A Hybrid High-Order method for the incompressible Navier--Stokes problem robust for large irrotational body forces

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A Hybrid High-Order method for the incompressible Navier–Stokes problem robust for large irrotational

body forces

Daniel Castanon Quiroz, Daniele Antonio Di Pietro

To cite this version:

Daniel Castanon Quiroz, Daniele Antonio Di Pietro. A Hybrid High-Order method for the incom-

pressible Navier–Stokes problem robust for large irrotational body forces. 2019. �hal-02151236�

A Hybrid High-Order method for the incompressible

Navier-Stokes problem robust for large irrotational body forces

Daniel Castanon Quiroz ^∗1 and Daniele A. Di Pietro ^†1

1

IMAG, Univ Montpellier, CNRS, Montpellier, France

June 7, 2019

Abstract

We develop a novel Hybrid High-Order method for the incompressible Navier–Stokes problem robust for large irrotational body forces. The key ingredients of the method are discrete versions of the body force and convective terms in the momentum equation formulated in terms of a globally divergence-free velocity reconstruction. Denoting by λ the L

²

-norm of the irrotational part of the body force, the method is designed so as to mimick two key properties of the continuous problem at the discrete level, namely the invariance of the velocity with respect to λ and the non- dissipativity of the convective term. The convergence analysis shows that, when polynomials of total degree ≤ k are used as discrete unknowns, the energy norm of the error converges as h

^k+1

(with h denoting, as usual, the meshsize), and the error estimate on the velocity is uniform in λ and independent of the pressure. The performance of the method is illustrated by a complete panel of numerical tests, including comparisons that highlight the benefits with respect to more standard HHO formulations.

Key words: Hybrid High-Order methods, incompressible Navier–Stokes equations, robust a priori error estimates

MSC 2010: 65N08, 65N30, 65N12, 35Q30, 76D05

1 Introduction

Let Ω ⊂ R

³

denote an open, bounded, simply connected polyhedral domain with Lipschitz boundary

∂ Ω . Let ν > 0 be a real number representing the kinematic viscosity of the fluid, and let f ∈ L

²

(Ω)

³

be a given vector field representing a body force. Setting U B H

¹

0

(Ω)

³

and P B

q ∈ L

²

(Ω) :

∫

Ω

q = 0 , we consider the steady incompressible Navier–Stokes problem: Find ( u, p) ∈ U × P such that

νa(u, v) + t(u, u, v) + b(v, p) = `( f , v) ∀ v ∈ U, (1a)

−b( u, q) = 0 ∀ q ∈ L

²

(Ω), (1b)

with bilinear forms a : U × U → R, b : U × L

²

(Ω) → R, and ` : L

²

(Ω)

^d

× U → R defined by a(w, v) B

∫

Ω

∇w : ∇v, b(v, q) B −

∫

Ω

(∇ · v)q, `( f , v) B

∫

Ω

f · v, (2)

∗

[email protected]

†

[email protected]

and trilinear form t : U × U × U → R such that t(w, v, z) B

∫

Ω

(∇×w) × v · z. (3)

Above, ∇· and ∇× denote, respectively, the divergence and curl operators, while × is the cross product of two vectors. The convective term in (3) is expressed in rotational form, so p is here the so-called Bernoulli pressure, which is related to the kinematic pressure p

_kin

by the equation p = p

_kin

+

¹₂

| u|

²

.

The domain Ω being simply connected, its first Betti number is zero, and we have the following (Helmholtz–)Hodge decomposition of the body force (see, e.g., [2, Section 4.3]):

f = g + λ∇ψ, (4)

where g is the curl of an element of H

0

( curl; Ω) B

v ∈ H( curl; Ω) : γ

_τ

v = 0 with γ

_τ

denoting the tangent trace operator on ∂Ω , ψ ∈ H

¹

(Ω) is such that k∇ψk

_L2(Ω)³

= 1, and λ ∈ R

⁺

. The goal of this paper is to design a Hybrid High-Order (HHO) discretization method for problem (1) robust with respect to large irrotational body forces, that is, for which velocity error estimates uniform in λ and independent of the pressure can be established. The robustness property should additionally be obtained without relying on the Hodge decomposition (4) of f , which is typically not available and whose numerical approximation may be computationally expensive to obtain.

The problem considered here is related to recent works pointing out the relevance of restoring at the discrete level the L

²

-orthogonality between irrotational and discretely divergence-free vector fields [30]; see also the bibliographic section therein for previous references on this subject. A lack of this orthogonality property may indeed result in poor approximations of the velocity field, whose error estimate has an adverse dependence on the pressure. In [30], the author proposes a modification of the original Crouzeix–Raviart scheme [13] where the test function in the right-hand side is replaced by an interpolate in the lowest-order Raviart–Thomas–Nédélec space [32, 34]. An extension of these ideas to arbitrary order HHO approximations of the Stokes problem is proposed in [16], where the authors derive error estimates for the velocity that are independent of the pressure and uniform in the kinematic viscosity. These ideas are further developed in [29], where the notion of discrete Helmholtz projector is introduced to achieve a similar goal in the context of classical Finite Element discretisations of the Stokes and Navier–Stokes problems; see also the recent paper [1].

In this work we extend the construction of [16] to the fully nonlinear Navier–Stokes problem by introducing a novel HHO discretisation of the convective trilinear form based on an H ( div; Ω) velocity reconstruction in the Raviart–Thomas–Nédélec space obtained working element by element.

We prove, for the proposed method, novel error estimates for the velocity that are robust with respect to large irrotational body forces in the sense made precise above, that is, they are uniform in λ and do not depend on the pressure. This result is achieved by mimicking at the discrete level two key properties of the continuous problem, namely the invariance of the velocity with respect to λ and the non-dissipativity of the convective term, and leveraging an a priori bound on the velocity uniform in λ . Specifically, we show that, when polynomials of total degree ≤ k at mesh elements and faces are used as velocity unknowns and polynomials of degree ≤ k inside elements are used as pressure unknowns, the energy norm of the velocity error converges as h

^k+1

(with h denoting, as usual, the meshsize).

A crucial point is that this error estimate is obtained under a data smallness assumption which

only involves g , and therefore persists in the limit λ → ∞ . An intermediate result of independent

interest used in the analysis are novel Sobolev inequalities for the Raviart–Thomas–Nédélec velocity

reconstruction. The proposed ideas apply to other hybrid methods for incompressible flows; see,

e.g., [7, 9, 12, 18, 23, 33, 36] and references therein. Replicating similar properties with virtually

divergence-free numerical methods [4, 5] (see also [3, 10]) may constitute an interesting path for

future research.

The rest of the paper is organised as follows. In Section 2 we discuss three properties of the continuous problem that will play a key role at the discrete level. In Section 3 we introduce the discrete setting, including the local reconstructions at the core of the HHO method. Section 4 contains the formulation of the discrete problem and includes, in particular, the definition and properties of the novel discrete convective trilinear form. The convergence analysis is carried out in Section 5, while a complete panel of numerical tests is provided in Section 6, including a comparison with the standard HHO scheme of [7].

2 Three key remarks

We start by highlighting three properties of the continuous problem that will play a key role in the design and analysis of the method.

2.1 Velocity invariance

Denote by n

Ω

the unit outward normal vector to ∂Ω . Recalling the Hodge decomposition (4), the first property is expressed by the following relation: For all v ∈ U ,

`(g + λ∇ψ, v) = `( g, v) +

∫

Ω

λ∇ψ · v

= `( g, v) −

∫

Ω

λψ (∇ · v) +

∫

∂Ω

λψ (v · n

Ω

) = `(g, v) + b(v, λψ), (5) where we have used the linearity of ` along with its definition in the first equality, an integration by parts together with the strongly enforced wall boundary condition in U to cancel the boundary term in the second equality, and we have concluded using the definition (2) of the bilinear form b . A straightforward consequence of (5) is that the velocity field is unaffected by the irrotational part of the body force, which is why we refer to this property as velocity invariance (with respect to λ ). Mimicking (5) at the discrete level will be crucial to cancel the pressure contribution from the discretization error in Theorem 10 below; see, in particular, (76). This is, in turn, a key point to achieve robustness in λ .

2.2 Non-dissipativity of the convective term

The integration by parts formula in the following proposition generalizes [25, Lemma 6.7].

Proposition 1 (Integration by parts) . Let X denote a simply connected open polyhedral subset of Ω.

For all v, w, z ∈ H

¹

(X )

³

, it holds

∫

X

(∇×w) × v · z =

∫

X

∇wv · z −

∫

X

∇w z · v, (6)

where we recall that, if y = (y

j

)

_1≤j≤3

∈ R

³

, ∇w y = Í

₃

j=1

y

j

∂

j

w.

Writing (6) for X = Ω and recalling the definition (3) of the trilinear form t , we obtain the second key property, namely: For any w, v ∈ H

¹

(Ω)

³

,

t(w, v, v) = ∫

Ω

∇wv · v −

∫

Ω

∇wv · v = 0 , (7)

which expresses the fact that the convective term is non-dissipative , i.e., it does not contribute to the

kinetic energy balance obtained taking v = u in (1a). In Section 4.4, we will leverage (6) with X

successively equal to the mesh elements to derive a reformulation of the convective term that will inspire the design of a consistent and non-dissipative discrete trilinear form. The discrete counterpart of (7), expressed by (46) below, will play a key role both in deriving an a priori bound on the discrete velocity uniform in λ (see Lemma 7) and in proving the error estimate of Theorem 10. For the sake of completeness, before proceeding, we give a proof of Proposition 1.

Proof of Proposition 1. First of all, observe that all the integrals in (6) are well-defined with the assumed regularity (use generalized Hölder inequalities with exponents ( 2 , 4 , 4 ) and the embedding H

¹

(X ) , → L

⁴

(X) ). It then suffices to prove (6) for v, w, z ∈ D(X )

³

, the space of restrictions to X of functions that are of class C

^∞

0

in an open set containing X , and conclude by density.

Recalling the vector identity (∇×w) × v = ∇wv − ∇(w · v) + (∇v)

^|

w (see, e.g., [25, Eq. (3.158)]), we can write

∫

X

(∇×w) × v · z =

∫

X

∇wv · z −

∫

X

∇(w · v) · z +

∫

X

(∇v)

^|

w · z C

∫

X

∇wv · z − T

₂

+ T

₃

. (8) For T

₂

, applying integration by parts we arrive at

T

₂

=

∫

∂T

(w · v)(z · n) −

∫

X

(w · v)(∇ · z). (9)

For T

₃

we can write, denoting by ⊗ the tensor product of vectors in R

³

, T

₃

= Õ

³

i=1

Õ

3

j=1

∫

T

∂

_i

v

_j

w

_j

z

_i

=

∫

T

∇v : w ⊗ z = −

∫

T

∇ · (w ⊗ z) · v +

∫

∂T

( z · n)(w · v), where the conclusion follows from an integration by parts. Using this last equation and the identity

∇ · (w ⊗ z ) = (∇ · z)w + ∇w z , we obtain T

₃

= −

∫

Ω

(∇ · z)(w · v) −

∫

Ω

∇w z · v +

∫

∂Ω

(z · n)(w · v). (10)

Plugging (9) and (10) into (8), we finally get (6).

2.3 Uniform a priori bound on the velocity

The third property is an a priori bound on the continuous velocity uniform in λ . Taking v = u in (1a), q = p + λψ in (1b), and summing the resulting relations, we can write

νa(u, u) + t(u, u, u) + b(u, p) − b(u, p + λψ) = `( f , u) = `( g, u) + b( u, λψ ),

where we have used the Hodge decomposition (4) of f followed by the velocity-invariance property (5) to conclude. Simplifying the terms involving the bilinear form b in the above expression, invoking the non-dissipativity property (7) to write t( u, u, u) = 0, and recalling the definition (2) of the bilinear form a , we can go on writing

ν|u |

²

H¹(Ω)^d

= νa(u, u) = `(g, u) ≤ k g k

_L2(Ω)^d

kuk

_L2(Ω)^d

≤ C

_P

kg k

_L2(Ω)^d

|u |

_H1(Ω)^d

, where C

_P

denotes a Poincaré constant in Ω . Simplifying, we arrive at

|u|

_H1(Ω)³

≤ ν

⁻¹

C

_P

kgk

_L2(Ω)³

. (11)

A crucial point is that, contrary to the classical estimate |u|

_H1(Ω)³

≤ ν

⁻¹

C

_P

k f k

_L2(Ω)³

obtained without

resorting to the velocity-invariance property (5), the a priori bound (11) persists in the limit λ → ∞ .

This bound, along with its discrete counterpart proved in Proposition 7, allows us to establish error

estimates under the smallness assumption (68), which only concerns the solenoidal part g of the body

force (see (4)).

Remark 2 (A priori bound on the pressure) . It cannot be expected, in general, to have an a priori bound on the kpk

_L2(Ω)

uniform in λ .

3 Discrete setting

In this section we establish the discrete setting.

3.1 Mesh

We consider a refined mesh sequence (T

_h

)

_h>0

where, for a given h > 0, T

_h

is a matching simplicial mesh characterized by the scalar h B max

T∈ T_h

h

_T

, with h

_T

denoting the diameter of the element T ∈ T

_h

. The mesh sequence is assumed to be shape-regular in the usual sense; see, e.g., [11, Eq.

(3.1.43)]. We denote by F

_h

the set collecting the faces of T

_h

, partitioned as F

_h

= F

_hⁱ

∪ F

_h^b

, with F

_hⁱ

collecting the interfaces contained in Ω and F

_h^b

the boundary faces contained in ∂ Ω . For any T ∈ T

_h

, we denote by F

_T

the set collecting the faces of F

_h

that lie on the boundary ∂T of T and, for any F ∈ F

_T

, we will refer with n

_{T F}

to the normal unit vector to F pointing outwards with respect to T .

To prevent the proliferation of generic constants we write, whenever possible, a . b in place of a ≤ Cb with C > 0 independent of ν , λ , h and, for local inequalities, also on the mesh element or face. The dependencies of the hidden constant will be further specified when relevant.

3.2 Local and broken spaces and projectors

Let X denote a mesh element or face and, for an integer l ≥ 0, denote by P

^l

(X ) the space spanned by the restrictions to X of polynomials in the space variables of total degree ≤ l . The L

²

-orthogonal projector π

^l_X

: L

¹

(X) → P

^l

(X) is such that, for all v ∈ L

¹

( X) ,

∫

X

(v − π

_X^l

v)w = 0 ∀ w ∈ P

^l

(X). (12)

Vector and matrix versions of the L

²

-orthogonal projector are obtained by applying π

^l_X

component- wise, and will both denoted with the bold symbol π

^l_X

in what follows. Optimal approximation properties for the L

²

-orthogonal projector are proved in [17, Appendix A.2] using the classical theory of [19] (cf. also [8, Chapter 4]). Specifically, let s ∈ { 0 , . . . , l + 1 } and r ∈ [ 1 , + ∞] . Then, it holds with hidden constant only depending on l , s , r , and the mesh regularity parameter: For all T ∈ T

_h

, all v ∈ W

^s,r

(T ) , and all m ∈ { 0 , . . . , s} ,

|v − π

_T^l

v|

_W^m,r_(T)

. h

_T^s−m

|v |

_Ws,r(T)

, (13a) and, if s ≥ 1 and m ≤ s − 1,

h

p1

T

|v − π

_T^l

v |

_Wm,r(F_T)

. h

_T^s−m

|v|

_W^s,r(T)

, (13b) where W

^m,p

(F

_T

) is the space spanned by functions that are in W

^m,p

(F) for all F ∈ F

_T

, endowed with the corresponding broken norm.

At the global level, the space of broken polynomial functions on T

_h

of total degree ≤ l is denoted by P

^l

(T

_h

) , and π

_h^l

is the corresponding L

²

-orthogonal projector such that, for all v ∈ L

¹

(Ω) , (π

_h^l

v )

_|T

B π

_T^k

v

|T

for all T ∈ T

_h

. Broken polynomial spaces form subspaces of the broken Sobolev spaces W

^s,r

(T

_h

) B

v ∈ L

^r

(Ω) : v

|T

∈ W

^s,r

(T ) ∀ T ∈ T

_h

, which will be used to express the regularity

requirements in consistency estimates. We additionally set, as usual, H

^s

(T

_h

) B W

^s,2

(T

_h

) .

3.3 Discrete spaces and norms

Let a polynomial degree k ≥ 0 be fixed. We define the following global space of discrete velocity unknowns:

U

^k_h

B

v

_h

= ((v

_T

)

_{T∈ Th}

, (v

_F

)

_{F∈ Fh}

) : v

_T

∈ P

^k

(T)

³

for all T ∈ T

_h

and v

_F

∈ P

^k

(F)

³

for all F ∈ F

_h

. For all v

_h

∈ U

^k_h

, we denote by v

_h

∈ P

^k_h

(T

_h

)

³

the vector-valued broken polynomial function obtained patching element-based unkowns, that is

(v

_h

)

_|T

B v

_T

∀ T ∈ T

_h

.

The restrictions of U

_h^k

and v

_h

∈ U

^k_h

to a generic mesh element T ∈ T

_h

are respectively denoted by U

_T^k

and v

_T

= (v

_T

, (v

_F

)

_{F∈ FT}

) . The vector of discrete variables corresponding to a smooth function over Ω is obtained via the global interpolation operator I

^k_h

: H

¹

(Ω)

³

→ U

^k_h

such that, for all v ∈ H

¹

(Ω)

³

,

I

^k_h

v B ((π

_T^k

v

|T

)

_T∈ T_h

, (π

^k_F

v

|F

)

_F∈ F_h

).

Its restriction to a generic mesh element T ∈ T

_h

is I

^k_T

: H

¹

(T)

³

→ U

_T^k

such that, for all v ∈ H

¹

(T )

³

, I

^k_T

v = (π

_T^k

v, (π

^k_F

v

|F

)

_{F∈ FT}

).

We furnish U

^k_h

with the discrete H

¹

-like seminorm such that, for all v

_h

∈ U

^k_h

, kv

_h

k

_1,h

B

Õ

T∈ T_h

kv

_T

k

²

1,T

!

¹₂

, (14)

where, for all T ∈ T

_h

, kv

_T

k

_1,T

B

k∇v

_T

k

²

L²(T)^3×3

+ | v

_T

|

²

1,∂T

¹

2

with | v

_T

|

_1,∂T

B Õ

F∈ F_T

h

⁻_F¹

kv

_F

− v

_T

k

²

L²(F)³

!

¹₂

. (15)

For further use, we note the following boundedness property of the global interpolator: For all v ∈ H

¹

(Ω)

³

,

kI

^k_h

v k

_1,h

≤ C

_I

|v |

_H1(Ω)³

, (16) with real number C

_I

> 0 independent of both h and v . Its proof relies on the stability properties of the L

²

-projectors on elements and faces proved in [17, Proposition 7.1].

The global spaces of discrete unknowns for the velocity and the pressure, respectively accounting for the wall boundary condition and the zero-average condition, are

U

^k_h,

0

B

v

_h

= ((v

T

)

_T∈ T_h

, (v

F

)

_F∈ F_h

) ∈ U

_h^k

: v

F

= 0 ∀ F ∈ F

_h^b

, P

_h^k

B P

^k

(T

_h

) ∩ P. (17) In the analysis, we need the following discrete Sobolev embeddings in U

^k_h,

0

(see [17, Proposition 5.4]): For all r ∈ [ 1 , 6 ] , it holds for all v

_h

∈ U

^k_h

kv

_h

k

_Lr(Ω)³

. kv

_h

k

_1,h

, (18)

where the hidden constant is independent of both h and v

_h

, but possibly depends on Ω , k , r , and the mesh regularity parameter. It follows from (18) that the map k·k

_1,h

defines a norm on U

^k_h,

0

. Classically, the corresponding dual norm of a linear form L

_h

: U

^k_h,

0

→ R is given by kL

_h

k

_1,h,∗

B sup

v_h∈U^k_h,0,kv_hk_1,h=1

L

_h

(v

_h

)

. (19)

3.4 Velocity reconstruction

Robustness with respect to λ hinges on the usage of a divergence-preserving velocity reconstruction in the discretization of the body force and convective terms. Let an element T ∈ T

_h

be fixed, and denote by RTN

^k

(T ) B P

^k

(T)

³

+ x P

^k

(T) the local Raviart–Thomas–Nédélec space of degree k ; see [32, 34]. We define the local velocity reconstruction operator R

^k_T

: U

^k_T

→ RTN

^k

(T ) such that, for all v

_T

∈ U

^k_T

,

∫

T

R

_T^k

v

_T

· w = ∫

T

v

_T

· w, ∀ w ∈ P

^k−1

(T )

³

, (20a) R

_T^k

v

_T

· n

T F

= v

F

· n

T F

∀ F ∈ F

_T

. (20b) Classically, the relations (20) identify R

^k_T

uniquely; see, e.g., [6, Proposition 2.3.4]. Moreover, for any v ∈ H

¹

(T)

³

, a direct computation shows that R

^k_T

I

_T^k

v = I

^k

RTN,T

v , where I

^k

RTN,T

is the local Raviart–Thomas–Nédélec interpolator. Finally, for all v

_T

∈ U

_T^k

, we have that

kR

_T^k

v

_T

− v

T

k

_L2(T)³

. Õ

F∈ F_T

h

1 2

F

k(v

F

− v

T

)· n

T F

k

_L2(F)

≤ Õ

F∈ F_T

h

1 2

F

kv

F

− v

T

k

_L2(F)³

≤ h

_T

|v

_T

|

_1,∂T

, (21) where the first bound follows from [16, Lemma 2], the second from a Hölder inequality with exponents ( 2 , ∞) along with k n

_{T F}

k

_L∞(F)³

= 1, and the third from the definition (15) of the |·|

_1,∂T

-seminorm together with h

_F

≤ h

_T

.

Let now RTN

^k

(T

_h

) B

v ∈ H ( div; Ω) : v

_|T

∈ RTN

^k

(T ) for all T ∈ T

_h

denote the global Raviart–

Thomas–Nédélec space on T

_h

. A global velocity reconstruction R

^k_h

: U

^k_h

→ RTN

^k

(T

_h

) is obtained patching the local contributions: For all v

_h

∈ U

^k_h

,

(R

^k_h

v

_h

)

_|T

B R

^k_T

v

_T

∀ T ∈ T

_h

.

Observe that R

^k_h

v

_h

is well-defined, since its normal components across each mesh interface are continuous as a consequence of (20b) combined with the single-valuedness of interface unknowns, and it holds, for any v ∈ H

¹

0

(Ω)

³

, R

^k_h

I

^k_h

v = I

^k_RTN,h

v . The following proposition contains novel Sobolev inequalities for the velocity reconstruction.

Proposition 3 (Sobolev inequalities for the velocity reconstruction) . It holds, for all r ∈ [ 1 , 6 ] and all v

_h

∈ U

_h,^k

0

,

kR

^k_h

v

_h

k

_Lr(Ω)³

. kv

_h

k

_1,h

, (22) where the hidden constant is independent of both h and v

_h

, but possibly depends on Ω, k, r , and the mesh regularity parameter.

Proof. Let a mesh element T ∈ T

_h

be fixed. Inserting ±v

_T

into the norm and using a triangle inequality, we can write

kR

^k_T

v

_T

k

_Lr(T)³

≤ kR

_T^k

v

_T

− v

T

k

_Lr(T)³

+ kv

T

k

_Lr(T)³

. (23) Let now an integer l ≥ 0 be fixed. From the discrete Lebesgue embeddings proved in [17, Lemma 5.1], it follows that, for all (α, β) ∈ [ 1 , + ∞] , all T ∈ T

_h

, and all v ∈ P

^l

(T ) ,

k v k

_Lα(T)

. h

α3−³_β

T

k v k

_Lβ(T)

, (24)

with hidden constant independent of h , T , and v , but possibly depending on l , α , β , and the mesh regularity parameter. Then, in the first term of (23) we get that

kR

^k_T

v

_T

− v

T

k

_Lr(T)³

. h

3 r−³

2

T

kR

_T^k

v

_T

− v

T

k

_L2(T)³

. h

3 r−¹

2

T

|v

_T

|

_1,∂T

, (25)

where we have used (24) with (α, β) = (r, 2 ) in the first inequality and the bound (21) in the second.

Thus, plugging (25) into (23), raising the resulting inequality to the r th power, using the inequality (a + b)

^r

. a

^r

+ b

^r

valid for any nonnegative real numbers a and b , and summing over T ∈ T

_h

, we get

kR

_T^k

v

_T

k

^r

L^r(Ω)³

. Õ

T∈ T_h

h

6−r 2 T

| v

_T

|

^r

1,∂T

+ kv

_h

k

^r

L^r(Ω)³

C T

₁

+ T

₂

. (26) To estimate the first term, we distinguish two cases. If r ∈ [ 1 , 2 ) , denoting by h

Ω

the diameter of Ω , we write

T

₁

= Õ

T∈ T_h

h

_T^r

h

3(2−r) 2 T

|v

_T

|

^r

1,∂T

≤ h

^r_Ω

Õ

T∈ T_h

h

3(2−r) 2 T

|v

_T

|

^r

1,∂T

≤ h

^r_Ω

Õ

T∈ T_h

h

_T³

!

^2−r

2

Õ

T∈ T_h

|v

_T

|

²

1,∂T

!

^r

2

. h

^r_Ω

|Ω|

^2−r2

kv

_T

k

^r

1,h

. kv

_T

k

^r

1,h

,

(27)

where we have used the fact that h

_T

≤ h

_Ω

to pass to the second line, a Hölder inequality with exponents

2r

,

²

2−r

on the sum over T ∈ T

_h

to pass to the third line, the mesh regularity to infer Í

T∈ T_h

h

_T³

. Í

T∈ T_h

|T | ≤ |Ω| along with the definitions (14) and (15) of the local and global discrete norms to pass to the fourth line, and h

^r_Ω

|Ω|

^2−r2

. 1 to conclude. If r ∈ [ 2 , 6 ] , on the other hand, we write

T

₂

= Õ

T∈ T_h

h

6−r 2

T

|v

_T

|

^r−2

1,∂T

| v

_T

|

²

1,∂T

≤ h

6−r 2

Ω

kv

_h

k

^r−2

1,h

Õ

T∈ T_h

|v

_T

|

²

1,∂T

!

. kv

_h

k

^r

1,h

, (28)

where we have used h

_T

≤ h

_Ω

along with

^6−r₂

≥ 0 and |v

_T

|

_1,∂T

≤ k v

_h

k

_1,h

along with r − 2 ≥ 0 in the first bound, and the definitions (14) and (15) of the local and global discrete norms together with h

Ω

. 1 to conclude. For the second term, on the other hand, the discrete Sobolev embeddings (18) readily give

T

₂

. kv

_h

k

_1,h

. (29) Plugging (29) and, depending on r , either (27) or (28) into (26), the conclusion follows.

3.5 Gradient reconstruction

Let an element T ∈ T

_h

be fixed. For any polyomial degree l ≥ 0, we define the local gradient reconstruction operator G

^l_T

: U

^k_T

→ P

^l

(T )

^3×3

such that, for all v

_T

∈ U

_T^k

and all τ ∈ P

^l

(T)

^3×3

,

∫

T

G

_T^l

v

_T

: τ = −

∫

T

v

T

· (∇ · τ) + Õ

F∈ F_T

∫

F

v

F

· τ n

T F

. (30) In (30), the right hand side is designed to resemble an integration by parts formula where the role of the function represented by v

_T

is played by v

T

in the volumetric term and by v

F

in the boundary term. This gradient reconstruction will be used with l = k in the viscous term (see Section 4.1) and with l = 2 (k + 1 ) in the convective terms (see Section 4.4). The following properties hold (see [18, Proposition 1]):

(i) Boundedness. For all v

_T

∈ U

_T^k

, it holds

kG

^l_T

v

_T

k

_L2(T)^3×3

. kv

_T

k

_1,T

. (31)

(ii) Consistency. For all v ∈ H

^m

(T )

³

with m = l + 2 if l ≤ k , m = k + 1 otherwise,

kG

^l_T

I

_T^k

v − ∇v k

_L2(T)^3×3

+ h

_T¹²

kG

^l_T

I

^k_T

v

_T

− ∇v k

_L2(∂T)^3×3

. h

_T^m−1

| v|

_H^m_(T)3

. (32) A global gradient reconstruction G

^l_h

: U

_h^k

→ P

^l

(T

_h

)

^3×3

can be defined setting, for all v

_h

∈ U

^k_h

, (G

_h^k

v

_h

)

_|T

B G

^k_T

v

_T

for all T ∈ T

_h

.

4 Discrete problem

In this section we discuss the discretization of the various term appearing in (1) along with the corresponding properties relevant for the analysis, we formulate the discrete problem, and we establish an a priori bound on the discrete velocity uniform in λ .

4.1 Viscous term

The viscous term is discretized by means of the bilinear form a

_h

: U

^k_h

× U

^k_h

→ R such that, for all w

_h

, v

_h

, ∈ U

^k_h

,

a

_h

(w

_h

, v

_h

) B

∫

Ω

G

^k_h

w

_h

: G

_h^k

v

_h

+ Õ

T∈ T_h

s

_T

(w

_T

, v

_T

), (33) where, for any T ∈ T

_h

, s

_T

: U

_T^k

× U

_T^k

→ R denotes a local stabilization bilinear form designed according to the principles of [15, Section 4.3.1.4], so that the following properties hold:

(i) Stability and boundedness. There exists C

_a

> 0 independent of h (and, clearly, also of ν and λ ) such that, for all v

_h

∈ U

^k_h

,

C

_a

kv

_h

k

²

1,h

≤ a

_h

(v

_h

, v

_h

) ≤ C

_a⁻¹

kv

_h

k

²

1,h

. (34)

(ii) Consistency. For all w ∈ H

¹

0

(Ω)

³

∩ H

^k+2

(T

_h

)

³

such that ∆w ∈ L

²

(Ω)

³

, it holds

kE

_a,h

(w ; ·)k

_1,h,∗

. h

^k+1

| w|

_Hk+2(T_h)³

, (35) where the linear form E

_a,h

(w ; ·) : U

^k_h

→ R representing the consistency error is such that

E

_a,h

(w ; v

_h

) B −

∫

Ω

∆w · v

_h

− a

_h

(I

^k_h

w, v

_h

). (36) A classical example of stabilization bilinear form along with the proofs of properties (34) and (35) can be found in [18], to which we refer for further details.

Remark 4 (Alternative formulation) . An alternative formulation with analogous properties is obtained expressing the consistent contribution in (33) in terms of a local velocity reconstruction in P

^k+1

(T) . For further details on this choice, considered in the numerical tests of Section 6, we refer to [7].

4.2 Pressure-velocity coupling

The pressure-velocity coupling hinges on the bilinear form b

_h

: U

^k_h

× P

^k

(T

_h

) → R such that, for all (v

_h

, q

_h

) ∈ U

^k_h

× P

^k

(T

_h

) ,

b

_h

(v

_h

, q

_h

) B −

∫

Ω

(∇ · R

^k_h

v

_h

) q

_h

. (37)

The bilinear form b

_h

enjoys the following properties:

(i) Consistency. It holds, for all v ∈ H

¹

0

(Ω)

³

,

b

_h

(I

^k_h

v, q

_h

) = b(v, q

_h

) ∀ q

_h

∈ P

^k

(T

_h

). (38) (ii) Stability. It holds, for all q

_h

∈ P

_h^k

, with P

_h^k

defined by (17),

kq

_h

k

_L2(Ω)

. sup

v_h∈U^k_h,0,kv_hk_1,h=1

b

_h

(v

_h

, q

_h

). (39)

Property (38) follows observing that, by construction, R

_h^k

I

^k_h

v = I

^k_RTN,h

v , hence ∇ · (R

^k_h

I

_h^k

v) =

∇ · (I

^k_RTN,h

v) = π

_h^k

(∇ · v) (see, e.g., [21, Lemma 3.7]), and the projector π

_h^k

can be removed since q

_h

∈ P

^k

(T

_h

) . Property (39) classically follows from (38) using Fortin’s argument (see, e.g., [21, Lemma 2.6]) after recalling the boundedness property (16) of the interpolator.

4.3 Body force

Denote by `

_h

: L

²

(Ω)

³

× U

^k_h

→ R the bilinear form such that, for any φ ∈ L

²

(Ω)

³

and any v

_h

∈ U

^k_h

,

`

_h

(φ, v

_h

) B

∫

Ω

φ · R

^k_h

v

_h

. (40)

This bilinear form has the following properties:

(i) Velocity invariance. Recalling the Hodge decomposition (4) of f , it holds

`

_h

( g + λ∇ψ, v

_h

) = `

_h

(g, v

_h

) + b

_h

(v

_h

, λπ

^k_h

ψ) ∀ v

_h

∈ U

^k_h,

0

. (41)

(ii) Consistency. For all φ ∈ L

²

(Ω)

³

∩ H

^k

(T

_h

)

³

,

kE

_`,h

(φ ; ·)k

_1,h,∗

. h

^k+1

|φ|

_Hk(T_h)³

. (42) where the linear form E

_`,h

(φ ; ·) : U

^k_h

→ R representing the consistency error is such that

E

_`,h

(φ ; v

_h

) B `

_h

(φ, v

_h

) −

∫

Ω

φ · v

_h

. (43)

The velocity invariance property can be proved writing

`

_h

( g + λ∇ψ, v

_h

) = `

_h

(g, v

_h

) +

∫

Ω

λ∇ψ · R

^k_h

v

_h

= `

_h

(g, v

_h

) −

∫

Ω

λψ (∇ · R

^k_h

v

_h

) +

∫

∂Ω

λψ (R

^k_h

v

_h

· n

Ω

)

= `

_h

(g, v

_h

) −

∫

Ω

λπ

_h^k

ψ (∇ · R

^k_h

v

_h

)

= `

_h

(g, v

_h

) + b

_h

(v

_h

, λπ

^k_h

ψ),

where we have used the linearity of `

_h

along with its definition (40) in the first line, we have integrated

by parts the second term and observed that the normal trace of R

^k_h

v

_h

vanishes on ∂ Ω as a consequence

of (20b) together with v

_F

= 0 for all F ∈ F

_h^b

in the second line, we have used the definition of the

global L

²

-orthogonal projector π

_h^k

after observing that ∇ · R

_h^k

v

_h

∈ P

^k_h

(T

_h

) in the third line, and we have

recalled the definition (37) of the pressure-velocity coupling bilinear form b

_h

to conclude. Property

(41) is the discrete counterpart of (5) and enables the cancellation of the terms involving the pressure

in the expression (76) of the discretization error. For the proof of the consistency property (43), we

refer the reader to [14, Chapter 8].

4.4 Convective term

The discrete counterpart of the convective trilinear form t defined by (3) is designed so as to approx- imate, for any w ∈ U , the quantity

`

h

((∇×w) × w, z

_h

) =

∫

Ω

(∇×w) × w · R

^k_h

z

_h

,

which naturally appears, with w = u , in the discretization error; see (76) below. Applying Proposition 1 with X successively equal to the mesh elements T ∈ T

_h

, we can reformulate this quantity as follows:

`

h

((∇×w) × w, z

_h

) = Õ

T∈ T_h

∫

T

(∇×w) × w · R

^k_T

z

_T

= Õ

T∈ T_h

∫

T

∇ww · R

^k_T

z

_T

− ∇wR

_T^k

z

_T

· w

. (44)

Starting from this expression, we obtain a discrete counterpart of t by replacing inside each element the continuous velocity and gradient by the corresponding reconstructions introduced in Sections 3.4 and 3.5, respectively. Thus, we introduce the global trilinear form t

_h

: U

^k_h

× U

^k_h

× U

^k_h

→ R such that

t

_h

(w

_h

, v

_h

, z

_h

) B Õ

T∈ T_h

t

_T

(w

_T

, v

_T

, z

_T

), (45a)

where, for any T ∈ T

_h

, t

_T

: U

_T^k

× U

_T^k

× U

_T^k

→ R is defined as t

_T

(w

_T

, v

_T

, z

_T

) B

∫

T

G

_T^2(k+1)

w

_T

R

^k_T

v

_T

· R

_T^k

z

_T

−

∫

T

G

²_T^(k+1)

w

_T

R

_T^k

z

_T

· R

_T^k

v

_T

. (45b) The choice of the polynomial degree l = 2 (k + 1 ) for the gradient reconstruction is justified in the following remark.

Remark 5 (Reformulation of t

_h

) . In the practical implementation, one does not need to compute the gradient reconstruction operators G

_T^2(k+1)

to evaluate t

_h

. As a matter of fact, expanding this operator in (45) according to its definition (30), we have that

t

_h

(w

_h

, v

_h

, z

_h

) = Õ

T∈ T_h

∫

T

∇w

_T

R

_T^k

v

_T

· R

^k_T

z

_T

−

∫

T

∇w

_T

R

^k_T

z

_T

· R

_T^k

v

_T

+ Õ

T∈ T_h

Õ

F∈ F_T

∫

F

(w

_F

− w

_T

) · R

_T^k

z

_T

R

^k_T

v

_T

· n

_{T F}

− Õ

T∈ T_h

Õ

F∈ F_T

∫

F

(w

_F

− w

_T

) · R

_T^k

v

_T

R

^k_T

z

_T

· n

_{T F}

.

The properties relevant for the analysis are summarized in the following lemma.

Lemma 6 (Properties of t

_h

) . The trilinear form t

_h

has the following properties:

(i) Non-dissipativity. For all w

_h

, v

_h

∈ U

^k_h

, it holds that

t

_h

( w

_h

, v

_h

, v

_h

) = 0 . (46) (ii) Boundedness. There exists a real number C

_t

> 0 independent of h (and, clearly, also of ν and

λ) such that, for all w

_h

, v

_h

, z

_h

∈ U

^k_h

,

|t

_h

(w

_h

, v

_h

, z

_h

)| ≤ C

_t

kw

_h

k

_1,h

kv

_h

k

_1,h

kz

_h

k

_1,h

. (47)

(iii) Consistency. It holds, for all w ∈ U ∩ W

^k+1,4

(T

_h

)

³

and all z

_h

∈ U

^k_h

,

kE

_t,h

(w ; ·)k

_1,h,∗

. h

^k+1

kwk

_W1,4(Ω)³

|w |

_Wk+1,4(T_h)³

, (48) where the linear form E

_t,h

(w ; ·) : U

^k_h

→ R representing the consistency error is such that, for all z

_h

∈ U

^k_h

,

E

_t,h

(w ; z

_h

) B `

h

((∇×w) × w, z

_h

) − t

_h

(I

^k_h

w, I

^k_h

w, z

_h

). (49) Proof. (i) Non-dissipativity. Immediate consequence of the definition (45) of t

_h

.

(ii) Boundedness. By (45), it suffices to prove that it holds, for all w

_h

, v

_h

, z

_h

∈ U

^k_h

, T B

Õ

T∈ T_h

∫

T

G

²_T^(k+1)

w

_T

R

_T^k

v

_T

· R

^k_T

z

_T

. kw

_h

k

_1,h

kv

_h

k

_1,h

kz

_h

k

_1,h

.

Using Hölder inequalities with exponents ( 2 , 4 , 4 ) , the bound (31), and again a discrete Hölder inequality on the sum over T ∈ T

_h

, we have that

T . Õ

T∈ T_h

kG

²_T^(k+1)

w

_T

k

_L2(T)^3×3

kR

_T^k

v

_T

k

_L4(T)³

kR

^k_T

z

_T

k

_L4(T)³

. Õ

T∈ T_h

kw

_T

k

_1,T

kR

_T^k

v

_T

k

_L4(T)³

kR

^k_T

z

_T

k

_L4(T)³

. k w

_h

k

_1,h

k R

^k_h

v

_h

k

_L4(Ω)³

k R

^k_h

z

_h

k

_L4(Ω)³

. k w

_h

k

_1,h

k v

_h

k

_1,h

k z

_h

k

_1,h

,

(50)

where, to bound the last two factors, we have invoked the discrete Sobolev embeddings (22) with r = 4.

(iii) Consistency. First of all, let us verify that the first term on the right hand side of (49) is well defined. By the assumed regularity, w ∈ W

^1,4

(Ω)

³

(combine the fact that the jumps of w vanish across interfaces since w ∈ H

¹

0

(Ω)

³

and the regularity w ∈ W

^1,4

(T

_h

)

³

), and we can write k(∇×w) × wk

_L2(Ω)³

≤ k∇×wk

_L4(Ω)³

kw k

_L4(Ω)³

. kwk

_W1,4(Ω)³

kw k

_H1(Ω)³

,

where we have concluded using the embedding H

¹

(Ω) , → L

⁴

(T ) . This shows that (∇×w) × w ∈ L

²

(Ω)

³

, so this quantity legitimately appear in the first argument of `

_h

.

Let now ˆ w

_h

B I

^k_h

w . Using (44) and the definition (45) of t

_h

, we have that E

_t,h

(w ; v

_h

) = Õ

T∈ T_h

∫

T

∇ww · R

_T^k

z

_T

− ∇wR

^k_T

z

_T

· w

+ Õ

T∈ T_h

−

∫

T

G

²_T^(k+1)

w ˆ

_T

R

_T^k

w ˆ

_T

· R

_T^k

z

_T

+

∫

T

G

²_T^(k+1)

w ˆ

_T

R

_T^k

z

_T

· R

^k_T

w ˆ

_T

.

Inserting into the right-hand side of this last equation the quantity

± Õ

T∈ T_h

∫

T

G

²_T^(k+1)

w ˆ

_T

w · R

^k_T

z

_T

+ G

_T^2(k+1)

w ˆ

_T

R

^k_T

z

_T

· w

,

we arrive at E

_t,h

(w ; v

_h

) = Õ

T∈ T_h

∫

T

(G

²_T^(k+1)

w ˆ

_T

− ∇w)R

_T^k

z

_T

· w

| {z }

T₁

+ Õ

T∈ T_h

∫

T

(∇w − G

_T^2(k+1)

w ˆ

_T

)w · R

_T^k

z

_T

| {z }

T₂

+ Õ

T∈ T_h

∫

T

G

_T^2(k+1)

w ˆ

_T

(w − R

_T^k

w ˆ

_T

) · R

^k_T

z

_T

| {z }

T₃

+ Õ

T∈ T_h

∫

T

G

_T^2(k+1)

w ˆ

_T

R

_T^k

z

_T

· (R

_T^k

w ˆ

_T

− w).

| {z }

T₄

(51) We next proceed to estimate the terms T

₁

, · · · , T

₄

.

(iii.A) Estimate of T

₁

. For the term T

₁

, we add ±π

_T⁰

w in the third factor, so we have that T

₁

= Õ

T∈ T_h

∫

T

(G

_T^2(k+1)

w ˆ

_T

− ∇w)R

_T^k

z

_T

· (w − π

_T⁰

w) + Õ

T∈ T_h

∫

T

( G

_T^2(k+1)

w ˆ

_T

− ∇w)R

_T^k

z

_T

· π

⁰_T

w C T

_1,1

+ T

_1,2

.

To bound T

_1,1

, we use Hölder inequalities with exponents ( 2 , 4 , 4 ) , the approximation properties (32) of G

²_T^(k+1)

with l = 2 (k + 1 ) and m = k + 1, and (13a) of the L

²

-projector with l = 0, m = 0, r = 4, and s = 1, so it holds that

|T

_1,1

| . Õ

T∈ T_h

kG

_T^2(k+1)

w ˆ

_T

− ∇wk

_L2(T)^3×3

kR

_T^k

z

_T

k

_L4(T)³

kw − π

⁰_T

wk

_L4(T)³

. h

^k+1

|w |

_Hk+1(T_h)³

kR

^k_h

z

_h

k

_L4(Ω)³

|w |

_W1,4(Ω)³

. h

^k+1

|w |

_Hk+1(T_h)³

kz

_T

k

_1,h

|w|

_W1,4(Ω)³

,

(52)

where, in the last step, we have used the discrete Sobolev embedding (22) with r = 4 and recalled that the assumed regularity implies w ∈ W

^1,4

(Ω)

³

.

For T

_1,2

, we start by observing that T

_1,2

= Õ

T∈ T_h

∫

T

(G

²_T^(k+1)

w ˆ

_T

− ∇w) : π

⁰_T

w ⊗ R

^k_T

z

_T

.

Integrating by parts the term involving ∇w and using, for each element T ∈ T

_h

, the definition (30) of G

_T^2(k+1)

with v

_T

= w ˆ

_T

and τ = π

⁰_T

w ⊗ R

^k_T

z

_T

(this is possible since π

⁰_T

w ⊗ R

_T^k

z

_T

∈ P

^k+1

(T )

^3×3

⊂ P

^2(k+1)

(T )

^3×3

), we get

T

_1,2

= − Õ

T∈ T_h

∫

T

(π

_T^k

w − w) · ∇ · (π

_T⁰

w ⊗ R

_T^k

z

_T

) + Õ

T∈ T_h

Õ

F∈ F_T

∫

F

(π

^k_F

w − w) · (π

⁰_T

w ⊗ R

_T^k

z

_T

n

_{T F}

). (53)

We have ∇ · (π

⁰_T

w ⊗ R

_T^k

z

_T

) = π

⁰_T

w(∇ · R

_T^k

z

_T

) ∈ P

^k

(T )

³

. Then, recalling the definition (12) of π

^k_T

, the first term of the right hand side of (53) vanishes. Moreover, π

⁰_T

w ⊗ R

^k_T

z

_T

n

_{T F}

= π

⁰_T

w( R

_T^k

z

_T

· n

_{T F}

) ∈ P

^k

(F)

³

(see (20b)). Then, by definition (12) of π

^k_F

, the second term of the right hand side of (53) vanishes as well, giving

T

_1,2

= 0 . Combining this result with (52), we conclude that

|T

₁

| ≤ h

^k+1

| w|

_H^k+1(T_h)³

kz

_T

k

_1,h

|w |

_W1,4(Ω)³

. (54)

(iii.B) Estimate of T

₂

. For the term T

₂

in (51), we insert ±π

_T⁰

w into the second factor, so we can write T

₂

= Õ

T∈ T_h

∫

T

(∇w − G

²_T^(k+1)

w ˆ

_T

)(w − π

_T⁰

w) · R

^k_T

z

_T

+ Õ

T∈ T_h

∫

T

(∇w − G

²_T^(k+1)

w ˆ

_T

)π

⁰_T

w · R

_T^k

z

_T

C T

_2,1

+ T

_2,2

.

(55)

We bound T

_2,1

similarly as done with T

_1,1

in (52), that is, we use Hölder inequalities with exponents ( 2 , 4 , 4 ) , the approximation properties of G

²_T^(k+1)

and π

_T⁰

, and (22) with r = 4, so it is inferred that

|T

_2,1

| . h

^k+1

|w|

_Hk+1(T_h)³

|w |

_W1,4(Ω)³

k z

_T

k

_1,h

, (56) To estimate T

_2,2

in (55), we integrate by parts the term involving ∇w and we use, for each element T ∈ T

_h

, the definition (30) of G

²_T^(k+1)

with v

_T

= w ˆ

_T

and τ = R

_T^k

z

_T

⊗ π

⁰_T

w (this is possible since

R

^k_T

z

_T

⊗ π

_T⁰

w ∈ P

^k+1

(T)

^3×3

⊂ P

^2(k+1)

(T)

^3×3

) to write T

_2,2

= − Õ

T∈ T_h

(( ∫ (( (( (( (( (( (( ( (

T

(w − π

_T^k

w) · ∇ · (R

_T^k

z

_T

⊗ π

⁰_T

w) + Õ

T∈ T_h

Õ

F∈ F_T

∫

F

(w − π

^k_F

w) · (R

_T^k

z

_T

⊗ π

_T⁰

w)n

_{T F}

, (57) where we have cancelled the first integral in (57) observing that ∇ · (R

^k_T

z

_T

⊗ π

⁰_T

w) = ∇R

_T^k

z

_T

π

_T⁰

w +

R

_T^k

z

_T

(∇ · π

⁰_T

w) ∈ P

^k

(T )

^d

and using the definition (12) of π

^k_T

. To bound the second term in (57), we add the quantity ±π

⁰_T

R

_T^k

z

_T

in its second factor, so we get that

∫

F

(w − π

^k_F

w) · (R

_T^k

z

_T

⊗ π

_T⁰

w)n

_{T F}

=

∫

F

(w − π

^k_F

w) ·

(R

_T^k

z

_T

− π

⁰_T

R

_T^k

z

_T

) ⊗ π

_T⁰

w n

_{T F}

+

(( ∫ (( (( (( (( (( (( (( ( (

F

(w − π

^k_F

w) · (π

_T⁰

R

^k_T

z

_T

⊗ π

_T⁰

w)n

_{T F}

,

(58)

where we have cancelled the second integral of the right hand side of (58) using the definition (12) of π

^k_F

after observing that (π

_T⁰

R

^k_T

z

_T

⊗ π

⁰_T

w)

_|F

n

_{T F}

∈ P

⁰

(F)

³

⊂ P

^k

(F)

³

. So, plugging (58) into (57) and applying Hölder inequalities with exponents ( 4 , 2 , 4 , ∞) along with k n

T F

k

_L∞(F)³

= 1, we can write

|T

_2,2

| ≤ Õ

T∈ T_h

Õ

F∈ F_T

kw − π

^k_F

w k

_L4(F)³

kR

_T^k

z

_T

− π

_T⁰

R

_T^k

z

_T

k

_L2(F)³

kπ

_T⁰

wk

_L4(F)³

. (59)

To estimate the first term in (59), we add ±π

_T^k

w , so we can write kw − π

^k_F

w k

_L4(F)³

≤ kπ

_T^k

w − π

^k_F

wk

_L4(F)³

+ kw − π

_T^k

w k

_L4(F)³

≤ kπ

^k_F

(π

_T^k

w − w)k

_L4(F)³

+ kw − π

_T^k

w k

_L4(F)³

. kw − π

_T^k

w k

_L4(F)³

,

where, in the last line, we have used the L

⁴

-boundedness of π

^k_F

(see [17, Lemma 3.2]). Thus, using this last inequality in (59), we get

|T

_2,2

| . Õ

T∈ T_h

Õ

F∈ F_T

kw − π

^k_T

wk

_L4(F)³

kR

_T^k

z

_T

− π

⁰_T

R

_T^k

z

_T

k

_L2(F)³

kπ

_T⁰

w k

_L4(F)³

. Õ

T∈ T_h

h

_T^k+1

| w|

_W^k+1,4(T)³

| R

_T^k

z

_T

|

_H1(T)³

kw k

_W1,4(T)³

. (60)

To pass from the first to the second line, we have used the approximation properties (13b) of the

L

²

-orthogonal projector, with l = k , m = 0, r = 4, and s = k + 1 for the first factor and with m = 0,