Proof. The discrete inf-sup condition from Assumption2.2.1gives that forξp,h ∈Qh, there exists a uniquev_h ∈V_h with

∇ ·vh =ξp,h, k∇v_hk₀≤ 1

β_hkξ_p,hk₀.

We subtract (3.1) from (2.9) and use (vh,0)∈V_h×Q_h as a test function. This leads to β_hk∇v_hk₀kξ_p,hk₀≤ kξ_p,hk²₀ = (ξ_p,h,∇ ·v_h) = (∂_tξ_u,h,v_h) + (ν∇ξ_u,h,∇v_h)

+cu(u;u,vh)−cu(uh;uh,vh) + (βgξθ,h,vh)−su(uh;uh,vh)−th(uh;uh,vh)

≤ k∇v_hk₀k∂_tξ_u,hk₋₁+νk∇ξ_u,hk₀+βkgk_∞kξ_θ,hk₋₁

+cu(u;u,vh)−cu(uh;uh,vh) +su(uh;ξ_u,h,vh)−su(uh;u,vh) +t_h(u_h;ξ_u,h,v_h),

where we used (∇ ·u, q) = 0 for all q∈L²(Ω) in the last term. We further have su(u_h;ξ_u,h,v_h)≤su(u_h;ξ_u,h,ξ_u,h)^1/2su(u_h;v_h,v_h)^1/2

≤ |||ξ_u,h|||_{LP S} max

M∈M_h{^qτ_M^u|u_M|}k∇v_hk₀,

−s_u(u_h;u,v_h)≤ ^X

M∈M_h

τ_M^u|u_M|²kκ^u_M(∇u)k²_0,M^1/2 max

M∈M_h{^qτ_M^u |u_M|}k∇v_hk₀, t_h(u_h;ξ_u,h,v_h)≤ |||ξ_u,h|||_{LP S} max

M∈Mh

{^pγ_Md}k∇v_hk₀.

Integration by parts of the convective terms, using (∇ ·u, q) = 0 for all q ∈ L²(Ω) and applying the Poincaré inequalityA.3.2yield:

c_u(u;u,v_h)−c_u(u_h;u_h,v_h)

=−(u−u_h,u· ∇v_h)−(u_h,(u−u_h)· ∇v_h) +1

2(∇ ·(u−u_h),u_h·v_h)

≤ kuk∞+ku_hk∞

kξ_u,hk₀k∇v_hk₀+C max

M∈Mh

{γ_M^−1/2ku_hk_∞,M}|||ξ_u,h|||_{LP S}k∇v_hk₀.

We obtain

βhkξ_p,hk₀≤ k∂_tξ_u,hk₀+βkgk_∞kξ_θ,hk₀+ kuk_∞+ku_hk_∞kξ_u,hk₀ +√

ν+C max

M∈M_h{γ_M^−1/2ku_hk_∞,M}+ max

M∈M_h{^qτ_M^u |u_M|}+ max

M∈M_h{^pγ_Md}|||ξ_u,h|||_{LP S}

+ ^X

M∈M_h

τ_M^u |u_M|²kκ^u_M(∇u)k²_0,M^1/2 max

M∈M_h{^qτ_M^u|u_M|}.

Squaring and integration in time result in the claim.

In the above estimate for the pressure error ξ_p,h, the results from Sections 3.2.1or 3.2.2 can be inserted if the respective requirements are met.

Remark 3.3.2. If the convective terms are estimated in a standard way (using Lemma A.3.7), the dependence onku_hk_∞ can be avoided, because we calculate

c_u(u;u,v_h)−c_u(u_h;u_h,v_h)

=cu(u−uh;u,vh)−cu(u−uh;u−uh,vh) +cu(u;u−uh,vh)

≤C kuk₁+kξ_u,hk₁kξ_u,hk₁k∇v_hk₀

≤C kuk₁+ν^−1/2|||ξ_u,h|||_{LP S}ν^−1/2|||ξ_u,h|||_{LP S}k∇v_hk₀. However, the resulting error bound is not stable for vanishingν.

Remark 3.3.3. The above a priori error estimate of the pressure suffers from the term k∂_tξ_u,hk_L2(0,t;L²(Ω)). In [DAL15], we show for the Oseen problem that an error reduction occurs. A similar result was obtained in [BF07] for an edge stabilization method of the Navier-Stokes problem.

This chapter is dedicated to the behavior of the fully discrete algorithm in terms of con-vergence in time and space where grad-div and LPS-SU stabilizations are incorporated.

We prove the stability of the fully discrete solution of the stabilized Oberbeck-Boussinesq model (3.1)-(3.2) and present a consistency analysis for the stabilized Navier-Stokes equa-tions. In Section2.3.1, we introduced the fully discrete scheme. For clarity, we repeat the important equations here and introduce some notations and simplifications we use in the following analysis.

Consider equidistant time steps of size ∆t. Let 0 =t₀< t₁ <· · ·< t_N =T. Foru,v ∈V and ψ,θ∈Θ, the LPS terms at time tn are written as

s_u(w,u,y,v) := ^X

M∈M_h

τ_Mⁿ(κ^u_M((w_M · ∇)u), κ^u_M((y_M · ∇)v))_M, sθ(w, ψ,y, θ) := ^X

L∈Lh

τ_Lⁿ((κ^θ_L(wL· ∇)ψ), κ^θ_L((y_L· ∇)θ))_L

with element-wise constant w_M, w_L, y_M, y_L ∈ R^d as introduced in Section 2.2.3 and stabilization parameters that coincide with the semi-discrete, time-continuous ones at all tn:

τ_Mⁿ =τ_M^u(t_n), τ_Lⁿ=τ_L^θ(t_n) forM ∈ M_h, L∈ L_h.

Note that this notation coincides with the definition from Section2.2if the first and third arguments are equal. For convenience, we assume a constant grad-div parameterγ :=γM

for all M ∈ M_h, i.e.,

t_h(u_eⁿ_ht;u_eⁿ_ht,v_h) =γ(∇ ·u_eⁿ_ht,∇ ·v_h),

and the casei_h≡0, since we use a continuous discrete pressure space for our simulations later. The extension to cell-wise linear and time dependent γ_Mⁿ =γ_M(t_n) can be proven easily.

Letθ^n∗_ht := 2θ_htⁿ⁻¹−θ_htⁿ⁻². For the analysis of the splitting algorithm, we consider the case of an incremental method, i.e.,χ^rot = 0. Then the fully discretized and stabilized scheme reads:

69

Find u_eⁿ_ht ∈V_h such that for allv_h ∈V_h: 3u_eⁿ_ht−4uⁿ⁻¹_ht +uⁿ⁻²_ht

2∆t ,v_h

!

+ν(∇u_eⁿ_ht,∇v_h) +c_u(u_eⁿ_ht;u_eⁿ_ht,v_h) +t_h(u_eⁿ_ht;u_eⁿ_ht,v_h) +su(u_eⁿ_ht,u_eⁿ_ht,u_eⁿ_ht,vh)−(pⁿ⁻¹_ht ,∇ ·vh) +β(g(tn)θ_ht^n∗,vh) = (fu(tn),vh),

ueⁿ_ht|_∂Ω = 0. (4.1) Find uⁿ_ht ∈V^div_h and pⁿ_ht∈Q_h such that for ally_h ∈Y_h andq_h∈Q_h:

3uⁿ_ht−3u_eⁿ_ht

2∆t +∇(pⁿ_ht−pⁿ⁻¹_ht ),y_h

= 0, (uⁿ_ht,∇q_h) = 0,

uⁿ_ht|_∂Ω= 0. (4.2)

Find θⁿ_ht∈Θ_h such that for allψh∈Θ_h:

(D_tθⁿ_ht, ψ_h) +α(∇θ_htⁿ,∇ψ_h) +c_θ(u_eⁿ_ht;θ_htⁿ, ψ_h) +s_θ(u_eⁿ_ht, θⁿ_ht,u_eⁿ_ht, ψ_h) = (f_θ(t_n), ψ_h), θⁿ_ht|_∂Ω = 0. (4.3) The above scheme provides the existence ofu_eⁿ_ht,uⁿ_ht,pⁿ_ht,θⁿ_htin every time step 1≤n≤N. This can be understood via induction. Given the solutions at timetn−1andtn−2, a solution ueⁿ_ht of (4.1) exists by standard theory. The velocity uⁿ_ht is given as the L²-projection of ueⁿ_ht onto V^div_h 6={0}. Ally_h ∈Y_h can be written asy_h =w_h+∇q_h∈Y_h =V^div_h ⊕ ∇Q_h. If we insertuⁿ_ht :=P_divu_eⁿ_ht into the projection equation (4.2), we have:

3uⁿ_ht−3u_eⁿ_ht

2∆t +∇(pⁿ_ht−pⁿ⁻¹_ht ),y_h= 3

2∆t(P_divu_eⁿ_ht−u_eⁿ_ht,w_h) + (∇(pⁿ_ht−pⁿ⁻¹_ht ),w_h) + 3

2∆t(uⁿ_ht−u_eⁿ_ht,∇q_h) + (∇(pⁿ_ht−pⁿ⁻¹_ht ),∇q_h)

= 3

2∆t(uⁿ_ht−u_eⁿ_ht,∇q_h) + (∇(pⁿ_ht−pⁿ⁻¹_ht ),∇q_h).

Thus, the projection equation becomes a Poisson problem for the pressurepⁿ_ht. Therefore, its existence is guaranteed and (4.2) is satisfied. Obviously, θⁿ_ht exists as well.

4.1. Stability of the Fully Discretized Quantities

In order to obtain stability for the fully discretized quantities, we are interested in suitable norms in which we want to control the solution and errors.

Definition 4.1.1 (Time-discrete norm).

Letvⁿ∈A be vector-valued andψⁿ∈B be scalar-valued quantities, whereA andB are normed spaces, 1≤n≤N. Let v= (v¹, . . . ,v^N)∈A^N and ψ= (ψ¹, . . . , ψ^N)∈B^N. The norms we want to control are defined by

kvk²_l2(0,T;A):= ∆t

Theorem 4.1.2 (Stability of the stabilized Oberbeck-Boussinesq model).

Let f_u ∈l²(0, T; [L²(Ω)]^d), fθ ∈l²(0, T;L²(Ω)) and g∈l^∞(0, T; [L^∞(Ω)]^d). With C >0,

provided 3∆tβkgk_l∞(0,T;L^∞(Ω))<1 and with a Gronwall constant C_G,OB behaving like C_G,OB∼T1−3∆tβkgk_l^∞_(0,T;L^∞_(Ω))⁻¹. (4.4) Proof. Due to the projection step (4.2) tested with y_h = ⁸₃(∆t)²∇pⁿ⁻¹_ht and the fact that uⁿ_ht ∈V^div_h , we have

−4∆t(u_eⁿ_ht,∇pⁿ⁻¹_ht ) +8(∆t)²

3 (∇(pⁿ_ht−pⁿ⁻¹_ht ),∇pⁿ⁻¹_ht ) = 0.

Testing withv_h = 4∆tu_eⁿ_ht in (4.1) yields due to skew-symmetry of c_u

23u_eⁿ_ht−4uⁿ⁻¹_ht +uⁿ⁻²_ht ,u_eⁿ_ht+ 4∆tν(∇u_eⁿ_ht,∇u_eⁿ_ht) + 4∆tγ(∇ ·u_eⁿ_ht,∇ ·u_eⁿ_ht) + 4∆ts_u(u_eⁿ_ht,u_eⁿ_ht,u_eⁿ_ht,u_eⁿ_ht)

=−4∆tc_u(u_eⁿ_ht;u_eⁿ_ht,u_eⁿ_ht) + 4∆t(f_u(t_n),u_eⁿ_ht)−4∆t(∇pⁿ⁻¹_ht ,u_eⁿ_ht)−4∆tβ(g(t_n)θ^n∗_ht,u_eⁿ_ht)

= 4∆t(f_u(t_n),u_eⁿ_ht)−4∆t(∇pⁿ⁻¹_ht ,u_eⁿ_ht)−4∆tβ(g(t_n)(2θⁿ⁻¹_ht −θⁿ⁻²_ht ),u_eⁿ_ht).

We add these two equations together and establish using Young’s inequality 23u_eⁿ_ht−4uⁿ⁻¹_ht +uⁿ⁻²_ht ,u_eⁿ_ht+ 4∆tνk∇u_eⁿ_htk²₀+ 4∆tγk∇ ·u_eⁿ_htk²₀

+ 4∆tsu(u_eⁿ_ht,u_eⁿ_ht,u_eⁿ_ht,u_eⁿ_ht) +8

3(∆t)²(∇(pⁿ_ht−pⁿ⁻¹_ht ),∇pⁿ⁻¹_ht )

= 4∆t(f_u(t_n),u_eⁿ_ht)−4∆tβ(g(t_n)(2θ_htⁿ⁻¹−θⁿ⁻²_ht ),u_eⁿ_ht)

≤ 5∆tβkg(t_n)k_∞

3 ku_eⁿ_htk²₀+ 12∆t

5βkg(t_n)k_∞kf_u(t_n)k²₀+4∆tβkg(t_n)k_∞ 3 ku_eⁿ_htk²₀ + 3∆tβkg(t_n)k∞k2θ_htⁿ⁻¹−θ_htⁿ⁻²k²₀

≤3∆tβkg(t_n)k_∞ku_eⁿ_htk²₀+ 12∆t

5βkg(t_n)k_∞kf_u(tn)k²₀+ 3∆tβkg(t_n)k_∞k2θ_htⁿ⁻¹−θⁿ⁻²_ht k²₀. (4.5) Denote δ_taⁿ := aⁿ−aⁿ⁻¹ and δ_ttaⁿ := δ_t(δ_taⁿ). The first term on the left-hand side is splitted according to

2(3u_eⁿ_ht−4uⁿ⁻¹_ht +uⁿ⁻²_ht ,u_eⁿ_ht) = 6(u_eⁿ_ht,u_eⁿ_ht−uⁿ_ht)

+ 2(u_eⁿ_ht−uⁿ_ht,3uⁿ_ht−4uⁿ⁻¹_ht +uⁿ⁻²_ht ) + 2(uⁿ_ht,3uⁿ_ht−4uⁿ⁻¹_ht +uⁿ⁻²_ht )

=I1+I2+I3

with I₁ := 3ku_eⁿ_htk²₀+ 3kuⁿ_ht−u_eⁿ_htk²₀−3kuⁿ_htk²₀, I₂ := 2(u_eⁿ_ht−uⁿ_ht,3uⁿ_ht−4uⁿ⁻¹_ht +uⁿ⁻²_ht ),

I₃ :=kuⁿ_htk²₀+k2uⁿ_ht−uⁿ⁻¹_ht k²₀+kδ_ttuⁿ_htk²₀− kuⁿ⁻¹_ht k²₀− k2uⁿ⁻¹_ht −uⁿ⁻²_ht k²₀,

where the following identities were taken advantage of:

2(a, a−b) =kak²₀+ka−bk²₀− kbk²₀, (4.6) 2(a,3a−4b+c) =kak²₀+k2a−bk²₀+ka−2b+ck²₀− kbk²₀− k2b−ck²₀. (4.7) The second termI₂ vanishes using (4.2) and due touⁿ_ht∈V^div_h :

3

4∆tI₂ = (∇(pⁿ_ht−pⁿ⁻¹_ht ),3uⁿ_ht−4uⁿ⁻¹_ht +uⁿ⁻²_ht )

=−(pⁿ_ht−pⁿ⁻¹_ht ,∇ ·(3uⁿ_ht−4uⁿ⁻¹_ht +uⁿ⁻²_ht )) = 0.

We deploy this splitting and apply identity (4.6) to ⁸₃(∆t)²(∇(pⁿ_ht−pⁿ⁻¹_ht ),∇pⁿ⁻¹_ht ) in (4.5):

3ku_eⁿ_htk²₀+ 3kuⁿ_ht−u_eⁿ_htk²₀−2kuⁿ_htk²₀+k2uⁿ_ht−uⁿ⁻¹_ht k²₀+4

3(∆t)²k∇pⁿ_htk²₀ + 4∆tνk∇u_eⁿ_htk²₀+ 4∆tγk∇ ·u_eⁿ_htk²₀+kδ_ttuⁿ_htk²₀+ 4∆ts_u(u_eⁿ_ht,u_eⁿ_ht,u_eⁿ_ht,u_eⁿ_ht)

≤ ku_eⁿ⁻¹_ht k²₀+k2uⁿ⁻¹_ht −uⁿ⁻²_ht k²₀+4

3(∆t)²k∇pⁿ⁻¹_ht k²₀+ 4

3(∆t)²k∇(pⁿ_ht−pⁿ⁻¹_ht )k²₀ + 3∆tβkg(t_n)k_∞ku_eⁿ_htk²₀+ 12∆t

5βkg(t_n)k_∞kf_u(tn)k²₀+ 3∆tβkg(t_n)k_∞k2θ_htⁿ⁻¹−θ_htⁿ⁻²k²₀. Then we use thatkuⁿ_htk₀≤ ku_eⁿ_htk₀ because uⁿ_ht=P_divu_eⁿ_ht is an orthogonalL²-projection:

ku_eⁿ_htk²₀+ 3kuⁿ_ht−u_eⁿ_htk²₀+k2uⁿ_ht−uⁿ⁻¹_ht k²₀+4

3(∆t)²k∇pⁿ_htk²₀

+ 4∆tνk∇u_eⁿ_htk²₀+ 4∆tγk∇ ·u_eⁿ_htk²₀+kδ_ttuⁿ_htk²₀+ 4∆ts_u(u_eⁿ_ht,u_eⁿ_ht,u_eⁿ_ht,u_eⁿ_ht)

≤ ku_eⁿ⁻¹_ht k²₀+k2uⁿ⁻¹_ht −uⁿ⁻²_ht k²₀+4

3(∆t)²k∇pⁿ⁻¹_ht k²₀+ 4

3(∆t)²k∇(pⁿ_ht−pⁿ⁻¹_ht )k²₀ + 3∆tβkg(t_n)k_∞ku_eⁿ_htk²₀+ 12∆t

5βkg(t_n)k_∞kf_u(tn)k²₀+ 3∆tβkg(t_n)k_∞k2θ_htⁿ⁻¹−θ_htⁿ⁻²k²₀. The projection equation (4.2) tested withy_h =∇(pⁿ_ht−pⁿ⁻¹_ht ) yields

2∆t

3 k∇(pⁿ_ht−pⁿ⁻¹_ht )k²₀ =−(uⁿ_ht−u_eⁿ_ht,∇(pⁿ_ht−pⁿ⁻¹_ht ))

≤ kuⁿ_ht−u_eⁿ_htk₀k∇(pⁿ_ht−pⁿ⁻¹_ht )k₀

⇒ 4

3(∆t)²k∇(pⁿ_ht−pⁿ⁻¹_ht )k²₀ ≤3kuⁿ_ht−u_eⁿ_htk²₀. We insert this in the previous estimate and obtain

ku_eⁿ_htk²₀+k2uⁿ_ht−uⁿ⁻¹_ht k²₀+4

3(∆t)²k∇pⁿ_htk²₀+ 4∆tνk∇u_eⁿ_htk²₀ + 4∆tγk∇ ·u_eⁿ_htk²₀+kδ_ttuⁿ_htk²₀+ 4∆ts_u(u_eⁿ_ht,u_eⁿ_ht,u_eⁿ_ht,u_eⁿ_ht)

≤ ku_eⁿ⁻¹_ht k²₀+k2uⁿ⁻¹_ht −uⁿ⁻²_ht k²₀+4

3(∆t)²k∇pⁿ⁻¹_ht k²₀+ 3∆tβkg(t_n)k_∞ku_eⁿ_htk²₀

+ 12∆t

5βkg(t_n)k_∞kf_u(t_n)k²₀+ 3∆tβkg(t_n)k_∞k2θ_htⁿ⁻¹−θ_htⁿ⁻²k²₀. (4.8) For the temperature, we test (4.3) with ψ_h = 4∆tθ_htⁿ and take advantage of the skew-symmetry ofcθ:

23θⁿ_ht−4θ_htⁿ⁻¹+θ_htⁿ⁻², θ_htⁿ+ 4∆tα(∇θⁿ_ht,∇θⁿ_ht) + 4∆ts_θ(u_eⁿ_ht, θⁿ_ht,u_eⁿ_ht, θ_htⁿ)

=−4∆tc_θ(u_eⁿ_ht;θⁿ_ht, θ_htⁿ) + 4∆t(f_θ(tn), θ_htⁿ)

= 4∆t(f_θ(t_n), θ_htⁿ)≤3∆tβkg(t_n)k_∞kθ_htⁿk²₀+ 4∆t

3βkg(t_n)k_∞kf_θ(t_n)k²₀. For the first term on the left-hand side, it holds due to identity (4.7):

2(3θ_htⁿ −4θⁿ⁻¹_ht +θⁿ⁻²_ht , θⁿ_ht) =kθⁿ_htk²₀+k2θⁿ_ht−θⁿ⁻¹_ht k²₀+kδ_ttθ_htⁿk²₀

+ ∆t

βkgk_l^∞_(0,T;L^∞_(Ω))

m

X

n=2

12

5 kf_u(t_n)k²₀+4

3kf_θ(t_n)k²₀

.

Provided 3∆tβkgk_l^∞_(0,T;L^∞_(Ω)) < 1, the discrete Gronwall Lemma A.3.6 can be applied forku_e^m_htk²₀+kθ^m_htk²₀+k2θ^m_ht−θ_ht^m−1k²₀. Hence, for 2≤m≤N:

ku_e^m_htk²₀+ (∆t)²k∇p^m_htk²₀+kθ^m_htk²₀+k2θ^m_ht−θ^m−1_ht k²₀ + ∆t

m

X

n=2

hνk∇u_eⁿ_htk²₀+γk∇ ·u_eⁿ_htk²₀+αk∇θ_htⁿk²₀ +s_u(u_eⁿ_ht,u_eⁿ_ht,u_eⁿ_ht,u_eⁿ_ht) +s_θ(u_eⁿ_ht, θⁿ_ht,u_eⁿ_ht, θ_htⁿ)ⁱ

≤e^CG,OBhku_e¹_htk²₀+k2u¹_ht−u⁰_htk²₀+ (∆t)²k∇p¹_htk²₀+kθ¹_htk²₀+k2θ¹_ht−θ_ht⁰ k²₀

+ C∆t

βkgk_l∞(0,T;L^∞(Ω)) m

X

n=2

kf_u(t_n)k²₀+kf_θ(t_n)k²₀ ⁱ

with a Gronwall constantCG,OB∼T1−3∆tβkgk_l^∞_(0,T;L^∞_(Ω))⁻¹.

Thus, the fully discrete velocity and temperature are stable with respect to the fully discrete norms as introduced in Definition4.1.1. We can prove stability of the pressure by taking advantage of the discrete inf-sup condition. Note that one immediately obtains a certain stability of the pressure as (∆t)²k∇p_htk²_l∞(0,T;L²(Ω)) ≤C.

4.2. Fully Discrete Convergence Results

A proof of convergence for the fully discretized model is not trivial. Even for the Navier-Stokes equations, we run into technical difficulties we want to discuss in this section. The main challenge here is to handle both the pressure term and the convective term in the advection-diffusion equation (4.1).

The rotational correction scheme is analyzed in [GS04] for the linear Stokes problem.

L²-convergence of the time-discretized velocity as (∆t)² is shown as expected. For the W^1,2-velocity and the L²-pressure errors, improved convergence results of order (∆t)^3/2 can be obtained. It is argued there that the nonlinear terms do not corrupt the order of convergence and can therefore be omitted.

In our considerations [AD15], we observe that indeed, the order of convergence stems from the linear problem, but nonlinear terms add technical problems for the rotational correc-tion scheme that cannot be handled easily. In order to extend the analysis of [GS04] to the nonlinear case - or worse the stabilized case -, one would need stability results and prior auxiliary estimates for the error that are not available for the rotational scheme, since one would need to control a term of the form (∇∇ ·u_eⁿ_ht,∇pⁿ_ht).

Therefore, we pursue a different ansatz for the nonlinear case considering the incremen-tal method. Our analysis is inspired by techniques from several authors who worked on the fully discretized Navier-Stokes equations: In [BC07], convergence results for a first or-der projection method for the fully discrete Navier–Stokes equations are given. The frac-tional step incremental projection method for the unstabilized Navier-Stokes equations with BDF2 time-discretization is analyzed in [Gue99]. Shen considers a different second order time-discretization scheme in [She96]. Neither author regards the dependence on constant problem parameters, in particular on ν. It is a stated aim of this thesis to point out the arising restrictions of our error estimates. Besides that, we take LPS SU and grad-div stabilization into account.

We desire to separately consider the errors produced by discretization in time and space.

So we bound the total error via the triangle inequality according to:

ku−u_ehtk_l^∞_(0,T;L2(Ω))≤ ku−uhk_l^∞_(0,T;L2(Ω))+ku_h−u_ehtk_l^∞_(0,T;L2(Ω)), ku−u_ehtk_l2(0,T;LP S)≤ ku−uhk_l2(0,T;LP S)+ku_h−u_ehtk_l2(0,T;LP S).

The errors resulting fromu−u_h can be bounded by a semi-discrete analysis, e.g., similar to Chapter 3, which provides a bound for the time-continuous norm as

ku−u_hk_L^∞_(0,T;L2(Ω))+ku−u_hk_L2(0,T;LP S).h^ku.

We use the fact that these norms provide an upper bound for the respective discrete norms;

see Section 4.2.1. The estimates for uh −u_eht are presented in Section 4.2.2. In Section 4.2.3, we combine the results and derive an error bound on kp−p_htk_l2(0,T;L²(Ω)) via the inf-sup stability of the ansatz spaces.

Assumption 4.2.1 (For error due to spatial discretization).

We require for some l∈ {1,2} (to be fixed later)

u∈W^l,2(0, T;LP S), u_h ∈W^l,2(0, T;LP S), (4.10) where W^l,2(0, T;LP S) consists of functions v ∈W^l,2(0, T; [W^1,2(Ω)]^d) such that all time derivatives v⁽ⁿ⁾ (n= 0, . . . , l) are bounded with respect to || · ||_{0,T;LP S}. Moreover, assume that the conditions for Corollary 3.2.13 hold for the velocity. This includes in particular

u∈L^∞(0, T; [W^1,∞(Ω)]^d)∩L²(0, T; [W^ku+1,2(Ω)]^d),

∂tu∈L²(0, T; [W^ku,2(Ω)]^d), p∈L²(0, T;W^kp+1,2(Ω)∩C(Ω)).

The latter conditions can be replaced by the premises of any other suitable semi-discrete a priori error estimate for the time dependent Navier-Stokes equations.

Assumption 4.2.2 (For linear error due to temporal discretization).

We require for the linear error due to time-discretization that there is a constant C > 0 (independent ofn) such that for all 1≤n≤N:

kRⁿ−Rⁿ⁻¹k²₀+k∇(p_h(t_n)−2p_h(tn−1) +p_h(tn−2))k²₀ ≤C(∆t)⁴, (4.11) k∇(p_h(tn)−ph(tn−1))k₀ ≤C∆t, (4.12)

∆t

N

X

n=1

kRⁿk²₋₁ ≤C(∆t)⁴, (4.13) whereRⁿ:=Dtu_h(tn)−∂_tu_h(tn) denotes the difference between of the time derivative and its BDF2-type discretization.

We remark that if u_h ∈ W^3,∞(0, T; [L²(Ω)]^d) and p_h ∈ W^2,∞(0, T;H¹(Ω)), conditions (4.11) and (4.12) can be shown via (generalized) Taylor expansion. (4.13) can be derived from

Z T 0

k∂_tttu_h(τ)k²₋₁+k∂_ttu_h(τ)k²₁+k∂_ttp_h(τ)k²₀dτ ≤C

using Taylor expansion. We refer to [She92] and [She96], where similar proofs are presented.

Assumption 4.2.3 (For nonlinear error due to temporal discretization).

For the nonlinear error due to temporal discretization, we require all conditions from As-sumptions 4.2.1 and4.2.2 as well as in addition

u∈L^∞(0, T; [W^2,2(Ω)]^d); (4.14) in particular, we assumekuk_L^∞_(0,T;W2,2(Ω))≤C withC independent of ν.

For the stabilization terms, we require the following properties. Let the cell-wise constant streamline direction be defined as

w_M = 1

|M|

Z

M

w(x)dx

for allM ∈ M_h. Furthermore, we require thats_u is linear in each argument; in particular, τ_Mⁿ must not depend nonlinearly on the arguments of s_u. Let i_h ≡0 and assume γ :=γ_M for all M ∈ M_h, i.e.,

t_h(u_eⁿ_ht;u_eⁿ_ht,v_h) =γ(∇ ·u_eⁿ_ht,∇ ·v_h).

Note that the above setting of w_M is in agreement with Assumption3.2.5that is needed for the semi-discrete analysis in Section 2.2.3. According to the grad-div parameter choice we discussed in the semi-discrete analysis, the assumption on t_h is reasonable.

Assumption 4.2.4 (For fully discrete error).

In addition to Assumption 4.2.3, let the following conditions be true for somep∈ {1,2}:

u∈L^∞(0, T; [W^ku+1,2(Ω)]^d), p∈W^1,2(0, T;L²(Ω)), u_h∈W^p,2(0, T; [L²(Ω)]^d).

4.2.1. Spatial Discretization of the Continuous Quantities

In order to make use of semi-discrete error estimates foru−u_h, we have to show equivalence of the time-discrete l²-norm and the time-continuous L²-norm. For this, we need some preparatory results. We do not show the proofs here; instead, we refer the reader to [AD15]. First, we state an interpolation result which is an extension of the Bramble-Hilbert Theorem.

Theorem 4.2.5 (Generalized Bramble-Hilbert Theorem).

Let X be a real, separable Hilbert space and m ≥ 1 an integer. Denote by P^m−1(a, b;X) the space of polynomials p: (a, b) ⊂ R → X of maximal order m−1 (with respect to time) and with values in X. Then there exists C > 0 such that for any bounded intervals (a, b) ⊂ (0, T) and any f ∈ H^m(a, b;X), there exists a polynomial q ∈ P^m−1(a, b;X) satisfying q(a) =f(a) and q(b) =f(b) and

kf−qk_Hk(a,b;X)≤C(b−a)^m−k|f|_Hm(a,b;X) ∀k∈N, k≤m. (4.15)

Lemma 4.2.6 (Equivalence of discrete and continuous norms).

Consider the set of points in timeN_T ={0 =t0, . . . , tN =T}, where we assume a constant time step size ∆t =T /N and tn = n∆t, n= 0, . . . , N. Let X be a Banach space. Then there exist constants c, C >0 such that the estimate

c∆t

N

X

i=0

kf(ti)k²_X ≤ kfk²_L2(0,T;X) ≤C∆t

N

X

i=0

kf(t_i)k²_X (4.16) holds true for all functions f : [0, T]→X that are piecewise linear with respect to N_T. We can use the semi-discrete errors as an upper bound of the errors with respect to the time-discrete norms according to the result below. For the proof, we refer to [AD15].

Theorem 4.2.7 (Time-continuous quantities in time-discrete norms). provided that the right-hand sides exist.

Under the conditions of Corollary 3.2.13, for example, we can derive an a priori estimate of the semi-discrete error ξ_u,h in the fully discrete norms. Here, all terms resulting from the non-isothermal coupling are omitted.

Corollary 4.2.8 (Spatial convergence in discrete norms).

If Assumption 4.2.1 holds, the semi-discrete error can be bounded by kξ_u,hk²_l∞(0,T;L²(Ω)) ≤ kξ_u,hk²_L∞(0,T;L²(Ω)) Proof. The right-hand sides in Theorem 4.2.7can be bounded by Corollary3.2.13:

kξ_u,hk²_L∞(0,T;[L²(Ω)]^d)+kξ_u,hk²_L2(0,T;LP S)

withC_G,h(u).1 +C|u|_L∞(0,T;W^1,∞(Ω))+Ch²|u|²_L∞(0,T;W^1,∞(Ω))+Ckuk²_L∞(0,T;L^∞(Ω)).

4.2.2. Temporal Discretization of the Space-Discrete Quantities

The time-discrete error u_h−u_eht is handled by introducing a solutionw_eht of an auxiliary linear problem as

ku_h−u_ehtk ≤ ku_h−w_ehtk+kw_eht−u_ehtk (4.17) with a suitable norm, where (w_eht,wht, rht)∈(Vh)^N×(V^div_h )^N×(Q_h)^N solves the problem:

Find w_eⁿ_ht∈V_h,wⁿ_ht∈V^div_h ,r_htⁿ ∈Q_h such that for allv_h ∈V_h,y_h ∈Y_h,q_h ∈Q_h 3w_eⁿ_ht−4wⁿ⁻¹_ht +wⁿ⁻²_ht

2∆t ,vh

!

+ν(∇w_eⁿ_ht,∇v_h) +γ(∇ ·w_eⁿ_ht,∇ ·vh) (4.18)

= (f_u(t_n),v_h)−(∇rⁿ⁻¹_ht ,v_h)−c_u(u_h(t_n);u_h(t_n),v_h)−s_u(u_h(t_n),u_h(t_n),u_h(t_n),v_h), weⁿ_ht|_∂Ω= 0,

3wⁿ_ht−3w_eⁿ_ht

2∆t +∇(rⁿ_ht−rⁿ⁻¹_ht ),y_h

= 0, (4.19)

(∇ ·wⁿ_ht, q_h) = 0, wⁿ_ht|_∂Ω= 0.

u_h−w_eht is called linear error and is estimated in Lemma 4.2.14, w_eht−u_eht denotes the so-called nonlinear error, see Lemma 4.2.16. Consistency estimates in time are obtained by combining the results of both auxiliary problems.

Definition 4.2.9 (Error splitting).

We denote the errors due to temporal discretization

ξⁿ_u :=u_h(t_n)−uⁿ_ht, ξ^en_u :=u_h(t_n)−u_eⁿ_ht, ξ_pⁿ:=p_h(t_n)−pⁿ_ht.

For the linear problem, we define the propagation operator δtaⁿ := aⁿ−aⁿ⁻¹ and the errors

ηⁿ_u :=u_h(t_n)−wⁿ_ht, η_eⁿ_u :=u_h(t_n)−w_eⁿ_ht, ηⁿ_p :=p_h(t_n)−rⁿ_ht. We introduce the nonlinear errors

eⁿ_u :=wⁿ_ht−uⁿ_ht, e_eⁿ_u :=w_eⁿ_ht−u_eⁿ_ht, eⁿ_p :=rⁿ_ht−pⁿ_ht.

Note that it holdseξⁿ_u =η_eⁿ_u+_eeⁿ_u,ξⁿ_u =ηⁿ_u+eⁿ_u and ξ_pⁿ=ηⁿ_p +eⁿ_p.

For convergence rates of the desired order, estimates of the initial errors are needed. For this, we cite [AD15].

Lemma 4.2.10 (Initialization).

The initial errors due to temporal discretization can be bounded by

keξ^m_uk²₀+ν(∆t)²keξ^m_uk²₁+ (∆t)²k∇ξ_p^mk²₀ ≤C(∆t)⁴ ∀m∈ {1,2}, provided the time step size satisfies

C∆t

ν³ +C∆t ν max

M∈M_h

(τ_M^m h^d_M

)

+C∆t ν max

M∈M_h

(τ_M^m h^d_M

)²

≤1 ∀m∈ {1,2}.

The initial linear errors can be bounded by

kη_e^m_uk²₀+ν(∆t)²kη_e^m_uk²₁+ (∆t)²k∇η_p^mk₀ ≤C(∆t)⁴ ∀m∈ {1,2}.

Proof. For the first time step, one takes advantage of the fact that the error at t₀ = 0 vanishes. For the next time steps, one uses the same techniques as for estimating the linear and nonlinear errors. See [AD15] for details.

The proofs for the linear error are a modification of the work in [GS04], where we work on the space-discrete level, add grad-div stabilization, handle the pressure term in a different way and do not consider the rotational correction.

Lemma 4.2.11 (Intermediate linear velocity error).

Let ∆t < ¹₂ and Assumption 4.2.2 be valid. For all 1≤m≤N, it holds

kη^m_u −η_e^m_uk²₀ ≤e^CG,lin(∆t)⁴ (4.20) withC_G,lin ∼T(1−2∆t)⁻¹.

Proof. The error equation due to the difference between the Navier-Stokes momentum equation and the advection-diffusion step (4.18) reads

3η_eⁿ_u−4ηⁿ⁻¹_u +ηⁿ⁻²_u

2∆t ,vh

!

+ν(∇η_eⁿ_u,∇v_h) +γ(∇ ·η_eⁿ,∇ ·vh)

= (Rⁿ,v_h)−(∇(p_h(t_n)−r_htⁿ⁻¹),v_h) ∀v_h∈V_h (4.21)

with Rⁿ :=D_tu_h(t_n)−∂_tu_h(t_n) and due to the projection step (4.19), we have the error equation

3ηⁿ_u−3η_eⁿ_u

2∆t +∇(rⁿ_ht−rⁿ⁻¹_ht ),y_h

= 0 ∀y_h∈Yh. (4.22) We consider the difference between two consecutive time steps of the advection-diffusion er-ror equation (4.21) and call this the propagation erer-ror equation for the advection-diffusion step. Since the propagation operatorδ_tis linear, we establish:

3δ_tη_eⁿ_u−4δ_tηⁿ⁻¹_u +δ_tηⁿ⁻²_u

2∆t ,vh

!

+ν(∇δ_tη_eⁿ_u,∇v_h) +γ(∇ ·δtηeⁿ_u,∇ ·vh)

= (δ_tRⁿ,v_h)−(∇δ_t(p_h(t_n)−r_tⁿ⁻¹),v_h) ∀v_h∈V_h. (4.23) The propagation error for the projection error equation (4.22) is similarly defined by

0 =

3δ_tηⁿ_u−3δ_tη_eⁿ_u 2∆t ,y_h

−(∇δ_t(rⁿ_ht−rⁿ⁻¹_ht ),y_h)

=

3δ_tηⁿ_u−3δ_tη_eⁿ_u 2∆t ,y_h

+ (∇δ_t(ηⁿ_p −ηⁿ⁻¹_p ),y_h)

−(∇δ_t(p_h(t_n)−p_h(tn−1)),y_h) ∀y_h∈Y_h. (4.24) Testing (4.23) withv_h= 4∆tδtηeⁿ_u gives

23δtηeⁿ_u−4δtηⁿ⁻¹_u +δtηⁿ⁻²_u , δtηeⁿ_u+ 4∆tν(∇δ_tη_eⁿ_u,∇δ_tη_eⁿ_u) + 4∆tγ(∇ ·δtηeⁿ,∇ ·δtηeⁿ_u)

= 4∆t(δ_tRⁿ, δ_tη_eⁿ_u)

−4∆t(∇δ_t(p_h(tn−1)−r_htⁿ⁻¹), δ_tη_eⁿ_u)−4∆t(∇δ_t(p_h(tn−1)−p_h(t_n)), δ_tη_eⁿ_u).

Now, we test the propagation error in the projection step (4.24) with y_h = ∇δ_tηⁿ⁻¹_p =

∇δ_t(p_h(tn−1)−rⁿ⁻¹_ht ) and get after integration by parts for the first term

− 3

2∆tδ_tη_eⁿ_u,∇δ_t(p_h(tn−1)−rⁿ⁻¹_ht )

=−(∇δ_t(η_pⁿ−η_pⁿ⁻¹),∇δ_tηⁿ⁻¹_p ) + (∇δ_t(p_h(t_n)−p_h(tn−1)),∇δ_tη_pⁿ⁻¹).

Combining these and using that δtηⁿ_u = PHδtηeⁿ_u, therefore kδ_tηⁿ_uk ≤ kδ_tη_eⁿ_uk, yield (in a similar way as in the proof of Theorem 4.1.2)

kδ_tη_eⁿ_uk²₀+ 3kδ_tηⁿ_u−δtηeⁿ_uk²₀+k2δ_tηⁿ_u−δtηⁿ⁻¹_u k²₀ +kδ_tttηⁿ_uk²₀− kδ_tη_eⁿ⁻¹_u k²₀− k2δ_tηⁿ⁻¹_u −δ_tηⁿ⁻²_u k²₀ + 4∆tνk∇δ_tη_eⁿ_uk²₀+ 4∆tγk∇ ·δtηeⁿk²₀+4

3(∆t)²k∇δ_tη_pⁿk²₀

≤ 4

3(∆t)²k∇δ_tηⁿ_p − ∇δ_tη_pⁿ⁻¹k²₀+ 4

3(∆t)²k∇δ_tη_pⁿ⁻¹k²₀ + 4∆t(δtRⁿ, δtηeⁿ_u)−4∆t(∇δ_t(ph(tn−1)−ph(tn)), δtηeⁿ_u) +8

3(∆t)²(∇δ_t(p_h(t_n)−p_h(tn−1)),∇δ_tη_pⁿ⁻¹). (4.25) In order to handle the first term on the right-hand side, the projection propagation error equation (4.24) is tested with y_h =∇δ_t(ηⁿ_p −η_pⁿ⁻¹):

2∆t

3 k∇δ_t(η_pⁿ−η_pⁿ⁻¹)k²₀ ≤ kδ_tηⁿ_u−δ_tη_eⁿ_uk₀k∇δ_t(η_pⁿ−ηⁿ⁻¹_p )k₀ +2∆t

3 (∇δ_t(p_h(t_n)−p_h(tn−1)),∇δ_t(η_pⁿ−η_pⁿ⁻¹))

≤ 3

4∆tkδ_tηⁿ_u−δ_tη_eⁿ_uk²₀+ ∆t

3 k∇δ_t(ηⁿ_p −η_pⁿ⁻¹)k²₀ +2∆t

3 (∇δ_ttp_h(t_n),∇δ_t(η_pⁿ−η_pⁿ⁻¹))

due to Young’s inequality. Therefore, after multiplication with 4∆t, 4

3(∆t)²k∇δ_t(ηⁿ_p −η_pⁿ⁻¹)k²₀≤3kδ_tηⁿ_u−δ_tη_eⁿ_uk²₀+8

3(∆t)²(∇δ_ttp_h(t_n),∇δ_t(η_pⁿ−η_pⁿ⁻¹)).

We insert this into (4.25) and use Young’s inequality:

kδ_tη_eⁿ_uk²₀+k2δ_tηⁿ_u−δ_tηⁿ⁻¹_u k²₀

+kδ_tttηⁿ_uk²₀− kδ_tη_eⁿ⁻¹_u k²₀− k2δ_tηⁿ⁻¹_u −δtηⁿ⁻²_u k²₀ + 4∆tνk∇δ_tη_eⁿ_uk²₀+ 4∆tγk∇ ·δ_tη_eⁿk²₀+4

3(∆t)²k∇δ_tηⁿ_pk²₀

≤ 4

3(∆t)²k∇δ_tη_pⁿ⁻¹k²₀+ 4∆t(δtRⁿ, δtηeⁿ_u) + 4∆t(∇δ_ttp_h(tn), δtηeⁿ_u) +8

3(∆t)²(∇δ_ttph(tn),∇δ_tη_pⁿ)

≤ 4

3(∆t)²k∇δ_tη_pⁿ⁻¹k²₀+ 4∆tkδ_tRⁿk²₀+ ∆tkδ_tη_eⁿ_uk²₀ + 4∆tk∇δ_ttph(tn)k²₀+ ∆tkδ_tη_eⁿ_uk²₀+8

3∆tk∇δ_ttph(tn)k²₀+2

3(∆t)³k∇δ_tηⁿ_pk²₀. Summing up fromn= 3 to m≤N gives

kδ_tη_e^m_uk²₀+k2δ_tη^m_u −δtη^m−1_u k²₀+4

3(∆t)²k∇δ_tη_p^mk²₀ +

m

X

n=3

kδ_tttηⁿ_uk²₀+ 4∆tνk∇δ_tη_eⁿ_uk²₀+ 4∆tγk∇ ·δtηeⁿk²₀

≤ kδ_tη_e²_uk²₀+k2δ_tη²_u−δtη¹_uk²₀+4

3(∆t)²k∇δ_tη_p²k²₀

+ ∆t This holds since the first three terms on the right-hand side denote initial errors (that can be bounded by Lemma 4.2.10). Moreover, Assumption 4.2.2ensures the estimate

kδ_tRⁿk²₀+k∇δ_ttp_h(t_n)k²₀ ≤C(∆t)⁴

with C independent ofn.C_G,lin ∼T(1−2∆t)⁻¹ denotes a Gronwall constant.

The intermediate result follows from the use of the projection error equation (4.22), from (4.26) and Assumption4.2.2:

Definition 4.2.12 (Grad-div stabilized inverse Stokes operator).

We define the grad-div stabilized space-discrete inverse Stokes operator S: V_h → V_h as the solution (Sv_h, r_h)∈V_h×Q_h of the problem

ν(∇Sv_h,∇w_h)−(rh,∇ ·wh) +γ(∇ ·Svh,∇ ·wh) = (vh,wh) ∀w_h∈Vh, (∇ ·Sv_h, q_h) = 0 ∀q_h∈Q_h,

Sv_h|_∂Ω = 0. (4.27)

Further, let |v_h|²_∗ := (vh, Svh) for any vh∈Vh.

Lemma 4.2.13 (Properties of the inverse Stokes operator).

Let ε >0 be arbitrary.S has the following properties:

k∇Sv_hk₀≤ 1 νkv_hk₀,

|v_h|²_∗= (v_h, Sv_h) =ν(∇Sv_h,∇v_h) +γ(∇ ·Sv_h,∇ ·v_h)

≥ 1−

2ν+γ ν

2 ε 4

!

kv_hk²₀− 1

εkv_h−v^∗_hk²₀ ∀v^∗_h∈V^div_h , (4.28)

|v_h|²_∗≤ 1 νkv_hk²₀.

Proof. By testing (4.27) symmetrically with w_h = v_h, we derive an estimate for the solution in theW^1,2-semi-norm

νk∇Sv_hk²₀+γk∇ ·Svhk²₀= (vh, Svh)−(rh,∇ ·Svh) = (vh, Svh)≤ kv_hk−1k∇Sv_hk₀

⇒ k∇Sv_hk₀≤ 1

νkv_hk₋₁≤ 1 νkv_hk₀

due to the fact that kv_hk₋₁ ≤ kv_hk₀. Thus, the upper bound for the semi-norm induced by the inverse Stokes operator can be derived as

|v_h|²_∗= (v_h, Sv_h)≤ kv_hk₋₁k∇Sv_hk₀ ≤ 1

νkv_hk²₋₁ ≤ 1 νkv_hk²₀.

Next, we are interested in a lower bound. If we add grad-div stabilization in [Gue99] (as in [AD15]), we can getk∇r_hk₀ ≤C 2 +^γ_νkv_hk₀ and calculate with this

|v_h|²_∗=ν(∇Sv_h,∇v_h) +γ(∇ ·Sv_h,∇ ·v_h) =kv_hk²₀+ (r_h,∇ ·v_h)

=kv_hk²₀−(∇r_h,v_h−v^∗_h)≥ kv_hk²₀− k∇r_hk₀kv_h−v^∗_hk₀

≥ 1−

2ν+γ ν

2 ε 4

!

kv_hk²₀− 1

εkv_h−v^∗_hk²₀ for all ε >0 and arbitraryv^∗_h ∈V^div_h .

Lemma 4.2.14 (Time convergence of the linear error).

If ∆t < ¹₂ and Assumption 4.2.2 are valid, it holds for all 1≤m≤N kη_e^m_uk²₀ ≤ C

ν²e^CG,lin(∆t)⁴, νk∇η_e^m_uk²₀+γk∇ ·η_e^m_uk²₀ ≤e^CG,lin(∆t)² withCG,lin ∼T(1−2∆t)⁻¹.

Proof. We test the advection-diffusion error equation (4.21) with the inverse Stokes oper-ator applied to 4∆tη_eⁿ_u and eliminate the terms containing ηⁿ⁻¹_u ,ηⁿ⁻²_u via the projection error equation (4.22) tested withSη_eⁿ_u. UsingSη_eⁿ_u ∈V^div_h gives:

2(3η_eⁿ_u−4η_eⁿ⁻¹_u +η_eⁿ⁻²_u , Sη_eⁿ_u) + 4∆tν(∇η_eⁿ_u,∇Sη_eⁿ_u) + 4∆tγ(∇ ·η_eⁿ_u,∇ ·Sη_eⁿ_u)

= 4∆t(Rⁿ, Sη_eⁿ_u) + 4∆t(∇−p_h(tn) +7

3rⁿ⁻¹−5

3rⁿ⁻²+1

3rⁿ⁻³, Sη_eⁿ_u) = 4∆t(Rⁿ, Sη_eⁿ_u).

For the first term, we use the identity

2(3η_eⁿ_u−4η_eⁿ⁻¹_u +η_eⁿ⁻²_u , Sη_eⁿ_u) =|η_eⁿ_u|²_∗+|2η_eⁿ_u−η_eⁿ⁻¹_u |²_∗+|δ_ttη_eⁿ_u|²_∗

− |η_eⁿ⁻¹_u |²_∗− |2η_eⁿ⁻¹_u −η_eⁿ⁻²_u |²_∗. This can be understood via Definition4.2.12 ofS: For vh,wh ∈Vh, it holds

(v_h, Sw_h) =ν(∇Sv_h,∇Sw_h) +γ(∇ ·Sv_h,∇ ·Sw_h) = (w_h, Sv_h) and thus with the definition|v_h|_∗ = (v_h, Sv_h) from Lemma 4.2.13

2(3η_eⁿ_u−4η_eⁿ⁻¹_u +η_eⁿ⁻²_u , Sη_eⁿ_u)

= (6η_eⁿ_u−4η_eⁿ⁻¹_u +η_eⁿ⁻²_u , Sη_eⁿ_u)−4(η_eⁿ_u, Sη_eⁿ⁻¹_u ) + (η_eⁿ_u, Sη_eⁿ⁻²_u )

= (η_eⁿ_u, Sη_eⁿ_u) + (2η_eⁿ_u−η_eⁿ⁻¹_u ,2Sη_eⁿ_u−Sη_eⁿ⁻¹_u ) + (η_eⁿ_u−2η_eⁿ⁻¹_u +η_eⁿ⁻²_u , Sη_eⁿ_u−2Sη_eⁿ⁻¹_u +Sη_eⁿ⁻²_u )

−(η_eⁿ⁻¹_u , Sη_eⁿ⁻¹_u )−(2η_eⁿ⁻¹_u −η_eⁿ⁻²_u ,2Sη_eⁿ⁻¹_u −Sη_eⁿ⁻²_u )

=|η_eⁿ_u|²_∗+|2η_eⁿ_u−η_eⁿ⁻¹_u |²_∗+|δ_ttη_eⁿ_u|²_∗− |η_eⁿ⁻¹_u |²_∗− |2η_eⁿ⁻¹_u −η_eⁿ⁻²_u |²_∗. With this, we get the following equation

|η_eⁿ_u|²_∗+|2η_eⁿ_u−η_eⁿ⁻¹_u |²_∗+|δ_ttη_eⁿ_u|²_∗+ 4∆tν(∇η_eⁿ_u,∇Sη_eⁿ_u) + 4∆tγ(∇ ·η_eⁿ_u,∇ ·Sη_eⁿ_u)

= 4∆t(Rⁿ, Sη_eⁿ_u) +|η_eⁿ⁻¹_u |²_∗+|2η_eⁿ⁻¹_u −η_eⁿ⁻²_u |²_∗.

Due to Lemma 4.2.13, the consistency error can be bounded as 4∆t(Rⁿ, Sη_eⁿ_u)≤ 4∆t

ν kRⁿk²₋₁+ ∆tνkSη_eⁿ_uk²₁ ≤ 4∆t

ν kRⁿk²₋₁+ ∆tkη_eⁿ_uk²₀.

Using (4.28) withε= 2_2ν+γ^{ν 2}, the diffusive term and the grad-div stabilization can be estimated by

4∆tν(∇η_eⁿ_u,∇Sη_eⁿ_u) + 4∆tγ(∇ ·η_eⁿ_u,∇ ·Sη_eⁿ_u)≥2∆tkη_eⁿ_uk²₀−c∆tkη_eⁿ_u−ηⁿ_uk²₀,

wherec= 2^2ν+γ_ν ² ≤C(1 +^ν_γ)². Combining these estimates and summing up fromn= 3 because of Lemma 4.2.11, Assumption 4.2.2 and initial error estimates: Lemma 4.2.13 implies that| · |_∗ can be bounded from above byk · k₀; the combination with Lemma4.2.10 gives the desired initial error bounds. In particular, we derive

kη_euk²_l2(0,T;L²(Ω))= ∆t

For the estimates for theW^1,2-semi-norm, we again use calculations from Lemma4.2.11.

Due to (4.26), we have

m

X

n=3

(νk∇δ_tη_eⁿ_uk²₀+γk∇ ·δ_tη_eⁿ_uk²₀)≤e^CG,lin(∆t)³ and therefore via triangle inequality and because ofN =T /∆t:

√νk∇η_e^m_uk₀+√

Lemma 4.2.15 (Stability of ∇u_h).

Set h:= max_M∈MhhM. Let Assumption 4.2.3 be valid and ∆t <1. Then we have ku_hk²_l∞(0,T;W^1,2(Ω)) ≤C+Ce^CG,h(u)h^2ku+h^2kp+2

ν∆t . (4.30)

If we additionally assume (h^2ku+h^2kp+2).e^−CG,h(u)ν∆t, it holds ku_hk²_l∞(0,T;W^1,2(Ω)) ≤C.

Proof. Thanks to Assumption4.2.3, we can apply Corollary4.2.8and establish ku−u_hk²_l2(0,T;W^1,2(Ω))≤Cku−u_hk²_L2(0,T;W^1,2(Ω))+ (∆t)^2l

≤ Ce^CG,h(u)

ν (h^2ku+h^2kp+2) +C(∆t)^2l, ku−uhk²_l∞(0,T;W^1,2(Ω))≤ C

∆tku−uhk²_l2(0,T;W^1,2(Ω))

≤Ce^CG,h(u)h^2ku+h^2kp+2

ν∆t +C(∆t)^2l−1. With this andu∈L^∞(0, T; [W^2,2(Ω)]^d) due to Assumption4.2.3, we derive

ku_hk²_l∞(0,T;W^1,2(Ω))≤Ckuk²_l∞(0,T;W^1,2(Ω))+Cku−uhk²_l∞(0,T;W^1,2(Ω))

≤C+Ce^CG,h(u)h^2ku+h^2kp+2

ν∆t +C(∆t)^2l−1 ≤C because (h^2ku+h^2kp+2).e^−CG,h(u)ν∆tand 2l≥1.

Now, let us turn our attention to the nonlinear error. The proof combines estimation strate-gies from [She96] with the handling of the discrete BDF2-type time derivative by [GS04] as well as adds grad-div and LPS stabilization and the thorough consideration ofν dependen-cies. Extra technical challenges matter since we do not requireu_h ∈L^∞(0, T; [W^2,2(Ω)]^d).

In addition to the previous lemma, we make use of the insights from the linear error estimate (Lemma 4.2.14).

Lemma 4.2.16 (Time convergence of the nonlinear error).

Denote

K_t,nl:=C∆tku_hk²_l∞(0,T;W^1,2(Ω))

ku_hk²_l∞(0,T;W^1,2(Ω))

ν³ + max

1≤n≤N max

M∈M_h

(τ_Mⁿ h^d_M

)

+

ku_hk²_l∞(0,T;L²(Ω))

ν max

1≤n≤N max

M∈M_h

(τ_Mⁿ h^d_M

)²! .

Under the conditions of Assumption 4.2.3, it holds with C_G,t ∼ T(1−K_t,nl)⁻¹, for all

Proof. Subtracting the advection-diffusion equations forw_eⁿ_htandu_eⁿ_htfrom each other gives 3_eeⁿ_u−4eⁿ⁻¹_u +eⁿ⁻²_u The advection-diffusion error equation (4.31) is tested symmetrically with 4∆t_eeⁿ_u and the resulting pressure term 4∆t(∇eⁿ⁻¹_p ,e_eⁿ_u) is handled via (4.32) tested with ⁸₃(∆t)²∇eⁿ⁻¹_p .

Equation (4.32) tested with ∇(eⁿ_p −eⁿ⁻¹_p ) gives 4

3(∆t)²k∇(eⁿ_p −eⁿ⁻¹_p )k²₀ ≤3keⁿ_u−_eeⁿ_uk²₀.

Due toe_eⁿ_u+η_eⁿ_u =u_h(tn)−u_eⁿ_ht, we calculate for the convective term using skew-symmetry (Qn,eeⁿ_u) =cu(η_eⁿ_u+_eeⁿ_u;uh(tn),eeⁿ_u) +cu(u_eⁿ_ht;η_eⁿ_u+e_eⁿ_u,eeⁿ_u)

=cu(η_eⁿ_u+_eeⁿ_u;uh(tn),_eeⁿ_u) +cu(uh(tn);η_eⁿ_u,_eeⁿ_u)

−cu(η_eⁿ_u;η_eⁿ_u,e_eⁿ_u)−cu(_eeⁿ_u;η_eⁿ_u,_eeⁿ_u)

and make use of Lemma A.3.7as well as the convergence results for the linear problem:

cu(e_eⁿ_u;u_h(tn),_eeⁿ_u)≤Ck_eeⁿ_uk^1/2₀ ku_h(tn)k₁k_eeⁿ_uk^3/2₁

≤ ν

32k∇_eeⁿ_uk²₀+ C

ν³ku_h(tn)k⁴₁ke_eⁿ_uk²₀, c_u(u_h(t_n);η_eⁿ_u,e_eⁿ_u) +c_u(η_eⁿ_u;u_h(t_n),e_eⁿ_u)

=c_u(u(t_n);η_eⁿ_u,e_eⁿ_u)−c_u(u(t_n)−u_h(t_n);η_eⁿ_u,_eeⁿ_u) +c_u(η_eⁿ_u;u(t_n),e_eⁿ_u)−c_u(η_eⁿ_u;u(t_n)−u_h(t_n),_eeⁿ_u)

≤Cku(t_n)k₂kη_eⁿ_uk₀ke_eⁿ_uk₁+Cku(t_n)−u_h(t_n)k₁kη_eⁿ_uk₁ke_eⁿ_uk₁

≤ ν

32k∇_eeⁿ_uk²₀+Cku(t_n)k²₂

ν kη_eⁿ_uk²₀+C

νkη_eⁿ_uk²₁ku(t_n)−uh(tn)k²₁, c_u(η_eⁿ_u;η_eⁿ_u,_eeⁿ_u)≤Ckη_eⁿ_uk²₁k_eeⁿ_uk₁≤ ν

32k∇e_eⁿ_uk²₀+C νkη_eⁿ_uk⁴₁, c_u(e_eⁿ_u;η_eⁿ_u,e_eⁿ_u)≤Ckη_eⁿ_uk₁ke_eⁿ_uk²₁.

From Lemma 4.2.14, we have that √

νkη_eⁿ_uk_l∞(0,T;W^1,2(Ω)) ≤exp(C_G,lin)∆t. Provided that C∆t≤ν^3/2/8, we can estimate the last term

c_u(_eeⁿ_u;η_eⁿ_u,_eeⁿ_u)≤Ckη_eⁿ_uk₁ke_eⁿ_uk²₁≤ ν

32k∇_eeⁿ_uk²₀. Taking kη_euk_l^∞_(0,T;L2(Ω)) ≤ ^exp(C_ν^G,lin)(∆t)² and √

νkη_eⁿ_uk_l^∞_(0,T;W1,2(Ω)) ≤ exp(CG,lin)∆t from Lemma4.2.14into account, we obtain in combination (with exp(C_G,lin) hidden inC)

(Qⁿ,_eeⁿ_u)≤ ν

8k∇_eeⁿ_uk²₀+ Cku_h(t_n)k⁴₁ ν³ ke_eⁿ_uk²₀ +Cku(t_n)k²₂+C

ν³ (∆t)⁴+ C

ν²ku(t_n)−u_h(t_n)k²₁(∆t)². (4.34)

Recall thats_u is supposed to be linear in each argument due to Assumption4.2.3. For the

According to LemmaA.3.8, Cauchy-Schwarz and Young’s inequality, the termsI5+I6 can be handled as

Summarizing these terms yields

Due to the estimates for the initial errors of the time-discretized problem and the linear auxiliary problem (see Lemma 4.2.10), the initial errors of the nonlinear problem also converge suitably

ke_e¹_uk²₀+k2e¹_u−e⁰_uk²₀+4

3(∆t)²k∇e¹_pk²₀≤C(∆t)⁴.

In addition, we consult Theorem 4.2.7and Corollary4.2.8in order to establish ku−u_hk²_l2(0,T;W^1,2(Ω))≤Cku(t_n)−u_h(t_n)k²_L2(0,T;W^1,2(Ω))+C(∆t)^2l

≤ 1

νku(t_n)−u_h(t_n)k²_L2(0,T;LP S)+C(∆t)^2l ≤ C

νe^CG,h(u)(h^2ku+h^2kp+2) +C(∆t)^2l.

Provided that (h^2ku+h^2kp+2).e^−CG,h(u)ν∆tand

application of the discrete Gronwall LemmaA.3.6forke_e^m_uk²₀ in (4.35) yields ke_e^m_uk²₀+ (∆t)²k∇e^m_p k²₀+

Now, we are prepared to state an estimate for the total error due to temporal discretization.

For this purpose, we combine Lemmas4.2.14and 4.2.16.

Theorem 4.2.17 (Time convergence of the semi-discrete quantities).

Under the assumptions of Lemmas 4.2.14 and 4.2.16, it holds keξ_uk²_l∞(0,T;L²(Ω))≤ C

with the abbreviation

Proof. For the linear error, Lemma4.2.14yields if ∆t < ¹₂ kη_e^m_uk²₀ ≤ C In combination, we establish the claim.