The mortar spectral element method in domains of operators Part … · The mortar spectral element method in domains of operators Part I: The divergence operator and Darcy’s equations

The mortar spectral element method

in domains of operators

Part I: The divergence operator and Darcy’s equations

by M. AZAIEZ1, F. BEN BELGACEM

2 and C. BERNARDI3

Abstract: The mortar spectral element method is a domain decomposition techniquethat allows for discretizing second- or fourth-order elliptic equations when set in stan-dard Sobolev spaces. The aim of this paper is to extend this method to some problemsformulated in spaces of square-integrable functions with square-integrable divergence. Adiscretization of the equations due to Darcy which model the flow in porous media isproposed. The numerical analysis of the discrete problem is performed and numericalexperiments are presented, they turn out to be in good coherency with the theoreticalresults.

Resume: La methode d’elements spectraux avec joints est une technique de decompositionde domaine permettant de discretiser des equations elliptiques d’ordre 2 ou 4 poses dansdes espaces de Sobolev usuels. Le but de cet article est d’etendre cette methode a cer-tains problemes variationnels formules dans des espaces de fonctions de carre integrable adivergence de carre integrable. On propose une discretisation des equations de Darcy quimodelisent l’ecoulement dans des milieux poreux, on en effectue l’analyse numerique et onpresente des experiences numeriques coherentes avec les resultats de l’analyse.

1Laboratoire TREFLE (UMR C.N.R.S. 8505), Site E.N.S.C.P.B.,

16 avenue Pey Berland, 33607 Pessac Cedex, France.2

Mathematiques pour l’Industrie et la Physique (UMR C.N.R.S. 5640), Universite Paul Sabatier,

118 route de Narbonne, 31062 Toulouse Cedex, France.3

Laboratoire Jacques-Louis Lions, C.N.R.S. & Universite Pierre et Marie Curie,

B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France.

1. Introduction.

The mortar element method, due to Bernardi, Maday and Patera [16], is a domaindecomposition technique which allows for working on general partitions of the domain,without conformity restrictions. It is specially important when combined with spectraltype discretizations, since handling complex geometries from very simple subdomains canbe performed with this method in a very efficient way. It can also be used to couple differentkinds of variational discretizations on the subdomains, such as finite elements or spectralmethods. So it leads to discrete problems which are most often non conforming in theHodge sense, which means that the discrete space is not contained in the variational one.It was firstly analyzed in the case of the two-dimensional Laplace equation [16][17] whichadmits a natural variational formulation in the usual Sobolev spaceH1(Ω) of functions withsquare-integrable first-order derivatives. We also refer to [9] for the first three-dimensionalresults. It was extended [6] to the bilaplacian equation where the variational space isthe standard space H2(Ω) of functions with square-integrable first-order and second-orderderivatives and also to the Stokes problem which is of saddle-point type, however it stillinvolves usual Sobolev spaces. We also quote [7] for an application of the mortar techniqueto weighted Sobolev spaces, in order to handle discontinuous boundary conditions for theNavier–Stokes equations.

However a number of interesting problems involve other types of Hilbert spaces whichare often domains of operators issued from mechanics and physics. Let us quote amongthem the spaces H(div,Ω) of square-integrable vector fields with square-integrable diver-gence and the space H(curl,Ω) of square-integrable vector fields with square-integrablecurl. Up to now, the mortar method has not yet been applied in this case, except when as-sociated with finite element discretizations [8][19][31]. The main difficulty for handling thisnew type of spaces is that the associated space of traces is made of non local distributions.

In this paper, we are interested in the mortar spectral element approximation offunctions in H(div,Ω). The problem that we consider is Darcy’s system which models theflow of a viscous incompressible fluid in a porous medium. It is also involved in Goda’sprojection algorithm [26] for the Navier–Stokes equations, so that analyzing its mortarelement discretization seems important. From its variational formulation in H(div,Ω),we construct a mortar discrete problem relying on spectral techniques (we refer to [4] forthe analysis of its spectral discretization without domain decomposition and to [3] for theanalysis on a conforming decomposition) and prove that it admits a unique solution.

In the next step, we investigate the approximation properties of divergence-free func-tions in the mortar discrete space. Indeed these properties are needed for the numericalanalysis of the discrete problem. Relying on this, we prove a priori error estimates of spec-tral type: the order of convergence only depends on the regularity of the solution, moreprecisely on its local regularity in each subdomain.

The implementation of the mortar technique mainly relies on an appropriate treatmentof the matching conditions on the interfaces that we briefly describe (we refer to [10] foranother way of handling these conditions). We write the resulting linear system and wepresent the algorithm which is used to solve it. Numerical experiments are described, wecheck that they are in good agreement with the theoretical results.

1

Part II of this work is devoted to the mortar spectral element discretization of problemsformulated in H(curl,Ω). The analysis of a vector potential problem is presently underconsideration, and the discretization relies on very similar ideas.

An outline of the paper is as follows.• In Section 2, we recall the main properties of the space H(div,Ω).• In Section 3, we present Darcy’s equations and prove the well-posedness of the equiva-lent variational problem.• Section 4 is devoted to the description of the mortar spectral element discretization ofthese equations, and the well-posedness of the discrete problem is also checked.• In Section 5, we derive some approximation properties of the mortar space, concerningmainly the approximation of divergence-free smooth functions by divergence-free piecewisepolynomial functions.• Error estimates between the exact and discrete solutions are derived from these prop-erties in Section 6.• Finally, we present some numerical experiments in Section 7.

2

2. Some properties of the space H(div,Ω).

Let Ω denote a bounded domain in Rd, d = 2 or 3, with a Lipschitz–continuousboundary. We denote by n the unit outward normal to Ω on ∂Ω. The generic point inΩ is denoted by x = (x, y) in the case d = 2, x = (x, y, z) in the case d = 3, while thecomponents of any vector field v in Rd, are denoted by vx and vy in the case d = 2, vx, vy

and vz in the case d = 3. However we most often state the notation and the proofs in thecase d = 3, since the corresponding results in dimension d = 2 are easier.

Let us first introduce the divergence operator in the case of dimension d = 3:

div v = ∂xvx + ∂yvy + ∂zvz , (2.1)

defined on all functions v in L2(Ω)3 in the distribution sense (as usual, D(Ω) stands forthe space of infinitely differentiable functions with a compact support in Ω):

∀ϕ ∈ D(Ω), 〈div v, ϕ〉 = −

∫

Ω

(vx ∂xϕ+ vy ∂yϕ+ vz ∂zϕ)(x) dx.

This definition is the same in dimension d = 2, when taking vz equal to zero and forgettingthe dependency with respect to z. With this operator, we can associate the space

H(div,Ω) =

v ∈ L2(Ω)d; div v ∈ L2(Ω)

,

provided with the natural norm

‖v‖H(div,Ω) =(

‖v‖2L2(Ω)d + ‖div v‖2

L2(Ω)

)12 . (2.2)

We note that H(div,Ω) is a Hilbert space and we recall from [25, Chap. I, Thm 2.4] or[32, Chap. 1, Thm 1.1] that the space D(Ω)d of restrictions of functions in D(Rd)d to Ωis dense in H(div,Ω). We also prove the trace theorem on H(div,Ω) in an obvious way.

Proposition 2.1. The trace operator: v 7→ v · n, defined from the formula

∀ϕ ∈ H1(Ω), 〈v · n, ϕ〉 =

∫

Ω

(

v · grad ϕ+ (div v)ϕ)

(x) dx, (2.3)

is continuous from H(div,Ω) onto the dual space H− 12 (∂Ω) of H

12 (∂Ω).

Proof: Thanks to the previous density result, we must only check that the trace mappingis onto. With any µ in H− 1

2 (∂Ω), we associate the solution ψ of the Laplace equation withNeumann boundary condition

−∆ψ = λ in Ω,∂nψ = µ on ∂Ω,

where the constant λ is chosen such that (here 〈·, 〉 denotes the duality pairing between

H− 12 (∂Ω) and H

12 (∂Ω))

λmeas (Ω) = −〈µ, 1〉.

3

Clearly, the function v = grad ψ belongs to H(div,Ω) and satisfies v · n = µ, whence theresult.

Remark: Let Γ be a connected part of ∂Ω with a positive measure. Since the extension

by zero is continuous from H1200(Γ) into H

12 (∂Ω), the trace operator: v 7→ v · n is also

continuous from H(div,Ω) into the dual space of H1200(Γ), which we denote by H

1200(Γ)′ (see

[27, Chap. 1, Thm 11.7] for the definition of H1200(Γ)).

We can now define the subspace

H0(div,Ω) =

v ∈ H(div,Ω); v · n = 0 on ∂Ω

.

Proposition 2.1 yields that it is also a Hilbert space. Moreover, it can be proven [25, Chap.I, Thm 2.6][32, Chap. 1, Thm 1.3] that D(Ω)d is dense in H0(div,Ω).

Let us also introduce the curl operator, in the three-dimensional case for brevity ofnotation (the results are the same in dimension d = 2 but simpler): it is defined on smoothfunctions by

curl v =

∂yvz − ∂zvy

∂zvx − ∂xvz

∂xvy − ∂yvx

.

We consider its domain

H(curl,Ω) =

v ∈ L2(Ω)3; curl v ∈ L2(Ω)3

.

Neither the space H(div,Ω) nor the space H0(div,Ω) nor the intersection H(div,Ω) ∩H(curl,Ω) has further regularity properties. But the intersection H0(div,Ω)∩H(curl,Ω)

is continuously imbedded in H12 (Ω)d [20] and, if the domain Ω is convex, in H1(Ω)d [1,

Thm 2.17]. Further results are known [21][22][23] when Ω is a polygon: a function u inH0(div,Ω) ∩H(curl,Ω) can be written

u = ur + gradS, (2.4)

where ur belongs toH1(Ω)2 and S is a linear combination of the singularities of the Laplaceequation provided with Neumann boundary conditions. We recall that each singularity inthe neighbourhood of a corner of the polygon with aperture ω has the form

rπω

(

ϕ(θ) + (log r)p ψ(θ))

,

where r is the distance to the corner, θ the corresponding angular variable, p is equal to0 except when π

ωis an integer where it is equal to 1. Finally, ϕ and ψ are combinations

of cosine functions, in order that gradS satisfies the same nullity condition as u on theboundary edges θ = 0 and θ = ω. More generally, any such function u which has thefurther property

div u ∈ Hs(Ω), curl u ∈ Hs(Ω)3, (2.5)

admits the expansion (2.4) with ur in Hs+1(Ω)2 for all s, 0 < s < 2πω

− 1.

4

3. The continuous Darcy’s equations.

As previously, Ω denotes a bounded domain in Rd, d = 2 or 3, with a Lipschitz–

continuous boundary. We consider Darcy’s equations

u+ grad p = f in Ω,

div u = 0 in Ω,

u · n = 0 on ∂Ω,

(3.1)

where the unknowns are the velocity u and the pressure p.

As noted in [4], system (3.1) admits several equivalent variational formulations, whichlead to different discrete problems. However, in order that the discretization describedbelow can be used in the projection part of Goda’s algorithm for the Stokes problem andwithout restriction for the application to porous media, we have rather work with thefollowing one:

Find (u, p) in H0(div,Ω) × L20(Ω) such that

∀v ∈ H0(div,Ω),

∫

Ω

u · v dx−

∫

Ω

p (div v) dx =

∫

Ω

f · v dx,

∀q ∈ L20(Ω), −

∫

Ω

q (div u) dx = 0,

(3.2)

where for simplicity we assume that the data f belong to L2(Ω)d and L20(Ω) is defined by

L20(Ω) =

q ∈ L2(Ω);

∫

Ω

q dx = 0

.

It follows from the density results quoted in Section 2 that problem (3.2) is equivalent tosystem (3.1).

Problem (3.2) is of saddle-point type. So, we introduce the kernel

V =

v ∈ H0(div,Ω); ∀q ∈ L20(Ω),

∫

Ω

q (div v) dx = 0

. (3.3)

It is readily checked that

V =

v ∈ H0(div,Ω); div v = 0 inΩ

,

so that the norms ‖·‖H(div,Ω) and ‖·‖L2(Ω)d are equivalent on V . This yields the ellipticityof the first bilinear form on V . Moreover, the inf-sup condition of the second bilinear formis an obvious consequence of the standard one (where H0(div,Ω) is replaced by H1

0 (Ω)d,see [25, Chap. I, Cor. 2.4]): There exists a positive constant β such that

∀q ∈ L20(Ω), sup

v∈H0(div,Ω)

−∫

Ωq (div v) dx

‖v‖H(div,Ω)≥ β ‖q‖L2(Ω). (3.4)

Combining all this leads to the well-posedness result.

5

Proposition 3.1. For any data f in L2(Ω)d, problem (3.2) has a unique solution (u, p)in H0(div,Ω) × L2

0(Ω).

The regularity of the solution of problem (3.1) is derived by taking the curl of the firstequation in (3.1), which yields

curl u = curl f in Ω.

Indeed, thanks to the results recalled in Section 2, for any data f in H(curl,Ω), thesolution (u, p) of problem (3.1) belongs to Hs(Ω)d × Hs+1(Ω) with s = 1

2in the general

case and s = 1 if Ω is convex. Moreover, when Ω is a polygon, u admits the expansion(2.4), and, if curl f belongs to Hs(Ω)2, 0 ≤ s < 2π

ω− 1, where ω denotes the largest

angle of Ω (equal to either π2 or 3π

2 in what follows), the regular part ur in this expansionbelongs to Hs+1(Ω)2.

6

4. Discretization of Darcy’s equations.

We now assume that Ω admits a disjoint decomposition into a finite number of (open)rectangles in dimension d = 2, rectangular parallelepipeds in dimension d = 3, denoted byΩk:

Ω =

K⋃

k=1

Ωk and Ωk ∩ Ωk′ = ∅, 1 ≤ k 6= k′ ≤ K. (4.1)

Note that, as indicated in [28], the extension to more complex subdomains leads to similarresults, however they involve very technical arguments that we prefer to avoid in this work.We make the further assumption that the intersection of each ∂Ωk with ∂Ω, if not empty,is a corner, a whole edge or a whole face of Ωk. For 1 ≤ k ≤ K, we denote by Γk,ℓ,1 ≤ ℓ ≤ L(k), the (open) edges in dimension d = 2, faces in dimension d = 3, of Ωk whichare not contained in ∂Ω. We denote by nk the unit outward normal vector to Ωk on ∂Ωk.Note that the decomposition is said to be conforming if the intersection of two differentdomains Ωk is either empty or a corner or a whole edge or face of both of them, howeverwe do not make this assumption since it is a priori not necessary for the mortar method.

Let us now introduce the skeleton S of the decomposition, S = ∪Kk=1∂Ωk \ ∂Ω. As

suggested in [16][17], we choose a disjoint decomposition of this skeleton into mortars γm:

S =M⋃

m=1

γm and γm ∩ γm′ = ∅, 1 ≤ m 6= m′ ≤M, (4.2)

where each γm = Γk(m),ℓ(m) is a whole edge in dimension d = 2, face in dimension d = 3,of a subdomain Ωk, denoted by Ωk(m).

To describe the discrete spaces, for each nonnegative integer n, we define on eachΩk, resp. Γk,ℓ, the space Pn(Ωk), resp. Pn(Γk,ℓ), of restrictions to Ωk, resp. to Γk,ℓ, ofpolynomials with d variables, resp. d − 1 variables (the tangential coordinates on Γk,ℓ),and degree ≤ n with respect to each variable. The discretization parameter δ is then aK–tuple of positive integers (N1, . . . , NK), with each Nk ≥ 2.

From Proposition 2.1, H0(div,Ω) coincides with the space of functions v such thattheir restrictions to each Ωk, 1 ≤ k ≤ K, belong to H(div,Ωk) and their normal tracesvanish on ∂Ω and are continuous through the skeleton S. In analogy with this definition,we define the corresponding discrete space Dδ(Ω) of functions vδ such that:• their restrictions vδ |Ωk

to each Ωk, 1 ≤ k ≤ K, belong to PNk(Ωk)d,

• their normal traces vδ · n vanish on ∂Ω,• the mortar function ϕ being defined on each γm, 1 ≤ m ≤M , by

ϕ|γm= vδ |Ωk(m)

· nk(m),

the following matching condition holds on each edge Γk,ℓ, 1 ≤ k ≤ K, 1 ≤ ℓ ≤ L(k), whichis not a mortar:

∀χ ∈ PNk−2(Γk,ℓ),

∫

Γk,ℓ

(vδ |Ωk· nk + ϕ)(τ )χ(τ ) dτ = 0. (4.3)

7

We also introduce the space Mδ(Ω) of discrete pressures:

Mδ(Ω) =

qδ ∈ L20(Ω); qδ |Ωk

∈ PNk−2(Ωk), 1 ≤ k ≤ K

. (4.4)

Note that there are many different possible choices for this space, however this one isjustified by two arguments: replacing Nk − 2 by Nk would give rise to spurious modeson the pressure (see [4, Lemme 4.1]), so that this choice must be eliminated. If Mδ(Ω) isdefined by (4.4) but with Nk−2 replaced by Nk−1, it does not contain any spurious modefor the pressure of Darcy’s equations, so it can be kept when solving only this equation,however it contains spurious modes for the pressure of the Stokes problem [15, Thm 24.1].So the previous choice seems reasonable, in order to be consistent with this problem whenusing Goda’s algorithm.

Starting from the standard Gauss–Lobatto formula on ]− 1, 1[, we define on each Ωk

and in each direction:• the nodes xk

i and yki , and the weights ρx,k

i and ρy,ki , 0 ≤ i ≤ Nk, in the case of dimension

d = 2,• the nodes xk

i , yki and zk

i , and the weights ρx,ki , ρy,k

i and ρz,ki , 0 ≤ i ≤ Nk, in the case of

dimension d = 3.A discrete product is then introduced on each Ωk by

(uδ, vδ)kδ =

∑Nk

i=0

∑Nk

j=0 uδ(xki , y

kj )vδ(x

ki , y

kj ) ρx,k

i ρy,kj for d = 2,

∑Nk

i=0

∑Nk

j=0

∑Nk

p=0 uδ(xki , y

kj , z

kp )vδ(x

ki , y

kj , z

kp ) ρx,k

i ρy,kj ρz,k

p for d = 3.

This leads to the global discrete product on Ω:

(uδ, vδ)δ =K

∑

k=1

(uδ, vδ)kδ , (4.5)

which coincides with the scalar product of L2(Ω) for all functions uδ annd vδ such thateach product (uδvδ)|Ωk

, 1 ≤ k ≤ K, belongs to P2Nk−1(Ωk). We also define, for 1 ≤ k ≤ K,

Ikδ as the Lagrange interpolation operator on all nodes (xk

i , ykj ), 0 ≤ i, j ≤ Nk, respectively

(xki , y

kj , z

kp ), 0 ≤ i, j, p ≤ Nk, with values in PNk

(Ωk), and finally the global operator Iδ by

(Iδv)|Ωk= Ik

δ v|Ωk, 1 ≤ k ≤ K. (4.6)

The discrete problem is now built from the variational formulation (3.2). For anycontinuous data f on Ω, it reads:

Find (uδ, pδ) in Dδ(Ω) × Mδ(Ω) such that

∀vδ ∈ Dδ(Ω), (uδ, vδ)δ − (pδ, div vδ)δ = (f , vδ)δ,

∀qδ ∈ Mδ(Ω), −(qδ, div uδ)δ = 0.(4.7)

Note however that, thanks to the exactness property of the quadrature formula, we have

∀vδ ∈ Dδ(Ω), ∀qδ ∈ Mδ(Ω), −(qδ, div vδ)δ = −

∫

Ω

qδ (div vδ) dx. (4.8)

8

We first prove the inf-sup condition, that relies on the arguments given in [4, Lemme4.2] and [12, Prop. 3.1] combined via the Boland and Nicolaides technique [18]. SinceDδ(Ω) is not contained in H(div,Ω) in the general case, we introduce the “broken” norm:

‖v‖H(div,∪Ωk) =(

K∑

k=1

‖v‖2H(div,Ωk)

)12

. (4.9)

Proposition 4.1. There exists an integer ND only depending on the decomposition of Ωsuch that, if all the Nk are ≥ ND, the following inf-sup condition

∀qδ ∈ Mδ(Ω), supvδ∈Dδ(Ω)

−∫

Ωqδ(div vδ) dx

‖vδ‖H(div,∪Ωk)≥ βD ‖qδ‖L2(Ω), (4.10)

holds for a positive constant βD depending on the decomposition of Ω but not on δ.

Proof: Let qδ be any function in Mδ(Ω). The idea consists in writing qδ as

qδ = qδ + qδ, with qδ |Ωk=

1

meas(Ωk)

∫

Ωk

qδ(x) dx, 1 ≤ k ≤ K.

This decomposition is orthogonal, in the sense that

‖qδ‖2L2(Ω) = ‖qδ‖

2L2(Ω) + ‖qδ‖

2L2(Ω). (4.11)

Next, since qδ |Ωkhas a null integral on Ωk, it follows from [4, Lemme 4.2] that there exists

a function vk in PNk(Ωk) ∩H0(div,Ωk) such that

−div vk = qδ |Ωkand ‖vk‖H(div,Ωk) ≤ c ‖qδ |Ωk

‖L2(Ωk). (4.12)

So, the function vδ defined by vδ |Ωk= vk, 1 ≤ k ≤ K, belongs to Dδ(Ω) (and even

to H0(div,Ω)). Concerning the function qδ (note that its integral on Ω is equal to theintegral of qδ , hence to zero), it follows from [12, Prop. 3.1] that there exists a function vδ

in H10 (Ω)d, with vδ |Ωk

in PND(Ωk)d, such that

−div vδ = qδ and ‖vδ‖H(div,Ω) ≤ c ‖qδ‖L2(Ω). (4.13)

This function vδ belongs to Dδ(Ω) when all the Nk are ≥ ND.We finally take: vδ = vδ + µvδ, for a positive parameter µ. Indeed, div vδ is orthogonalto qδ (this comes from the Stokes formula applied on each Ωk), so that, from (4.12) and(4.13),

−

∫

Ω

qδ(div vδ) dx ≥ ‖qδ‖2L2(Ω) + µ ‖qδ‖

2L2(Ω) − µ ‖div vδ‖L2(Ω) ‖qδ‖L2(Ω).

Using once more (4.13) yields

−

∫

Ω

qδ(div vδ) dx ≥ ‖qδ‖2L2(Ω) + µ ‖qδ‖

2L2(Ω) − µ c ‖qδ‖L2(Ω) ‖qδ‖L2(Ω)

≥1

2‖qδ‖

2L2(Ω) + µ(1 −

µ c2

2) ‖qδ‖

2L2(Ω).

9

So choosing µ = 1/c2 and using (4.11) gives

−

∫

Ω

qδ(div vδ) dx ≥ c ‖qδ‖2L2(Ω).

Next, from (4.12) and (4.13) combined with (4.11), it is readily checked that

‖vδ‖H(div,∪Ωk) ≤ c ‖qδ‖L2(Ω) + µ c ‖qδ‖L2(Ω) ≤ c ‖qδ‖L2(Ω),

which ends the proof.

Remark: As explained in [12, form. (3.3) & (3.5)], the integer ND can be computedexplicitly as a function of the geometry of decomposition at least in dimension d = 2.

The well-posedness of problem (4.7) follows from Proposition 4.1.

Proposition 4.2. Assume that all the Nk, 1 ≤ k ≤ K, are ≥ ND. For any data fcontinuous on Ω, problem (4.7) has a unique solution (uδ, pδ) in Dδ(Ω) × Mδ(Ω).

Proof: Problem (4.7) results into a square linear system, hence the existence of a solutionfollows from its uniqueness. So we now take f equal to zero. Choosing vδ equal to uδ

yields that(uδ,uδ)δ = 0,

or equivalently, since the weights of the Gauss–Lobatto formula are positive, that eachuδ |Ωk

vanishes in the (Nk + 1)d different points of a tensorized grid. Since it belongs to

PNk(Ωk)d, it is zero. Next Proposition 4.1 yields that pδ is also equal to zero, which ends

the proof.

To conclude, we introduce the discrete kernel

Vδ(Ω) =

vδ ∈ Dδ(Ω); ∀qδ ∈ Mδ(Ω),

∫

Ω

qδ (div vδ) dx = 0

. (4.14)

As usual, it plays a key role in the numerical analysis of problem (4.7). Moreover it is notcontained in the space V introduced in (3.3) for the discretization that we have chosen.

10

5. Approximation of divergence-free functions.

We now intend to estimate the distance of a divergence-free function to the spaceVδ(Ω) introduced in (4.14). Here, we consider separately the cases of dimension d = 2 anddimension d = 3. For each δ = (N1, . . . , NK), we define the parameter µδ

• equal to 1 in the case of a conforming decomposition,• equal to the maximum of the ratios Nk/Nk′ for all adjacent subdomains Ωk and Ωk′ ,1 ≤ k, k′ ≤ K (here, “adjacent” means that Ωk and Ωk′ share a part of an edge in dimensiond = 2, of a face in dimension d = 3).

Proposition 5.1. Assume the function u in V such that each u|Ωk, 1 ≤ k ≤ K, belongs

to Hsk(Ωk)d, sk ≥ 12. In the case of dimension d = 2, there exists a constant c independent

of δ such that

infvδ∈Vδ(Ω)

‖u− vδ‖L2(Ω)d ≤ c µ12

δ

K∑

k=1

N−sk

k ‖u‖Hsk (Ω)d . (5.1)

Proof: Since u is divergence-free, there exists a stream–function ψ in H1(Ω) such thatu = curl ψ, with each ψ|Ωk

in Hsk+1(Ωk) and ψ constant on each connected componentof ∂Ω. It is also known [16, Appendix B] that there exists a function ψδ:• which is equal to ψ on ∂Ω,• which is continuous in all the corners of the Ωk,• such that each ψδ |Ωk

belongs to PNk(Ωk) and satisfies on each Γk,ℓ (here, ϕ stands for

the mortar function associated with ψδ, equal to ψδ |Ωk(m)on each γm)

∀χ ∈ PNk−2(Γk,ℓ),

∫

Γk,ℓ

(ψδ − ϕ)χdτ = 0,

• and finally which satisfies

‖ψ − ψδ‖H1(∪Ωk) ≤ c µ12

δ

K∑

k=1

N−sk

k ‖ψ‖Hsk+1(Ω)d .

Taking vδ = curl ψδ gives estimate (5.1). So, we only must check that it belongs to Dδ(Ω),more precisely that it satisfies the matching condition (4.3). We first observe that, if τdenotes the unit vector directly orthogonal to nk(m), the mortar function associated withvδ is defined by, for 1 ≤ m ≤M ,

ϕ = vδ |Ωk(m)· nk(m) = ∂τψδ |Ωk(m)

= ϕ′.

So, we have on each Γk,ℓ which is not a mortar (recall that ψδ |Ωk− ϕ vanishes at each

endpoint of Γk,ℓ)

∀χ ∈ PNk−2(Γk,ℓ),

∫

Γk,ℓ

(vδ |Ωk· nk + ϕ)χdτ = −

∫

Γk,ℓ

(∂τψδ |Ωk− ϕ′)χdτ

=

∫

Γk,ℓ

(ψδ |Ωk− ϕ)χ′ dτ = 0,

11

which is the desired matching condition. Moreover, in the case of a conforming decom-position, there exists [16, §3.1] a function ψδ in H1(Ω) which satisfies all the previousproperties with µδ = 1.

Now, we are interested in the case of dimension d = 3, which is much more complex.The proof mainly relies on special interpolation operators, which has been introduced byNedelec [29] in the finite element framework and extended to spectral methods in [11].We begin by proving approximation results involving the mortar space Cδ(Ω) which is theanalogue of Dδ(Ω) for the approximation of functions in H(curl,Ω). More precisely andwith the notation of Section 4, there also in the case of dimension d = 3, it is the space offunctions vδ such that:• their restrictions vδ |Ωk

to each Ωk, 1 ≤ k ≤ K, belong to PNk(Ωk)3,

• their tangential traces vδ × n vanish on ∂Ω,• the mortar function ϕ being defined on each γm by

ϕ|γm= vδ |Ωk(m)

× nk(m),

the following matching condition holds on each edge Γk,ℓ, 1 ≤ k ≤ K, 1 ≤ ℓ ≤ L(k), whichis not a mortar:

∀χ ∈ PNk−2(Γk,ℓ)3,

∫

Γk,ℓ

(vδ |Ωk× nk + ϕ)(τ ) · χ(τ ) dτ = 0. (5.2)

We first consider the case of a conforming decomposition, with some restrictions on thechoice of the Nk.

Lemma 5.2. Assume that the function u is equal to curl ξ for a function ξ in H(curl,Ω),with null tangential traces on ∂Ω, and is such that each u|Ωk

, 1 ≤ k ≤ K, belongs to

Hsk(Ωk)d, sk ≥ 32 . In the case of dimension d = 3, if the decomposition is conforming and

if moreover for each mortar γm, 1 ≤ m ≤M , which is a face of both Ωk(m) and Ωk, Nk(m)

is ≥ Nk, there exists a constant c independent of δ such that

infξδ∈Cδ(Ω)

‖u− curl ξδ‖L2(Ω)3 ≤ cK

∑

k=1

N−sk

k ‖u‖Hsk (Ω)d . (5.3)

Proof: It follows from [11, Thm 4.9] that there exists, for 1 ≤ k ≤ K and for sk ≥ 32 , a

polynomial ξkδ in PNk

(Ω)3 such that ξkδ × n vanishes on ∂Ω ∩ ∂Ωk and that

‖curl ξ − curl ξkδ‖L2(Ωk)3 ≤ cN−sk

k ‖u‖Hsk (Ωk)3 . (5.4)

Moreover, this function ξkδ satisfies, for 1 ≤ ℓ ≤ L(k),

∀q ∈ PNk−2(Γk,ℓ)3,

∫

Γk,ℓ

(ψ − ξkδ ) × n · q dτ = 0. (5.5)

In the case of a conforming decomposition and thanks to the assumption on the Nk, equa-tion (5.5) implies that the function ξδ defined by ξδ |Ωk

= ξkδ , 1 ≤ k ≤ K, belongs to

12

Cδ(Ω), whence the result.

In the case of a nonconforming decomposition, the previous approximation ξδ does notbelong to Cδ(Ω) but can be used in the first step of the construction of an approximationin this space. However proving this result seems to require the same restrictions on thedecomposition as for more standard approximation results in H1(Ω), see [10, Chap. 3]or [13]. Moreover the final result does not take into account the local resularity of thefunction u. So we have rather not present the result here.

A further argument seems necessary to prove the desired approximation result in thecase d = 3.

Proposition 5.3. Assume the function u in V such that each u|Ωk, 1 ≤ k ≤ K, belongs

to Hsk(Ωk)d, sk ≥ 32. In the case of dimension d = 3, if the decomposition is conforming

and if moreover for each mortar γm, 1 ≤ m ≤ M , which is a face of both Ωk(m) and Ωk,Nk(m) is ≥ Nk, there exists a constant c independent of δ such that

infvδ∈Vδ(Ω)

‖u− vδ‖L2(Ω)d ≤ cK

∑

k=1

N−sk

k ‖u‖Hsk (Ωk)d , (5.6)

Proof: It follows from [1, §3.b] that there exists a vector potential ξ such that u = curl ξ

(it does not necessarily satisfy the boundary condition ξ×n = 0 on ∂Ω when Ω is multiply-connected). However the arguments of Lemma 5.2 still hold in this case, and estimate (5.3)yields that the function vδ = curl ξδ satisfies the desired property (5.6). Moreover, it canbe checked [11, §4] that, on each face Γ of Ωk, curl ξδ · n is equal to the orthogonalprojection (in L2(Γ)) of u · n onto PN−1(Γ). So the jump of curl ξδ · n through Γ (orits trace if Γ is contained in ∂Ω) is zero, and the function vδ belongs to Vδ(Ω).

Note that, for a conforming decomposition, the condition Nk(m) ≥ Nk is not at allrestrictive since the choice of the mortar side is fully arbitrary. However enforcing theconformity of the decomposition is a little disappointing and the conditions sk ≥ 3

2 arenot satisfied by all solutions (it is likely but not proven that estimate (5.6) holds for allsk ≥ 0).

13

6. Error estimates.

We prove an error estimate, first for the velocity, second for the pressure. Let wδ beany function in the kernel Vδ(Ω). Multiplying the first line of (3.1) by wδ gives

∫

Ω

u · wδ dx−

∫

Ω

p(div wδ) dx =

∫

Ω

f · wδ dx−

∫

S

[wδ · n] p dτ ,

where the notation [·] means the jump across S. Using the definition of Vδ(Ω) thus implies,for any qδ in Mδ(Ω),

∫

Ω

u · wδ dx−

∫

Ω

(p− qδ)(div wδ) dx =

∫

Ω

f · wδ dx−

∫

S

[wδ · n] p dτ . (6.1)

Next, we recall from the standard property of the Gauss–Lobatto formula [15, Rem. 13.3]that

∀zδ ∈ PNk(Ωk), ‖zδ‖

2L2(Ωk) ≤ (zδ, zδ)

kδ ≤ 3d ‖zδ‖

2L2(Ωk). (6.2)

So, we have for any vδ in Vδ

‖uδ − vδ‖2L2(Ω)d ≤ (uδ − vδ,uδ − vδ)δ.

Adding (6.1) with wδ = uδ − vδ and subtracting the first line of (4.7) leads to

‖uδ − vδ‖2L2(Ω)d ≤

∫

Ω

(u− vδ) · (uδ − vδ) dx

+

∫

Ω

vδ · (uδ − vδ) dx− (vδ,uδ − vδ)δ

−

∫

Ω

(p− qδ)(div (uδ − vδ)) dx

+ (f ,uδ − vδ)δ −

∫

Ω

f · (uδ − vδ) dx

+

∫

S

[(uδ − vδ) · n] p dτ .

The last idea consists in integrating once more by parts the term in the third line,which gives

−

∫

Ω

(p− qδ)(div (uδ − vδ)) dx =

∫

Ω

(uδ − vδ) · grad (p− qδ) dx

−

∫

S

[(uδ − vδ) · n] (p− qδ) dτ −

∫

S

(uδ − vδ) · n [p− qδ] dτ ,

where, in the decomposition of the jump in the last line, p− qδ is taken on the non-mortarside of S. Inserting this in the previous line and noting that the jump [(uδ − vδ) · n] is

14

orthogonal to qδ by definition of Dδ(Ω) yields

‖uδ − vδ‖2L2(Ω)d ≤

∫

Ω

(u− vδ) · (uδ − vδ) dx

+

∫

Ω

vδ · (uδ − vδ) dx− (vδ,uδ − vδ)δ

+

∫

Ω

(uδ − vδ) · grad (p− qδ) dx

+ (f ,uδ − vδ)δ −

∫

Ω

f · (uδ − vδ) dx

−

∫

S

(uδ − vδ) · n [p− qδ] dτ .

(6.3)

Next, let Πδ− be defined by (Πδ−z)|Ωk= Πk

δ−z|Ωk, where Πk

δ− stands for the orthog-onal projection operator from L2(Ωk) onto PNk−1(Ωk). By adding and subtracting thefunction Πδ−u, we deduce from the exactness of the quadrature formula and (6.2) that

∫

Ω

vδ · (uδ − vδ) dx− (vδ,uδ − vδ)δ

≤ (1 + 3d) (‖u− vδ‖L2(Ω)d + ‖u− Πδ−u‖L2(Ω)d) ‖uδ − vδ‖L2(Ω)d .

(6.4)

Similarly, we also have

(f ,uδ − vδ)δ −

∫

Ω

f · (uδ − vδ) dx

≤ (1 + 3d) (‖f − Iδf‖L2(Ω)d + ‖f − Πδ−f‖L2(Ω)d) ‖uδ − vδ‖L2(Ω)d .

(6.5)

We refer to [15, Thm 7.1] and [15, Thm 14.2] for the approximation properties of theoperators Πδ− and Iδ, respectively.

To estimate the last term in (6.3), we need a lemma. For a while, we work on the modeldomain Σ =] − 1, 1[d and we use the Legendre polynomials Ln, n ≥ 0: each polynomialLn has degree n, is orthogonal to the other ones in L2(−1, 1) and satisfies

Ln(±1) = (±1)n,

∫ 1

−1

L2n(ζ) dζ =

1

n+ 12

. (6.6)

Lemma 6.1. There exists a constant c independent of N such that, for any polynomialwN in PN (Σ) and for any (d− 1)–face Γ of Σ,

‖wN‖L2(Γ) ≤ cN ‖wN‖L2(Σ). (6.7)

Proof: We check this inequality in the case d = 3, since the case d = 2 is simpler. First,we write any wN in PN (Σ) as

wN (x, y, z) =N

∑

m=0

N∑

n=0

N∑

p=0

αmnp Lm(x)Ln(y)Lp(z),

15

so that

‖wN‖2L2(Σ) =

N∑

m=0

N∑

n=0

N∑

p=0

α2mnp

(m+ 12 )(n+ 1

2 )(p+ 12 ). (6.8)

On the other hand, on the face Γ of equation x = ±1 for instance,

wN |Γ(y, z) =

N∑

m=0

N∑

n=0

N∑

p=0

(±1)mαmnp Ln(y)Lp(z),

so that

‖wN‖2L2(Γ) =

N∑

n=0

N∑

p=0

(

N∑

m=0

(±1)mαmnp

)2 1

(n+ 12)(p+ 1

2).

Using a Cauchy–Schwarz inequality then implies

‖wN‖2L2(Γ) =

(

N∑

m=0

(m+1

2))

N∑

n=0

N∑

p=0

(

N∑

m=0

α2mnp

m+ 12

) 1

(n+ 12 )(p+ 1

2 ),

which, when compared with (6.8), yields the desired result.

Applying Lemma 6.1 leads to, for the parameter µδ introduced at the beginning ofSection 5,

|

∫

S

(uδ − vδ) · n [p− qδ] dτ | ≤ c µδ

(

K∑

k=1

L(k)∑

ℓ=1

Nk ‖p− qδ‖L2(Γk,ℓ)‖uδ − vδ‖L2(Ωk)d

)

,

whence

|

∫

S

(uδ − vδ) · n [p− qδ] dτ | ≤ c µδ

(

K∑

k=1

L(k)∑

ℓ=1

N2k ‖p− qδ‖

2L2(Γk,ℓ)

)12 ‖uδ − vδ‖L2(Ω)d . (6.9)

Inserting (6.4), (6.5) and (6.9) into (6.3) and using a triangle inequality yields

‖u− uδ‖L2(Ω)d ≤ c(

‖u− vδ‖L2(Ω)d + ‖u− Πδ−u‖L2(Ω)d

+(

K∑

k=1

‖p− qδ‖2H1(Ωk) + µ2

δ

L(k)∑

ℓ=1

N2k ‖p− qδ‖

2L2(Γk,ℓ)

)12

+ ‖f − Iδf‖L2(Ω)d + ‖f − Πδ−f‖L2(Ω)d

)

.

(6.10)

Next, we choose qδ by: qδ |Ωk= Π1,k

δ p, where Π1,kδ stands for the orthogonal projection

operator from H1(Ωk) onto PNk−2(Ωk) defined by

∀qkδ ∈ PNk−2(Ωk),

∫

Ωk

grad (p−Π1,kδ p) · grad qk

δ dx = 0 and

∫

Ωk

(p−Π1,kδ p) dx = 0.

16

Indeed, the following approximation properties of this operator are well-known [15, Thm7.3]: if the function p|Ωk

belongs to Hsk+1(Ωk), sk ≥ 0,

‖p− Π1,kδ p‖H1(Ωk) +Nk ‖p− Π1,k

δ p‖L2(Ωk) ≤ cN−sk

k ‖p‖Hsk+1(Ωk), (6.11)

while estimating the norm of ‖p − Π1,kδ p‖L2(Γk,ℓ) results from a duality argument and is

proven in the following lemma.

Lemma 6.2. For 1 ≤ k ≤ K, there exists a constant c independent of N such that, if thefunction p|Ωk

belongs to Hsk+1(Ωk), sk ≥ 0, and for 1 ≤ ℓ ≤ L(k),

‖p− Π1,kδ p‖L2(Γk,ℓ) ≤ cN

−sk−12

k ‖p‖Hsk+1(Ωk). (6.12)

Proof: The duality argument reads

‖p− Π1,kδ p‖L2(Γk,ℓ) = sup

g∈L2(Γk,ℓ)

∫

Γk,ℓ(p− Π1,k

δ p)g dτ

‖g‖L2(Γk,ℓ).

So, for any g in L2(Γk,ℓ), denoting by g its extension by zero to ∂Ωk, we consider thefollowing Neumann problem

−∆q = − 1meas(Ωk)

∫

Γk,ℓg(τ ) dτ in Ωk,

∂nq = g on ∂Ωk.

This problem admits a solution q in H1(Ωk) ∩ L20(Ωk). Moreover, it belongs to H

32 (Ωk)

[23] and satisfies‖q‖

H32 (Ωk)

≤ c ‖g‖L2(Γk,ℓ). (6.13)

So, the desired estimate follows from the duality formula

∫

Γk,ℓ

(p− Π1,kδ p)g dτ =

1

meas(Ωk)

∫

Γk,ℓ

g(τ ) dτ

∫

Ωk

(p− Π1,kδ p) dx

+

∫

Ωk

grad (p− Π1,kδ p) · grad (q − Π1,k

δ q) dx,

together with (6.11) and (6.13).

Remark: In the case of a conforming decomposition, a continuous approximation qδ of p,i.e. in Mδ(Ω) ∩H1(Ω), can be built by standard arguments [16, §3.1], which satisfies

‖p− qδ‖H1(Ω) ≤ cK

∑

k=1

N−sk

k ‖p‖Hsk+1(Ωk). (6.14)

Of course, with this choice, we have

∫

S

(uδ − vδ) · n [p− qδ] dτ = 0, (6.15)

17

so that estimate (6.10) can be replaced by the simpler one

‖u− uδ‖L2(Ω)d ≤ c(

‖u− vδ‖L2(Ω)d + ‖u− Πδ−u‖L2(Ω)d

+ ‖p− qδ‖H1(Ω) + ‖f − Iδf‖L2(Ω)d + ‖f − Πδ−f‖L2(Ω)d

) (6.16)

We are in a position to write the first error estimate, by inserting (6.11) and (6.12) into(6.10) or (6.14) into (6.16) and using the approximation properties stated in Propositions5.1 and 5.3.

Theorem 6.3. Assume the data f such that each f|Ωk,, 1 ≤ k ≤ K, belongs to Hσk(Ωk)d,

σk >d2 , and the solution (u, p) of problem (3.1) such that each (u|Ωk

, p|Ωk), 1 ≤ k ≤ K,

belongs to Hsk(Ωk)d ×Hsk+1(Ωk), sk ≥ d− 32 . Then, in the two cases

(i) in dimension d = 2,(ii) in dimension d = 3, if the decomposition is conforming and if moreover for each mortarγm, 1 ≤ m ≤M , which is a face of both Ωk(m) and Ωk, Nk(m) is ≥ Nk,the following error estimate holds between this velocity u and the velocity uδ of problem(4.7):

‖u−uδ‖L2(Ω)d

≤ c

K∑

k=1

(

λk µδ N−sk

k (‖u‖Hsk (Ωk)d + ‖p‖Hsk+1(Ωk)) +N−σk

k ‖f‖Hσk (Ωk)d

)

,(6.17)

where each λk, 1 ≤ k ≤ K, is equal to 1 if the decomposition is conforming, to N12

k

otherwise.

Estimate (6.17) is not fully optimal, except in the case of a conforming decompositionand with the further assumption that the mortars are chosen on the side of the smallestNk in dimension 3. However, when µδ is bounded (this is most often the case in practical

situations), the lack of optimality is only N12

k .

In the two-dimensional case of a polygon Ω, a more explicit estimate can be deducedfrom the previously quoted regularity results. We briefly explain the main arguments ofthe proof and refer to [14] for details.

Corollary 6.4. In the two-dimensional case of a polygon Ω, assume the data f such thateach f|Ωk

, 1 ≤ k ≤ K, belongs to Hσk(Ωk)2, σk > 1. Then, the following error estimateholds between the velocity u of problem (3.1) and the velocity uδ of problem (4.7):

‖u− uδ‖L2(Ω)2 ≤ c µδ

K∑

k=1

λk Ek ‖f‖Hσk (Ωk)2 , (6.18)

for the λk introduced in Theorem 6.3 and where the Ek, 1 ≤ k ≤ K, are given by

Ek =

N−σk

k if Ωk contains no corner of Ω,

maxN−σk

k , N−4k (logNk)

32 if Ωk contains a corner of Ω

but no nonconvex corner,

maxN−σk

k , N− 4

3

k (logNk)12 if Ωk contains a nonconvex corner of Ω.

(6.19)

18

Proof: Going back to inequality (6.10), we keep the same estimate for the terms involvingf , however we further investigate the approximation of u and p. Indeed,1) when Ωk contains no corner of Ω, the regularity of (u, p) on Ωk only depends on the dataf , namely it belongs to Hσk(Ωk)2 ×Hσk+1(Ωk). So the desired approximation propertyfollows from the same arguments as previously.2) when Ωk contains a corner of Ω with angle ω, we derive from (2.4) that

u|Ωk= ur + gradS, p|Ωk

= pr − S,

where (ur, pr) belongs to Hs(Ωk)2 × Hs+1(Ωk) for s ≤ σk and s < 2πω

. The idea nowconsists in looking for separate approximations of ur and pr and of S. Indeed, the approx-imation properties of ur and pr are the same as stated previously, say for instance

‖pr − Π1,kδ pr‖H1(Ωk) ≤ cN−s

k ‖pr‖Hs+1(Ωk).

However when σk is ≥ 2πω

, we note that the constant in the previous line is independentof s, use the further estimate, for 0 < ε < 1

2 ,

‖pr‖H

2πω

−ε(Ωk)≤ c ε−

12 ‖f‖Hσk (Ωk)2 ,

and take ε = (logNk)−1, which leads to

‖pr − Π1,kδ pr‖H1(Ωk) ≤ cN

− 2πω

k (logNk)12 ‖f‖Hσk (Ωk)2 .

Moreover, as explained in Section 2, the restriction of the function S to a small enoughneighbourhood of the largest corner of Ω contained in Ωk can be written

rπω

(

ϕ(θ) + (log r)p ψ(θ))

,

with p equal to 1 or 0 according as ω is equal to π2 or 3π

2 . The local approximationproperties of these functions are well-known [14, §3]

‖S − Π1,kδ S‖H1(Ωk) ≤ cN

− 2πω

k (logNk)p+ 12 . (6.20)

This, combined with the same duality argument as in Lemma 6.2, an analogous evaluationof u−Πδ−u and the construction of a global function vδ in Vδ(Ω) (which is rather simplein dimension d = 2, see Proposition 5.1), leads to the desired estimate.

Estimating the error on the pressure is now easy.

Theorem 6.5. If the assumptions of Theorem 6.3 are satisfied and in cases (i) and (ii) ofthis theorem, the following error estimate holds between the pressure p of problem (3.1)and the pressure pδ of problem (4.7):

‖p− pδ‖L2(Ω) ≤ c

K∑

k=1

(

λk µδ N−sk

k (logNk)ν2 (‖u‖Hsk (Ωk)d + ‖p‖Hsk+1(Ωk))

+N−σk

k ‖f‖Hσk (Ωk)d

)

,

(6.21)

19

for the λk introduced in Theorem 6.3, with ν equal to zero in the case of a conformingdecomposition and to 1 otherwise.

Proof: From the inf-sup condition (4.10), we derive that, for any qδ in Mδ(Ω),

βD ‖pδ − qδ‖L2(Ω) ≤ supvδ∈Dδ(Ω)

−∫

Ω(pδ − qδ)(div vδ) dx

‖vδ‖H(div,∪Ωk).

In order to evaluate −∫

Ω(pδ − qδ)(div vδ) dx, we first use the discrete problem (4.7):

−

∫

Ω

(pδ − qδ)(div vδ) dx = (f , vδ)δ − (uδ, vδ)δ +

∫

Ω

qδ(div vδ) dx.

Next, we apply equation (3.1) to the function vδ, integrate by parts and add it to theprevious line. This yields

−

∫

Ω

(pδ − qδ)(div vδ) dx = (f, vδ)δ −

∫

Ω

f · vδ dx

+

∫

Ω

(u− uδ) · vδ dx+

∫

Ω

uδ · vδ dx− (uδ, vδ)δ

−

∫

Ω

(p− qδ)(div vδ) dx+

∫

S

[vδ · n]p dx.

Using the same arguments as in (6.4) and (6.5) together with a triangle inequality yields

‖p− pδ‖L2(Ω) ≤ c(

‖p− qδ‖L2(Ω) + ‖u− uδ‖L2(Ω)d + ‖u− Πδ−u‖L2(Ω)d

+ ‖f − Iδf‖L2(Ω)d + ‖f − Πδ−f‖L2(Ω)d

+ supvδ∈Dδ(Ω)

∫

S[vδ · n]p dx

‖vδ‖H(div,∪Ωk)

)

.

(6.22)

All the terms in the right-hand side have been estimated previously, except the last onewhich represents the consistency error. To evaluate it, we use a partition of S into nonmortar edges or faces. Indeed, on each Γk,ℓ which is not a mortar and in the case of anon-conforming decomposition, we derive from the matching condition (4.3) that, for anyrδ in PNk−2(Γk,ℓ) and for any positive and small enough ε,

∫

Γk,ℓ

[vδ · n]p dx =

∫

Γk,ℓ

[vδ · n](p− rδ) dx

≤ c ε−12

(

∑

k′∈Ck,ℓ

‖vδ · n‖H

−12+ε(∂Ωk′)

)

‖p− rδ‖H

12−ε(Γk,ℓ)

,

where Ck,ℓ is the set of indices k′, 1 ≤ k′ ≤ K, such that the intersection of ∂Ωk′ and Γk,ℓ

has a positive measure. The term c ε−12 represents the maximum norm of the extensions

by zero from H12−ε(Γk,ℓ) into H

12−ε(∂Ωk′) (this can be established with c independent of

ε by using intrinsic norms on these spaces). Thanks to formula (2.3) combined with aninverse inequality, we have

‖vδ · n‖H

−12+ε(∂Ωk′ )

≤ ‖vδ‖Hε(Ωk′)d + ‖div vδ‖L2(Ωk′) ≤ cN2εk′ ‖vδ‖H(div,Ωk′).

20

Next, we choose rδ equal to the image of p by the projection operator in H12−ε(Γk,ℓ), which

gives‖p− rδ‖

H12−ε(Γk,ℓ)

≤ N−ε−sk

k ‖p‖Hsk+1(Γk,ℓ).

Combining all this and taking ε equal to 1log Mk,ℓ

, where Mk,ℓ denotes the maximum of

the Nk′ , k′ ∈ Ck,ℓ, yields the desired estimate. The same arguments hold in the case of a

conforming decomposition with ε = 0 and c ε−12 replaced by c.

We skip the proof of the corollary which is now obvious.

Corollary 6.6. In the two-dimensional case of a polygon Ω, if the assumptions of Corollary6.4 are satisfied, the following error estimate holds between the pressure p problem (3.1)and the pressure pδ of problem (4.7):

‖p− pδ‖L2(Ω) ≤ c µδ

K∑

k=1

λk Ek (log Nk)ν2 ‖f‖Hσk (Ωk)2 , (6.23)

for the λk and ν introduced in Theorems 6.3 and 6.5 and the Ek defined in (6.19).

Estimates (6.17) and (6.21) are fully optimal in the case of a conforming decompo-

sition, and the fact that each u|Ωkmust belong to Hd− 3

2 (Ωk) is not at all restrictive indimension d = 2, see Section 2, however weakening this condition in dimension d = 3 is aninteresting question (note however that the solution (u, p) satisfies the desired regularityproperty at least for smooth data and for all Ωk such that ∂Ωk does not intersect an edgeor a corner of Ω). The results are a little more disppointing in the case of a non-conformingdecomposition in dimension d = 2. However, if µδ is bounded independently of δ (this is

the case in most practical situations), the lack of optimality is only of order (Nk log Nk)12 .

No convergence result is proven in the case of dimension 3 and with a non-conformingdecomposition. However numerical experiments show this convergence in the case ofsecond-order elliptic problems, see [24].

21

7. Numerical algorithms and experiments.

We briefly explain how to write the discrete problem described in Section 4 as asquare linear system, and we propose an algorithm for solving it. Although the matchingconditions through the skeleton are most often handled via the introduction of a Lagrangemultiplier, see [10], we have chosen to construct a basis of the discrete velocity space madeof functions that satisfy these conditions.

The whole unknowns of the discrete system are given by• the vector U of the values of uδ at all nodes (xk

i , ykj ), 0 ≤ i, j ≤ Nk, 1 ≤ k ≤ K, in

dimension d = 2, (xki , y

kj , z

kp ), 0 ≤ i, j, p ≤ Nk, 1 ≤ k ≤ K, in dimension d = 3,

• the vector P of the values of pδ at all nodes (xki , y

kj ), 1 ≤ i, j ≤ Nk − 1, 1 ≤ k ≤ K,

in dimension d = 2, (xki , y

kj , z

kp ), 1 ≤ i, j, p ≤ Nk − 1, 1 ≤ k ≤ K, in dimension d = 3,

minus one node (pδ is assumed to be zero in this node, in order to “fix” the constant andis enforced to belong to L2

0(Ω) in a postprocessing step).Next, the coefficients of u in the basis of Dδ(Ω) are now computed via the introduction ofa rectangular matrix Q in order to enforce the boundary and matching conditions, whichleads to a new vector QU . Problem (4.7) is now equivalent to the following square linearsystem

(

QTAQ QTBBTQ 0

) (

UP

)

=

(

QTAF0

)

. (7.1)

The matrix A is fully diagonal, its diagonal terms are the ρx,ki ρy,k

j or the ρx,ki ρy,k

j ρz,kp

according to the dimension. The matrix B is only block-diagonal, with K blocks Bk onthe diagonal, one for each Ωk. Each Bk represents the local discrete gradient, and BT

k thelocal discrete divergence.

System (7.1) is solved by using the Schwarz algorithm of Dirichlet-Neumann typefor handling the domain decomposition, see [30, §1.3]. Each local problem is invertedvia Uzawa algorithm, and a direct Choleski factorization is used to compute the discretepressure (see [2]).

For the first numerical experiment, we consider the two-dimensional L-shaped domainΩ =]− 1, 3[2\[1, 3[2, partitioned into 3 subdomains illustrated in Figure 1 (left panel). Foran integer N ≥ 6, the N1 associated with the middle square subdomain Ω1 =] − 1, 1[2 isequal to N while N2 and N3 are equal to N − 5. We use our spectral method to copputean approximation of the analytical solution (u, p) given by

u(x, y) =

(

− sin(πx) cos(πy)cos(πx) sin(πy)

)

, p(x, y) = sin(

π(x+ y))

. (7.2)

Figure 1 (right panel) depicts, in a semi-logarithmic scale, the curves of the errors‖u − uδ‖L2(Ω)2 and ‖p − pδ‖L2(Ω) as a function of N , for N varying from 6 to 22. Ascan be previewed from estimates (6.17) and (6.21), the convergence rate here is exponen-tial despite the nonconformity of the discretization.

22

−1 0 1 2 3−1

0

1

2

3

6 8 10 12 14 16 18 20 22N

10−15

10−12

10−9

10−6

10−3

100

log(e

rror)

eL2(velocity)eL2(pressure)

Figure 1: The nonmatching grids for N=22 and the velocity

and pressure error curves for a smooth solution.

−1 0 1 2 3−1

0

1

2

3

8 16 24 32 40 48 5664log(N)

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

log(

erro

r)

eL2(velocity)eL2(pressure)

Figure 2: The matching grids for N=32 and the velocity

and pressure error curves for a nonsmooth solution

Next, we are interested in the computation of a nonmooth solution in the same domainΩ with the same decomposition. We take all the Nk equal to N , as illustrated in Figure 2(left part). The data f being given by

f(x, y) =

(

y0

)

. (7.3)

We calculate the reference solution (ur, pr) on the conforming partition with N equal to128, which is represented in Figures 3 and 4. Figure 2 (right part) presents the curvesof the errors ‖ur − uδ‖L2(Ω)2 and ‖pr − pδ‖L2(Ω) as a function of N , in bilogarithmicscales and for N varying from 8 to 64. The convergence order expected by the theory,see estimates (6.18) and (6.23), is N− 4

3 . The slopes of the curves are −2.1 and −4.5, sothey are better than the theoretical prediction (we refer to [5] for the first observation of

23

this superconvergence phenomenon). The same numerical test is performed in [3] usingstaggered grids and similar trends are observed

−1 0 1 2 3−1

0

1

2

3

−3.18

−2.52

−1.85

−1.19

−0.527

0.135

0.798

1.46

2.12

2.79

3.45

−1 0 1 2 3−1

0

1

2

3

−3.18

−2.52

−1.85

−1.19

−0.527

0.135

0.798

1.46

2.12

2.79

3.45

Figure 3: The isovalues of the two components of the velocity

−1 0 1 2 3−1

0

1

2

3

−4.22

−3.43

−2.65

−1.87

−1.08

−0.302

0.481

1.26

2.05

2.83

3.61

Figure 4: The isovalues of the pressure

We finally consider the square Ω =]0, 2[2, with a nonconforming decomposition intotwo squares Ω1 =]0, 1[2 and Ω2 =]1, 2[×]0, 1[ and a rectangle Ω3 =]0, 2[×]1, 2[. For aninteger N ≥ 8, we take all the Nk equal to N , see Figure 5.

We work successively with the solutions (u, p) given by

u(x, y) =

(

− sin(πx) cos(πy)cos(πx) sin(πy)

)

, p(x, y) = exp 2x(2 − x)(x+ 1) exp(−2y), (7.4)

u(x, y) =

(

− sin(πx) cos(πy)cos(πx) sin(πy)

)

, p(x, y) = (x− 1)x32 (2 − x)

32 exp(−2y). (7.5)

In Figure 6 are plotted, the curves of the errors ‖u−uδ‖L2(Ω)2 and ‖p− pδ‖L2(Ω) for bothcases as a function of N . For the smooth solution, a linear/logarithmic scale is used andwe observe that the exponential dacaying of the error is preserved despite the nonconform-ing domain decompsotion. For the nonsmooth soution rather a full logarithmic scale is

24

adopted, we observe the good convergence of the discretization. Due to the nonconformityof the discretization, the expected rate is 7

2, up to some logarithmic terms, see estimates

(6.18) and (6.23), while the experimental one is approximatively 3.4. The results can thenbe considered in good agreement with the theoretical predictions.

0 1 20

1

2

Figure 5: The nonmatching grids for a nonconforming decomposition with N=32

4 8 12 16 20 24 28 32 36N

10−12

10−10

10−8

10−6

10−4

10−2

100

102

log(

erro

r)

eL2(velocity)eL2(pressure)

8 16 24 32 40 48 56 64log(N)

10−7

10−6

10−5

10−4

10−3

10−2

log(

erro

r)

eL2(velocity)eL2(pressure)

Figure 6: The error curves for a smooth solution (left panel) and a nonsmooth solution (right panel).

25

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