vorticity-velocity-pressure formulation for the stokes problem

MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2002; 25:1091–1119 (DOI: 10.1002/mma.328)MOS subject classi�cation: 65N 30

Vorticity–velocity–pressure formulation for the Stokes problem

Fran�cois Dubois1;2;∗†

1Applications Scienti�ques du Calcul Intensif; UPR-CNRS No. 9029; bat. 506; B.P. 167; F-91403 Orsay;Union Europ�eenne

2Conservatoire National des Arts et M�etiers; Equipe de recherche associ�ee No. 3196; 15 rue Marat;F-78 210 Saint Cyr l’Ecole; Union Europ�eenne

Communicated by J. C. Nedelec

SUMMARY

We present a new variational formulation of Stokes problem of �uid mechanics that allows to take intoaccount very general boundary conditions for velocity, tangential vorticity or pressure. This formulationconducts a well posed mathematical problem in a family of particular cases. Copyright ? 2002 JohnWiley & Sons, Ltd.

KEY WORDS: �uid mechanics; Stokes equation; mixed �nite elements; inf-sup condition; vector �elddecomposition; boundary layer

1. PHYSICAL AND NUMERICAL MOTIVATION

Let � be a bounded connected domain of RN (N =2 or 3) with a regular boundary 9�=�.The Stokes problem modelizes the stationary equilibrium of an incompressible viscous �uidwhen the velocity u is su�ciently small in order to neglect the non-linear terms (see e.g.[1]). From a mathematical point of view, this problem is the �rst step in order to considerthe non-linear Navier–Stokes equations of incompressible �uids, as proposed e.g. by Lions[2], Temam [3] or Girault–Raviart [4]. The Stokes problem can be classically written withprimal formulation involving velocity u and pressure p:

−��u+∇p=f in � (1)

div u=0 in � (2)

u=0 on � (3)

where �¿0 is the kinematic viscosity and f is the datum of external forces.

∗ Correspondence to: F. Dubois, Applications Scienti�ques du Calcul Intensif, UPR-CNRS No. 9029, bat. 506, B.P.167, F-91403 Orsay, Union Europ�eenne

† E-mail: [email protected]

Copyright ? 2002 John Wiley & Sons, Ltd. Received 4 October 2001

1092 F. DUBOIS

Figure 1. Marker and cell discretization on a Cartesian mesh.

Our motivation comes from the numerical simulations in computational �uid dynamics.The marker and cell (MAC) method proposed by Harlow and Welch [5] (see also the C-grid of Arakawa [6]) contains staggered grids relative to velocity and pressure and is stillvery popular when used in industrial computer softwares as Flow3d of Harper et al. [7] orPhoenics developed by Patankar and Spalding [8]. This discretization is founded on the useof a Cartesian mesh (Figure 1): velocity is de�ned with the help of �uxes on the faces of themesh and pressure is supposed to be constant in each cell. We try to generalize these degreesof freedom to arbitrary meshes that respect the usual topological constraints associated with�nite elements (see e.g. [9]) and in particular to triangles (Figure 2) or tetrahedra. Some yearsago, Nicolaides [10] proposed a new interpretation of the MAC-Cgrid method with the helpof dual �nite volumes for triangular meshes. An analysis of the MAC scheme as a numericalquadrature for �nite elements has also been proposed by Girault and Lopez [11].From the point of view of numerical analysis, this MAC-Cgrid discretization can be seen as

the search for an approximation of velocity �eld conforming in the H(div;�) Sobolev spacewith the help of the Raviart–Thomas [12] (and N�ed�elec [13] when N =3) �nite element ofdegree one. On the other hand, the approximation of pressure �eld in space L2(�) is associatedwith discontinuous �nite elements of degree zero. But this vision also adopted by Nicolaides,is a variational crime for the Stokes problem (1)–(3), where velocity classically belongs to�nite dimensional linear spaces that are included in the Sobolev space Hl(�) (see e.g. [14]).Also note a completely dierent approach proposed by Ern et al. [15] for Stokes problemwith vorticity and velocity vector �elds in R3 and associated with a philosophy of classicalconforming continuous linear �nite elements.In this paper we recall the variational formulation that we have previously proposed [16; 17]

involving the three �elds of vorticity, velocity and pressure. A particularity of this formula-tion is that boundary conditions can be considered in a very general way; previous work of

Copyright ? 2002 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2002; 25:1091–1119

VORTICITY–VELOCITY–PRESSURE FORMULATION 1093

Figure 2. Degrees of freedom on a triangular mesh.

Beghe et al. [18] and Girault [19] appears as particular cases of what we obtain and �nally,boundary condition (3) can in our sense be seen as a mixed Dirichlet–Neumann bound-ary condition. We develop an abstract approach that makes the technical inf-sup hypothesesevident, which are su�cient to satisfy in order to prove that with this triple formulation, theStokes problem conducts a mathematically well-posed problem with a continuous dependenceon the solution from data. These conditions are completely non-trivial for a general tridimen-sional domain � that is bounded, connected, non-simply connected and with a non-connectedboundary. For proving it, we have conducted a new general representation theorem for vector�elds that generalizes previous results summarized in Bendali et al. [20].

2. VORTICITY–VELOCITY–PRESSURE FORMULATION

The basic idea of our formulation is the same as the one used in stream–function–vorticityformulation ([21–23] in R2; [24; 25] in R3): a solenoidal vector �eld u (satisfying div u=0)can a priori be represented as the curl of some stream function : u=curl . For the completegenerality of the approach that we have here chosen, we do not represent the solenoidalvelocity �eld u with a stream function for multiple reasons: �rst, any representation of thetype u=curl precludes �ows with sinks and sources [26] and moreover, this representationis in the numerical practice restricted to two dimensional domains even if, following the ideaintroduced in [24; 27] has done �rst tentatives in three dimensional domains with N�ed�elec’svectorial �nite elements [13] conforming in space H(curl;�).Recall that if �ijk is the notation for the complete antisymmetric tensor of order 3, (�ijk is

equal to 1 if (i; j; k) is a direct permutation of (1; 2; 3), �ijk equal to −1 if the permutation is


1094 F. DUBOIS

even and �ijk is null in the other cases), then we have

div v=3∑

j=1

9vj9xj

; v∈H(div;�)

(curl’)i =3∑

j=1

3∑k=1

�ijk9’k

9xj; i=1; 2; 3; ’∈H(curl;�); �⊂R3

curl’=9’19x2

− 9’29x1

; ’∈H(curl;�); �⊂R2

With Duvaut–Lions [28], Hilbert spaces H(div;�) and H(curl;�) are de�ned by

H(div;�)= {v∈(L2(�))N ; div v∈L2(�)} (4)

H(curl;�)= {’∈(L2(�))N ; curl’∈(L2(�))2N−3} (5)

and the associated norms will be denoted by ‖ • ‖div;� and ‖ • ‖curl;� and are de�ned from theL2 norm ‖ • ‖0;� by the relations

‖v‖div;� = N∑

j=1

‖vj‖20;� + ‖div v‖20;�

1=2 (6)

‖’‖curl;� = N∑

j=1

‖’j‖20;� +2N−3∑k=1

‖(curl’)k‖20;�

1=2 (7)

In the two dimensional case, variables ! or ’ are scalar valued functions and belongto Sobolev space H 1(�). This space can also be considered in H(curl;�′) for some threedimensional domain �′ such that �∩ (R2×{0})⊂�′, with !=(0; 0; !3) and ’=(0; 0; ’3).The adaptation of these results in the two dimensional case is classical and we refer toGirault–Raviart [4] or our work with Salaun and Salmon [29]. In the following, all notationsand formulae are supposed to be correct when � is a three-dimensional domain.We introduce vorticity:

!=curl u in � (8)

where !∈R2N−3 and we re-write Equation (1) under the form

� curl!+∇p=f in � (9)

We multiply Equation (8) (respectively, Equation (1), Equation (2)) by some test vector func-tion ’ in space H(curl;�) (respectively, v in space H(div;�), q in space L2(�)), we integrate



by parts and we denote by (•; •) (respectively 〈•; •〉) the L2 scalar product of functions indomain � (respectively on the boundary �). Then we obtain

!∈H(curl;�); u∈H(div;�); p∈L2(�) (10)

(!;’)− (u; curl’) = 〈n× u; ’〉; ∀’∈H(curl;�) (11)

�(curl!; v)− (p; div v) = (f; v)− 〈p; v • n〉; ∀v∈H(div;�) (12)

(div u; q) = 0; ∀q∈L2(�) (13)

We must give a mathematical sense to the boundary terms 〈n× u; ’〉 and 〈p; v • n〉. We�rst prove for surface traces of H(div;�) and H(curl;�) functions (see also [30–33]).

De�nition 1 (Scalar and vectorial functions on the boundary)Let �1;�2 be a partition of boundary �:

�=�1 ∪�2 with �1 ∩�2 = ∅ (14)

and � be the trace operator from H 1(�) onto H 1=2(�) or from (H 1(�))N onto (H 1=2(�))N (seeLions–Magenes [33]). We denote by H 1=2

00 (�1) and TH 1=200 (�1) the following spaces of scalar

and tangential vector functions that are null on the component �2 of the boundary:

H 1=200 (�1) = {��; �∈H 1(�); ��=0 on �2} (15)

TH 1=200 (�1) = {n× �’× n; ’∈(H 1(�))N ; �’× n=0 on �2} (16)

where n is the normal external to the boundary 9�.

Proposition 1 (Trace theorem for functions of H(div;�) and H(curl;�))There exist two continuous mappings �• and �× satisfying the following conditions:

�• :H(div;�)→ (H 1=200 (�1))

′; �•v= v • n when v is regular (17)

�× :H(curl;�)→ (TH 1=200 (�1))

′; �×’=’× n when ’ is regular (18)

where (H 1=200 (�1))

′ (respectively, (TH 1=200 (�1))

′) is the linear space of continuous linear formsacting on H 1=2

00 (�1) (respectively, on TH 1=200 (�1)).

Proof of Proposition 1Let v∈H(div;�) and �∈D( ��) be a regular �eld null on �2. We de�ne �•v acting on � bythe relation

〈�•v; �〉=(v;∇�) + (div v; �) (19)

and we have the following continuity property:

|〈�•v; �〉|62‖v‖div;�‖�‖1;� (20)


1096 F. DUBOIS

By Green formula we have clearly �•v= v•n when v is regular. On the other hand, 〈�•v; �〉depends only on surface values �� of function � since the right-hand side of relation (19) isnull when � belongs to space D(�). So we deduce

|〈�•v; �〉|62‖v‖div;� inf�∈H 1(�); ��=��

‖�‖1;�

Using lifting theorem of Lions (see [30]), we have

inf�∈H 1(�); ��=��

‖�‖1;�6C‖��‖1=2;�

for some constant C independent of �. We deduce that 〈�•v; �〉 is in fact de�ned for �∈H 1=200 (�1)

and from the previous relations we have the estimate

|〈�•v; �〉|6C‖v‖div;�‖�‖1=2;� (21)

The �rst step (17) of the proposition is established.The proof is analogous for surface traces in H(curl;�). Let ’ be given in H(curl;�) and

�∈D( ��)N be a regular vector �eld such that �× n=0 on �2. We set

〈�×’; �〉=(’; curl �)− (curl’; �) (22)

and we have by Cauchy–Schwarz inequality

|〈�×’; �〉|62‖’‖curl;�‖�‖1;� (23)

The Green formula for regular vector �elds in R3

(’; curl �)= (curl’; �) + 〈’× n; �〉 (24)

shows that �×’=’× n when ’ is regular. Therefore, 〈�×’; �〉 depends only on tangentialsurface values n× �× n of vector �eld � since 〈�×’; �〉=0 if � is a vector �eld such that�× n is null on 9�. We deduce that

|〈�×’; �〉|62‖’‖curl;� inf�∈(H 1(�))N ; ��=��

‖�‖1;�

Using again the Lions lifting theorem, we get

inf�∈(H 1(�))N ; ��=��

‖�‖1;�6C‖��‖1=2;�

where C is a constant independent of �. By the density of traces of D( ��)N in TH 1=200 (�1), the

previous inequalities remain valid for ’∈H(curl; �) and �∈TH 1=200 (�1). We deduce

|〈�×’; n× �× n〉|6C‖’‖curl;�‖n× �× n‖1=2;� (25)

Then �×’ is a continuously linear form acting on TH 1=200 (�1). This establishes the second part

(8) of Proposition 1.



We give now a mathematical sense to the boundary terms 〈p; v • n〉 and 〈n× u; ’〉. Wesuppose for a while that v • n is null on some part �m of the boundary and that ’× n is alsonull on another subset �� of the boundary i.e.

�•v=0 in (H 1=200 (�m))

′ (26)

�×’=0 in (TH 1=200 (��))

′ (27)

It is completely natural to introduce the complements �p and �t of �m and ��, respectively:

� = �m ∪�p with �m ∩�p= ∅ (28)

�=�� ∪�t with �� ∩�t = ∅ (29)

Consider now two �elds �0 and �0 in H 1=200 (�p) and TH 1=2

00 (�t) respectively. The boundaryconditions

p=�0 in space H 1=200 (�p) (30)

n× u× n= �0 in space TH 1=200 (�t) (31)

can be written in the form

〈p; v • n〉= 〈�0; �•v〉H 1=200 (�p); (H

1=200 (�p))

′ (32)

〈n× u; ’〉= 〈�0; �×’〉TH 1=200 (�t); (TH

1=200 (�t))

′ (33)

and the boundary conditions (30)–(31) are included in the variational formulation. For thisreason, we will indicate in the following two independent Neumann boundary conditionson pressure and tangential velocity. In a similar way, two independent Dirichlet boundaryconditions for tangential component of vorticity and normal component of velocity naturallyappear:

u • n= g0 on �m (34)

!× n= �0 on �� (35)

where g0 and �0 are a priori given in spaces (H1=200 (�m))

′ and (TH 1=200 (��))

′, respectively.If we restrict ourselves to homogeneous Dirichlet boundary conditions, then we can intro-

duce appropriate Sobolev spaces:

W ={’∈H(curl;�); ’× n = 0 on �� i:e: �×’=0 in (TH 1=2

00 (��))′}

(36)

X = {v∈H(div;�); v • n=0 on �m; i:e: �•v=0 in (H 1=200 (�m))

′} (37)


1098 F. DUBOIS

Y =

L2(�) if meas(�p) �=0{q∈L2(�); (q; 1)=0} if meas(�p)=0

(38)

We can re-write the Stokes problem (10)–(13) with non-homogeneous Neumann boundaryconditions (30)–(31) and homogeneous Dirichlet boundary conditions (34)–(35) (with g0 = 0and �0 = 0) under the following variational form:

!∈W; u∈X; p∈Y (39)

(!;’)− (u; curl’) = 〈�0; �×’〉; ∀’∈W (40)

�(curl!; v)− (p; div v) = (f; v)− 〈�0; �•v〉; ∀v∈X (41)

(div u; q) = 0; ∀q∈Y (42)

3. AN ABSTRACT RESULT

In the previous section, we have shown that with the vorticity–velocity–pressure formulation,the Stokes problem takes the form (39)–(42). This formulation is a ‘triple mixed’ variationalformulation and classical results of Babu ska [34] or Brezzi [35] cannot be applied in astraightforward manner. For this reason we have �rst developed [36] general approach thatmakes appropriate inf-sup hypotheses evident in order to ensure that the Stokes problem iswell posed. We present in this section a generalization of this result. Following a remark ofRaviart [35], it seems possible to reduce this approach to classical ones with an appropriatechoice of product spaces but with a dierent set of hypotheses.

Theorem 1 (Triple mixed variational formulation)Let Y and Z be two real Hilbert spaces equipped with their scalar product (•;•)Y and (•;•)Z ;the associated norm is, respectively, denoted by ‖•‖Y and ‖•‖Z :

(q; q)Y ≡ (q; q)Y = ‖q‖2Y ; q∈Y (43)

(z; z)Z ≡ (z; z)= ‖z‖2Z ; z∈Z (44)

Let W⊂Z and X ⊂Z be two subspaces of Z that are moreover equipped with a structureof Hilbert space associated to scalar products (•;•)W and (•;•)X and to norms ‖•‖W and ‖•‖Xsatisfying

‖’‖2W ≡ (’;’)W¿(’;’)Z ; ∀’∈W (45)

‖v‖2X ≡ (v; v)X¿(v; v)Z ; ∀v∈X (46)

We suppose that there exist two continuous mappings R :W →Z and D :X →Y ; for eachkernel ker R and kerD we de�ne two dierent orthogonal spaces (ker R)⊥; (ker R)�; (kerD)⊥,



(kerD)�, and the orthogonal of the range (ImR)� by the following relations:

(ker R)⊥ = {’∈W; (’;w)W =0;∀w∈ker R} (47)

(ker R)� = {z∈Z; (z; w)Z =0;∀w∈ker R} (48)

(kerD)⊥ = {v∈X; (v; x)X =0;∀x∈kerD} (49)

(kerD)� = {z∈Z; (z; x)Z =0;∀x∈kerD} (50)

(ImR)� = {z∈Z; (z; R’)Z =0;∀’∈W} (51)

We make the following hypotheses:

∃a¿0; infq∈Y; q �=0

supv∈X; v �=0

(q;Dv)Y‖q‖Y ‖v‖X ¿a (52)

∃b¿0; infv∈ker D; v �=0

sup’∈W; ’ �=0

(v; R’)Z‖v‖X ‖’‖W ¿b (53)

∃d¿0; ∀’∈ker R; (’;’)Z¿d‖’‖2W (54)

Then for each �=(; ; �)∈W ′ ×X ′ ×Y ′, the problem

!∈W; u∈X; p∈Y (55)

(!;’)Z + (u; R’)Z = 〈; ’〉; ∀’∈W (56)

(R!; v)Z + (p;Dv)Y = 〈; v〉; ∀v∈X (57)

(Du; q)Y = 〈�; q〉; ∀q∈Y (58)

has a unique solution (!(�); u(�); p(�))∈W ×X ×Y which continuously depends on datum �:{∃C¡0; ∀�∈W ′ ×X ′ ×Y ′

‖!(�)‖W + ‖u(�)‖X + ‖p(�)‖Y6C‖�‖W ′×X ′×Y ′(59)

We �rst introduce canonical injections i :W →Z and j :X →Z

W �’ �→ i’=’∈Z (60)

X �v �→ jv= v∈Z (61)

that are continuous according to inequalities in (45) and (46). Second, Riesz isomorphismsk :Z →Z ′ and l :Y →Y ′ are de�ned according to

Z�z �→ kz=(Z�w �→ 〈kz; w〉Z′ ; Z =(z; w)Z ∈R)∈Z ′ (62)

Y �q �→ lq=(Y �y �→ 〈lq; w〉Y ′ ; Y =(w; y)Y ∈R)∈Y ′ (63)


1100 F. DUBOIS

Figure 3. Relations between Hilbert spaces for abstract version of Stokes problem.

and are isometries from Z onto Z ′ and from Y onto Y ′, respectively. Moreover, we introducedual operators D′ :Y ′ →X ′ and R′ :Z ′ →W ′ of D and R, respectively, by the classical relations

〈D′�; x〉X ′ ; X = 〈�; Dx〉Y ′ ; Y ; ∀�∈Y ′ and x∈X (64)

〈R′�; ’〉W ′ ;W = 〈�; R’〉Z′ ; Z ; ∀�∈Z ′ and ’∈W (65)

The scalar products in (56)–(58) can be rewritten in terms of previous operators; we have

(!;’)Z = 〈i′ki!;’〉W ′ ;W ; ∀!∈W; ∀’∈W (66)

(u; R’)Z = 〈R′kju; ’〉W ′ ;W ; ∀u∈X; ∀’∈W (67)

(R!; v)Z = 〈j′kR!; v〉X ′ ; X ; ∀!∈W; ∀v∈X (68)

(p;Dv)Y = 〈D′lp; v〉X ′ ; X ; ∀p∈Y; ∀v∈X (69)

(Du; q)Y = 〈lDu; q〉Y ′ ; Y ; ∀u∈X; ∀q∈Y (70)

A global map of all the relations evocated above is proposed in Figure 3. The proof ofTheorem 1 is based on a classical result derived by Girault and Raviart (see e.g. [4]) that werecall.

Proposition 2 (A classical result)Let T and M be two Hilbert spaces and T ×M �(t; ) �→ b(t; )∈R be a continuous bilinearform acting on the product space T ×M . We de�ne left kernel V of bilinear form b(•;•) andits polar space V 0 by the relations

V = {t∈T; b(t; )=0;∀∈M} (71)

V 0 = {�∈T ′; 〈�; t〉T ′ ; T =0;∀t∈V} (72)

and linear operators B :T → M ′ and B0 :M →T ′ according to

T � t �→ Bt=(M � �→ 〈Bt; 〉M ′ ;M = b(t; )∈R)∈M ′ (73)

M � �→ B0=(T � t �→ 〈B0; t〉T ′ ; T = b(t; )∈R)∈T ′ (74)



The following three conditions are equivalent:

∃�¿0; inf∈M; �=0

supt∈T; t �=0

b(t; )‖t‖T‖‖M¿� (75)

B0∈ Isom(M;V 0) (76)

B∈ Isom(V⊥; M ′) (77)

where V⊥ is the orthogonal of kernel V in Hilbert space T.

Proposition 3 (Interpretation of hypotheses (51)–(53))Under the hypotheses (51), (52) and (53) of Theorem 1, we have the following isomorphisms:

lD ∈ Isom(kerD)⊥; Y ′) (78)

D′l∈ Isom(Y; (kerD)0) (79)

j′kR ∈ Isom((ker R)⊥; (kerD)′) (80)

R′kj ∈ Isom(kerD; (ker R)0) (81)

where (kerD)0⊂Y ′ and (ker R)0⊂W ′ are polar spaces of kernels kerD and ker R, respectively.

Proof of Proposition 3We introduce �rst the result given in Proposition 1 with the notations proposed for hypothe-sis (52). Due to relation (69) and (70), we have

M =Y; T =X; B= lD; B0 =D′l and V =kerD (82)

due to the fact that l is an isomorphism. Then (77) implies (78) and (76) implies (79).The same mechanism occurs for second inf-sup hypothesis (53). We have

M =kerD; T =W; B= j′kR; B0 =R′kj (83)

due to relations (67) and (68). The evaluation of kernel V can be explicated in this particularcase as follows:

V = {’∈W; ∀v∈kerD; (v; R’)Z =0} (84)

and if ’ belongs to V , then R’ belongs to (kerD)� introduced in (50). According tohypothesis (52), the space (kerD)� is included in (ImR)�. Then

R’∈(ImR)∩ (ImR)� (85)

Moreover, we have the following classical inclusion:

(ImR)∩ (ImR)� ⊆ adhZ(ImR)∩ (adhZ(ImR))�


1102 F. DUBOIS

where adhZ denotes the adherence relative to topology in space Z . Then R’ is null. Inconsequence,

V =ker R (86)

and (80) and (81) are, respectively, consequences of (77) and (76).

Proof of Theorem 1We introduce operator A :W ×X ×Y →W ′ ×X ′ ×Y ′ by the following matrix of operators:

A=

i′ki R′kj 0

j′kR 0 D′l

0 lD 0

(87)

and Equations (56)–(58) can be explicated in the produce space W ′ ×X ′ ×Y ′ as follows:

i′ki!+ R′kju= (88)

j′kR!+D′lp= (89)

lDu= � (90)

where (!; u; p)∈W ×X ×Y is the unknown and �=(; ; �)∈W ′ ×X ′ ×Y ′ is the datum ofthe problem.Now we �rst prove that A is an injective operator. Consider in consequence the particular

case =0∈W ′; =0∈X ′; �=0∈Y ′ as right-hand sides of Equations (88)–(90). We �rsttake as a test function v∈kerD in Equation (89) with =0. Then 〈D′lp; v〉=(p;Dv)Y =0and Equation (89) implies that j′kR!=0 in (kerD)′. Taking into account the isomorphism(80), we deduce that !∈ker R and

R!=0 (91)

We report this last property in Equation (89). Then isomorphism (79) shows that p=0 in Y .We test Equation (88) against ’∈ker R. Then 〈R′kju; ’〉=(u; R’)Z is null and the particular

choice ’=! (which is allowed due to relation (91)) states that 〈i′ki!;!〉= ‖!‖2Z =0. So!=0.Now Equation (90) and isomorphism (78) show clearly that u∈kerD. Moreover, R′kju=0

due to Equation (88) considering =0. The isomorphism (81) shows that u=0. The operatorA is injective.Now we establish that A is a surjective operator. We �rst look to Equation (89). The right

hand side is equal to ∈X ′ ⊂ (kerD)′. Then when we apply the left-hand side for v∈kerD,we have clearly 〈D′lp; v〉=(p;Dv)Y =0. Then − D′lp belongs to (kerD)′ and does notdepend on p∈Y . Consequently, the isomorphism (80) shows the existence of !1∈(ker R)⊥independent of p∈Y such that

j′kR!1 = −D′kp in (kerD)′ (92)

We remark that for each !2∈ker R, the research of vorticity ! under the form

!=!1 +!2; !1∈(ker R)⊥; !2∈ker R (93)



satis�es j′kR!= j′kR!1. Then modulo this (unknown) choice of !2∈ker R, Equation (89)can now be written under the form

D′lp= − j′kR!≡ � (94)

The right-hand side does not depend on !2 is then completely known and is null on sub-space kerD by construction of !2 i.e. �∈(kerD)0. The isomorphism denotes that there ex-ists some (unique) p∈Y such that Equation (94) is satis�ed. At this step of the proof,Equation (89) has been entirely solved; unknowns p∈Y and !1∈(ker R)⊥ are �xed and theother component !2 of representation (94) remains completely free.Equation (90) and isomorphism (78) give a (unique) u1 lying in (kerD)⊥ such that lDu1 = �

and as previously, we remark that a decomposition of vector �eld u under the form

u= u1 + u2; u1∈(kerD)⊥; u2∈kerD (95)

is always the solution of Equation (90).We report the decompositions (93) and (95) in Equation (88) that becomes

i′ki!2 + R′kju2 = − i′ki!1 − R′kju1≡ � (96)

and �∈W ′ ⊂ (ker R)′ is completely known. When we test Equation (96) over ’∈ker R, thenthe second term on the left-hand side 〈R′kju2; ’〉W ′ ;W =(u2; R’)Z becomes null. Then for eachchoice of u2∈kerD, linear form �≡ �− R′kju2 remains equal to � in dual space (ker R)′ andEquation (96) takes the particular form{

!2∈ker R;(!2; ’)Z = 〈�; ’〉; ∀’∈ker R

(97)

Hypothesis (54) shows that the Z-scalar product is d-elliptic on subspace ker R. Lax–Milgramlemma [38] allows to conclude that Equation (97) admits a (unique) solution !2∈ker R.We report this information in Equation (96) that can now be written as

R′kju2 = �− i′kj!2 (98)

and (�− i′ki!2)∈(ker R)0 by the construction of variable !2. Then isomorphism (81) showsthat a (unique) u2∈kerD is solution of problem (98). This step achieves the proof of surjec-tivity of operator A.The proof of Theorem 1 is a direct consequence of the fact the operator A is one to one

and of Banach isomorphism theorem.

4. GENERAL REPRESENTATION OF VECTOR FIELDS

Let � be a connected bounded domain of RN (N =2 or 3) with a smooth boundary 9�: Wesuppose that 9� is of class at least C2 in order to manage continuous curvatures on 9�, seee.g. [39]. We suppose that 9� is decomposed into a partition composed by two subsets �1and �2 such that

9�=�1 ∪�2; �1 ∩�2 = ∅ (99)


1104 F. DUBOIS

and we suppose, moreover, that the intersection

�=�1 ∩�2 (100)

is a �nite set of points if N =2 or a regular curve pictured on the boundary 9� if N =3.We denote by H 1

0 (�; �1) the subspace of Sobolev space H 1(�) composed by scalar functionswhose trace is null on �1 or with a null integral if meas(�1)=0:

H 10 (�; �1)=

{{�∈H 1(�); ��=0 on �1} if meas(�1)¿0

{�∈H 1(�); (�; 1)=0} if meas(�1)=0(101)

and H 10 (�; �1;�2) is by de�nition the subspace of (H

1(�))N composed by vector �elds thatare tangent on �1 and parallel to the normal direction n on �2:

H 10 (�; �1;�2)= {’∈(H 1(�))N ; �•’=0 on �1; �×’=0 on �2} (102)

Proposition 4 (Density)Let C2(�) be the space of regular scalar functions two times continuously derivable on theadherence of domain �. Then C2( ��)∩H 1

0 (�; �1) is dense in space H 10 (�; �1) and (C

2( ��))N ∩H 10 (�; �1;�2) is dense in space H 1

0 (�; �1;�2).

Proof of Proposition 4The �rst result is clear if meas(�1)=0. If it is not the case, let ∈H 1

0 (�; �1) be a scalarfunction orthogonal to space C2( ��)∩H 1

0 (�; �1) for the H 1 scalar product, i.e.{ ∈H 1

0 (�; �1)

( ; ’) + (∇ ;∇’)=0; ∀’∈C2( ��)∩H 10 (�; �1)

(103)

If we restrict ourselves to functions ’ that belong to space D(�), we deduce from secondline of (103) that

−� =0 in the sense of distributions (104)

and function � belongs to space L2(�). Then we multiply relation (104) by an arbitraryfunction �∈H 1

0 (�; �1) and after integrating by parts, we deduce

( ; �) + (∇ ;∇�)−⟨9 9n ; ��

⟩=0; ∀�∈H 1

0 (�; �1) (105)

Consider now a given boundary function ∈H 1=200 (�2). Then the function is de�ned by an

extension of to �1 by nullity, i.e.

=

{0 on �1

on �2(106)

belongs to space H 1=2(�) and by classical density result (see [33]) there exists a sequencek ∈D(�2) such that k converges towards in space H 1=2

00 (�2) and moreover, k belongs tospace D(�):

k ∈D(�2); k → in H 1=200 (�2); k ∈D(�) (107)



Moreover, consider the solution �k of the Dirichlet Laplace problem{−��=0 in �

��k = k on �(108)

By successive applications of regularity theorems [40] and Sobolev embedding injections [14],function �k belongs to space C2( ��)∩H 1

0 (�; �1) and can be used in both relations (103) and(105). We obtain by dierence⟨

9 9n ; k

⟩�=⟨9 9n ; k

⟩(H 1=2

00 (�2))′ ; H 1=2

00 (�2)= 0 (109)

moreover, 9 =9n is a continuous linear form acting on space H 1=200 (�2) and k converges to

in this space and is arbitrary. Then 9 =9n is null on �2:

9 9n =0 in space (H 1=2

00 (�2))′ (110)

and due to (105), function is the solution of the following variational problem:{ ∈H 1

0 (�; �1)

( ; �) + (∇ ;∇�)=0; ∀�∈H 10 (�; �1)

(111)

and is then identically equal to zero, which establishes the �rst point of Proposition 4.The second result of Proposition 4 concerns vector �elds. In the analogy with previous

case, consider a vector valued function in space H 10 (�; �1;�2) and H 1-orthogonal to vector

space (C2( ��))N ∩H 10 (�; �1;�2):{

∈H 10 (�; �1;�2)

( ; ’) + (∇ ;∇’)=0; ∀’∈(C2( ��))N ∩H 10 (�; �1;�2)

(112)

First by taking ’∈(D(�))N , we have −� =0 in the sense of distributions and in conse-quence � ∈(L2(�))N . For an arbitrary vector �eld �∈H 1

0 (�; �1;�2), we integrate (−� ; �)by parts and obtain as in �rst step:

( ; �) + (∇ ;∇�)−⟨9 9n ; ��

⟩=0; ∀�∈H 1

0 (�; �1;�2) (113)

The boundary term 〈9 =9n; ��〉 is decomposed with the help of tangential and normal com-ponents of vector �elds:⟨

9 9n ; ��

⟩=

⟨99n ( × n); �×�

⟩(TH 1=2

00 (�1))′ ; TH 1=2

00 (�1)+

⟨99n ( • n); � • �

⟩(H 1=2

00 (�2))′ ; H 1=2

00 (�2)(114)

Consider in consequence �∈TH 1=200 (�1) and ∈H 1=2

00 (�2). By density of space of regular func-tions in Sobolev spaces with exponent 1=2, there exist two sequences �k ∈(D(�1))N and


1106 F. DUBOIS

k ∈D(�2) converging towards � and in spaces TH 1=200 (�1) and H 1=2

00 (�2), respectively, andwith notation ‘tilda’ de�ned by the relation

�=

{� on �1

0 on �2(115)

for tangent vector �eld � and by (106) for scalar �eld ; moreover we have �k ∈(D(�))Nand k ∈D(�): �k ∈(D(�1))N ; �k → � in TH 1=2

00 (�1); �k ∈(D(�))N

k ∈D(�2); k → in H 1=200 (�2); k ∈D(�)

(116)

Then we solve the following two auxiliary pure Dirichlet problems for Laplace equations:{−�k =0 in �

�k = �k on �(117)

{−��k =0 in �

��k = kn on �(118)

and the solution k of (117) and �k of (118) belongs to the space (C2( ��))N ∩H 10 (�; �1;�2)

thanks to regularity of solution of Laplace equation on a domain with a smooth boundary. Weconsider the �rst choice �= k inside relations (112) and (113) and we make the dierencebetween the two equations:⟨

99n ( × n); �k

⟩(TH 1=2

00 (�1))′ ; TH 1=2

00 (�1)= 0; ∀k∈N (119)

and due to the �rst line of (116) and the fact that � is arbitrary, 9=9n( × n) is null on �1.We proceed in an analogous way with �= �k and we get using the same argument⟨

99n ( • n); k

⟩(H 1=2

00 (�2))′ ; H 1=2

00 (�2)= 0; ∀k∈N (120)

from the second line of (116) and the fact that is arbitrary, we deduce that 9=9n( • n) isnull on �2. Due to (114), the boundary term in (113) is null and we get{

∈H 10 (�; �1;�2)

( ; �) + (∇ ;∇�)=0; ∀�∈H 10 (�; �1;�2)

(121)

Then vector �eld is null and Proposition 4 is established.

Proposition 5 (Peetre–Tartar lemma)Let E0; E1; E2 be three Banach spaces, A1 :E0→E1 and A2 :E0→E2 be two linear continuousmappings such that A2 is compact and such that

∃C¿0; ∀v∈E0; ‖v‖E06C(‖A1v‖E1 + ‖A2v‖E2) (122)



Then we have the following two properties:

ker A1 is �nite dimensional; ImA1 is closed (123)

∃C0¿0; ∀v∈E0; infw∈ker A1

‖v+ w‖E06C0‖A1v‖E1 (124)

Proposition 5 is classical (see [41; 42]) and this result is necessary for the establishment ofProposition 6.

Proposition 6 (An equivalent norm on space H 10 (�; �1;�2))

Under the previous general hypotheses done on domain �, there exists some constant C¿0such that ‖’‖1;�6C(‖’‖20;� + ‖div’‖20;� + ‖curl’‖20;�)1=2

∀’∈H 10 (�; �1;�2)

(125)

Proof of Proposition 6Let (’; ) be a pair of regular vector �elds on domain �. We �rst develop the followingintegral:

(−�’; )=∫�(curl(curl’)−∇(div)’; ) d� (126)

by integrating by parts of each of the three terms of second order, the following is obtained:(∇’;∇ )=(curl’; curl ) + (div’; div ) +

⟨9’9n ;

⟩�

+ 〈curl’; × n〉� − 〈div’; • n〉�(127)

Following precise geometric work done by Bendali [43], it is possible to majorate the boundaryterms of relation (127) when we set =’ and in the two particular cases when ’ • n=0 or’× n=0. If ’ belongs to space (C2( ��))N ∩H 1

0 (�; �1;�2), then we get in the �rst case theexistence of a constant C independent of ’ such that

∣∣∣∣∫�1

[(9’9n ; ’

)+ (curl’;’× n)

]d�∣∣∣∣6C

∫�1|’|2d�

∀’∈(C2( ��))N such that ’ • n=0 on �1

(128)

In the second case there exists another constant also named C in order to satisfy∣∣∣∣∫�2

[(9’9n ; ’

)+ (div’;’ • n)

]d�∣∣∣∣6C

∫�2|’|2d�

∀’∈(C2( ��))N such that ’× n=0 on �2

(129)


1108 F. DUBOIS

By summation of relations (128) and (127) in (127), we obtain the estimate‖∇’‖20;�6‖div’‖20;� + ‖curl’‖20;� + C

∫9�

|’|2d�

∀’∈(C2( ��))N ∩H 10 (�; �1;�2)

(130)

and by the density established from Proposition 4, relation (130) holds in space H 10 (�; �1;�2)

if we replace in the integral on the boundary 9� in the right-hand side of (130), the value ’by the trace �’∈(H 1=2(�))N

De�nition of the H 1 norm and relation (130) show that condition (122) of Proposition 5can be used with the following particular data:

E0=H 10 (�; �1;�2)

E1=(L2(�))N × (L2(�))N × (L2(�))2N−3

E2=(L2(�))N

(131)

{A1’=(’; div’; curl’)

A2’=�’:(132)

If the injection (H 1=2(�))N � � �→ �∈ (L2(�))N is compact (see e.g. [33] then the mapping A2is a compact operator and Peetre–Tartar lemma can be applied. The �rst consequence (relation(123)) gives no particular information because, following (131), we have clearly ker A1 = {0}.On the contrary, taking into account the previous point, it is inferred from relation (125) that{∃C0¿0; ∀’∈H 1

0 (�; �1;�2)

‖’‖1;�6C0(‖’‖0;� + ‖div’‖0;� + ‖curl’‖0;�)(133)

and after elementary algebraic details, relation (133) is equivalent to relation (125), thatestablishes the proposition.

Proposition 7 (Second equivalent norm on space H 10 (�; �1;�2))

Under the same hypotheses concerning domain �, the subspace M 1(�; �1;�2) of H 10 (�; �1;�2)

obtained by annulation of div and curl operators, i.e.{M 1(�; �1;�2)={’∈(H 1(�))N ; div’=0; curl’=0

�’ • n=0 on �1; �’× n=0 on �2}(134)

is of �nite dimension. If �1�1 ;�2 denotes the orthogonal projector onto space M 1(�; �1;�2)relative to the L2(�)N scalar product, then there exists a constant C¿0 such that{‖’‖1;�6C(‖�1�1 ;�2’‖20;� + ‖div’‖20;� + ‖curl’‖20;�)1=2

∀’∈H 10 (�; �1;�2)

(135)



Proof of Proposition 7We apply again Peetre–Tartar lemma, this time with the particular choice

E0=H 10 (�; �1;�2) with the equivalent norm

‖’‖E0 ≡‖’‖20;� + ‖div’‖20;� + ‖curl’‖20;�)1=2

E1=(L2(�))× (L2(�))2N−3

E2=(L2(�))N

(136)

{A1’=(div’; curl’)

A2’=’(137)

Due to Rellich theorem and the fact that the domain � is bounded, the injection (H 1(�))N ,→(L2(�))N is compact, then by composition it is also the case for operator A2. We observe alsothat the initial estimate (122) in Proposition 5 is exactly the one as in (125) established atProposition 6. Finally, we remark that ker A1 =M 1(�; �1;�2) and the �rst point of Proposition 7is established.We decompose an arbitrary vector �eld ’∈H 1

0 (�; �1;�2) under the form of two orthogonal�elds in L2(�)N :

’=�1�1 ;�2’+ (’−�1�1 ;�2’) (138)

and taking into account (125) and (134), we haveinf

�∈M 1(�;�1 ;�2)(‖’− �‖20;� + ‖div(’− �)‖20;� + ‖curl(’− �)‖20;�)1=2

= (‖’−�1�1 ;�2’‖20;� + ‖div’‖20;� + ‖curl’‖20;�)1=2(139)

Then we have

‖’‖1;�6 ‖�1�1 ;�2’‖1;� + ‖’−�1�1 ;�2’‖1;�

6C‖�1�1 ;�2’‖0;�

+C(‖’−�1�1 ;�2’‖20;� + ‖div’‖20;� + ‖curl’‖20;�)1=2

6C‖�1�1 ;�2’‖0;� + C inf�∈M 1(�;�1 ;�2)

(‖’− �‖20;�

+ ‖div(’− �)‖20;� + ‖curl(’− �)‖20;�)1=2 due to (139)

6C‖�1�1 ;�2’‖0;� + CC0(‖div’‖0;� + ‖curl’‖0;�)

due to relation (124) in Peetre–Tartar lemma and relation (135) is established.


1110 F. DUBOIS

We de�ne space M 0(�; �1;�2) of vector �elds ’ lying in space L2(�)N , whose div and curlare identically null and, due to Proposition 1, such that the normal trace �•’ is null in dualspace (H 1=2

00 (�1)N )′ and tangential trace �×’ is identically null in dual space (TH 1=2

00 (�2))′:{

M 0(�; �1;�2)= {’∈(L2(�))N ; div’=0; curl’=0�•’=0 in (H 1=2

00 (�1)N )′ �×’=0 in (TH 1=2

00 (�2))′}

(140)

We observe the inclusion

M 1(�; �1;�2)⊂M 0(�; �1;�2) (141)

Proposition 8 (Closed partition)Let � be a connected bounded domain of RN (N =2 or 3) and (�1;�2) be a partition of itsboundary � satisfying general hypotheses done at the beginning of Section 4. Then spaceM 0(�; �1;�2) de�ned in (140) is closed in L2(�)N and we denote by �0�1 ;�2 the orthogonalprojector (L2(�))N →M 0(�; �1;�2).

Proof of Proposition 8Let (’k)k∈N be a sequence lying in space M 0(�; �1;�2) and converging in space L2(�)N tosome function ’∈(L2(�))N . Then sequences (div’k)k∈N and (curl’k)k∈N converge in thesense of distributions towards div’ and curl’, respectively. We deduce from nullity of pre-vious sequences that we have necessarily div’=0 and curl’=0 and (’k)k∈N is convergingtowards ’ in space H (div;�)∩H (curl;�).The normal and tangential traces �•’ and �×’ are continuously de�ned H (div;�)→

(H 1=200 (�1)

N )′ and H (curl;�)→ (TH 1=200 (�2))

′, respectively, and sequences (�•’k)k∈N and(�×’k)k∈N are identically equal to zero due to the hypothesis ’k ∈M 0(�; �1;�2). Then thereare others converging to zero in their respective Hilbert spaces and �•’=0, �×’=0 because(’k)k∈N converges towards ’ in spaces H (div;�)∩H (curl;�). Thus, we have established that’∈M 0(�; �1;�2) and Proposition 8 is proven.

In the case where the boundary 9� is su�ciently regular (or convex, see [30]), we knowthat spaces M 0(�; ∅;�) and M 1(�; ∅;�) on one hand, M 0(�; �; ∅) and M 1(�; �; ∅) on the otherhand, are exactly equal (see e.g. [26; 43; 32]). More precisely, dimension of space M 0(�; ∅;�)is exactly equal to the number of connected components of the boundary, minus one, andspace M 0(�; �; ∅) parameterizes the second cohomology group of the open set � related tothe number of holes in the domain. The main di�culty of our generalization is that the mixedboundary condition proposed in (140), i.e. formally ’•n=0 on �1 and ’× n=0 on �2 is thepossible presence of bidimensional singularities if n=2 (see [24]) or tridimensional if n=3(see [45]).

Theorem 2 (Representation of vector �elds)Let � be a connected bounded domain of RN (N =2 or 3) and (�1;�2) be a partition of itsboundary � satisfying general hypotheses done at the beginning of Section 4. Let u∈L2(�)N

be a vector �eld. Then there exists two potentials ’ and satisfying the condition{’∈H 1

0 (�; �1)

’∈H 10 (�; �1;�2)

(142)



uniquely and continuously de�ned if we impose the supplementary one following conditionsto vector potential when n=3:

div =0 in �; �1�1 ;�2 =0 (143)

∃C¿0; ‖’‖1;�6C‖u‖0;�; ‖ ‖1;�6C‖u‖0;� (144)

and chosen in such a way that u admits the following orthogonal decomposition in spaceL2(�)N :

u=∇’+ curl +�0�2 ;�1u (145)

In order to simplify the notations, we introduce the space H 100(�; �1;�2) of vector �elds

lying in space H 10 (�; �1;�2) and satisfying, moreover, the two conditions (143):H 100(�; �1;�2)={’∈(H 1(�))N ; �•’=0 on �1

�×’=0 on �2; div’=0; �1�1 ;�2’=0}(146)

Proof of Theorem 2We �rst remark that conditions (142) and (143) have been chosen in order to assume thatdecomposition (145) is orthogonal in space L2(�)N . If we use generical letter � for traces,then we �rst have

(∇’; curl ) = 〈�’; �(curl • n)〉H 1=200 (�2); (H

1=200 (�2))

′

= 〈�’; div�(�× )〉H 1=200 (�2); (H

1=200 (�2))

′

due to the classical relation curl’ • n=div�(’× n) valid for regular �elds (see e.g.[47]) and easily extended by duality. We observe now that �× =0 in TH 1=2

00 (�2) if ∈H 10 (�;

�1;�2) then div�(�× ) is null in space (H 1=200 (�2))

′ (see [45] or [48] concerning the fact thatif vector �eld belongs to space H (curl;�), tangent �eld × n belongs to TH−1=2(�) andin addition scalar boundary �eld div�( × n) belongs to space H−1=2(�)) and we have

(∇’; curl )=0 (147)

Note also that relation (145) can also be obtained by integrating by parts on the other side.We have

(∇’; curl ) = 〈�(∇’)× n; � 〉(TH 1=200 (�2))

′ ; TH 1=200 (�2)

= 〈�×(∇’); n× � × n〉(TH 1=200 (�2))

′ ; TH 1=200 (�2)

= 0

because tangential gradient of ’ is identically null on �1 and tangential component of vector is null on �2. Global coherence between spaces in duality shows that the formal way ofmaking the calculus is justi�ed.We have in an analogous way

(∇’;�0�2 ;�1u)= 〈�’; �•(�0�2 ;�1u)〉H 1=200 (�2); (H

1=200 (�2))

′


1112 F. DUBOIS

and �•�≡ 0 in dual space (H 1=200 (�2))

′ if �∈M 0(�; �2;�1). Then we deduce

(∇’;�0�2 ;�1u)=0 (148)

Finally, we have

(curl ;�0�2 ;�1u)=−〈�× ; �(�0�2 ;�1u)〉TH 1=200 (�1); (TH

1=200 (�1))

′

and we decompose the trace of vector �∈M 0(�; �2;�1) into tangential and normal components

��= n× (�×�) + (�•�)n (149)

the �rst one is null on �1 if �∈M 0(�; �2;�1) because �×�=0 in space (TH 1=200 (�1))

′ and thesecond one is normal to the boundary. Then we have

(curl ;�0�2 ;�1u)=0 (150)

Taking into account orthogonality relations (147) and (148) developed at the previous pointand representation (145), scalar potential ’ is necessarily a solution of the following varia-tional problem: {

’∈H 10 (�; �1)

(∇’;∇�)= (u;∇�); ∀�∈H 10 (�; �1)

(151)

De�nition (101) of space H 10 (�; �1) and Rellich theorem shows classically that the H 1 semi-

norm is elliptic on space H 10 (�; �1):

∃ ¿0; ∀�∈H 10 (�; �1); ‖∇�‖20;�¿ ‖�‖21;� (152)

Then Lax–Milgram theorem [38] shows the existence and uniqueness of scalar potential ’satisfying (151), and in consequence the �rst parts of conditions (142) and (144).In an analogous way, Proposition 7 and in particular relation (135) show that the bi-

linear form (H 1(�))2N−3× (H 1(�))2N−3 � (’; �) �→ (curl’; curl �) ∈ R is elliptic on spaceH 100(�; �1;�2):

∃�¿0; ∀�∈H 100(�; �1;�2); ‖curl �‖20;�¿�‖�‖21;� (153)

Independently, orthogonal decomposition (145), orthogonalities (147) and (150) and gaugeconditions (143) show that vector potential is necessarily solution of the problem{

∈H 100(�; �1;�2)

(curl ; curl �)= (u; curl �); ∀�∈H 100(�; �1;�2)

(154)

Due to ellipticity (153) and Lax–Milgram theorem, vector potential is uniquely de�ned byproblem (154) and continuously depends on datum u. Then inequalities (144) are completelyestablished.We have now to characterize the residual vector �eld � de�ned by the relation

�= u−∇’− curl (155)



that clearly belongs to space L2(�)N , admits a divergence and a curl equal to zero and issuch that the right-hand sides of relation (150) and (154) associated to this �eld are equal tozero: {

(�;∇�)=0; ∀�∈H 10 (�; �1)

(�; curl �)=0; ∀� ∈ H 100(�; �1;�2)

(156)

Let g∈H 1=200 (�2) be an arbitrary scalar �eld and �∈H 1

0 (�; �1) be the variational solution ofthe problem

��=0 in �

�=0 on �1

�= g on �2

(157)

After integrating by parts the expression (div �; �), the �rst relation in (156) shows that

〈�•�; g〉(H 1=200 (�2))

′ ; H 1=200 (�2)

= 0 (158)

i.e. in a common way of speaking, � • n=0 on �2.In an analogous way, let ∈TH 1=2

00 (�1) be a given tangential vector �eld and �∈TH 100

(�; �1;�2) be the variational solution of the problem

−��=0 in �

div �=0 on 9�

n× �× n= on �1

�× n=0 on �2

�1�1 ;�2�=0

(159)

that can also be written under the form{�∈H 1

00(�; �1;�2)

(curl �; curl’)=−〈; �×’〉; ∀’∈H 100(�; �1;�2)

(160)

Relation (156), interpretation (159) of variational formulation (160) and Green formula(curl �; �)= (�; curl �) + 〈n× �; �〉 show that

〈�×�; 〉(TH 1=200 (�1))

′ ; TH 1=200 (�1)

= 0 (161)

which means that �× n=0 on �1. We have established that �∈M 0(�; �2;�1). Moreover, takinginto account (155), we have, due to orthogonality relations (148) and (150),

(u− �; �)=0; ∀�∈M 0(�; �2;�1) (162)

In consequence, the relation �=�0�2 ;�1 is completely established and Theorem 2 is proven.


1114 F. DUBOIS

5. A FIRST EXISTENCE AND UNIQUENESS RESULT

We prove in this section that the abstract result that mathematically modelizes the Stokesproblem under vorticity–velocity–pressure formulation can eectively be used. The letter �still represents a connected bounded domain of R2 or R3 with a smooth boundary �≡ 9�which is, as in relations (28) and (29), supposed to have been partitionated in two ways,(�m;�p) on the one hand and (��;�t) on the other hand. We suppose, in this section, that wehave the particular condition that �� ≡�m and �t ≡�p.

9�=�m ∪�p; �m ∩�p= ∅ (163)

��=�m and �t =�p (164)

Moreover we suppose the following technical hypothesis, which is quite strong, to be precisedin a geometrical point of view, in the future.

Hypothesis 1. No special functions between �m and �p

M 0(�; �m;�p)= {0} (165)

We also introduce the three Hilbert spaces W;X and Y for vorticity, velocity and pressure,respectively, de�ned in relations (36)–(38), i.e.

W = {−’∈H (curl;�); ’× n=0 on ��; i:e: �×’=0 in (TH 1=200 (��))

′} (166)

X = {v ∈ H (div;�); v • n=0 on �m; i:e: �•v=0 in (H 1=200 (�m))

′} (167)

Y =

{L2(�) if meas (�p) �=0{q ∈ L2(�); (q; 1)=0} if meas (�p)=0

(168)

and datum � according to{�=(�0; �0; f)∈H 1=2

00 (�p)×TH 1=200 (�t)× (L2(�))N

‖�‖data ≡‖�0‖1=2;�p + ‖�0‖1=2;�t + ‖f‖0;�(169)

We can set and prove the following theorem with homogeneous Dirichlet boundary condi-tions and non-homogeneous Neumann boundary conditions when formulated with help of thethree �elds of vorticity, velocity and pressure.

Theorem 3 (A particular Stokes problem)Let � be a connected bounded domain satisfying hypotheses recalled in the beginning ofthis section, a partition (�m;�p) of its boundary 9� satisfying relation (163) and a secondpartition of the boundary (��;�t) chosen according to (164). Moreover we suppose that thepair (�m;�p) satis�es Hypothesis 1.



We consider datum � de�ned in relation (169). The Stokes problem under vorticity–velocity–pressure formulation

!− curl u=0 in �

curl!+∇p=f in �

div u=0 in �

u • n=0 on �m

!× n=0 on �m

p=�0 on �p

n× u× n=�0 on �p

(170)

admits the following variational formulation:

!∈W; u∈X; p∈Y (171)

(!;’)− (u; curl’) = 〈�0; �×’〉; ∀’∈W (172)

(curl!; v)− (p; div v) = (f; v)− 〈�0; �•v〉;∀v∈X (173)

(div u; q) = 0; ∀q∈Y (174)

Problem (171)–(174) admits a unique solution (!(�); u(�); p(�))∈W ×X ×Y that continu-ously depends on datum � de�ned in (169):

∃C¿0; ‖!(�)‖W + ‖u(�)‖X + ‖p(�)‖Y6C‖�‖data (175)

Proof of Theorem 3Taking into Account all the work done in Section 2 to obtain variational formulation (39)–(42) written here under the form (171)–(174), we just have to apply Theorem 1, i.e. verifywhether the four hypotheses (52)–(54) of this abstract result are true. The letters W;X andY represent relative to Theorem 1 the objects with the names introduced in (166)–(168). Wealso set Z =L2(�)N ; Dv=div v for all v∈X; R’= − curl’ for all ’∈W . Then it is clear fromde�nitions (4) and (5) of H (div;�) and H (curl;�) norms that properties (43) and (44) hold.We establish now the equality

kerD=ImR (176)

that clearly implies relation (53). We �rst have the inclusion ImR⊂ kerD. If ’∈W , wehave ’× n=0 on �m ≡��, i.e. �×’=0 in (TH 1=2

00 (�m))′ and in consequence (curl’)•n≡ div�

(’× n)=0 on �m. Then the �rst inclusion ImR⊂ kerD is established.On the other hand and according to Theorem 2, let v∈L2(�)N be decomposed under the

form

v=∇�+ curl + � (177)


1116 F. DUBOIS

with scalar potential �, vector potential and special vector �eld � chosen according to�∈H 1

0 (�; �p)

∈H 100(�; �p;�m)

�∈M 0(�; �m;�p)

(178)

From relation (157), we know that scalar potential � is the solution of the following problem:{�∈H 1

0 (�; �p)

(∇�;∇)= (v;∇); ∀∈H 10 (�; �p)

(179)

But we have also the fact that ∈H 10 (�; �p) implies naturally that the trace � belongs to

space H 1=200 (�m). In consequence, due to de�nition (167) of space X and if moreover v∈kerD,

we have

(v;∇)= − (div v; ) + 〈�•v; �〉(H 1=200 (�m))

′ ;H 1=200 (�m)

= 0

Then left-hand side of (179) is identically null and � is identically equal to zero. In conse-quence, v belongs to ImR because relation (165) of Hypothesis 1 implies �≡ 0.We consider now second hypothesis (52) of Theorem 1:

∃a¿0; infq∈Y; q �=0

supv∈X; v �=0

(q; div v)‖q‖Y ‖v‖X ¿a (180)

For doing this, we consider the following auxiliary variational problem:{�∈H 1

0 (�; �m)

(∇�;∇)= − (q; ); ∀∈H 10 (�; �m)

(181)

and even if meas (�m)=0, problem (181) has a unique solution in space H 10 (�; �m) satis�ying

the continuity relative to datum q:

‖�‖1;�6C‖q‖0;� (182)

where C¿0 only depends on domain � and on sub-boundary �m. We set

v=∇� (183)

and this �eld satis�es relation

div v=��= q∈L2(�) (184)

then v belongs to space H (div;�). Moreover, taking into account the variational formulation(181), we deduce

�•v= 9�9n =0 in (H 1=200 (�m))

′ (185)



We deduce from (184) and (185) that vector �eld v belongs to space X . We have also, dueto inequality (182) and characterization (184),

‖v‖2div;�6‖�‖21;� + ‖q‖20;�6(1 + C2)‖q‖20;� = (1 + C2)((q; div v)‖q‖0;�

)2that establishes inf-sup condition (180) with the particular choice for constant a:

a=1√

1 + C2(186)

Second inf-sup condition (53) is written in the following form:

∃b¿0; infv∈ker D; v �=0

sup’∈W;’ �=0

(v; curl’)‖v‖X ‖’‖W ¿b (187)

Here we take into account the fact that kerD=ImR (relation (176)). If v∈kerD is given,then there exists ’∈H 1

00(�; �p;�m)=H 100(�; �t ;��) such that

v=curl’ (188)

Moreover, due to (144), there exists some constant C1 which is only a function of � and �msuch that

‖’‖1;�6C1‖v‖0;� (189)

We deduce from (188) and (189) the following set of inequalities:

‖’‖curl;�6‖’‖1;�6C1‖v‖0;� =C1(v; curl’)‖v‖0;� =C1

(v; curl’)‖v‖div;�

which establishes (187) with

b=1C1

(190)

The last hypothesis (54) of Theorem 1, i.e.

∃d¿0; ∀’∈ker R; (’;’)¿d‖’‖2W (191)

is straightforward to deduce from algebraic relation (7) that de�nes the norm in spaceH (curl;�). Then Theorem 3 is established.

6. CONCLUSION AND ACKNOWLEDGEMENTS

We have proposed in this work a new formulation of the Stokes problem for mechanics ofincompressible �uids. The key point of this formulation is to consider a velocity �eld thatbelongs to Hilbert space H (div;�) of vector �elds. We have explored the natural choice forvorticity, i.e. the hypothesis that vorticity belongs to space H (curl;�) and this Hilbert spacecoincides with Sobolev space H 1(�) when domain � is included in R2. We have developed anabstract result for triple-mixed formulation, a new theorem for the representation of squarely


1118 F. DUBOIS

integrable vector �elds and have �nally obtained a positive result that states that the Stokesproblem is well posed in a particular case of boundary conditions. In the general case, vorticitydoes not belong to space H (curl;�) (see [49] for two-dimensional domains) and we proposein [50] a weaker form of our vorticity–velocity–pressure formulation.The author thanks Tou�c Abboud, Mohamed Amara, Claude Bardos, Abderrahmane Ben-

dali, Christine Bernardi, Carlos Conca, Monique Dauge, Marie Farge, Vivette Girault, JeanGiroire, Jean-Fran�cois Maitre, Sylvie Mas-Gallic, Mohand Moussaoui, Jean-Claude N�ed�elec,Arnaud Poitou, Pierre-Arnaud Raviart and last but not the least Michel Salaun and St�ephanieSalmon, for their stimulating discussions and helpful comments on �rst draft of this article.

REFERENCES

1. Landau L, Lifchitz E. Fluid mechanics. Nauka: Moscow, 1953.2. Lions JL. Quelques M�ethodes de R�esolution des Probl�emes aux Limites Nonlin�eaires. Dunod: Paris, 1969.3. Temam R. Theory and Numerical Analysis of the Navier–Stokes Equations. North-Holland: Amsterdam, 1977.4. Girault V, Raviart PA. Finite Element Mehods for Navier–Stokes Equations, Theory and Applications. SpringerVerlag, 1996.

5. Harlow F, Welch J. Numerical calculation of time-dependent viscous incompressible �ow of �uid with freesurface. Physics of Fluids 1965; 8:2182–2189.

6. Arakawa A. Computational design for long term numerical integration of the equations of �uid motion. Journalof Computational Physics 1996; 1:119–143.

7. Harper, RP, Hirt CW, Sicilian JM. Flow2d: a computer program for transient, two-dimensional �ow analysis.Flow Science Inc., Report FSI-83-00-01, 1983.

8. Patankar SW, Spalding DB. A calculation procedure for heat, mass and momentum transfer in three-dimensionalparabolic �ows. International Journal of Heat Mass Transfer 1972; 15:1787–1806.

9. Ciarlet PG. The Finite Element Method for Elliptic Problems. North-Holland: Amsterdam, 1978.10. Nicolaides RA. Flow discretization by complementary volume schemes. AIAA Paper No. 89-1978, 1989.11. Girault V, Lopez H. Finite element error estimates for the MAC scheme. IMA Journal of Numerical Analysis

1996; 16:347–379.12. Raviart PA, Thomas JM. A mixed �nite element method for 2nd order elliptic problems. In Lectures Notes in

Mathematics, vol. 306, Dold, Eckmann (eds). Springer: Berlin, 1977; 292–315.13. N�ed�elec JC. Mixed �nite elements in R3. Numerische Mathematik 1980; 35:315–341.14. Adams R. Sobolev Spaces. Academic Press: New York, 1975.15. Ern A, Guermond JL, Quartapelle L. Vorticity–velocity formulations of the Stokes problem in 3D. Mathematical

Methods in the Applied Sciences, 1999; 22:531–546.16. Dubois F. Une formulation tourbillon-vitesse-pression pour le probl�eme de Stokes. C.R. Acad. Sci. Paris, t.

314, S�erie 1, 1992; 277–280.17. Dubois F. A vorticity-velocity-pressure formulation for the Stokes problem. Ninth International Conference on

Finite Elements in Fluids, Venice (Italy), October 1995.18. Begue C, Conca C, Murat F. Pironneau O. C.R. Acad. Sciences Paris, Serie 1 1987; 304:23–28.19. Girault V. Incompressible �nite element methods for Navier–Stokes equations with nonstandard boundary

conditions in R3. Mathematics of Computation 1988; 51(183):55–74.20. Bendali A, Dominguez JM, Gallic S. A variational approach for the vector potential formulation of the Stokes

and Navier–Stokes problems arising in three dimensional domains. Journal of Mathematical Analysis andApplications 1985; 107:537–580.

21. Glowinski R. Approximations externes par �el�ements �nis de Lagrange d’ordre un et deux du probl�eme deDirichlet pour l’op�erateur biharmonique et m�ethode it�erative de r�esolution des probl�emes approch�es. In Topicsin Numerical Analysis, Miller J (ed.). Academic Press: New York, 1973; 123–171.

22. Ciarlet PG, Raviart PA. A mixed �nite element method for the biharmonic equation. In Mathematical Aspectsof Finite Elements in Partial Di�erential Equations, de Boor C (ed.). Academic Press: New York, 1974;125–145.

23. Girault V. A Combined Finite Element and Marker and Cell Method for Solving Navier–Stokes Equations.Numerische Mathematik 1976; 26(183):39–59.

24. N�ed�elec JC. El�ements �nis mixtes incompressibles pour l’�equation de Stokes dans R3. Numerische Mathematik1982; 39:97–112.

25. Amara M, Barucq H, Dulou�e M. Une formulation mixte convergente pour les �equations de Stokestridimensionnelles. C.R. Acad. Sci. Paris, t. 328, S�erie 1, 1999; 935–938.



26. Foias C, Temam R. Remarques sur les �equations de Navier–Stokes stationnaires et les ph�enom�enes successifsde bifurcation, Annali Della Scuola Normale Superior de Pisa Science 1978; 5(1):29–63.

27. Roux FX. Seminar, Paris 6 University, March 1984.28. Duvaut G, Lions JL. Les In�equations en M�ecanique et an Physique. Dunod: Paris, 1972.29. Dubois F, Salaun M, Salmon S. Aspects num�eriques de la formulation tourbillon-vitesse-pression. Congr�es

National d’Analyse Num�erique, mai 1997.30. Bernardi C. Optimal �nite element interpolation on curved domains. SIAM Journal of Numerical Analysis

1989; 26:1212–1240.31. Levillain V. Couplage �el�ements �nis-�equations int�egrales pour la r�esolution des �equations de Maxwell en milieu

h�et�erog�ene. Thesis, Ecole Polytechnique, Palaiseau, 1991.32. Amrouche C, Bernardi C, Dauge M, Girault V. Vector potentials in three-dimensional nonsmooth domains.

Mathematical Methods in the Applied Sciences 1998; 21(9):823–864.33. Lions JL, Magenes E. Problemes aux Limites Non Homog�enes et applications. Dunod: Paris, 1968.34. Babu ska I. Error-bounds for �nite element method. Numerische Mathematik 1971; 16:322–333.35. Brezzi F. On the existence, uniquencess and approximation of saddle point problems arising from Lagrange

multipliers. Mathematical Modelling and Numerical Analysis (Rairo), 1974; 2:129–151.36. Dubois F. Une formulation tourbillon-vitesse-pression pour le probl�eme de Stokes. Internal Report Aerospatiale

Espace & Defense, ST/S No. 206, October 1991.37. Raviart PA. Personal communication, 1991.38. Lax P, Milgram N. Parabolic equations. In Contributions to the Theory of Partial Dierential Equations. Annals

of Mathematical Studies, No 33, Princeton, 1954; 167–190.39. Choquet-Bruhat Y. G�eom�etrie Di��erentielle et Syst�emes Ext�erieurs. Dunod: Paris, 1968.40. Agmon S, Douglis A, Nirenberg L. Estimates near the boundary for solutions of elliptic partial dierential

equations satisfying general boundary conditions I. Communications in Pure and Applied Mathematics 1959;12:623–727, II, Communications in Pure and Applied Mathematics 1964; 17:35–92.

41. Peetre J. An other approach to elliptic boundary value problems. Communications in Pure and AppliedMathematics 1961; 14:711–731.

42. Tartar L. Nonlinear partial dierential equations using compactness method. In Technical Summary Report No1584. University of Wisconsin, Madison, 1976.

43. Bendali A. Approximation par �el�ements �nis de surface de probl�emes de diraction des ondes�electromagn�etiques, Th�ese d’Etat, Universit�e Paris 6, Janvier 1984.

44. Grisvard P. Elliptic problems in Nonsmooth Domains. Pitman, 1985.45. Dauge M. Elliptic boundary value problems on corner domains. Lecture Notes in Mathematics No. 1341.

Springer: Berlin, 1988.46. Dubois F. Discrete vector potential representation of a divergence-free vector �eld in three-dimensional domains:

numerical analysis of a model problem. SIAM Journal of Numerical Analysis 1990; 27(5):1103–1141.47. Abboud T, Starling F. Diraction d’ondes �electromagn�etiques par un �ecran mince, Internal Report No. 246,

Centre de Math�ematiques Appliqu�ees de l’ Ecole Polytechnique, Palaiseau, France, June 1991.48. Terrasse I. R�esolution math�ematique et num�erique des �equations de Maxwell instationnaires par une m�ethode

de potentiels retard�es. Thesis, Ecole Polytechnique, Palaiseau, January 1993.49. Bernardi C, Girault V, Maday Y. Mixed spectral element approximation of the Navier–Stokes Equations in a

stream function and vorticity formulation. IMA Journal for Numerical Analysis 1992; 12:565–608.50. Dubois F, Salaun M, Salmon S. Vorticity–velocity–pressure and vorticity stream function formulations for the

Stokes problem. Internal Report No. 353 Institut A�ero-Technique, 2001.


vorticity-velocity-pressure formulation for the stokes problem

Documents