Stabilization of a non standard FETI-DP mortar method for the Stokes problem

ESAIM: M2AN 48 (2014) 285–304 ESAIM: Mathematical Modelling and Numerical Analysis DOI: 10.1051/m2an/2013102 www.esaim-m2an.org STABILIZATION OF A NON STANDARD FETI-DP MORTAR METHOD FOR THE STOKES PROBLEM E. Chac´ on Vera1and T. Chac´ on Rebollo2 Abstract. In a recent paper [E. Chac´on Vera and D. Franco Coronil, J. Numer. Math. 20 (2012) 161–182.] a non standard mortar method for incompressible Stokes problem was introduced where the use of the trace spaces H1/2and H1/2 00 and a direct computation of the pairing of the trace spaces with their duals are the main ingredients. The importance of the reduction of the number of degrees of freedom leads naturally to consider the stabilized version and this is the results we present in this work. We prove that the standard Brezzi–Pitkaranta stabilization technique is available and that it works well with this mortar method. Finally, we present some numerical tests to illustrate this behaviour. Mathematics Subject Classification. 65N30, 65N55. Received July 26, 2012. Revised June 13, 2013. Published online January 10, 2014. 1. Introduction The starting point in many domain decomposition methods is to split the computational domain Ω(Ω⊂R2 bounded polygonal domain to ease presentation) into non overlapping (open) polygonal subdomains {Ωs}swith the purpose of working locally and recover a global solution after some iteration process. Following Girault and Raviart [10], Grisvard [11]orAdams[1] we introduce some well known notation to ease the presentation: Denote by Hr(Ω) the usual Sobolev space endowed with the Sobolev norm · r,Ω and by H1 0(Ω) the closure in H1(Ω) of all the smooth functions with support inside Ω. Then, we split Ω,the computational domain, as follows Ω=∪S s=1Ωs,Ω s∩Ωt=∅for s=t. (1.1) Next, denote by ∂Ωsthe boundary of any of the (open) polygonal subdomains, suppose that Γs,t =∂Ωs∩∂Ωt Keywords and phrases. Incompressible Stokes problem, non-standard FETI-DP mortar method. 1Dpto. Matem´aticas, Facultad de Matem´aticas, Universidad de Murcia, Campus Espinardo, 30100 Murcia, Spain. [email protected] 2Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, Tarfia sn. 41012 Sevilla, Spain. [email protected] Article published by EDP Sciences c EDP Sciences, SMAI 2014 286 E. CHAC´ ON VERA AND T. CHAC ´ ON REBOLLO is either an edge (i.e., a segment), a crosspoint or empty and, finally, consider E0={Γe}e=1,...,E (1.2) the sorted set of all edges inside Ω, also known as the skeleton of the decomposition. A common idea to devise a parallel computation of a function u∈H1 0(Ω) is to use a variational formulation where the restriction to subdomains of the original equation and some interface terms show up. Usually, this process is performed via integration by parts and boundary terms appear as dualities. How to handle this dualities is the core of the following arguments. A standard space for mortar methods is the product Hilbert space Xδ={v∈L2(Ω); vs=v|Ωs∈H1(Ωs)∩H1 0(Ω),1≤s≤S}(1.3) endowed with the natural norm v1,δ =S  s=1 vs2 1,Ωs1/2 . However, this space Xδhas the disadvantage that it does not control the jumps [v]Γeacross interfaces Γe;the internal crosspoints (final points of internal edges) are the trouble. As a consequence, it seems that the following strict subspace of Xδ X={v∈Xδ,[v]Γe∈H1/2 00 (Γe),∀Γe∈E 0}.(1.4) gives a better framework because it considers only the functions whose jumps belong to the trace spaces H1/2 00 (Γe), which are the correct ones for neighboring subdomains. Endowed with the graph norm vX=S  s=1 vs2 1,Ωs+ E  e=1 [v]Γe2 1/2,00,Γe1/2 (1.5) Xis a Hilbert space. Here, · 1/2,00,Γeis the norm induced by the scalar product (·,·)1/2,00,Γeon H1/2 00 (Γe) given by (w,v)1/2,00,Γe=(w,v)1/2,Γe+Γe w(x)v(x) d(x, ∂Γe)dx where d(x, ∂Γe) is the distance of xto the boundary of Γeand (w,v)1/2,Γe=Γe w(x)v(x)dx+ΓeΓe (w(x)−w(y)) (v(x)−v(y)) |x−y|2dxdy. To simplify notation, we write {w,v}Γe=(w,v)1/2,00,Γe,∀w, v ∈H1/2 00 (Γe).(1.6) Clearly, Xis not a Hilbert space with respect to · 1,δ because the norm in H1/2 00 (Γe) is strictly stronger than that on H1/2(Γe). Moreover, H1 0(Ω)XXδ.(1.7) STABILIZATION OF A NON STANDARD FETI-DP MORTAR METHOD FOR THE STOKES PROBLEM 287 Remark 1.1. Poincare’s inequality and the control we have on the jumps, interconnecting all subdomains, allow the use on Xof the equivalent norm |v|2 X=(v,v)X,where(·,·)Xis the scalar product given for any u, v ∈Xby (u, v)X= S  s=1 (∇us,∇vs)Ωs+ E  e=1 {[u]Γe,[v]Γe}Γe, what is to say, |v|2 X= 0 implies v=0. As far as we know, the use of the space Xδwas introduced by Raviart and Thomas [13]andXwas considered by Braess, Dahmen and Wieners [2] and by Ben Belgacem [3]. Following their analysis, one sees that H1 0(Ω) can be identify as a subspace of Xand, also, as a subspace of Xδby means of appropriate linear restrictions: •In the case of Xδ, just one linear restriction on Xδthat acts globally on the skeleton E0is enough. The global action guarantees that the local functions are correctly glued together, see the characterization (2.12) of H1 0(Ω)inRaviartandThomas[13]. •On the other hand, for Xwe need as many linear restrictions as interfaces. But, as the coupling is already present in Xby definition, all of these linear restrictions are independent from each other and their local action is just to guarantee the nullity of the jump across each interface. In both cases, these linear restrictions are identified with Lagrange multipliers. The most important issue here is the nature of these multipliers: •in the case of Xδ, they are a properly chosen sum of the duality pairings H−1/2(∂Ωs)−H1/2(∂Ωs). •while in the case of X, they are the pairing H−1/2 00 (Γe)−H1/2 00 (Γe) for each of the local Lagrange multipliers. Many popular methods for solving elliptic problems are based on the idea of forcing the nullity of the jumps across interfaces in some way. For instance, mortar methods do it in a weak sense while FETI methods impose pointwise continuity at some interface nodes. A proper formulation of the continuous problem will guarantee the stability of the discrete version. Consequently, it is interesting to discuss some of the characteristics of these two above decompositions: First, the obtention of the jumps at the interfaces from the first approach is not possible at the continuous level. That is because at corner points the normal vector is not defined and it is not possible to split locally the action of the normal derivative, considered as a linear form, see Grisvard [11]. Therefore, the second approach seems to be safer because jumps naturally show up. Moreover, the control of the jumps only in terms of the gradient norms is impossible due to internal crosspoints. Again, the use of Xseems to be more interesting. Secondly, the Lagrange multipliers are usually implemented by means of the scalar L2product on interfaces and end up representing the normal fluxes across these interfaces, usually some physical quantity of interest. But this procedure makes sense only when these normal fluxes are smooth enough to be identified with L2 functions. Even when the true solution is smooth enough to guarantee smooth fluxes (something that usually comes from the regularity of the data), the norms involved in the existence analysis are usually weaker and do not see this extra regularity for the true solution. As a consequence, we are faced with stability issues that are delicate to solve. In fact, the continuous version of the Lagrange multiplier action on each interface Γeis the duality H−1/2 00 (Γe)−H1/2 00 (Γe) which means that the regularity expected for the Lagrange multiplier is H−1/2 00 . As a consequence, assuming an L2regularity for this multiplier and writing this duality in terms of the scalar product in L2(Γe) requires stabilization techniques. Although the analysis in the case of elliptic problems has been done on a discrete level using mesh dependent norms, see for instance the works by Ben Belgacem and Maday [3] or Bernardy, Maday and Patera [5], a continuous study face the question of controlling all the jumps and the difficulty of handling the duality pairing for discrete functions without wasting information. In the previous works by Bernardi et al. [4,5]andChac´on Vera et al. [6], these questions are handled by using the space Xand the (Riesz) identification of the Hilbert space H−1/2 00 (Γe) with its dual H1/2 00 (Γe): the 288 E. CHAC´ ON VERA AND T. CHAC ´ ON REBOLLO duality H−1/2 00 (Γe)−H1/2 00 (Γe) is represented in terms of the scalar product in H1/2 00 (Γe). As a consequence, the duality is computable and the regularity of the Lagrange multiplier now is H1/2 00 instead of H−1/2 00 . The stability questions are solved but the disadvantage is that the Lagrange multiplier loses its physical meaning, although the normal fluxes can still be recovered from the computed solution. A similar effort can be found in the paper by Lee and Park [12]. According to these ideas, in [4,6] a continuous framework is proposed for elliptic equations in terms of a saddle point problem that resembles the standard primal hybrid formulation and, sort of, falls into the FETIDP mortar familly of methods: FETI-DP alike because only continuity at cross points is imposed and mortar alike because nonmatching meshes at interfaces are allowed, although we use a different scalar product for the mortaring process. In this method, all of the previously mentioned technical difficulties are avoided and the continuous analysis holds for internal approximations even in the case of non conforming meshes, i.e.,weobtain mesh independent inf-sup and continuity conditions at the discretization level. Another remarkable property is that the stability of the discretization is independent of the varying mesh sizes, i.e., it does not matter which side is the mortar. In [7] this effort is extended to the case of the incompressible Stokes equations with mixed finite elements showing that the same results hold. In this work we study the use of stabilization techniques to make the computation less expensive. Our analysis will be presented in the two dimensional setting to simplify the presentation. These main ideas extend easily to the three dimensional situation although a detailed study is required. Finally some numerical tests are shown as a conclusion. 2. Incompressible stokes equations The splitting of the computational subdomain for the pressure is not easy because the zero average imposes a global restriction. To fix this difficulty, we introduce a new variable in the Stokes equations that sets free the pressure space from this restriction. Incompressible Stokes equations with homogeneous boundary conditions amount to find u∈H1 0(Ω)= (H1 0(Ω))2and p∈L2(Ω) such that (∇u, ∇v)Ω−(p, div(v))Ω=(f,v)Ω,∀v∈H1 0(Ω) −(q,div(u))Ω=0,∀q∈L2(Ω) Ω p=0. We add a new scalar unknown that takes the role of the pressure average as follows: we look for a pair of values (u, τ)∈H1 0(Ω)×Rand p∈L2(Ω) such that (∇u, ∇v)Ω−(p, div(v))Ω+t(τ−Ω p)=(f,v)Ω,∀(v,t)∈H1 0(Ω)×R −(q,div(u))Ω−τΩ q=0,∀q∈L2(Ω). Evidently, q≡1 implies τ=0andv≡0 implies Ωp= 0. This formulation can be seen as a saddle point problem if we pair the uand τvariables. Set W=H1 0(Ω)×Rand let v=(v,t)∈Wany element of W.We norm Wby using v2 W=(v,t)2 W=∇v2 0,Ω +t2 and let (·,·)W:W×W→ Rbe the scalar product on W,i.e., (u,v)W=((u, τ),(v,t))W=(∇u, ∇v)Ω+τt. STABILIZATION OF A NON STANDARD FETI-DP MORTAR METHOD FOR THE STOKES PROBLEM 289 Take b:W×L2(Ω)→ Rgiven by b(q,(v,t)) = −(q,div(v))Ω−tΩ q. Then, we look for u=(u, τ)∈Wand p∈L2(Ω) such that for all v=(v,t)∈Wand q∈L2(Ω) (u,v)W+b(p, v)=(f,v)Ω,(2.1) b(q,u)=0.(2.2) While the ellipticity for this saddle point problem is trivial, we need an inf-sup condition for b. The following result is claimed but not fully proved in [7], we give it now: Lemma 2.1. For any p∈L2(Ω)there exists a positive constant β>0with sup (v,t)∈W b(p, (v,t)) (v,t)W ≥sup v∈H1 0(Ω),t∈R b(p, (v,t)) (∇v2 0,Ω +t2)1/2≥βp0,Ω.(2.3) As a consequence, problem (2.1)−(2.2)is well posed and its solution is that of the original Stokes problem with Dirichlet homogeneous boundary conditions. Proof. Given p∈L2(Ω), for any (v,t)∈Wwe have b(p, (v,t)) = −(p, div(v))Ω−tΩ p =−(p−pΩ,div(v))Ω−t|Ω|pΩ where we use that Ωdiv(v)dx= 0 and take pΩ=1 |Ω|Ω p. Now we choose (v,t) properly: as p−pΩ∈L2 0(Ω) there exists a function v∈H1 0(Ω) and a constant Csuch that −(p−pΩ)=div(v),∇v0,Ω ≤Cp−pΩ0,Ω. Then b(p, (v,t)) = −(p−pΩ,div(v))Ω−t|Ω|pΩ =p−pΩ2 0,Ω −t|Ω|pΩ. On the other hand, p−pΩ2 0,Ω =p2 0,Ω +pΩ2 0,Ω −2(p, pΩ)0,Ω =p2 0,Ω −|Ω|p2 Ω. Then b(p, (v,t)) = p2 0,Ω −|Ω|p2 Ω−t|Ω|pΩ. and the choice t=−pΩgives b(p, (v,t )) = p2 0,Ω.(2.4) 290 E. CHAC´ ON VERA AND T. CHAC ´ ON REBOLLO Finally, we see that (v,t )W≤Cp0,Ω.Weusethat p2 0,Ω =p−pΩ2 0,Ω +|Ω|p2 Ω. (v,t )2 W=∇v2 0,Ω +(t)2≤Cp−pΩ2 0,Ω +p2 Ω ≤max{C, 1} {p−pΩ2 0,Ω +p2 Ω}. Then, straightforward bounds give the existence of a positive constant β>0 that only depends on the domain Ωsuch that (v,t )W≤β−1p0,Ω and the inf-sup condition (2.3) follows with this constant β>0 b(p, (v,t )) = p2 0,Ω ≥βp0,Ω(v,t )W. Remark 2.2. The solution of the standard Stokes equations is associated to the saddle point (u, p)∈ (H1 0(Ω))2×L2 0(Ω) of the Lagrangean functional over (H1 0(Ω))2×L2 0(Ω)givenby L(v,q)=1 2Ω ∇v2 0,Ω −Ω fv−Ω qdiv(v) or, equivalently, with the constrained minimization of the energy problem min v∈(H1 0(Ω))21 2Ω ∇v2 0,Ω −Ω fv  subject to div(v)=0. We have taken V=(H1 0(Ω))2×Rand considered the saddle point ((u, τ),p)∈V×L2(Ω) of the Lagrangean functional over V×L2(Ω)givenby L((v,t),q)=1 2Ω ∇v2 0,Ω +1 2t2−Ω fv−Ω qdiv(v)−tΩ q, or, equivalently, with the constrained minimization of the new energy problem min v∈(H1 0(Ω))21 2t2+1 2Ω ∇v2 0,Ω −Ω fv  subject to div(v)+t=0. As a consequence, we have relaxed the energy functional in a simple way and this small change removes the zero average restriction on the pressure. 2.1. Multisubdomain formulation Next we split Ω=∪S s=1Ωsas in (1.1)−(1.2) such that each Ωsis of area O(H2) and shape regular while each Γeis of length O(H)forsomefixedH>0, see Figure 1for example. This means that there exists positive constants c1,c 2and H>0 such that for any e=1,...,E and s=1,...,S we have c1H≤|Γe|≤c2H, c1H2≤|Ωs|≤c2H2, where |·|is the corresponding Lebesgue measure. We also assume that E0={Γe}e=1,...,E STABILIZATION OF A NON STANDARD FETI-DP MORTAR METHOD FOR THE STOKES PROBLEM 291 Figure 1. Example of domain decomposition where cross points are marked with big dots and the skeleton is the set of all the Γifor i=1,2,3,...,10. In the case of Dirichlet Boundary conditions we have ΓN=∅. contains the sorted set of all edges inside Ω. This is now the skeleton of the decomposition. We let Cbe the set of all vertices of the polygonal subdomains Ωsthat are not on ∂Ω; these will be called cross points. On the other hand, any open line segment, without corners or junctions with other lines, on the boundary of a subdomain between two consecutive crosspoints will be referred to as an edge, or interface, see Figure 1. Finally, we denote by [v]Γethe jump across any interface Γe. Once all the interfaces and subdomains are sorted these jumps are always computed following the same pattern. We propose to relax jumps across interfaces and add the correct multipliers for each of these jumps. First we define our decomposition for the velocities. We take Xδ={v∈L2(Ω); vs=v|Ωs∈H1(Ωs)∩H1 0(Ω),1≤s≤S},(2.5) and consider Xgiven by X={v∈Xδ,[v]Γe∈H1/2 00 (Γe),∀Γe∈E 0}(2.6) endowed with the norm |v|2 X=(v,v)X,where(·,·)Xis the scalar product given for any u, v ∈Xby (u, v)X= S  s=1 (∇us,∇vs)Ωs+ E  e=1 {[u]Γe,[v]Γe}Γe. Then, we set the product hilbert space X=X×Xand observe that H1 0(Ω) is the subspace of elements of X with zero jumps across all the internal edges inside Ω. Following the idea of taking a scalar value to handle the pressure on the whole domain Ω,weconstruct V=X×Rand represent by v=(v,t) any element of Vwhere v∈Xand t∈R. Obviously, Vis Hilbert space with norm v2 V=|v|2 X+t2. Now for the pressure space we consider M=S s=1 L2(Ωs)(≈L2(Ω)) and represent the incompressibility restriction in terms of the continuous bilinear form b:M×V→ Rgiven by b(q,v)=− S  s=1 (qs,div(vs))Ωs−t S  s=1 Ωs qs(2.7) 292 E. CHAC´ ON VERA AND T. CHAC ´ ON REBOLLO where each qs∈L2(Ωs). Next, for each Γe∈E 0we take H1/2 00 (Γe)=(H1/2 00 (Γe))2, and handle the lagrange multipliers for the jumps with the space N= E  e=1 H1/2 00 (Γe) endowed with the scalar product for all λ, μ ∈Ngiven by (λ, μ)N= E  e=1 {λe,μ e}Γe.(2.8) A trivial characterization for H1 0(Ω) as a subspace of Xis provided by the continuous bilinear form c:N×X→ R given by c(λ, v)= E  e=1 {λe,[v]Γe}Γe(2.9) in the form H1 0(Ω)={v∈X;c(λ, v)=0,∀λ∈N}. The new formulation we propose at the continuous level is: find u=(u, τ)∈V,p={ps}s∈Mand λ={λe}e∈ Nsuch that S  s=1 (∇us,∇vs)Ωs+ E  e=1 {[u]Γe,[v]Γe}Γe+τt − S  s=1 (ps,div(vs))Ωs−t S  s=1 Ωs ps+ E  e=1 {λe,[v]Γe}Γe= S  s=1 (f,vs)Ωs, − S  s=1 (qs,div(us))Ωs−τ S  s=1 Ωs qs=0, E  e=1 {μe,[u]Γe}Γe=0 for all v=(v,t)∈V,q={qs}s∈Mand μ={μe}e∈N. We see that we added the jumps to the elliptic terms to guarantee the control on the internal jumps and replaced the pairings H−1/2 00 (Γ)−H1/2 00 (Γ) for the normal fluxes on the edges by the scalar product in H1/2 00 (Γ), hence all terms are suitable to compute in a Galerkin approach. Remark 2.3. The solution to this problem satisfies u∈H1 0(Ω),p∈L2(Ω)andτ=Ωp; using any constant qs≡qfor all swe also obtain τ=0andthenp∈L2 0(Ω). Finally, using test functions v∈H1 0(Ω), it gives the solution of the incompressible Stokes equations on Ω. In a more compact form, taking F(v)=S s=1(f,vs)Ωs,the problem is ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Find (u,p,λ)∈V×M×Nsuch that (u,v)V+b(p, v)+c(λ, v)=F(v),(a) b(q,u)=0,(b) c(μ, u)=0,(c) for all (v,q,μ)∈V×M×N (2.10) STABILIZATION OF A NON STANDARD FETI-DP MORTAR METHOD FOR THE STOKES PROBLEM 293 where (u,v)V= S  s=1 (∇us,∇vs)Ωs+ E  e=1 {[u]Γe,[v]Γe}Γe+τt, b(p, v)=− S  s=1 (ps,div(vs))Ωs−t S  s=1 Ωs ps, c(μ, u)=c(μ, u)= E  e=1 {μe,[u]Γe}Γe. The bilinear forms b:M×V→ Rand c:N×V→ Rare continuous with norms b≤1andc≤1and there exists positive constants β>0andγ>0 such that inf q∈Msup v∈V b(p, v) qMvV ≥β>0,(2.11) inf μ∈Nsup v∈V c(μ, v) μNvV ≥γ>0.(2.12) Condition (2.12) was proved in Lemma 2.9 of [3] and in Theorem 2 in [6] and condition (2.11) follows from the global inf-sup condition (2.3). Remark 2.4. A key result is that the inf-sup for the bilinear form bis reached for a function v∈Vwith zero jumps, recall (2.3), sup (v,t)∈V b(p, (v,t)) (v,t)V ≥sup v∈H1 0(Ω),t∈R b(p, (v,t)) (∇v2 0,Ω +t2)1/2≥βp0,Ω, while the inf-sup for bilinear form cis reached for functions v∈Vwith non-zero jumps, see Lemma 2.9 of [3] for instance. As a consequence, formulation (2.10) has a unique solution that is the one of (2.1)−(2.2). 2.2. Setting of the dual problem Next, if we denote by primal variables velocity and pressure on the subdomains and dual variables the Lagrange multipliers, we eliminate the primal variables in terms of the dual variables, i.e., we obtain a dual problem that once solved will give the correct boundary data for the primal variables. The writing of problem (2.10) in terms of operators will further simplify our analysis. For a Hilbert space H, we denote by (·,·)Hits scalar product, by Hits dual space and represent its duality by ·,·H;moreover,for a linear operator Twe denote by Tis adjoint. Then, (2.10)is ⎧ ⎪ ⎨ ⎪ ⎩ Find (u,p,λ)∈V×M×Nsuch that Au+Bp+Cλ=Fin V, Bu =0 inM, Cu=0 inN, (2.13) where Ais the Riesz isomorphism between Vand Vthat satisfies Au,vV=(u,v)V=Av,uV,∀u,v ∈V, Bis a linear operator from Vinto Mand Bv,pM=b(p, v)=Bp, vV∀v∈V,∀p∈M. 300 E. CHAC´ ON VERA AND T. CHAC ´ ON REBOLLO Figure 2. Triangulations for the two subdomain test with conforming and nonconforming meshes, curved interface. Figure 3. Global solution on fine mesh, pressure and velocity fields for first test. 4. Some numerical tests All the numerical tests that we present have been performed with FreeFem++ [8]. 4.1. Academic test 1: two subdomains We first consider Ω=(0,2) ×(0,1) and split it into two subdomains through a curved interface using conforming and nonconforming meshes; on Figure 2we show a typical conforming and nonconforming mesh configuration in the case h=1/10, in the nonconforming. We move the flow via the boundary conditions u(x, 1) = x(2 −x),u(x, 0) = −x(2 −x),u(0,y)=u(2,y)=0 which somehow resemble the driven lid test and Figure 3shows the computed solution in the global domain. On Figure 4we show the decay of the residuals for the CG method measure in the H1/2 00 norm while on Figure 5 STABILIZATION OF A NON STANDARD FETI-DP MORTAR METHOD FOR THE STOKES PROBLEM 301 0 2 4 6 8 10 12 10 −6 10 −4 10 −2 10 0Conforming meshes: Error Lagrange multipliers vs. number of iterations 0 1 2 3 4 5 6 7 8 9 10 −3 10 −2 10 −1 10 0Nonconforming meshes: Error Lagrange multipliers vs. number of iterations h=1/10 h=1/20 h=1/40 h=1/10 h=1/20 h=1/40 Figure 4. Log in base 10 plot of the residuals for the CG method with respect to the number of iterations. Residuals measured in the H1/2 00 norm for first test and with different values of the mesh size h. In the nonconforming case the resolution of the interface from both subdomains is different but similar. Figure 5. Computed velocity field and pressure for test 1 with non the conforming mesh shown on Figure 2. we show the computed solution via our domain decomposition method. We have found similar results for the conforming and non conforming configuration. 4.2. Academic test 2: a floating subdomain As a second and final test we consider a disk Ωcentered at (0,0) with radius R=2withafloatingsubdomain, another disk centered at the point (0.5,25) with radius r=0.75. In this last configuration there are no cross points and we just work with the H1/2scalar product on the internal disk boundary. We let the flow rotate via 302 E. CHAC´ ON VERA AND T. CHAC ´ ON REBOLLO Figure 6. Global pressure and velocity solution for second test. Figure 7. Computed pressure solution for second test with conforming triangulations. the boundary conditions u(x, y)=−xy/2,v(x, y)=x/2,on Γ1={(x, y)∈∂Ω, x > 0}, u(x, y)=0,v(x, y)=0 on ∂Ω \Γ1. Figures 6to 8show the global and computed solution with conforming triangulations after six iterations, here the convergence results and ratios are similar to the previous example. To conclude, Figures 9to 11 show the results using nonconforming triangulations after two iterations. 5. Conclusions The use of the H1/2 00 norm allows to obtain an abstract framework for a FETI-DP mortar method which is simple to analyze in continuous and in discrete formulation. We studied the use of this tool on several differential STABILIZATION OF A NON STANDARD FETI-DP MORTAR METHOD FOR THE STOKES PROBLEM 303 Figure 8. Computed velocity field solution for second test with conforming triangulations. Figure 9. 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