p-Multilevel preconditioners for HHO discretizations of the Stokes equations with static condensation

HAL Id: hal-02951823

https://hal.archives-ouvertes.fr/hal-02951823

Preprint submitted on 29 Sep 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

p-Multilevel preconditioners for HHO discretizations of the Stokes equations with static condensation

Lorenzo Botti, Daniele Antonio Di Pietro

To cite this version:

Lorenzo Botti, Daniele Antonio Di Pietro. p-Multilevel preconditioners for HHO discretizations of the Stokes equations with static condensation. 2020. �hal-02951823�

p-Multilevel preconditioners for HHO discretizations of the Stokes equations with static condensation

Lorenzo Botti¹and Daniele A. Di Pietro²

1Department of Engineering and Applied Sciences, University of Bergamo, Italy,lorenzo.botti@unibg.it 2IMAG, Univ Montpellier, CNRS, Montpellier, France,daniele.di-pietro@umontpellier.fr

September 28, 2020

Abstract

We propose ap-multilevel preconditioner for Hybrid High-Order discretizations (HHO) of the Stokes equation,

numerically assess its performance on two variants of the method, and compare with a classical Discontinuous

Galerkin scheme. We specifically investigate how the combination ofp-coarsening and static condensation influences

the performance of theV-cycle iteration for HHO. Two different static condensation procedures are considered, resulting

in global linear systems with a different number of unknowns and non-zero elements. An efficient implementation is

proposed where coarse level operators are inherited usingL²-orthogonal projections defined over mesh faces and the

restriction of the fine grid operators is performed recursively and matrix-free. The various resolution strategies are thoroughly validated on two- and three-dimensional problems.

1 Introduction

In this work we develop and numerically validate p-multigrid solution strategies for nonconforming polytopal dis- cretizations of the Stokes equations, governing the creeping flow of incompressible fluids.

For the sake of simplicity, we focus on a Newtonian fluid with uniform density and unit kinematic viscosity. Given a polygonal or polyhedral domainΩ ⊂R^d,d ∈ {2,3}, with boundary∂Ω, the Stokes problem consist in finding the velocity fieldu:Ω→R^d, and the pressure fieldp:Ω→R, such that

−∆u+∇p= f inΩ, (1a)

∇ ·u=0 inΩ, (1b)

u=g_D on∂Ω_D. (1c)

−n· ∇u+pn=g_N on∂Ω_N, (1d)

wherendenotes the unit vector normal to∂Ωpointing out ofΩ,g_Dandg_Ndenote, respectively, the prescribed velocity on the Dirichlet boundary∂Ω_D⊂∂Ωand the prescribed traction on the Neumann boundary∂Ω_NB∂Ω\∂Ω_D, while f :Ω→ R^d is a given body force. For the sake of simplicity, it is assumed in what follows that both∂Ω_Dand∂Ω_N have non-zero(d−1)-dimensional Hausdorff measure (otherwise, additional closure conditions are needed).

Our focus is on new generation discretization methods for problem (1) that support general polytopal meshes and high-order: Hybrid High-Order (HHO) and Discontinuous Galerkin (DG) methods.

Hybrid High-Order discretizations of the Stokes equations have been originally considered in [2] and later extended in [34] to incorporate robust handling of large irrotational body forces. Other extensions include their application to the Brinkman problem, considered in [18], and to the full Navier–Stokes equations [17,35,36]; see also [30, Chapters 8 and 9] for further details. In this work, we consider two HHO schemes that are novel variations of existing schemes

with improved features. The first scheme, based on a hybrid approximation of the velocity along with a discontinuous approximation of the pressure, is a variation of the one considered in [30, Chapter 8] including two choices for the polynomial degree of the element velocity unknowns in the spirit of [21] (see also [30, Section 5.1]). The second scheme, inspired by the Hybridizable Discontinuous Galerkin (HDG) method of [47], hinges on hybrid approximations of both the velocity and the pressure and includes, with respect to the above reference, a different treatment of viscous terms that results in improved orders of convergence. In both cases, the Dirichlet condition on the velocity is enforced weakly in the spirit of [17].

Since the pioneering works [23,25–28] dating back to the late 1980s, DG methods have gained significant popularity in computational fluid mechanics, boosted by the 1997 landmark papers [12,13] on the treatment of viscous terms. The extension of DG methods to general polyhedral meshes was systematically considered in [31] and [32]. Crucially, this extension paved the way to adaptive mesh coarsening by agglomeration, a strategy proposed in [8] and exploited in [9, 14] in practical CFD applications to provide high-order accurate geometry representation with arbitrarily coarse meshes.

More recent developments, includinghp-versions and the support of meshes with small faces, can be found in [3,4];

see also the recent monograph [19]. Our focus is on an equal-order approximation with stabilized pressure-velocity coupling in the spirit of [24] and a treatment of the viscous term based on the Bassi–Rebay 2 (BR2) method of [13].

Related works include [10,29]; see also [32, Chapter 6] and references therein.

p-Multilevel solvers are well suited for both HHO and DG methods because the process of building coarse level operators based on polynomial degree reduction is straightforward and inexpensive. The purpose of applying iterative solvers to coarse problems is twofold: on one hand, a coarser operator translates into a global sparse matrix of smaller size with fewer non-zero entries, resulting in cheaper matrix-vector products; on the other hand, coarse level iterations are best suited to smooth out the low-frequency components of the error, that are hardly dumped by fine level iterations.

In the context of DG discretizations,p-multilevel solvers have been fruitfully utilized in practical applications see, e.g., [11,38,39,44,48].h-,p- andhp-Multigrid solvers for DG discretizations of elliptic problems have been considered in [5], where uniform convergence with respect to the number of levels for the W-cycle iteration has been proved, and in [16]. Multigrid solvers for HDG discretizations of scalar elliptic problems were considered in [22] and, more recently, in [37, 42], where a comparison with DG is carried out. p-Multivel solvers for HDG methods with application to compressible flow simulations have been recently considered in [40]. Preconditioners for DG and HDG discretizations of the Stokes problem have been considered in [1,6,15,20,41] and [46], respectively. Finally, anh-multigrid method for HHO discretizations of scalar diffusion problems has been recently proposed in [43]. The main novelty consists, in this case, in the use of the local potential reconstruction in the prolongation operator.

In this work we propose and numerically assess p-multilevel solution strategies for HHO discretizations of the Stokes equations. We specifically investigate how the combination ofp-coarsening and static condensation influences the performance of theV-cycle iteration. To this end, we compare different static condensation strategies. In order to preserve computational efficiency, statically condensed coarse level operators are inherited using localL²-orthogonal projections defined over mesh faces. Restriction of fine grid operators is performed recursively and matrix-free, relying on L²-orthogonal basis functions to further reduce the computational burden. Performance assessment is based on accuracy and efficiency of p-multilevel solvers considering DG discretizations as a reference for comparison. High- order accurate solutions approximating smooth analytical velocity and pressure fields are computed over standard and severely gradedh-refined mesh sequences in both two and three space dimensions. Interestingly, the static condensation strategy plays a crucial role in case of graded meshes.

The rest of this work is organized as follows. In Section 2 we state the HHO and DG schemes considered in the numerical tests. Thep-multilevel strategy is discussed in Section 3 and computational aspects are discussed in Section 4. Section 5 contains an extensive panel of numerical results that enable one to assess and compare several solution strategies. Finally, some conclusions are drawn in Section 6.

2 Three nonconforming methods for the Stokes problem

In this section we describe two HHO and one DG methods for the approximation of problem (1) that will be used to assess the performance of the p-multilevel preconditioner. In order to lay the ground for future works on the full

nonlinear Navier–Stokes equations, the corresponding discrete problems are formulated in terms of the annihilation of residuals.

2.1 Discrete setting

We consider meshes of the domain Ωcorresponding to couples M_h B (T_h,F_h), where T_h is a finite collection of polygonal (ifd =2) or polyhedral (ifd=3) elements such thathBmax

T∈ Th

h_T >0 withh_T denoting the diameter ofT, whileF_his a finite collection of line segments (ifd=2) or polygonal faces (ifd =3). For the sake of brevity, in what follows the term “face” will be used in both two and three space dimensions. It is assumed henceforth that the mesh M_h matches the geometrical requirements detailed in [30, Definition 1.4]. This covers, essentially, any reasonable partition ofΩinto polyhedral sets, not necessarily convex. For each mesh elementT ∈ T_h, the faces contained in the element boundary∂T are collected in the setF_T, and, for each mesh face F ∈ F_h,T_F is the set containing the one or two mesh elements sharingF. We define three disjoint subsets of the setF_T: the set of Dirichlet boundary faces F_T^D B{F ∈ F_T : F ⊂ ∂Ω_D}; the set of Neumann boundary facesF_T^N B{F ∈ F_T : F ⊂ ∂Ω_N}; the set of internal facesF_Tⁱ BF_T\ F_T^D∪ F_T^N

. For future use, we also letF_T^i,DBF_Tⁱ∪ F_T^D. For allT ∈ T_hand allF ∈ F_T,nT Fdenotes the unit vector normal toFpointing out ofT.

Hybrid High-Order methods hinge on local polynomial spaces on mesh elements and faces. For given integers

` ≥0 andn≥1, we denote byP<sup>`</sup>nthe space ofn-variate polynomials of total degree≤`(in short, of degree`). ForX mesh element or face, we denote byP^{`</sup>(X)the space spanned by the restriction toXof functions inP<sup>`}_d. WhenXis a mesh face, the resulting space is isomorphic toP^`_d−1(see [30, Proposition 1.23]). At the global level, we will need the broken polynomial space

P^`(T_h)B

q∈L²(Ω):q_|T ∈ P^`(T)for allT ∈ T_h .

Let againXdenote a mesh element or face. The localL²-orthogonal projectorπ^{`</sup><sub>X</sub> :L<sup>2</sup>(X) → P<sup>`}(X)is such that, for allq∈L²(X),

∫

X

(q−π^{`</sup><sub>X</sub>q)r=0 ∀r ∈ P<sup>`}(X).

Notice that, above and in what follows, we omit the measure from integrals as it can always be inferred from the context.

TheL²-orthogonal projector onP^{`</sup>(X)<sup>d</sup>, obtained applyingπ<sup>`}_Xcomponent-wise, is denoted byπ^`_X.

2.2 Local reconstructions and face residuals

The HHO discretizations of the Stokes problem considered in this work hinge on velocity reconstructions devised at the element level and obtained assembling diffusive potential reconstructions component-wise. In what follows, we let a mesh elementT ∈ T_hbe fixed, denote byk≥0 the degree of polynomials attached to mesh faces, and byk⁰∈ {k,k+1} the degree of polynomials attached to mesh elements.

2.2.1 Scalar potential reconstruction

The velocity reconstruction is obtained leveraging, for each component, thescalar potential reconstructionoriginally introduced in [33] in the context of scalar diffusion problems (see also [21] and [30, Section 5.1] for its generalization to the case of different polynomial degrees on elements and faces). Define the local scalar HHO space

V_T^k0,k B n

v_T = v_T,(v_F)_{F∈ FT}

:v_T ∈ P^k0(T)andv_F ∈ P^k(F)for allF∈ F_To

. (2)

The scalar potential reconstruction operatorp_T^k+1: V^k_T^0,k → P^k+1(T)maps a vector of polynomials ofV_T^k0,k onto a polynomial of degree (k +1) overT as follows: Given v_T ∈ V_T^k0,k,p_T^k+1v_T is the unique polynomial in P^k+1(T)

satisfying

∫

T

∇p_T^k+1v_T· ∇w_T =∫

T

∇v_T· ∇w_T + Õ

F∈ FT

∫

F

(v_F−v_T) ∇w_T·nT F ∀w_T ∈ P^k+1(T),

∫

T

p_T^k+1v_T =∫

T

v_T.

Computingp_T^k+1 for eachT ∈ T_h requires to solve a small linear system. This is an embarrassingly parallel task that can fully benefit from parallel architectures.

2.2.2 Velocity reconstruction

Define, in analogy with (2), the following vector-valued HHO space for the velocity:

V_T^k0,k B n

v_T = v_T,(v_F)_{F∈ FT}

:v_T ∈ P^k0(T)^dandv_F ∈ P^k(F)^dfor allF∈ F_T o .

Thevelocity reconstructionP_T^k+1:V^k_T^0,k → P^k+1(T)^dis obtained setting P_T^k+1v_T B p_T^k+1v_T,i

i=1,...,d,

where, for all i = 1, . . . ,d, v_T,i ∈ V_T^k0,k is obtained gathering theith components of the polynomials in v_T, i.e., v_T,i B v_T,i,(v_F,i)_F∈ F_T

ifvT =(v_T,i)_i=1,...,dandvF=(v_F,i)_i=1,...,dfor allF∈ F_T. 2.2.3 Face residuals

LetT ∈ T_handF∈ F_T. The stabilization bilinear form for the HHO discretization of the viscous term in the momentum equation (1a) hinges on theface residualR_{T F}^k :V_T^k0,k → P^max(k0,k)(F)^dsuch that, for allv_T ∈V^k_T^0,k,

R_T^k0,kv_T B r_{T F}^k0,kv_T,i

i=1,...,d, where the scalar face residualr^k0,k

T F :V^k_T^0,k → P^max(k0,k)(F)is such that, for allv_T ∈V_T^k0,k, r^k0,k

T Fv_T Bπ_F^k v_F−p_T^k+1v_T

−π_T^k0 v_T −p_T^k+1v_T.

2.3 HHO schemes

We consider two HHO schemes based, respectively, on discontinuous and hybrid approximations of the pressure. In both cases, the Dirichlet boundary condition is enforced weakly, considering a symmetric variation of the method discussed in [18].

2.3.1 An HHO scheme with discontinuous pressure

Let againk≥0 andk⁰∈ {k,k+1}denote the polynomial degrees of the face and element unknowns, respectively, and let a mesh elementT ∈ T_hbe fixed. Given(u_T,p_T) ∈V_T^k0,k× P^k(T), the local residualsr_I^mnt_,T((u_T,p_T);·):V_T^k0,k →Rof the discrete momentum conservation equation andr^cnt_I,T(u_T;·):P^k(T) →Rof the discrete mass conservation equation

are such that, respectively: For allv_T ∈V^k_T^0,k and allqT ∈ P^k(T), r_I,T^mnt((u_T,p_T);v_T)B

∫

T

∇P_T^k+1u_T :∇P_T^k+1v_T+ Õ

F∈ FT

1 hF

∫

F

R_{T F}^k u_T ·R_{T F}^k v_T

− Õ

F∈ FD T

∫

F

nT F· ∇P_T^k+1u_T

·vF+uF· nT F· ∇P_T^k+1v_T + Õ

F∈ FD T

η h_F

∫

F

uF·vF

−

∫

T

p_T(∇·v_T) − Õ

F∈ F_T

∫

F

p_T(v_F−v_T) ·n_{T F}+ Õ

F∈ F_T^D

∫

F

p_T(v_F·n_{T F})

− Õ

F∈ F_T^D

∫

F

g_D·

nT F· ∇P_T^k+1v_T + η h_FvF

− Õ

F∈ F_T^N

∫

F

g_N·vF−

∫

T

f ·vT,

(3a)

r_I,T^cnt(u_T;q_T)B −

∫

T

(∇·u_T)q_T− Õ

F∈ FT

∫

F

(u_F−u_T) ·n_{T F}q_T + Õ

F∈ F_T^D

∫

F

(u_F·n_{T F})q_T

− Õ

F∈ F_T^D

∫

F

g_D·nT Fq_T.

(3b)

In the expression ofr_I^mnt_,T((u_T,p_T);·),η >0 is a user-dependent parameter that has to be taken large enough to ensure coercivity. The penalty term where the parameterηappears, along with the consistency terms in the second line and the term involving the boundary datumg_Din the fourth line, are responsible for the weak enforcement of the Dirichlet boundary condition for the velocity.

Define the global vector HHO space V^k_h^0,k B

n

v_h = (vT)_T∈ T_h,(vF)_F∈ F_h

:vT ∈ P^k0(T)^dfor allT ∈ T_handvF ∈ P^k(F)^dfor allF ∈ F_h o . For all v_h ∈ V_h^k0,k and all T ∈ T_h, we denote by v_T ∈ V_T^k0,k the restriction of v_h toT. The global residuals r_I^mnt_,h (u_h,p_h);·

:V^k_h^0,k →Randr^cnt_I,h(u_h;·):P^k(T_h) →Rare obtained by element-by-element assembly, i.e.: For all v_h ∈V^k_h^0,k and allq_h ∈ P^k(T_h),

r_I,h^mnt (u_h,p_h);v_h B

Õ

T∈ Th

r_I^mnt_,T (u_T,p_T);v_T, r^cnt_I,h(u_h;q_h)B Õ

T∈ Th

r_I,T^cnt(u_T;q_h|T). (4)

Scheme I(HHO-dp: HHO scheme with discontinuous pressure). Find(u_h,ph) ∈V^k_h^0,k× P^k(T_h)such that r_I,h^mnt (u_h,p_h);v_h =0 ∀v_h∈V_h^k0,k,

r_I,h^cnt(u_h;q_h)=0 ∀q_h ∈ P^k(T_h). (5) 2.3.2 An HHO scheme with hybrid pressure

An interesting variation of Scheme I is obtained combining the HHO discretization of the viscous term withk⁰=k+1 with a hybrid approximation of the pressure inspired by [47]. LetT ∈ T_h. Given(u_T,p

T) ∈V^k+1,k_T ×V^k_T^,k, the local residualsr^mnt_{I I,T}((u_T,p

T);·):V^k+1,k_T →Rof the discrete momentum andr_{I I}^cnt_,T(u_T;·):V^k,k_T →Rof the discrete mass

conservation equations for the HHO scheme with hybrid pressure are such that, for allv_T ∈V^k+1_T ^,k and allq

T ∈V^k_T^,k, r_{I I,T}^mnt((u_T,p

T);v_T)B

∫

T

∇P_T^k+1u_T :∇P_T^k+1v_T+ Õ

F∈ F_T

1 h_F

∫

F

R_{T F}^k v_T · R_{T F}^k v_T

− Õ

F∈ F_T^D

∫

F

nT F· ∇P_T^k+1u_T

·vF+uF· nT F· ∇P_T^k+1v_T + Õ

F∈ F_T^D

η hF

∫

F

uF·vF

−

∫

T

pT(∇ ·vT)+ Õ

F∈ FT

∫

F

pF(vT−vF) ·nT F + Õ

F∈ F_T^D

∫

F

pF(vF·nT F)

− Õ

F∈ F_T^D

∫

F

g_D·

nT F · ∇P_T^k+1v_T+ η h_FvF

− Õ

F∈ F_T^N

∫

F

g_N·vF−

∫

T

f ·vT,

r_{I I,T}^cnt (u_T;q

T)B−

∫

T

(∇ ·u_T)q_T+ Õ

F∈ FT

∫

F

(u_T−u_F) ·n_{T F}q_F.

As before,η >0 is a penalty parameter that has to be taken large enough to ensure coercivity. The boxed terms are the ones that distinguish the local residuals on the momentum and mass conservation equations for the HHO scheme with hybrid pressure from Scheme I withk⁰=k+1.

Define the global scalar HHO space V^k_h^,k B

n

qh = (q_T)_T∈ T_h,(q_F)_F∈ F_h

: q_T ∈ P^k(T)for allT ∈ T_handq_F ∈ P^k(F)for allF ∈ F_ho . The global residualsr_{I I,h}^mnt((u_h,p

h);·):V^k_h^+1,k →Randr_{I I}^cnt_,h(u_h;·):V^k_h^,k →Rare obtained by element-by-element assembly of the local residuals.

Scheme II(HHO-hp: HHO scheme with hybrid pressure). Find(u_h,p

h) ∈V^k+1,k_h ×V_h^k,ksuch that r_{I I,h}^mnt((u_h,p

h);v_h)=0 ∀v_h ∈V^k+1,k_h , r_{I I,h}^cnt (u_h;q

h)=0 ∀q

h ∈V^k,k_h . (7)

The HHO method (7) yields a velocity approximation that is pointwise divergence free (as can be checked adapting the argument of [47, Proposition 1]) and improves by one order theh-convergence rates of the HDG method proposed in [47], since it relies on an HHO discretization of the viscous term (cf. the discussion in [21] and also [30, Section 5.1.6]).

A key point consists in using element unknowns for the velocity one degree higher than face unknowns. Notice that seeking the velocity in the spaceV^k+1_T ^,k as opposed toV^k_T^,k does not alter the number of globally coupled unknowns, as all velocity degrees of freedom attached to the mesh elements can be removed from the global linear system by static condensation procedures similar to the ones discussed in Section 4.1.2.

2.4 DG scheme

The third approximation of the Stokes problem is based on discontinuous approximations of both the velocity and the pressure. Specifically, we use the BR2 formulation for the vector Laplace operator (see [13] and also [32, Section 5.3.2]) together with a stabilized equal order pressure-velocity coupling. Fix a polynomial degreek≥1 and letT ∈ T_h. We define thelocal discrete gradientG^k_T :H¹(T_h)^d → P^k(T)^d×dsuch that, for allv∈H¹(T_h)^d,

∫

T

G_T^k(v):τ=∫

T

∇v_|T :τ− Õ

F∈ F_T^i,D

1 2

∫

F

(n_{T F}⊗nvoT F):τ ∀τ∈ P^k(T)^d×d,

where, for anyF ∈ F_T^i,D, the jump ofvacrossFis defined as nvoT F B

(v|T−v|T⁰ ifF ∈ F_Tⁱ∩ F_Tⁱ0withT,T⁰∈ T_h,T ,T⁰, 2(vT−g_D) ifF ∈ F_T^D.

Introducing, for allF ∈ F_T^i,D, the jump lifting operatorL^k_FT :L²(F)^d → P^k(T)^d×dsuch that, for allϕ ∈L²(F)^dand allτ∈ P^k(T)^d×d,

∫

T

L^k_FT(ϕ):τ=1 2

∫

F

(nT F ⊗ϕ):τ, it holds, for allv∈H¹(T_h)^d,

G^k_T(v)=∇v_|T− Õ

F∈ F_T^i,D

L^k_FT(nvoT F).

Given(u_h,p_h) ∈ P^k(T_h)^d× P^k(T_h), the local residualr_{I I I}^mnt_,T((u_h,p_h);·) : P^k(T)^d → Rof the discrete momentum equation andr^cnt_{I I I,T}((uh,p_h);·): P^k(T) →Rof the discrete mass equation are such that, for allvT ∈ P^k(T)^d and all q_T ∈ P^k(T),

r_{I I I}^mnt_,T((uh,p_h);vT)B

∫

T

G^k_T(uh):∇vT − Õ

F∈ F_T^i,D

∫

F

nT F· {∇uT−η_FL^k_FT(nuhoT F)}_F

·vT

−

∫

T

p_T(∇ ·vT)+ Õ

F∈ F_T^i,D

∫

F

{p_h}_F(vT·nT F)

−

∫

T

f ·v_T − Õ

F∈ F_T^N

∫

F

g_N·v_T,

r_{I I I}^cnt_,T((uh,p_h);q_T)B

∫

T

uT· ∇q_T− Õ

F∈ F_T^i,D

∫

F

{uh}_F·nT Fq_T+ Õ

F∈ F_Tⁱ

h_F

∫

F

np_hoT Fq_T,

where, for allϕ∈H¹(T_h)and allF ∈ F_h, {ϕ}_FB









 1

2 ϕ_|T+ϕ_|T⁰

ifF∈ F_Tⁱ ∩ F_Tⁱ0withT,T⁰∈ T_h,T ,T⁰,

ϕ_|F otherwise,

with the understanding that the average operator acts componentwise when applied to vector and tensor functions, and η_F >

(

max card(F_T),card(F_T⁰)

ifF ∈ F_Tⁱ∩ F_Tⁱ0withT,T⁰∈ T_h,T,T⁰, card(F_T) ifF ∈ F_T^D.

The global residualsr_{I I I,h}^mnt ((uh,p_h);·):P^k(T_h)^d →Randr_{I I I}^cnt_,h((uh,p_h);·):P^k(T_h) →Rare obtained by element- by-element assembly of local residuals.

Scheme III(DG: DG scheme). Find(uh,p_h) ∈ P^k(T_h)^d× P^k(T_h)such that r_{I I I}^mnt_,h (u_h,p_h);v_h= Õ

T∈ T_h

r^mnt_{I I I,T} (u_h,p_h);v_h|T =0 ∀v_h ∈ P^k(T_h)^d, r_{I I I}^cnt_,h (uh,p_h);q_h)= Õ

T∈ T_h

r^cnt_{I I I,T} (uh,p_h);q_h|T)=0 ∀q_h ∈ P^k(T_h).