Mortar spectral element discretization of the Stokes problem in axisymmetric domains

Mortar spectral element discretization of the Stokes problem

in axisymmetric domains

Saloua Mani Aouadi

, Christine Bernardi

, and Jamil Satouri

Abstract

The Stokes problem in a tridimensional axisymmetric domain results into a countable family of two-dimensional problems when using the Fourier coeffi- cients with respect to the angular variable. Relying on this dimension reduction, we propose and study a mortar spectral element discretization of the problem.

Numerical experiments confirm the efficiency of this method.

R´esum´e

L’utilisation des coefficients de Fourier par rapport `a la variable angulaire permet de r´eduire le probl`eme de Stokes dans un ouvert tridimensionnel axisym´etrique `a une famille d´enombrable de probl`emes bidimensionnels. Grˆace `a cette r´eduction de dimension, nous proposons une discr´etisation de ce probl`eme par la m´ethode d’´el´ements spectraux avec joints et nous en effectuons l’analyse num´erique. Des exp´eriences num´eriques confirment l’int´erˆet de cette m´ethode.

Key words: Axisymmetric domains, mortar method, spectral element method, Stokes equation.

1Faculty of Sciences of Tunis, University of Tunis El Manar, 2060 Tunis, Tunisia.

2Laboratoire Jacques-Louis Lions, C.N.R.S. & Universit´e Pierre et Marie Curie, Boˆıte courrier 187, 4 place Jussieu, 75252 Paris Cedex 05, France.

e-mails: [email protected], [email protected], [email protected]

1 Introduction

The Stokes system







−4u˘+grad p˘=f˘ in ˘Ω, divu˘= 0 in ˘Ω,

˘

u = ˘g on ∂Ω,˘

(1.1) models the laminar flow of a viscous incompressible fluid in a domain ˘Ω, when subjected to a density of forcesf˘and with boundary datag, the unknowns being˘ the velocity u˘ and the pressure ˘p of the fluid. We are specifically interested in the case where ˘Ω is tri-dimensional and axisymmetric, i.e. invariant by rotation around an axis. Note that this type of geometry appears in a large number of realistic situations, for instance for the flow in a cylindrical pipe or around a spherical obstacle.

The main idea for handling three-dimensional problems in such geometries consists in using the Fourier coefficients of the data and the solution with re- spect to the angular variable: Indeed, the three-dimensional problem is then reduced to a countable family of uncoupled two-dimensional problems, one for each Fourier coefficient, in the meridian domain. The drawback is that the vari- ational formulation of each problem involves weighted Sobolev spaces, as fully investigated in [2] in a general framework.

The discretization is then performed in two steps. First, we use Fourier truncation, i.e. we only solve a finite number of two-dimensional problems and recall the estimate of the corresponding error from [2, Thm IX.1.9]. Second, we consider a discretization of each two-dimensional problem. Even if finite element discretizations have already been studied in this context, see [4] for instance, we have chosen here to use spectral type methods in order to preserve the accuracy of Fourier truncation. More precisely, in order to handle the possible complexity of the two-dimensional domain Ω, we consider a mortar spectral element dis- cretization of each problem. Indeed, handling such geometries is one of the first applications of the mortar element method as introduced in [7]. We prove the well-posedness of each discrete problem and also establish optimal error esti- mates (see [11] for the first results in this direction). We present some numerical experiments which confirm the interest of this discretization.

The outline of the paper is as follows:

• In Section 2, we recall from [2] the variational formulation of the two- dimensional problems and also the error issued from Fourier truncation.

• In Section 3, we describe the discrete problems constructed from the mortar spectral element method and we prove their well-posedness.

• Section 4 is devoted to the numerical analysis of these problems.

• Numerical experiments are presented in Section 5.

2 The two-dimensional problems

We first make precise some notation about the geometry of the axisymmet- ric domain and introduce the weighted spaces which are needed on the two- dimensional domain. Next we write the variational formulation of the two- dimensional problems and recall their well-posedness. We conclude with an estimate for the error due to Fourier truncation.

2.1 About the geometry

With a point in R³, we associate its Cartesian coordinates (x, y, z) and its cylin- drical ones (r, θ, z) with

x=rcosθ, y =rsinθ, r∈R+, θ∈[−π, π[.

We denote by R²+ the product R⁺ ×R and consider a polygon Ω in R²+ with boundary ∂Ω made of a finite number of segments Γ_i, 1≤i≤I. The endpoints of these segments are known as corners of Ω: We call c₁, c₂, ...c_p the corners of the polygon which are on the axis r= 0,and e₁, e₂, ...e_j the other corners of Ω.

Let Γ₀ be the intersection of∂Ω with the axis r= 0 and Γ =∂Ω\Γ₀.

Let ˘Ω be the domain of R³ obtained by rotation of Ω around the axis r= 0.

The set Ω is then called meridian domain and we have Ω =˘

(r, θ, z)∈R³, (r, z)∈Ω∪Γ0, −π ≤θ < π .

In Figure 1, we illustrate some examples of domains ˘Ω which we treat in our numerical experiments.

Figure 1: Examples of domains Ω and ˘Ω

2.2 Weighted Sobolev spaces

We define the Hilbert spacesL²₁(Ω),L²₋₁(Ω) andH₁^m(Ω), for any positive integer m by:

L²_±1(Ω) = n

v : Ω−→Cmeasurable;

kvk_L2

±1(Ω) = Z

Ω

|v²(r, z)|r^±1dr dz¹₂

<+∞o ,

H₁^m(Ω) =n

v ∈L²₁(Ω); kvk_Hm 1 (Ω)= (

m

X

k=0 k

X

`=0

||∂_r^{`</sup>∂<sub>z</sub><sup>k−`}v||²_L2

1(Ω))¹² <+∞o . For a positive real number s, the spaceH₁^s(Ω) is deduced in a standard way by interpolation between the space H₁^[s](Ω) and H₁^[s]+1(Ω), where [s] stands for the integral part of s. We also need the Hilbert space V₁¹(Ω) = H₁¹(Ω)∩L²₋₁(Ω), and we provide it with the norm

||w||_V¹

1(Ω) = (||w||²_H1

1(Ω)+||w||²_L2

−1(Ω))¹².

Remark 2.1 In the monodimensional case of an edge Λ of Ω, the spacesL²_±1(Λ), H₁^s(Λ) andV₁¹(Λ) are defined in the same way as in the two-dimensional case by using the measure dτ =r dr if Λ is perpendicular to the axis (Oz) and dτ =dz if it is parallel to this axis. For more details see [2, Chap. II].

Indeed, with any scalar function ˘v in L²( ˘Ω), we associate its Fourier coeffi- cients v^k, k ∈Z, given by

v^k(r, z) = 1

√2π Z π

−π

˘

v(r, θ, z)e^−ikθdθ. (2.1) It is readily checked that each v^k then belongs toL²₁(Ω).

Similarly, for each vector fieldv˘inL²( ˘Ω)³, we consider its cylindrical compo- nenta ˘v_r, ˘v_θ and ˘v_z and the associated Fourier coefficient (v^k_r, v_θ^k, v_z^k) defined by the analogue of (2.1) which now belong toL²₁(Ω)³. It is proved in [2, Thm II.3.6]

that the Fourier transform: v˘→ (v_r^k, v^k_θ, v_z^k)_k maps H¹( ˘Ω)³ onto Q

k∈ZH¹_(k)(Ω) with

H¹_(k)(Ω) =















V₁¹(Ω)×V₁¹(Ω)×H₁¹(Ω) if k= 0, n

(v_r, v_θ, v_z)∈H₁¹(Ω)×H₁¹(Ω)×V₁¹(Ω); v_r+ik v_θ∈L²₋₁(Ω)o if |k|= 1, V₁¹(Ω)×V₁¹(Ω)×V₁¹(Ω) if |k| ≥2.

More general results exist for the spaces H^s( ˘Ω)³, see [2, Chap. II], we do not state them for simplicity.

2.3 Variational formulation of the problems

In order to take into account the boundary conditions, we introduce the spaces H₁¹ (Ω) =n

v ∈H₁¹(Ω); v = 0 on Γo ,

V₁¹(Ω) =V₁¹(Ω)∩H₁¹ (Ω), H¹_(k)(Ω) =H¹_(k)(Ω)∩H₁¹ (Ω)³. We also need the spaces

L²_(k)(Ω) =

(nq∈L²₁(Ω); R

Ωq(r, z)r dr dz = 0o

if k = 0,

L²₁(Ω) if |k| ≥1.

We use a lifting of the boundary data g˘that we still denote by g˘for simplicity.

Then, it is readily checked that, if (˘u,p) is the solution of problem (1.1) with˘ data (f˘,g) in˘ L²( ˘Ω)³×H¹( ˘Ω)³, the Fourier coefficients u^k = (u^k_r, u^k_θ, u^k_z), p^k are the solutions of the following variational problems, for all k ∈Z:

Find(u^k, p^k) in H¹_(k)(Ω)×L²_(k)(Ω), with u^k−g^k inH¹_(k)(Ω), such that

∀v ∈H¹_(k)(Ω), A_k(u^k,v) +B_k(v, p^k) =hf^k,vi, (2.2)

∀q ∈L²₁(Ω), Bk(u^k, q) = 0,

where the rather complex sesquilinear forms A_k(·,·) and B_k(·,·) are defined by A_k(u,w) =a₀(u_r, w_r) +a₀(u_θ, w_θ) +a₀(u_z, w_z)

+ Z

Ω

1 +k²

r² (u_rw_r+u_θw_θ) + 2ik

r² (u_θw_r−u_rw_θ) + k² r²u_zw_z

r dr dz, with

a₀(u, w) = Z

Ω

(∂_ru∂_rw+∂_zu∂_zw)(r, z)r drdz, and

Bk(w, q) =− Z

Ω

q

∂rwr+ 1

r(wr+ik wθ) +∂zwz

r dr dz.

The Hermitian product h·,·i is given by hf,vi=

Z

Ω

f(r, z) · v(r, z)r drdz.

From now on, the space H¹_(k)(Ω) is equipped with the norm kvk_H¹

(k)(Ω) =A_k(v,v)¹²

(indeed, the quantity A_k(v,v) is real and nonnegative) and the space L²_(k)(Ω) is equipped with the norm k · k_L²

1(Ω). Then, appropriate properties of the forms A_k(·,·) andB_k(·,·) on these spaces are derived in [2, Prop. IX.1.3], which leads to the following result.

Proposition 2.2 For any dataf^k inL²₁(Ω)³andg^k inH¹_(k)(Ω)satisfying more- over in the case k = 0 the null flux condition

Z

Γ

(g_rn_r+g_zn_z)(τ)r(τ)dτ = 0, (2.3) problem (2.2) has a unique solution (u^k, p^k) in H¹_(k)(Ω) ×L²_(k)(Ω). Moreover, this solution satisfies, for a constant c only depending onΩ,

ku^kk_H¹

(k)(Ω)+kp^kk_L²

1(Ω) ≤c(kf^kk_L²

1(Ω)³ +kg^kk_H¹

(k)(Ω)

. (2.4)

Remark 2.3 The data f˘and ˘g are said to be axisymmetric if all functions ˘f_r, f˘_θ and ˘f_z, ˘g_r, ˘g_θ and ˘g_z are independent ofθ. In this case, only problem (2.2) for k = 0 has a non-zero solution. Moreover it results into the set of two uncoupled problems

Find uθ inV₁¹(Ω), withuθ−gθ inV₁¹(Ω), such that

∀v ∈V₁¹(Ω), a₁(u_θ, v) = hf_θ, vi (2.5) and

Find (ur, uz, p)in V₁¹(Ω)×H₁¹(Ω)×L²₍₀₎(Ω),

with u_r−g_r inV₁¹(Ω) and u_z −g_z inH₁¹ (Ω), such that

∀(v_r, v_z)∈V₁¹(Ω)×H₁¹ (Ω),

a₁(u_r, v_r) +a₀(u_z, v_z) +b(v_r, v_z;p) =hf_r, v_ri+hf_z, v_zi, (2.6)

∀q∈L²₍₀₎(Ω) , b(ur, uz;q) = 0,

where the sesquilinear forms a₁(·,·) and b(·,·) are now defined by a₁(u, w) = a₀(u, w) +

Z

Ω

u(r, z)w(r, z)r⁻¹drdz, b(w, q) =−

Z

Ω

q

∂rwr+ 1

rwr+∂zwz

r dr dz.

These problems seem much simpler and their discretization is considered sepa- rately in the next section.

2.4 Fourier truncation

Of course, we intend to discretize only a finite number of problems (2.2). So, we chose a positive integer K and, in analogy with the formula

˘

u(r, θ, z) = 1

√2π X

k∈Z

u^k(r, z)e^ikθ, p(r, θ, z) =˘ 1

√2π X

k∈Z

p^k(r, z)e^ikθ,

we define an approximation of the solution (˘u,p) of problem (1.1) by˘

˘

u_K(r, θ, z) = 1

√2π X

|k|≤K

u^k(r, z)e^ikθ,

˘

p_K(r, θ, z) = 1

√2π X

|k|≤K

p^k(r, z)e^ikθ. (2.7)

Indeed, the following result can be found in [2, Thm IX.1.9].

Proposition 2.4 For any s ≥ 0, if the data (f˘,g)˘ belong to H^s−1( ˘Ω)³ × H^s+1(Ω)³, the following estimate holds between the solution (u,˘ p)˘ of problem (1.1) and its approximation (˘u_K,p˘_K) defined in (2.7)

k˘u−u˘_Kk_H1( ˘Ω)³ +kp˘−p˘_Kk_L2( ˘Ω) ≤c K^−s

kf˘k_Hs−1( ˘Ω)³ +k˘gk_Hs+1( ˘Ω)³

. (2.8) Remark 2.5 When Fourier truncation is used, the Fourier coefficients of the data f^k and g^k,|k| ≤K, are usually computed by a quadrature formula: With θ_m= _2K+1^2mπ , the approximate Fourier coefficients are given for |k| ≤K by

f^k∗(r, z) =

√2π 2K+ 1

X

|m|≤K

f˘(r, θm, z)e^−ikθm,

g^k∗(r, z) =

√2π 2K+ 1

X

|m|≤K

˘

g(r, θm, z)e^−ikθm. We do not take this modification into account in the next section, since the final error estimates are exactly the same.

3 The discrete problems

We first recall the decomposition of the domain and the approximation spaces that are required for the mortar spectral element method. Next, we write the corresponding discrete problems, first in the case of axiysmmetric data, second in the general case, and prove their well-posedness.

3.1 About the mortar element method

In view of the discretization, we consider a decomposition of Ω into L open rectangles Ω<sub>`</sub>, 1≤` ≤L, such that

Ω =¯ ∪^L

`=1

Ω¯_{`</sub> and Ω<sub>`}∩Ω_m =∅, 1≤` < m≤L. (3.1) Note that the edges of the Ω` are either parallel or orthogonal to the axis (Oz).

For any two-dimensional domainOand nonnegative integerN,PN(O) stands for the space of restrictions to O of polynomials on R² with degree ≤ N with respect to each variable r and z. In view of the discretization, we define a L-tuple of positive integers δ = (N₁, ..., N_L). Indeed, the idea of the mortar spectral element method is to approximate the discrete solutions in a subspace of

Yδ(Ω) =n

v_δ∈L²₁(Ω); v_δ|<sub>Ω`</sub> ∈PN`(Ω<sub>`</sub>),1≤`≤Lo .

Let also Yδ(Ω) stand for the space of functions in Y^δ(Ω) vanishing on Γ.

To define this subspace, we introduce the skeletonS of the domain decompo- sition, equal to ∪^L

`=1∂Ω<sub>`</sub>\∂Ω. It admits a partition without overlap into mortars S¯=^M

+

µ=1∪ γ_µ⁺, with γ_µ⁺∩γ_µ⁺0 =∅, 1≤µ < µ⁰ ≤M⁺,

eachγ_µ⁺being a whole edge of one of Ω_`, which is then denoted by Ω⁺_µ. Note that the choice of this decomposition is not unique, however it is decided a priori for all the discretizations we work with. Once it is fixed, we have another partition of the skeleton into non-mortars:

S¯= ^M

−

m=1∪ γ_m⁻, with γ_m⁻∩γ_m⁻0 =∅, 1≤m < m⁰ ≤M⁻,

where each γ_m⁻ is a whole edge of one of Ω_`, then denoted by Ω⁻_m (if there exists an index µsuch that γ_m⁻ and γ_µ⁺ coincide, Ω⁻_m is different of Ω⁺_µ).

Next, with all v_δ in Yδ(Ω), we associate the mortar function φ_vδ in L²₁(S) defined by φ_vδ|_γ⁺

µ = (v_δ|_Ω⁺

µ)|_γ⁺

µ, 1≤µ≤M⁺. With obvious definition of N_m⁻, we

define our fundamental discrete space Xδ by:

Xδ(Ω) =n

v_δ ∈Yδ;

∀ψ ∈PNm⁻−2(γ_m⁻), Z

γm⁻

(v_δ|_Ω⁻

m−φ_vδ)(τ)ψ(τ)dτ = 0, 1≤m≤M⁻o

(3.2) with dτ =r dr if the non-mortarγ_m⁻ is parallel to the axis (Or) and dτ =dz if γ_m⁻ is parallel to the axis (Oz). We also need its subspaces defined as follows:

(i) Xδ is the space of functionsv_δ vanishing on Γ;

(ii) X^∗δ is the space X^∗δ(Ω) =

v_δ∈Xδ(Ω); v_{`</sub> =v<sub>δ</sub>|<sub>Ω`} ∈L²₋₁(Ω<sub>`</sub>), 1≤`≤L , (3.3) and X⁰δ is the intersection of Xδ and X^∗δ.

Finally, we introduce the space

Xδ(k)(Ω) =















X^∗δ(Ω)×X^∗δ(Ω)×X^δ(Ω) if k = 0, n

(v_r, v_θ, v_z)∈Xδ(Ω)×Xδ(Ω)×X^∗δ(Ω); v_r+ik v_θ∈X^∗δ(Ω)o if |k|= 1, X^∗δ(Ω)×X^∗δ(Ω)×X^∗δ(Ω) if |k| ≥2,

(3.4) and its intersection X_δ(k)(Ω) with Xδ(Ω)³. All these spaces are needed for the approximation of the velocity.

The spaces for the approximation of the pressure are more simpler, they are defined by

M^δ(Ω) = n

vδ ∈L²₁(Ω); vδ|Ω<sub>`</sub> ∈P<sup>N</sup>`−2(Ω`),1≤`≤L o

,

M^δ(k)(Ω) =M^δ(Ω)∩L²_(k)(Ω). (3.5) To conclude, we introduce quadrature formulas. Let (ξj, ρj), 0≤j ≤N, de- note the nodes and weights of the Gauss-Lobatto quadrature formula on [−1,1]

for the measure dζ and (ζ_i, ω_i), 1 ≤i≤N + 1, their analogues for the measure (1 + ζ)dζ, see [2, Section VI.1] for a more explicit definition. We denote by (Ω_{`</sub>)1≤`≤L₀ the rectangles such that ∂Ω_{`</sub> ∩Γ<sub>0</sub> 6= ∅ and by (Ω<sub>`})_L0+1≤`≤L those such that ∂Ω<sub>`}∩Γ₀ =∅. If Ω_{`</sub> is equal to ]0, r<sup>0</sup><sub>`}[×]z_{`</sub>, z<sup>0</sup><sub>`}[ for 1≤ ` ≤L<sub>0</sub> and to ]r`,r<sub>`</sub><sup>0</sup>[×]z`, z<sub>`</sub><sup>0</sup>[ for L0+ 1≤`≤L, we use the following definitions:

(i) For 1≤`≤L<sub>0</sub> and with N =N<sub>`</sub>, ζ_i<sup>`</sup> = r<sub>`</sub>⁰

2 (ζ_i+ 1), ω_i<sup>`</sup>=ω<sub>i</sub>r<sup>02</sup><sub>`</sub>

4 , 1≤i≤N_`+ 1;

(ii) For L₀+ 1≤` ≤Land still with N =N<sub>`</sub>, ξ_i<sup>(r)`</sup> = (r<sub>`</sub>⁰ −r_`)

2 ξ_i+ (r⁰_{`</sub>+r<sub>`})

2 , ρ<sup>(r)`</sup><sub>i</sub> =ρ<sub>i</sub>r<sup>0</sup><sub>`</sub>−r_`

2 , 0≤i≤N<sub>`</sub>; (iii) For 1 ≤`≤L and once more with N =N_`,

ξ_j<sup>`</sup> = (z<sup>0</sup><sub>`</sub>−z_`)

2 ξ_i+(z_{`</sub><sup>0</sup> +z<sub>`})

2 , ρ<sup>`</sup><sub>j</sub> =ρ<sub>j</sub>z<sub>`</sub>⁰ −z_`

2 , 0≤j ≤N_`.

We are thus in a position to define the discrete scalar product: For all functions u and v such that u_{`</sub> =u|<sub>Ω`} and v_{`</sub> =v|<sub>Ω`} are continuous on Ω<sub>`</sub>, 1≤`≤L,

(u, v)_δ =

L0

X

`=1 N<sub>`</sub>+1

X

i=1 N_`

X

j=0

u_{`</sub>(ζ<sub>j</sub><sup>`</sup>, ξ_i<sup>`</sup>)v<sub>`}(ζ_j^{`</sup>, ξ<sub>i</sub><sup>`})ω_i^{`</sup>ρ<sup>`}_j

+

L

X

`=L0+1 N`

X

i=0 N`

X

j=0

u_{`</sub>(ξ<sub>i</sub><sup>(r)`</sup>, ξ_j<sup>`</sup>)v<sub>`}(ξ^{(r)`</sup><sub>i</sub> , ξ<sub>j</sub><sup>`})ξ_i^{(r)`</sup>ρ<sup>(r)`}_i ρ^`_j.

We denote byI_{`</sub><sup>+</sup>andI<sub>`}the Lagrange interpolation operators associated with the nodes (ζ_j^{`</sup>, ξ<sub>i</sub><sup>`}) for 1≤` ≤L0 and with (ξ<sub>i</sub><sup>(r)`</sup>, ξ<sup>`</sup><sub>j</sub>) for L0+ 1≤`≤L, respectively, with values in PN`(Ω<sub>`</sub>). Let also I_δ stand for the global interpolation operator with values in Yδ(Ω).

3.2 The discrete problems for axisymmetric data

As standard for spectral methods, the discrete problems are constructed by the Galerkin method with numerical integration. The problem associated with (2.5) reads

Findu_θ,δ inX^∗δ(Ω), withu_θ,δ − I_δg_θ inYδ(Ω), such that

∀v_δ ∈X⁰δ(Ω), a_1δ(u_θ,δ, v_δ) = (f_θ, v_δ)_δ (3.6) where the bilinear form a_1δ(·,·) is defined by

a_0δ(u_δ, w_δ) = (∂_ru_δ, ∂_rw_δ)_δ+ (∂_zu_δ, ∂_zw_δ)_δ,

a_1δ(u_δ, w_δ) =a_0δ(u_δ, w_δ) + (r⁻¹u_δ, r⁻¹w_δ)_δ. From now on, we omit the study of this problem and we refer to [1] for its detailed numerical analysis.

The problem associated with (2.6) reads

Find (u_r,δ, u_z,δ, p_δ)in X^∗δ(Ω)×Xδ(Ω)×Mδ(0)(Ω),

with u_r,δ− I_δg_r in Yδ(Ω) and u_z,δ− I_δg_z inYδ(Ω), such that

∀(v_r,δ, v_z,δ)∈X⁰δ(Ω)×Xδ(Ω),

a1δ(ur,δ, vr,δ) +a0δ(uz,δ, vz,δ) +bδ(vr,δ, vz,δ;pδ)

= (f_r, v_r,δ)_δ+ (f_z, v_z,δ)_δ, (3.7)

∀q_δ ∈Mδ(0)(Ω), b_δ(u_r,δ, u_z,δ;q_δ) = 0, where the form b_δ(·,·) is defined by

b_δ(w_δ, q_δ) = −(q_δ, ∂_rw_r,δ+r⁻¹w_r,δ+∂_zw_z,δ)_δ.

In order to investigate the well-posedness of problem (3.7), we now establish some properties of the sesquilinear forms which are involved in it. The continuity of the forms a_0δ(·,·) and a_1δ(·,·) follow from the positivity and boundedness of the Gauss–Lobatto formulas, see [6, Remark 13.3] and [2, Lemma VI.1.4]. This also yields the ellipticity of a_1δ(·,·). However, a further and now well-known argument is needed to prove the ellipticity of a_0δ(·,·), we refer to [5, Chap IV, Lemma 3.2] for the complete proof. All discrete spaces are equipped with the broken norms that results from their definition, namely

kvk_H¹

1D(Ω) =

L

X

`=1

kvk²_H1 1(Ω`)

¹₂

, kvk_V¹

1D(Ω) =

L

X

`=1

kvk²_V1 1(Ω`)

¹₂ .

Lemma 3.1 Let N_D denote the maximal number of corners of theΩ_` which are inside one of the non-mortars γ_µ⁻, 1≤µ≤M⁻.

(i) The form a1δ(·,·)is continuous onX^∗δ(Ω)×X^∗δ(Ω)and elliptic onX⁰δ(Ω), with norm and ellipticity constant independent of δ.

(ii) If all the N_` satisfy

N<sub>`</sub> ≥N<sub>D</sub> + 2, 1≤` ≤L, (3.8)

the form a_0δ(·,·) is continuous on Xδ(Ω)×Xδ(Ω) and elliptic on Xδ(Ω), with norm and ellipticity constant independent of δ.

Next, we observe that, due to the definition of Mδ(Ω), the forms b(·,·) and b_δ(·,·) coincide on X^∗δ(Ω)× Xδ(Ω)× Mδ(Ω), whence the continuity of b_δ(·,·).

However proving the second part of the next lemma is more difficult.

Lemma 3.2 The form b_δ(·,·) is continuous on X^∗δ(Ω)×Xδ(Ω)×Mδ(Ω), with norm independent of δ. Moreover, for any q_δ ∈ M^δ(0), there holds the inf-sup condition

sup

wδ∈X⁰_δ(Ω)×X_δ(Ω)

b_δ(w_δ, q_δ) kwδk_V1

1D(Ω)×H_1D¹ (Ω)

≥β_δkq_δk_L2

1(Ω), (3.9)

where

β_δ =cN¯⁻

1 2

δ (log ¯N_δ)⁻¹ with N¯_δ= max{N<sub>`</sub>,1≤` ≤L}. (3.10) Proof. For already explained reasons, we only establish the second part of the lemma and prove it with b_δ(·,·) replaced by b(·,·). Let q_δ belong to M_δ(0). We take

q_δ = ¯q_δ+ ˜q_δ with q¯_δ|_{Ω`</sub> = 1 meas(Ω<sub>`})

Z

Ω`

q_`(r, z)r dr dz.

1) On each Ω`, we remark that ˜q` = ˜qδ|Ω_{`</sub> belongs to P<sup>N</sup>`−2(Ω`) and has a null weighted integral on Ω<sub>`}. Let P⁰N`(Ω<sub>`</sub>) stand for the subspace of P⁰N(Ω_{`</sub>) made of polynomials vahising on ∂Ω<sub>`}. Thus, according to [2, Proposition X.2.5]

and [6, Sections 24 and 25], there exists w˜` in P<sup>0</sup>N<sub>`</sub>(Ω`)× P<sup>0</sup>N<sub>`</sub>(Ω`) such that b(w˜<sub>`</sub>,q˜_{`</sub>) =k˜q<sub>`}k²_L2

1(Ω`) and kw˜_δk_V1

1(Ω`)×H<sup>1</sup><sub>1</sub>(Ω`) ≤˜cp

N_{`</sub>logN<sub>`}k˜q<sub>`</sub>k<sub>L</sub>2 1(Ω`).

We take w˜_δ such that w˜_δ|_{Ω`</sub> = w˜<sub>`}. It is readily checked that w˜_δ belongs to X⁰δ(Ω)². Moreover we have

b(w˜δ,q˜δ) = k˜qδk²_L2

1(Ω) and kw˜δk_V¹

1D(Ω)×H¹_1D(Ω) ≤˜cN¯

1 2

δ log ¯Nδk˜qδk_L²

1(Ω) (3.11) 2) Since ¯q_δis constant on each subdomain Ω_` and according to [2, Lemma XI.1.1]

(see also [2, Prop. XI.1.7]), there exists w¯_δ inX⁰δ(Ω)×Xδ(Ω) such that b(w¯_δ,q¯_δ) = k¯q_δk²_L2

1(Ω) and kw¯_δk_V¹

1D(Ω)×H_1D¹ (Ω) ≤¯ckq¯_δk_L2

1(Ω). (3.12) 3) We now use the Boland and Nicolaides argument [9] and take : w_δ=w˜_δ+λw¯_δ for a positive constant λ. Sinceb( ˜w_δ,q¯_δ) = 0, we have

b_δ(w_δ, q_δ) = k˜q_δk²_L2

1(Ω)+λk¯q_δk²_L2

1(Ω)−¯cλkq˜_δk_L²

1(Ω)k¯q_δk_L²

1(Ω)

≥(1−c¯²η²

2 )k˜qδk²_L2

1(Ω)+λ(1− λ

2η²)kq¯δk²_L2 1(Ω), for any η >0. When taking η= ¹_¯c and λ=η² we deduce:

b_δ(w_δ, q_δ) ≥inf{ 1 2¯c²,1

2} kq_δk²_L2

1(Ω). (3.13)

By using (3.11) and (3.12), we obtain kw_δk

Z ≤cN¯

1 2

δ (log ¯N_δ)kq_δk_L²

1(Ω). (3.14)

Finally by combining (3.13) and (3.14) we obtain the inf-sup condition (3.9).

Even if condition (3.9) is not optimal, it is well-known that it cannot be improved, see [2, Prop. X.2.5] or [6, Thm 25.5]. In any case, it is sufficient for proving the next result.

Proposition 3.3 If condition(3.8)holds, for any data (f_r, f_z)and(g_r, g_z)con- tinuous on Ω and satisfying the null flux condition (2.3), problem (3.7) has a unique solution (u_r,δ, u_z,δ, p_δ) in X^∗δ(Ω)×Xδ(Ω)×Mδ(0)(Ω). Moreover, in the case of homogeneous boundary conditions g_r =g_z = 0, this solution satisfies, for a constant conly depending on Ω and its decomposition (3.1),

ku_r,δk_V¹

1D(Ω)+ku_z,δk_H¹

1D(Ω)+β_δkp_δk_L²

1(Ω) ≤c(kI_δf_rk_L²

1(Ω)+kI_δf_zk_L²

1(Ω)

. (3.15) We prefer not to state the stability property in the general case since it is more complex.

3.3 The discrete problems in the general case

There also, the discrete problems are constructed by the Galerkin method with numerical integration. For all k6= 0, they read

Find (u^k_δ, p^k_δ) inX^δ(k)(Ω)×M^δ(Ω), with u^k_δ − I_δg^k in Y(Ω)³, such that

∀v_δ∈X_δ(k)(Ω), A_k,δ(u^k_δ,v_δ) +B_k,δ(v_δ, p^k_δ) = (f^k,v_δ)_δ, (3.16)

∀q_δ∈Mδ(Ω), B_k,δ(u^k_δ, q_δ) = 0,

where the sesquilinear forms A_k,δ(·,·) and B_k,δ(·,·) are now defined by Ak,δ(uδ,wδ) =a0δ(ur,δ, wr,δ) +a0δ(uθ,δ, wθ,δ) +a0δ(uz,δ, wz,δ)

+ (1 +k²)(r⁻¹u_r,δ, r⁻¹w_r,δ)_δ+ (1 +k²)(r⁻¹u_θ,δ, r⁻¹w_θ,δ)_δ

+ 2ik(r⁻¹u_θ,δ, r⁻¹w_r,δ)_δ−2ik(r⁻¹u_r,δ, r⁻¹w_θ,δ)_δ+k²(r⁻¹u_z,δ, r⁻¹w_z,δ)_δ, and

B_k,δ(w_δ, q_δ) =−

q_δ, ∂_rw_r,δ+r⁻¹(w_r,δ+ik w_θ,δ) +∂_zw_z,δ

δ

.

Despite the complex aspect of the forms A_k,δ(·,·) and B_k,δ(·,·), proving the well-posedness of the previous problem is simpler than for problem (3.7). Indeed, the continuity and ellipticity ofA_k,δ(·,·) immediately follows from the properties of the Gauss–Lobatto formulas, see [6, Remark 13.3] and [2, Lemma VI.1.4], due to the definition of the norm k · k_H¹

(k)(Ω) introduced in Section 2 (see also [2, Section X.1]). We hide here an obvious definition for the norm k · k_H¹

(k)D(Ω). Lemma 3.4 The formA_k,δ(·,·)is continuous onXδ(k)(Ω)×Xδ(k)(Ω)and elliptic on Xδ(k)(Ω), with norm and ellipticity constant independent of δ.

The continuity of the form Bk,δ(·,·) also follows from the properties of the Gauss–Lobatto formulas. Moreover, since no global or matching condition ap- pears in the definition of Mδ(Ω), the inf-sup condition is easily derived from local ones. We refer to [2, Eq. X.2.32] for this.

Lemma 3.5 The form B_k,δ(·,·) is continuous on Xδ(k)(Ω)×Mδ(Ω), with norm independent of δ. Moreover, for all k 6= 0 and for any q_δ ∈ Mδ(Ω), there holds the inf-sup condition

sup

wδ∈X_δ(k)(Ω)

B_k,δ(w_δ, q_δ) kw_δk_H¹

(k)D(Ω)

≥βδ(k)kq_δk_L²

1(Ω) (3.17)

where

β_δ(k)=c|k|⁻¹N¯⁻

1 2

δ (log ¯N_δ)⁻¹, (3.18) with N¯_δ defined in (3.10).

All this leads to the well-posedness property.

Proposition 3.6 For all k 6= 0 and for any data f^k and g^k continuous on Ω, problem (3.16) has a unique solution (u^k_δ, p^k_δ) in X^δ(k)(Ω)×M^δ(Ω). Moreover, in the case of homogeneous boundary conditions g^k =0, this solution satisfies, for a constant conly depending on Ω and its decomposition (3.1),

ku^k_δk_H¹

(k)D(Ω)+β_δ(k)kp^k_δk_L²

1(Ω)≤ckI_δf^kk_L²

1(Ω)³. (3.19)

4 Error estimates

We prove the a priori error estimates concerning first the solution of problem (3.7), second the solution of problem (3.16). We conclude with an estimate on the whole domain ˘Ω.

4.1 The case of axisymmetric data

For simplicity, we denote byZδthe productX^∗δ(Ω)×Xδ(Ω) and byZδthe product X⁰δ(Ω)×Xδ(Ω), by k · k_Z the norm on the product space V_1D¹ (Ω)×H_1D¹ (Ω). We define the space V^δ by :

Vδ =n

w_δ ∈Zδ; ∀q_δ∈Mδ(Ω), b_δ(w_δ, q_δ) = 0o .

In the case of homogeneous boundary data gr = gz = 0, to prove an estimate between the solutions u of problem (2.6) and u_δ of problem (3.7), we first use the Strang Lemma: With obvious notation for the bilinear form A,

ku−u_δk_Z ≤c

v_δinf∈Vδ

ku−v_δk_Z + inf

qδ∈Mδ(Ω)kp−q_δk_L2 1(Ω)

+ sup

wδ∈Zδ

A(v_δ,w_δ)− A_δ(v_δ,w_δ) kwδk_Z

+ sup

z_δ∈Zδ

R

Ωf(r, z)·z_δ(r, z)r dr dz−(I_δf,z_δ)_δ kz_δk_Z

+ sup

y_δ∈Zδ

PM⁻ µ=1

R

γµ⁻

∂u

∂nµ +pn

(τ) · [y_δ](τ)dτ ky_δk_Z

, where cis a positive constant independent of δ.

Even if the previous estimate seems rather complex, it can be noted that the terms due to numerical integration, i.e. on the second and third lines, can be evaluated separately on each Ω`. So bounding them relies on the exactness properties of the quadrature formula and standard approximation and interpo- lation results [2, Sections 5.2 and 6.3]. Similarly, due to the definition ofMδ(Ω), the same arguments yield, for any s` ≥0,

inf

qδ∈Mδ(Ω)kp−q_δk_L2

1(Ω) ≤c

L

X

`=1

N<sub>`</sub><sup>−s`</sup>kpk_Hs`

1 (Ω`). (4.1) So it remains to bound the approximation error inf

vδ∈Vδ

ku−v_δk_Z and the con- sistency term in the fourth line of Strang’s inequality.

Proposition 4.1 Assume that the part u = (u_r, u_z) of the solution of problem (2.6) with homogeneous boundary conditions is such that each u<sub>|Ω`</sub> belongs to H<sub>1</sub><sup>s`+1</sup>(Ω`)<sup>2</sup>, with s` > ³₂ for 1≤`≤L0 and s` > ¹₂ otherwise. If condition (3.8) holds, there exists a function v_δ in Vδ such that:

ku−v_δk_Z ≤cλ

3 4

δ L

X

`=1

N<sub>`</sub><sup>−s`</sup>kuk_Hs`+1

1 (Ω`)². (4.2) where

λ_δ = max{N_µ⁺ N_m⁻,N_m⁻

N_µ⁺} (4.3)

the maximum being taken on all mortars γ_µ⁺, 1 ≤ µ ≤ M⁺, and non-mortars γ_m⁻, 1≤m≤M⁻, such that γ_µ⁺∩γ_m⁻ has a positive measure.

Proof. Since it is very complex, we only give an abridged version and refer to [11, Prop. 3.2.3] for details (see also [3] for similar arguments). The function v_δ is built as the sum v¹_δ+v²_δ+v³_δ.

1) Construction of v¹_δ: Since u is divergence-free in the sense

∂_ru_r,δ+r⁻¹u_r,δ+∂_zu_z,δ = 0 on Ω, there exists a function ψ inH₁²(Ω) such that

u =Rot_a(ψ) = (∂_zψ,−1

r∂_r(rψ)).

Setting ψ_{`</sub> = ψ|<sub>Ω`}, the idea is to take v¹_δ such that v¹_δ|_Ω` = Rot_a( ˜Π^∗,2_N

`ψ<sub>`</sub>), where each ˜Π^∗,2_N

` is an appropriate projection operator fromH<sub>1</sub><sup>2</sup>(Ω<sub>`</sub>) ontoPN`(Ω<sub>`</sub>) preserving the nullity of the function and its normal derivative on the edges of Ω_{`</sub>, and also the values at the corners of the Ω<sub>`}. Then, the function v¹_δ is still divergence-free on each Ω_`.

2) Construction of v²_δ: For 1 ≤ µ ≤ M⁺, let a^µ_p, 1 ≤ p ≤ P_µ, be the corners of the Ω_` which are inside γ_µ⁺. With each of them, we associate a tensorized polynomial η_p inPNµ⁺(Ω⁺_µ) which vanishes on ∂Ω⁺_µ \γ_µ⁺ and moreover satisfies

η_p(a_p) = 1 and η_p(a_p⁰) = 0, 1≤p⁰ ≤P_µ, p⁰ 6=p.

Note that this requires condition (3.8). Thus, we set η^∗_µ,p=

(ηp in Ω⁺_µ,

0 in Ω\Ω¯⁺_µ. (4.4)

The idea is to take, with obvious notation, z_δ =

M⁺

X

µ=1 Pµ

X

p=1

(ψ−Π˜^∗,2_δ ψ)(a_p)η^∗_µ,p, v_δ²|_{Ω`</sub> =Rot<sub>a</sub>(z<sub>δ</sub>|<sub>Ω`}).

3) Construction of v³_δ: Setting z_δ¹²|_Ω` = ˜Π^∗,2_N

`ψ<sub>`</sub>+z_δ|_Ω`, we define σ_γ⁻

m = ˜π^∗,2

Nm⁻−2 [z_δ¹²]_γ⁻

m

, σ_γⁿ−

m =∂_n ˜π^∗,2

Nm⁻−2([z_δ¹²]_γ⁻

m

, where [·]_γ⁻

mstands for the jump throughγ_m⁻ (with the right sign) and the operator

˜ π^∗,2

Nm⁻−2 is an appropriate projection operator ontoPNm⁻−2(γ_m⁻). Next, on eachγ_m⁻, we use a lifting operator R^2,γ−m of the trace and normal derivative and set

v³_δ =

M⁻

X

m=1

Rot_a◦ R^2,γm−(σ_γ⁻

m, σⁿ_γ− m).

The function v_δ =v¹_δ+v²_δ+v³_δ now belongs to Vδ. Estimate (4.2) is proved in [11, Prop. 3.2.3].

We say that the decomposition is conforming if the intersection of two dif- ferent domains Ω_` is either empty or a common vertex or a whole edge of both of them. The result of Proposition 4.1 can be improved in this case, since the step 2 of its proof, i.e. the construction of v²_δ, can be omitted.

Corollary 4.2 Assume that the part u = (u_r, u_z) of the solution of problem (2.6) with homogenous boundary conditions is such that each u<sub>|Ω`</sub> belongs to H<sub>1</sub><sup>s`+1</sup>(Ω`), with s` > ³₂ for 1≤ ` ≤ L0 and s` > ¹₂ otherwise. In the case of a conforming decomposition, there exists a function v_δ in Vδ such that:

ku−v_δk_Z ≤c

L

X

`=1

N<sub>`</sub><sup>−s`</sup>ku<sub>`</sub>k<sub>H</sub>s`+1

1 (Ω`)². (4.5)

Unfortunately, the previous estimates do not extend to the case of nonhomo- geneous boundary conditions, and we are led to use the formula (see [10, Chap.

II, Eq. (1.16)] for instance) in this case:

vδinf∈Vδ

kw−v_δk_Z ≤β_δ⁻¹ inf

zδ∈Z_δ

kw−z_δk_Z.

Since βδ is not bounded independently of δ, see (3.10), this leads to a lack of optimality. We now treat the consistency error.

Proposition 4.3 For any function ϕ such that each ϕ|_{Ω`</sub>, 1 ≤ ` ≤ L, belongs to H₁<sup>s`</sup>(Ω<sub>`}), with s<sub>`</sub> > 1 for 1 ≤ ` ≤ L₀ and s_` > 0 otherwise, the following estimate holds for all wδ in X^δ

X

γ⁻m∈S

Z

γ⁻m

ϕ(τ)[w_δ](τ)dτ

≤cX^L

`=1

N<sub>`</sub><sup>−s`</sup>(logN<sub>`</sub>)<sup>%`</sup>kϕk_Hs`

1 (Ω`)

kw_δk_H1 1D(Ω),

(4.6) where %_{`</sub> is equal to 1 if one of the sides of Ω<sub>`} is a γ_m⁻ and intersects at least two subdomains Ω¯<sub>`</sub><sup>0</sup>, `⁰ 6=`, and 0 otherwise.