Stability and optimal convergence of unfitted extended finite element methods with Lagrange multipliers for the Stokes equations

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Stability and optimal convergence of unfitted extended finite element methods with Lagrange multipliers for the

Stokes equations

Michel Fournié, Alexei Lozinski

To cite this version:

Michel Fournié, Alexei Lozinski. Stability and optimal convergence of unfitted extended finite element

methods with Lagrange multipliers for the Stokes equations. UCL workshop :Geometrically Unifitted

Finite Element Methods and Applications, 2016, Londres, United Kingdom. �hal-01978126�

extended finite element methods with Lagrange multipliers for the Stokes equations

Michel Fourni´e and Alexei Lozinski

Abstract We study a fictitious domain approach with Lagrange multipliers to dis- cretize Stokes equations on a mesh that does not fit the boundaries. A mixed finite element method is used for fluid flow. Several stabilization terms are added to im- prove the approximation of the normal trace of the stress tensor and to avoid the inf-sup conditions between the spaces of the velocity and the Lagrange multipliers.

We generalize first an approach based on eXtended Finite Element Method due to Haslinger-Renard [12] involving a Barbosa-Hughes stabilization and a robust re- construction on the badly cut elements. Secondly, we adapt the approach due to Burman-Hansbo [4] involving a stabilization only on the Lagrange multiplier. Mul- tiple choices for the finite elements for velocity, pressure and multiplier are con- sidered. Additional stabilization on pressure (Brezzi-Pitkranta, Interior Penalty) is added, if needed. We prove the stability and the optimal convergence of several vari- ants of these methods under appropriate assumptions. Finally, we perform numerical tests to illustrate the capabilities of the methods.

1 Introduction

Let D ⊂ R ^d , d = 2 or 3, be a bounded polygonal (polyhedral) domain. We are inter- ested in the Stokes equations in a setting motivated by the fluid-structure interaction, especially by simulations of particulate flows. We thus assume that D is decomposed into the fluid domain F and the solid one S. The domains F and S are separated by the interface Γ , cf. Fig. 1. We also denote Γ wall = ∂ D and assume, for simplicity, that Γ and Γ wall are disjoint. Consider the problem

Michel Fourni´e

Name, Address of Institute e-mail: [email protected] Alexei Lozinski

Laboratoire de Math´ematiques de Besanc¸on, UMR CNRS 6623, Univ. Bourgogne Franche-Comt´e, e-mail: [email protected]

1

−2 divD(u) + ∇p = f in F, (1)

div u = 0 in F , (2)

u = g on Γ , (3)

u = 0 on Γ wall , (4)

for u and p, respectively, the velocity and the pressure of the fluid filling F. Here D(u) = ¹

2 ∇u + ∇u ^T

and the viscosity has been set to 1 for simplicity. In appli- cations we have in mind, i.e. simulations of the motion of rigid or elastic particles flowing in the fluid, the interface Γ is moving in time while the outer boundary Γ _wall is immobile. In this chapter, we shall study Finite Element (FE) discretizations of the problem above on a mesh fixed on D which is thus fitted to Γ _wall but is cut in an arbitrary manner by interface Γ . The interest of these methods in the context of fluid-structure interaction is that it allows one to avoid remeshing when the interface advances with time.

Fig. 1: The fluid domain F, the interface Γ and the outer boundary Γ

.

Introducing the force exerted by the fluid on the solid at each point of Γ

λ = −2D(u)n + pn, on Γ (5)

with n the unit normal looking outside from F, and interpreting λ as the Lagrange multiplier associated with the Dirichlet condition (3), we can write the weak formu- lation of (1)–(4) as

Find (u, p, λ ) ∈ H _wall ¹ (F) ^d × L ² ₀ (F) × H ⁻

(Γ ) ^d such that

A(u, p,λ ; v, q, µ) = L(v, µ), ∀(v, q, µ) ∈ H _wall ¹ (F) ^d ×L ² ₀ (F) × H ⁻

(Γ ) ^d (6) where

A(u, p, λ ;v, q, µ) = 2 Z

F

D(u) : D(v)−

Z

F

(p divv +q divu) + Z

Γ

(λ · v + µ · u) L(v , µ) =

Z

F f · v + Z

Γ

g · µ

and H _wall ¹ (F) is the space of H ¹ functions on F vanishing on Γ _wall . The FE methods studied in this chapter will be based on the variational formulation (6). They shall thus discretize the Lagrange multiplier λ , alongside u and p, thus giving a natural approximation of the force exerted by the fluid on the solid.

As mentioned above, our FE methods will rely on a ”background” fixed mesh T _h that lives on the fluid-structure domain D ⊃ F (the boundary of D is Γ _wall and is well fitted by T _h ). In the actual computations, the elements of T _h having no intersection with F will be discarded and the FE spaces for velocity and pressure will be defined on the mesh T _h ^e := T _h ⁱ ∪ T _h ^Γ where T _h ^Γ is the union of elements of T _h that are cut by Γ and T _h ⁱ is the union of elements of T _h inside F. The FE space for the Lagrange multiplier will live only on the cut elements T _h ^Γ , cf. Fig. 2.

Fig. 2: The meshes T

= T

∪ T

: the triangles of T

are marked by • and those of T

are marked by • ; triangles marked by • are not used.

Denoting by F _h ^e (resp. F _h ⁱ , F _h ^Γ ) the domain covered by mesh T _h ^e (resp. T _h ⁱ , T _h ^Γ ) we introduce three FE spaces

V _h ⊂ H _wall ¹ (F _h ^e ) ^d , Q _h ⊂ L ² (F _h ^e )∩ L ² ₀ (F), W _h ⊂ L ² (F _h ^Γ ) ^d (7) to approximate velocity, pressure and Lagrange multiplier respectively. Several choices of FE spaces V _h , Q _h , and W _h will be considered, but we restrict ourselves in this chapter to triangular (tetrahedral) quasi-uniform meshes T _h and to the standard continuous piecewise polynomial FE-spaces P k (k ≥ 1) or the piecewise constant space P 0 on such a mesh. ¹ Our FE spaces will be always based on meshes inherited from T _h : T _h ^e for V _h , Q _h , and T _h ^Γ for W _h . Note that velocity and pressure are approx- imated on a domain F _h ^e slightly larger than F but all the integrals in the discretized problem will be calculated on F or Γ . Note also that we choose the FE space for λ on a domain F _h ^Γ rather than on the surface Γ to avoid the complicated issue of meshing a surface.

A straightforward Galerkin approximation of (6) is not stable in general (al- though it often works in practice, as will be seen in the numerical experiments at the end of this chapter). Several stabilization techniques were therefore proposed in

1

The case of regular non-quasi-uniform meshes can also be easily treated at the expense of some

technicalities. However, in applications, one will typically use a simplest possible mesh on D (for

example, structured Cartesian) so that the quasi-uniformity restriction seems quite acceptable.

the literature, using either Lagrange multipliers [12, 4] or a Nitsche-like method [6]

to take into account the boundary conditions on Γ . We shall be concerned in this chapter only with the methods based on Lagrange multipliers. Firstly, we adapt the method of Haslinger-Renard (cf. [12] for the Poisson problem) to Stokes equations.

The method is based on a Barbosa-Hughes stabilization [1] on Γ with additional local treatment on badly cut mesh elements. An extension to Stokes equations was already presented in [8] but the analysis there relied on a number of hypotheses, difficult to verify. In this paper, we present a complete theoretical analysis in two cases :

1. LBB-unstable velocity-pressure FE pairs, namely, P 1 − P 1 or P 1 − P 0 elements.

A stabilization is needed in this case even on a fitted mesh. We shall show, that adding the well known stabilization terms such as Brezzi-Pitk¨aranta [3] for P 1 − P 1 elements (or interior penalty for P 1 − P 0 elements) to a Haslinger-Renard fictitious domain method, as in [8], makes it stable and optimally convergent.

2. LBB-stable velocity-pressure FE pairs, namely, P k − P k−1 Taylor-Hood ele- ments. We show that a version of the method above (with and additional pressure stabilization on Γ but avoiding stabilization over the whole domain F ) is also stable and optimally convergent. Our proofs are presented here only in the 2D case and under some additional assumptions on the mesh.

We generalize moreover a method by Burman-Hansbo [4] to Stokes equations.

This is also a fictitious domain method with Lagrange multipliers. Unlike the method by Haslinger-Renard (where the stabilization comes by enforcing (5) on Γ and thus involves all the variables u, p, λ ), one stabilizes here only the multi- plier λ by enforcing its continuity in some sense, so that the structure of resulting matrices is simpler. Fortunately, much of the theory outlined above can be reused for the analysis of this method. We are thus able to prove the stability and optimal convergence for the same choices of the FE spaces as above.

The chapter is concluded by numerical experiments aiming at comparing differ- ent stabilizations and choices of of FE spaces.

Nomenclature.

Domains: F is the fluid domain where the problem (1)–(4) is posed while F _h ⁱ , F _h ^e , F _h ^Γ are the domains occupied by the meshes T _h ⁱ , T _h ^e , T _h ^Γ respectively. We have thus F _h ⁱ ⊂ F ⊂ F _h ^e and F _h ^Γ = F _h ^e \ F _h ⁱ .

Meshes: T _h ⁱ , T _h ^e , T _h ^Γ are submeshes of a background mesh T _h so that T _h ⁱ = {T ∈ T _h : T ⊂ F }, T _h ^Γ = {T ∈ T _h : T ∩Γ 6= ∅ } and T _h ^e := T _h ⁱ ∪ T _h ^Γ .

E _h ^e and E _h ^Γ stand for the sets of interior edges of T _h ^e and T _h ^Γ respectively.

F _T (resp. Γ T ) denotes T ∩ F (resp. T ∩ Γ ) for any cut element T ∈ T _h ^Γ .

Norms: k · k _k,ω stands for the norm in H ^k (ω) where ω can be a domain in R ^d or a (d −1)-dimensional manifold. We identify H ⁰ (ω ) with L ² (ω).

| · | _k,ω stands for the semi-norm in H ^k (ω ), k > 0.

k · k ∞,ω stands for the norm in L ^∞ (ω).

2 Methods `a la Haslinger-Renard

The starting point for the construction of the Haslinger-Renard method (proposed in [12] for the Poisson equation) is to add to the variational formulation (6) the Barbosa-Hughes stabilization [1], which enforces the relation λ + 2D(u)n − pn = 0 on Γ . These terms take the form

−γ 0 h Z

Γ

(λ + 2D(u)n − pn)· (µ +2D(v)n −qn) (8) with a mesh-independent γ 0 > 0. This idea, at least in the context of the Poisson equation as in [12], produces a stable and optimally convergent approximation pro- vided the mesh elements are cut by Γ in a certain way so that F ∩ T is a big enough portion of T for any T ∈ T _h ^Γ . If, for some elements, this is not the case the method can be still cured by replacing the approximating polynomial in such “bad elements”

by the polynomial extended from from adjacent “good elements”. The relation be- tween bad and good elements is made precise in the following

Assumption A. We fix a threshold θ _min ∈ (0, 1] and declare any T ∈ T _h ^Γ a good element (resp. bad element) if ^|F _|T

_| ^| ≥ θ _min (resp. ^|F _|T|

^| < θ _min ). We assume that one can choose for any bad element T a “good neighbor” T ⁰ ∈ T _h ^e , ^|T _|T

^∩F|

| ≥ θ _min , such that T and T ⁰ share at least one node, cf. Fig. 3.

Remark 1. Typically, Assumption A will hold true even for θ min = 1 if the mesh is sufficiently refined. One could also relax the notion of a neighbor (at the expense of some complication of the forthcoming proofs) to the requirement dist(T, T ⁰ ) ≤ Ch with a mesh-independent C > 0.

Fig. 3: Good element T

and bad element T

We now define a “robust reconstruction” on F _h ^Γ for the FE functions on F _h ^e Definition 1. For any v _h ∈ V _h set v b _h on any T ∈ T _h ^Γ as

• v b _h = v _h on T if T is a good element,

• ( v b _h ) _|T = (v _h ) _|T

if T is a bad element. Here T ⁰ is the good neighbor of T from

Assumption A and the relation should be understood in the sense that v b _h on T is

taken as the same polynomial as the polynomial giving v _h on T ⁰ .

For any q _h ∈ Q _h , one constructs q b _h in the same way.

We shall show in the subsequent paragraphs that adding stabilization (8) to (6) and replacing u,v in these terms (sometimes also p, q) by their robust reconstructions produces indeed a stable approximation to the Stokes equations. We end this general introduction to the Haslinger-Renard method by a Proposition illustrating the use- fulness of the selection criterion for good elements, showing that the L ² norm on the cut portion of an element T controls L ^∞ (and hence any other) norm on the whole element with an equivalence constant depending on the relative measure of the cut portion, followed by a list of interpolation error estimates that shall be needed in the forthcoming analysis.

Proposition 1. Let p be a polynomial of degree ≤ k and θ ∈ (0, 1]. Then for any T ∈ T _h and any measurable set S ⊂ T with |S| ≥ θ|T | one has

kpk _∞,T ≤ C

h ^d/2 kpk _0,S (9)

with a constant C > 0 depending only on θ , k and mesh regularity.

Proof. By scaling, it is sufficient to prove (9) on a reference element. We thus fix a simplex T ∈ R ^d of diameter h = 1 and consider for any p ∈ P k

N _θ (p) = inf

S⊂T,|S|≥θ|T| kpk _0,S

It is easy to see that N _θ is a continuous function on the finite-dimensional space P k . Consequently, it attains a minimum on the set Σ 1 := {p ∈ P k , k pk _∞,T = 1}, i.e. ∃α ≥ 0 and p _α ∈ Σ ₁ such that N _θ (p) ≥ N _θ (p _α ) = α for all p ∈ Σ ₁ . It remains to prove α > 0. To this end, let m(δ ) = meas{x ∈ T : | p _α (x)| ≤ δ } for any δ ≥ 0. Since m(δ ) is decreasing down to 0 as δ → 0, one can find ε > 0 s.t. m(ε) ≤ ^θ ₂ |T |. We observe now k p _α k ² _0,S ≥ ε ² |S| − ^θ ₂ |T |

for any S ⊂ T , hence α ² = N _θ ² (p _α ) ≥ ε ^{2 θ} ₂ |T | > 0.

By homogeneity, this also proves N _θ (p) ≥ αk pk _∞,T for all p ∈ P k entailing (9) with C = _α ¹ (recall that the proof is done on the reference element with h = 1). u t

We are going to establish interpolation estimates on the cut domain. To this end, we introduce

Assumption B. Ω is a Lipschitz domain and there exist constants c Γ ,C _Γ > 0 such that for any T ∈ T _h ^Γ

1. |Γ _T | ≤ C Γ h ^d−1 with Γ _T := T ∩Γ ;

2. there exists a unit vector χ _T ∈ R ^d such that χ _T · n ≥ c Γ a.e. on Γ _T where n is the unit normal looking outward from F.

Remark 2. The bound on |Γ _T | in the first part of the Assumption B is automatically

satisfied on Lipschitz domain. We prefer however to write this bound explicitly in

order to emphasize that some of the estimates below will depend on the constant C _Γ ,

so that Γ should be supposed not too oscillating. The second part of the Assumption

B is not too restrictive either. Typically, one can take χ _T as the normal n at the

middle point of Γ _T if Γ _T is smooth or as the average between the two normals if Γ _T is the union of two segments (in the case when Ω is a 2D polygon). Such choices will suffice on a sufficiently refined mesh.

Proposition 2. Let V _h , Q _h ,W _h be (respectively) P k

, P k

, P k

FE spaces on meshes T _h ^e , T _h ^e , T _h ^Γ as in (7). Under Assumptions A and B, there exist interpolation opera- tors I _h ^u : H _wall ¹ (F) ^d → V _h , I _h ^p ∈ L ² ₀ (F) → Q _h , I _h ^λ : H

(Γ ) ^d → W _h s.t. for any suffi- ciently smooth u, p, λ

1

h ku − I _h ^u uk 0,F + |u − I _h ^u u| 1,F + 1

√

h ku − I _h ^u uk _0,Γ ≤ Ch ^s

|u| _s

+1,F (10) (for all integer s u : 0 ≤ s u ≤ k u ) k∇u − ∇I _h ^u uk _0,Γ + k∇u − ∇ I c _h ^u uk _0,Γ

≤ Ch ^s

⁻

|u| _s

+1,F (11) (for all integer s _u : 1 ≤ s _u ≤ k _u ) 1

h k p −I _h ^p pk 0,F +| p − I _h ^p p| 1,F

+ 1

√ h

kp − I _h ^p pk _0,Γ + k p − I c _h ^p pk _0,Γ

≤ Ch ^s

| p| _s

+1,F (12) (for all integer s _p : 0 ≤ s _p ≤ k _p )

√ 1

h kλ − I _h ^λ λ k _0,Γ ≤ Ch ^s

|λ | _s

λ

+

,Γ (13)

(for all integer s _λ : 0 ≤ s _λ ≤ k _λ ) with C > 0 depending only on the constants in Assumptions A, B, and on the mesh regularity, and k _u ≥ 1 in the case of estimate (11). Moreover, operator I _h ^λ can be extended to I _h ^λ : H _wall ¹ (F) ^d → W _h s.t. for any λ ˜ ∈ (H ^s

⁺¹ (F )∩H _wall ¹ ) ^d and an integer s _λ , 0 ≤ s _λ ≤ k _λ

1

h k λ ˜ −I _h ^λ λ ˜ k _0,F

h

+ | λ ˜ −I _h ^λ λ ˜ | _1,F

h

+ 1

√ h k λ ˜ − I _h ^λ λ ˜ k _0,Γ ≤ Ch ^k

| λ ˜ | _k

λ

+1,F (14) Proof. We start with the construction of I _h ^u . Extension theorems for Sobolev spaces guarantee for any u ∈ H ^s

⁺¹ (F) ^d existence of ˜ u ∈ H ^s

⁺¹ (F _h ^e ) ^d with k uk ˜ _s

+1,F

h

≤

Ckuk _s

+1,F and ˜ u = u on F . Let ˜ I _h : H _wall ¹ (F _h ^e ) ^d → V _h be a Cl´ement-type interpola- tion operator [9] satisfying

1

h k u− ˜ I ˜ _h uk ˜ _0,T +| u ˜ − I ˜ _h u| ˜ _1,T + 1

√

h k u ˜ − I ˜ _h uk ˜ _{0,∂ T} +

√

hk∇( u ˜ − I ˜ _h u)k ˜ _{0,∂ T} ≤Ch ^s

| u| ˜ _s

_+1,ω

on any T ∈ T _h ^e with ω T begin the patch of elements of T _h ^e touching T . Let I _h ^u u =

I ˜ _h u| ˜ _F . Summing the estimates above over all the mesh elements yields immediately

the estimates in L ² (F ) and H ¹ (F) in (10). Now, on any element T ∈ T _h ^Γ

c Γ ku −I _h ^u uk ² _0,Γ

T

≤

Z

Γ

( u ˜ − I ˜ _h u) ˜ ² χ T ·n = Z

F

div(( u ˜ − I ˜ _h u) ˜ ² χ T )−

Z

F∩∂ T

( u ˜ − I ˜ _h u) ˜ ² χ T ·n since ∂ F _T = Γ _T ∪ (F ∩ ∂ T ). Developing and applying the interpolation estimates above gives

c _Γ ku − I _h ^u uk ² _0,Γ

T

≤

Z

F

2( u ˜ − I ˜ _h u)∇( ˜ u ˜ − I ˜ _h u) ˜ · χ T +k u ˜ − I ˜ _h uk ˜ ² _{0,F ∩∂ T}

≤ 2k u ˜ − I ˜ _h uk ˜ _0,T | u ˜ − I ˜ _h u| ˜ _1,T + k u ˜ − I ˜ _h uk ˜ ² _{0,∂ T} ≤ Ch ^2s

⁺¹ (| u| ˜ ² _s

u

+1,ω

) Summing this over all the elements in F _h ^Γ yields the L ² (Γ )-estimate in (10).

If s _u ≥ 1, we have moreover on any T ∈ T _h ^e h| u ˜ − I ˜ _h u| ˜ _2,T + √

hk∇( u ˜ − I _h u)k ˜ _{0,∂ T} ≤ Ch ^s

| u| ˜ _s

_+1,ω

This, by the same argument as above, gives the L ² (Γ ) estimate on ∇(u − I _h ^u u) in (11). In order to extend this to ∇(u − I c _h ^u u) consider a bad element T and its good neighbor T ⁰ . Both T and T ⁰ belong to the patch ω _T

and examining the derivation of interpolation estimates for the Cl´ement interpolator ˜ I _h reveals that the polynomial ( I ˜ _h u) ˜ _|T

gives actually an optimal approximation of ˜ u on the whole ω _T

, i.e.

|u − I c _h ^u u| 1,T = | u ˜ − ( I ˜ _h u) ˜ _|T

| 1,T ≤ | u ˜ −( I ˜ _h u) ˜ _|T

| 1,ω

≤ Ch

|u| _s

+1,ω

Similarly, ¹ _h ku − I c _h ^u uk _0,T + ^{√ 1}

h ku − I c _h ^u uk _{0,∂ T} ≤ Ch

|u| _s

_+1,ω

T0

. Thus, the same argu- ment as above gives the L ² (Γ ) estimate on ∇(u − I c _h ^u u) in (11).

The remaining estimates (12), (13) and (14) are proved in a similar manner. We skip the details and make only the following remarks:

• The operator I _h ^p should preserve the restriction that pressure is of zero mean on F. We thus define it as I _h ^p p = I ˜ _h p ˜ − i _h (p) where ˜ I _h is the Cl´ement interpolation operator on T _h ^e , ˜ p is an extension of p to F _h ^e , and i _h (p) = ^R _F I ˜ _h p. The correction ˜ i _h (p) can be bounded as

|i _h (p)| = Z

F

( I ˜ _h p ˜ − p)

≤ |F |

k p ˜ − I ˜ _h pk ˜ _0,F

h

≤ Ch ^s

⁺¹ | p| ˜ _s

_+1,F

h

and thus it does not perturb the estimates (12).

• Concerning the interpolation of λ , we note that (13) is in fact an easy corol- lary to (14). Indeed, for any λ ∈ H ^k

⁺

(Γ ) ^d there exists (by the trace theorem) λ ˜ ∈ H ^k

⁺¹ (F _h ^Γ ) ^d satisfying ˜ λ | _Γ = λ and | λ ˜ | _k

λ

+1,F

≤ C|λ | _k

λ

+

,Γ . We can thus define I _h ^λ λ := I _h ^λ λ ˜ and observe that (14) entails (13).

u

t

2.1 P 1 − P 1 velocity-pressure spaces with Brezzi-Pitk¨aranta stabilization.

Let us choose P 1 FE spaces for both V _h and Q _h , add Brezzi-Pitk¨aranta-like stabiliza- tion for the pressure and the Barbosa-Hughes-like stabilization on the interface as described above. We choose to introduce the robust reconstruction from Definition 1 in the last terms only for the velocity in this case (on both trial function u _h and test function v _h ). The method thus reads

Find (u _h , p _h ,λ _h ) ∈ V _h × Q _h ×W _h such that

A ^HR−BP (u _h , p _h ,λ _h ; v _h ,q _h , µ _h ) = L(v _h , µ _h ), ∀(v _h , q _h , µ _h ) ∈ V _h × Q _h ×W _h (15) where

A ^HR−BP (u, p, λ ; v , q, µ) = A(u, p, λ ; v , q, µ)

− γ 0 h Z

Γ

(λ + 2D( b u)n − pn) · (µ + 2D( b v)n − qn) −θ h ² Z

F

∇p · ∇q

V _h , Q _h are continuous P 1 FE spaces on mesh T _h ^e and W _h is P 1 or P 0 FE space on mesh T _h ^Γ , cf. (7).

We remind that the Brezzi-Pitk¨aranta stabilization (the last term above) should be present on F to compensate the lack of the discrete inf-sup in P1-P1 velocity- pressure FE spaces. In addition, in our fictitious domain situation, it is extended to the larger domain F _h ^e thus helping to ensure stability near Γ .

In the following propositions, Assumptions A and B are implicitly implied and the constants C may vary from line to line and depend on c Γ ,C _Γ > 0 from Assump- tion B, θ min from Assumption A, and on the mesh regularity.

Proposition 3. For all v _h ∈ V _h one has

hk∇ v b _h k ² _0,Γ ≤ C|v _h | ² _1,F (16) Proof. Taking any T ∈ T _h ^Γ and denoting its good neighbor by T ⁰ we observe

k∇ v b _h k _0,Γ

≤ p

|Γ _T |k∇ v b _h k _L

(T ) ≤ C p

|Γ _T |k∇v _h k _L

(T

) ≤ C p |Γ _T |

h ^d/2 k∇v _h k _0,F

T0

The last inequality above holds by Proposition 1 with a constant dependent on θ min . The last but one inequality is easily proven by scaling given that T and T ⁰ are neigh- bors. Using the bound |Γ _T | ≤ C _Γ h ^d−1 and summing over all T ∈ T _h ^Γ yields (16).

u t

Proposition 4. For all q _h ∈ Q _h one has hkq _h k ² _0,Γ ≤ C

kq _h k ² _0,F + h ² |q _h | ² _1,F

Proof. Using the notations T, T ⁰ as in the preceding proof and assuming that these two elements share a node x, we observe

kq _h k _0,Γ

≤ p

|Γ _T |kq _h k _L

(T ) ≤ p

|Γ _T |(|q _h (x)| +hk∇q _h k _L

(T) )

≤ p

|Γ _T |(kq _h k _L

(T

) + hk∇q _h k _L

(T ) ) ≤ C p |Γ _T |

h ^d/2 (kq _h k 0,F

+ hk∇q _h k 0,T ) We have used again Proposition 1 on the good element T ⁰ . We conclude thanks to

|Γ _T | ≤ C _Γ h ^d−1 from Assumption B and the summation over all T ∈ T _h ^Γ . u t We shall also need a special interpolation operator adapted to functions vanishing on Γ , the idea of which goes to [13].

Proposition 5. There exists an interpolation operator I _h ⁰ : H ₀ ¹ (F) ^d → V _h such that kv −I _h ⁰ vk _0,F ≤ Ch|v| _1,F , |I _h ⁰ v| _1,F ≤ C|v| _1,F

and I _h ⁰ v = 0 on F _h ^Γ (and consequently I _h ⁰ v = 0 on Γ ) for any v ∈ H ₀ ¹ (F ) ^d with a mesh-independent constant C > 0.

Proof. The construction of I _h ⁰ will be based on the interpolator I _h ^u from Proposition 2 with k u = 1. For any v ∈ H ₀ ¹ (F) ^d , let us put I _h ⁰ v(x) = I _h ^u (x) at all the interior nodes x of T _h ⁱ (i.e. excepting the nodes lying on ∂ T _h ⁱ ) and I _h ⁰ v(x) = 0 on all the nodes of T _h ^Γ . Since I _h ⁰ v is the piecewise linear function on T _h ^e , this uniquely defines it everywhere on F _h ^e . Moreover, I _h ⁰ v = 0 on F _h ^Γ .

Let us denote, for a mesh edge E lying on ∂ F _h ⁱ , the adjacent element from T _h ^Γ by T ^Γ and the union of all the elements from T _h ⁱ sharing at least a node with E by ω _E ⁱ . By scaling

kI _h ^u v− I _h ⁰ vk _0,ω

E

≤ C √

hkI _h ^u vk _0,E ≤ C(kI _h ^u vk _0,T

+h|I _h ^u v| _1,T

)

Summing over all such edges and introducing the extension ˜ v to F _h ^e as in the proof of Proposition 2 yields

kI _h ^u v−I _h ⁰ vk _0,F

h

≤C(kI _h ^u vk _0,F

h

+h|I _h ^u v| _1,F

h

) ≤ C(k v−I ˜ _h ^u vk _0,F

h

+k vk ˜ _0,F

h

+h|I _h ^u v| _1,F

) Since I _h ⁰ v = 0 on F _h ^Γ this entails

kv− I _h ⁰ vk 0,F ≤ kvk 0,F + kv − I _h ^u vk _0,F

h

+ kI _h ^u v− I _h ⁰ vk _0,F

≤ C(k vk ˜ _0,F

h

+k v ˜ −I _h ^u vk _0,F

h

+h|I _h ^u v| _1,F

) We now employ the bound k vk ˜ _0,F

h

≤ Ch| v| ˜ _1,F

h

, which is valid since F _h ^Γ is a band of thickness h around Γ and ˜ v = 0 on Γ . Moreover,

1

h k v ˜ −I _h ^u vk _0,F

h

+|I _h ^u v| _1,F

h

≤ C| v| ˜ _1,F

h

as follows from the proof of Proposition 2, cf. (10) with s _u = 0. Since | v| ˜ 1,F

h

≤

C|v| 1,F by the extension theorem, this proves the announced estimate of kv − I _h ⁰ vk 0,F .

The estimate for the H ¹ norm of I _h ⁰ v follows using the inverse inequality and the L ² error estimates proved above:

|I _h ⁰ v| 1,F = |I _h ⁰ v| _1,F

h

≤ |I _h ⁰ v−I _h ^u v| _1,F

h

+|I _h ^u v| _1,F

h

≤ C

h kI _h ⁰ v−I _h ^u vk _0,F

h

+|I _h ^u v| _1,F

h

≤C|v| 1,F

u t

Lemma 1. Under Assumption A and B, taking γ 0 > 0 small enough and any θ > 0, there exists a mesh-independent constant c > 0 such that

inf

(u

,p

,λ

)∈V

×Q

×W

sup

(v

,q

,µ

)∈V

×Q

×W

A ^HR−BP (u _h , p _h , λ h ; v _h , q _h , µ h )

|||u _h , p _h , λ _h ||| |||v _h , q _h , µ _h ||| ≥ c where the triple norm is defined by

|||u, p, λ ||| =

|u| ² _1,F +k pk ² _0,F + h ² |p| ² _1,F

h

+ hkλ k ² _0,Γ + 1 h kuk ² _0,Γ

1/2

Proof. We observe, using Proposition 3, A ^HR−BP (u _h , p _h ,λ _h ; u _h , −p _h , −λ _h )

= 2kD(u _h )k ² _0,F −4γ ₀ hkD( u b _h )k ² _0,Γ + γ ₀ hkλ _h − p _h nk ² _0,Γ + θh ² |p _h | 1,F

h

≥ 2kD(u _h )k ² _0,F −Cγ ₀ |u _h | ² _1,F + γ 0 hkλ _h − p _h nk ² _0,Γ + θh ² |p _h | _1,F

h

≥ 1

K |u _h | ² _1,F +γ ₀ hkλ _h − p _h nk ² _0,Γ + θ h ² | p _h | _1,F

h

We have used in the last line the assumption that γ 0 is sufficiently small and Korn inequality

|v| ² _1,F ≤ KkD(v)k ² _0,F , ∀v ∈ H _wall ¹ (F) (17) Note that the inequality is valid in this form because the functions from H _wall ¹ (F) vanish on Γ wall , i.e. on a part of the boundary ∂ F with non zero measure.

The continuous inf-sup condition [10] implies for all p _h ∈ Q _h there exists v _p ∈ H ₀ ¹ (F) ^d such that

− Z

F p _h divv _p = k p _h k ² _0,F and |v _p | _1,F ≤ Ck p _h k _0,F . (18)

Recalling that v p = I _h ⁰ v p = 0 on Γ we can write

− Z

F p _h div(I _h ⁰ v p ) = kp _h k ² _0,F − Z

F p _h div(I _h ⁰ v p −v _p )

= kp _h k ² _0,F − Z

F ∇p _h · (v _p − I _h ⁰ v _p ) ≥ kp _h k ² _0,F −Ch|p _h | 1,F

h

|v _p | 1,F

≥ kp _h k ² _0,F −Ch|p _h | _1,F

h

k p _h k 0,F (19)

where we have used the bounds from Proposition 5 and (18). Combining this with Young inequality we obtain

A ^HR−BP (u _h , p _h ,λ _h ; I _h ⁰ v _p ,0, 0) ≥ −kD(u _h )k 0,F kD(I _h ⁰ v _p )k 0,F + k p _h k ² _0,F −Ch|p _h | 1,F

h

k p _h k 0,F

≥ 1

2 k p _h k ² _0,F −C|u _h | ² _1,F −Ch ² |p _h | ² _1,F

Recall interpolation operator I _h ^λ from Proposition 2 and observe, using Proposi- tion 3 with Young inequality,

A ^HR−BP (u _h , p _h ,λ _h ; 0, 0, 1

h I _h ^λ u _h ) = 1 h Z

Γ

u _h · I _h ^λ u _h −γ ₀ Z

Γ

(2D( b u _h )n − p _h n+λ _h )·I _h ^λ u _h

≥ 1

2h ku _h k ² _0,Γ − 1

2h ku _h −I _h ^λ u _h k ² _0,Γ −γ ₀ C

√

h |u _h | 1,F + kλ _h − p _h nk 0,Γ

ku _h k 0,Γ + ku _h − I _h ^λ u _h k 0,Γ

≥ 1

4h ku _h k ² _0,Γ − C

h kI _h ^λ u _h − u _h k ² _0,Γ −C|u _h | ² _1,F −Chkλ _h − p _h nk ² _0,Γ

≥ 1

4h ku _h k ² _0,Γ −C|u _h | ² _1,F −Chkλ _h − p _h nk ² _0,Γ In the last line, we have used the bound ku _h − I _h ^λ u _h k _0,Γ ≤ C √

h|u _h | 1,F , i.e. (14) with s _λ = 0.

Combining the above inequalities, we can obtain for any κ , η > 0, A ^HR−BP (u _h , p _h ,λ _h ; u _h +κI _h ⁰ v _p , −p _h , −λ _h + η

h I _h ^λ u _h ) ≥ 1

K |u _h | ² _1,F + κ

2 k p _h k ² _0,F + η 4h ku _h k ² _0,Γ + (θ −Cκ)h ² |p _h | _1,F

h

+ (γ ₀ −Cη )hkλ _h − p _h nk ² _0,Γ −C(κ + η)|u _h | ² _1,F (20) In order to split p _h and λ _h inside kλ _h − p _h nk 0,Γ we establish the following bounds with any t > 0 and use finally Proposition 4

k p _h n − λ h k ² _0,Γ ≥ k p _h k ² _0,Γ +kλ _h k ² _0,Γ −(t + 1)kp _h k ² _0,Γ − 1

t +1 kλ _h k ² _0,Γ

= t

t +1 kλ _h k ² _0,Γ −tk p _h k ² _0,Γ ≥ t

t + 1 kλ _h k ² _0,Γ − Ct h

kp _h k ² _0,F + h ² | p _h | ² _1,F

(21)

Substituting this into inequality (20) and assuming γ ₀ , κ , η , t sufficiently small, we

obtain finally

A ^HR−BP (u _h , p _h ,λ _h ; u _h +κ I _h ⁰ v p , −p _h , −λ _h + η

h I _h ^λ u _h ) (22)

≥ c

|u _h | ² _1,F + k p _h k ² _0,F + h ² |p _h | ² _1,F

h

+ hkλ _h k ² _0,Γ + 1 h ku _h k ² _0,Γ

= c|||u _h , p _h , λ _h ||| ² . On the other hand, the estimates of Propositions 2 and 5 give immediately

|||u _h + κI _h ⁰ v _p , −p _h , −λ _h + η

h I _h ^λ u _h ||| ≤ C|||u _h , p _h , λ _h ||| (23) Dividing (22) by (23) yields the result of the Lemma. u t

Theorem 1. Under Assumptions A, B, γ 0 > 0 small enough, any θ > 0, and (u, p, λ ) ∈ H ² (F) ^d × L ₀ ² (F) × H

(Γ ), the following a priori error estimates hold for method (15):

|u − u _h | 1,F + kp − p _h | 0,F + √

hkλ − λ h k _0,Γ ≤ Ch(|u| 2,F + |p| 1,F + |λ | _1/2,Γ ) (24) Moreover, assuming the usual elliptic regularity for the Stokes problem in F, i.e. the bound (28) for the solution to (27), one has ∀ϕ ∈ H ^3/2 (Γ )

Z

Γ

(λ −λ _h )ϕ

≤ Ch ² (|u| 2,F +| p| 1,F +|λ | _1/2,Γ )|ϕ | _3/2,Γ (25) Proof. Use Galerkin orthogonality (taking b u = u for the exact solution u and ex- tending p from F to F _h ^e )

A ^HR−BP (u _h − u, p _h − p, λ h −λ ; v _h , q _h , µ h ) = θh ² Z

F

∇p · ∇q _h (26) to conclude

A ^HR−BP (u _h − I _h ^u u, p _h − I _h ^p p, λ _h − I _h ^λ λ ; v _h , q _h , µ _h ) = 2 Z

F

D(u −I _h ^u u) : D(v _h )

− Z

F ((p − I _h ^p p) divv _h + q _h div(u − I _h ^u u)) + Z

Γ

((λ − I _h ^λ λ )· v _h + µ _h · (u − I _h ^u u))

− γ 0 h Z

Γ

(λ −I _h ^λ λ + 2D(u − I c _h ^u u)n − (p − I _h ^p p)n) · (µ _h + 2D( v b _h )n − q _h n) +θ h ²

Z

F

h

∇I _h ^p p · ∇q _h All the terms in the right-hand side can be bounded thanks to Proposition 2 with s _u = 1, s _p = s _λ = 0 so that

A ^HR−BP (u _h −I _h ^u u, p _h −I _h ^p p, λ _h −I _h ^λ λ ; v _h , q _h , µ _h ) ≤Ch(|u| _2,F +|p| _1,F +|λ | _1/2,Γ )|||v _h , q _h , µ _h |||

The inf-sup lemma 1 now gives (24).

To prove (25), choose any ϕ ∈ H ^3/2 (Γ ) and take v, q solution to

−2 divD(v) + ∇q = 0, divv = 0 on F, v = ϕ on Γ (27) as well as µ = −(2D(v)n − qn)| _Γ . Integration by parts gives

2 Z

F

D(u −u _h ) : D(v) − Z

F q div(u −u _h ) + Z

Γ

(u − u _h )µ = 0 Subtracting this from Galerkin orthogonality relation (26) gives

Z

Γ

(λ −λ _h )·ϕ = 2 Z

F D(u−u _h ) : D(v−v _h )−

Z

F

((p − p _h )div(v−v _h )+(q −q _h ) div(u −u _h )) +

Z

Γ

((λ −λ _h ) · (v− v _h ) + (µ − µ h )· (u − u _h ))

− γ 0 h Z

Γ

(λ − λ _h + 2D(u − u b _h )n − (p − p _h )n) · (µ − µ _h + 2D(v − v b _h )n − (q − q _h )n)

− θh ² Z

F

h

∇p _h · ∇q _h

Taking v _h = I _h ^u v, q _h = I _h ^p q, µ h = I _h ^λ µ, applying Proposition 2 with s _u = 1, s _p = s _λ = 0 and recalling that

(|v| 2,F +|q| 1,F +|µ| _1/2,Γ ) ≤ C|ϕ | _3/2,Γ (28) thanks to the elliptic regularity of the Stokes problem, yields (25). u t

2.2 P 1 − P 0 velocity-pressure spaces with interior penalty stabilization.

Let us now choose P 1 FE for V _h and P 0 for Q _h and add interior penalty (IP) stabi- lization to the Haslinger-Renard method. The method becomes:

Find (u _h , p _h , λ h ) ∈ V _h × Q _h ×W _h such that

A ^HR−IP (u _h , p _h , λ h ;v _h , q _h , µ h ) = L(v _h , µ h ), ∀(v _h ,q _h , µ h ) ∈ V _h ×Q _h ×W _h , (29) where

A ^HR−IP (u, p,λ ; v, q, µ) = A(u, p,λ ;v, q, µ)

−γ ₀ h Z

Γ

(λ + 2D( u)n b − pn) · (µ +2D( v)n b − qn) − θh ∑

E∈E

h

Z

E

[p][q]

V _h is continuous P 1 FE space on mesh T _h ^e , Q _h is P 0 FE space on mesh T _h ^e , and W _h

is P 0 FE space on mesh T _h ^Γ , cf. (7).

Note that the IP stabilization is applied to the pressure in the interior on F as well as on the cut elements. The analysis of this method is similar to that of the previous one (15) and we give immediately the final result:

Theorem 2. Under Assumptions A and B, γ 0 > 0 small enough, any θ > 0, and (u, p, λ ) ∈ H ² (F) ^d × L ² ₀ (F) × H

(Γ ), the a priori error estimates (24) and (25) hold for method (29).

Proof. We shall not repeat all the technical details but only point out some important changes that should be made in Propositions 3–5 and the inf-sup lemma from the preceding section in order to adapt them to the the analysis of method (29):

• The estimate of Proposition 4 should be changed to hkq _h k ² _0,Γ ≤ C

kq _h k ² _0,F +h ∑ E∈E

R

E [q _h ] ²

This can be proved observing on any bad element T ∈ T _h ^Γ sharing an edge E with its good neighbor T ⁰

kq _h k _0,Γ

= p

|Γ _T | |(q _h ) _|T | ≤ p

|Γ _T |(|[q _h ] _E | + |(q _h ) _|T

|)

= p

|Γ _T | 1

p |E | k[q _h ]k 0,E + 1

p |T ⁰ | kq _h k _0,T

!

≤C

k[q _h ]k 0,E + 1

√

h kq _h k _0,T

)

The case of a bad element that does not share an edge with its good neighbor can be treated similarly by introducing a chain of elements connecting T to T ⁰ . The case when T ∈ T _h ^Γ is “good” itself is trivial.

• The term h ² | p| 1,F

h

in the triple norm in Lemma 1 should be replaced by h ∑ E∈E

h

R E [ p] ²

• The treatment (19) of the velocity-pressure term inside the proof of Lemma 1 is now replaced by

− Z

F

p _h divI _h ⁰ v _p = k p _h k ² _0,F + Z

F

p _h div(v _p − I _h ⁰ v _p )

= k p _h k ² _0,F + ∑

E∈E

h

Z

E∩F

[p _h ]n · (v _p − I _h ⁰ v _p )

and the bound ∑ E∈E

kv _p − I _h ⁰ v p k ² _0,E ≤ Ch|v _p | _1,F which is proved as in Proposi- tion 5.

u t

2.3 Taylor-Hood spaces.

We now choose P k (resp. P k−1 ) FE space with k ≥ 2 for V _h (resp. Q _h ). These are well

known Taylor-Hood spaces which satisfy the discrete inf-sup conditions in the usual

setting and thus no stabilization for pressure “in the bulk” is needed. Intuitively, some extra stabilization should be now added for the pressure on the cut triangles.

We thus propose the following modification of the Haslinger-Renard method for Taylor-Hood spaces:

Find (u _h , p _h , λ _h ) ∈ V _h ×Q _h ×W _h such that

A ^{HR−T H} (u _h , p _h , λ _h ; v _h , q _h , µ _h ) = L(v _h , µ _h ), ∀(v _h , q _h , µ _h ) ∈ V _h × Q _h ×W _h , (30) where

A ^{HR−T H} (u, p, λ ; v , q, µ) = A(u, p, λ ;v, q, µ)

−γ ₀ h Z

Γ

(λ +D( u)n b − pn) b · (µ +D( b v)n − b qn) V _h is continuous P k FE space on mesh T _h ^e , Q _h (resp. W _h ) is continuous P k−1 FE space on mesh T _h ^e (resp. T _h ^Γ ) for k ≥ 2, cf. (7). The notation b · stands here again for the “robust reconstruction” from Definition 1. We emphasize that it is applied here not only to the velocity, but also to pressure, unlike versions of the method (15) and (29) studied above.

The analysis of this method will be done under more restrictive assumptions than that of the previous ones:

Assumption C. The dimension is d = 2, F _h ⁱ contains at least 3 triangles, Γ is a curve of class C ² , and the mesh T _h is sufficiently fine (with respect to the curvature of Γ ).

Remark 3. Assumption C covers Assumption B, cf Remark 2.

We shall tacitly assume Assumption C in all the Propositions until the end of this Section. Proposition 3 will be reused in the analysis of the present case but Proposition 4 should be replaced with the following

Proposition 6. For all q _h ∈ Q _h one has

hk q b _h k ² _0,Γ ≤ Ckq _h k ² _0,F .

The proof is a straight-forward adaptation of Proposition 3 to the pressure space.

Another important ingredient in our analysis will be the discrete inf-sup condi- tion, cf. Proposition 10 below. We recall first a well-known auxiliary result:

Proposition 7. The exists a mesh independent constant β > 0 such that for any q _h ∈ Q _h

β h|q _h | _1,F

h

≤ sup

v

∈V

R

F q _h div v _h

|v _h | _1,F

(31) where V _h ⁱ = V _h ∩ (H ₀ ¹ (F _h ⁱ )) ^d .

This result is customarily applied to the analysis of FE discretization of the

Stokes equations via the Verf¨urth trick [9]. The proof in the 2D case under the as-

sumption that the mesh contains at least 3 triangles can be found in [2]. We note in

passing that a 3D generalization in a similar context is presented in [11].

Let B ^Γ _h := F \ F _h ⁱ and note that the boundary of B ^Γ _h consists of ∂ F _h ⁱ and Γ . Proposition 8. Let p _h ∈ Q _h and v ∈ H ¹ (B ^Γ _h ) vanishing on Γ . Then

Z

∂ F

h

|p _h v| ≤ Ck p _h k _0,B

|v| _1,B

h

(32)

Proof. Take any triangle T ∈ T _h ^Γ such that one of its sides E is an edge on ∂ F _h ⁱ . Introduce the polar coordinates (r, ϕ) centered at the vertex O of T opposite to side E (thus O lies outside F). The part of T inside F can be represented in these coor- dinates as

F _T = {(r , ϕ) such that α < ϕ < β , r _Γ (ϕ) < r < r _i (ϕ)}

with r _Γ (ϕ ) and r _i (ϕ) representing, respectively, Γ and E ⊂ ∂ F _h ⁱ . In view of As- sumption C, r _Γ (ϕ) is a C ² function and there are positive numbers r _min and r _max such that r _min ≤ r Γ (ϕ ) < r _i (ϕ ) ≤ r max for all ϕ ∈ [α, β ]. There are 2 options: ei- ther Γ T is very close to edge E so that ^r _r

min

≤ ρ, or F _T covers a significant portion of T so that |F _T | ≥ θ|T |. The positive numbers ρ and θ here can be chosen in a mesh-independent manner.

We start with the first option: ^r _r

min

≤ ρ. Using the notations above and recalling v = 0 at r = r Γ (ϕ) gives

Z

E

|p _h v| ≤ C Z β

α

(|p _h v|r) _r=r

_(ϕ) dϕ = C Z β

α Z r

(ϕ)

r

(ϕ)

∂ |p _h vr|

∂ r drdϕ

≤ C Z β

α Z r

(ϕ)

r

(ϕ)

∂ p _h

∂ r v

rdrdϕ + k p _h k 0,F

k∇vk 0,F

+ 1 r min

kp _h k 0,F

kvk 0,F

We set l(ϕ) = r _i (ϕ) − r _Γ (ϕ) and bound the first integral above using, for any ϕ fixed, an inverse inequality for p _h on the interval (r _Γ (ϕ), r _i (ϕ)) and Poincar´e in- equality for v on the same interval (recall that v = 0 at r = r _Γ (ϕ ))

Z β α

Z r

(ϕ) r

(ϕ)

∂ p _h

∂ r v

rdrdϕ ≤ r _max Z β

α

Z r

(ϕ) r

(ϕ)

∂ p _h

∂ r 2

dr

!

_{Z r}

i

(ϕ) r

(ϕ)

v ² dr

dϕ

≤ Cr _max Z β

α

1 l(ϕ)

_{Z r}

_(ϕ)

r

(ϕ)

p ² _h dr

× l(ϕ ) Z r

(ϕ)

r

(ϕ)

∂ v

∂ r 2

dr

!

d ϕ

≤ C r _max

r _min k p _h k 0,F

k∇vk 0,F

(33) Recalling the bound on ^r _r

min

(which implies, in particular, r _min ≥ _ρ ^h ) we conclude

Z

E

| p _h v| ≤ C

kp _h k _0,F

k∇vk _0,F

+ 1

h k p _h k _0,F

kvk _0,F

(34) On the other hand, if |F _T | ≥ θ |T |, extending v by 0 outside F, applying Propo- sition 1 and an inverse inequality (valid on the whole triangle T ) also yields (34):

Z

E

|p _h v| ≤ √

hk p _h vk _{0,∂ T} ≤ C(k p _h vk _0,T + h| p _h v| _1,T )

≤ C(k p _h k _∞,T kvk 0,F

+ hk∇p _h k _∞,T kvk 0,F

+ hk p _h k _∞,T k∇vk 0,F

)

≤ C

k p _h k 0,F

k∇vk 0,F

+ 1

h kp _h k 0,F

kvk 0,F

Summing (34) over all T ∈ T _h ^Γ having a side on ∂ F _h ⁱ yields Z

∂ F

h

| p _h v| ≤ Ckp _h k _0,B

k∇vk _0,B

+ 1 h kvk _0,B

h

Recall that v = 0 on Γ and the width of B ^Γ _h is of order h, so that kvk _0,B

h

≤

Chk∇vk _0,B

h

by a Poincar´e inequality. We have thus proved (32). u t

Proposition 9. Under Assumption C, there exists a continuous piecewise linear vector-valued function ψ _h on mesh T _h ^e such that ψ _h · n ≥ 0 on Γ , div ψ _h ≥ δ 0 on all the triangles of T _h ^Γ , and divψ _h ≥ −δ ₁ h on all the triangles of T _h ⁱ with some constants δ 0 , δ 1 > 0. Moreover, there is a constant C > 0 such that for any p _h ∈ Q _h

|p _h ψ _h | 1,F + 1

√

h kp _h ψ _h k _0,Γ ≤ Ck p _h k 0,F (35) Proof. Let B _ζ = {x ∈ R ² / dist(x, Γ ) < ζ } for any ζ > 0. Thanks to the smoothness of Γ , one can introduce orthogonal coordinates (ξ ₁ , ξ 2 ) on B _η with some mesh- independent η > 0 such that ξ ₂ = 0 on Γ and ξ ₂ < 0 on F ∩ B _η . Let e _i denote the basis vectors of these coordinates (e _i = ∂ r/∂ ξ _i ). Assuming η > h, let us introduce the vector-valued function given on B _η by ψ = ξ 2 e 2 for |ξ ₂ | < h, ψ = −h ^ξ _η−h

^+η e 2

for −η < ξ ₂ < −h, left undefined for h < ξ ₂ < η, and extended by 0 on F \ B _η . Let ψ _h = I _h ψ where I _h is the standard nodal interpolation operator to continuous P 1 FE space on T _h ^e .

Clearly, divψ = 1 on Γ , hence divψ ≥ ¹ ₂ on B _h by continuity for sufficiently small h. Since B _h ⊃ F _h ^Γ , one observes on all the triangles of T _h ^Γ

div ψ h ≥ 1

2 −div(ψ − ψ h ) ≥ 1

2 −Chkψk _W

(B

) = δ 0 > 0

since h is sufficiently small and ψ is sufficiently smooth thanks to the hypothesis on

Γ . Turning to the triangles of T _h ⁱ we make the following observation: if Γ were a

straight line, ψ · e ₁ would vanish, and ψ · e ₂ would be linear on −η < ξ 2 < −h with

the slope − _η−h ^h so that divψ _h ≥ − _η−h ^h on the triangles of T _h ⁱ . The actual geometry

of Γ introduces the corrections of order O(h ² ) into this argument so that one still has div ψ _h ≥ −δ ₁ h on these triangles. We also observe, by construction of ψ _h ,

kψ _h k _∞,F

h

≤ Ch and k∇ψ _h k _∞,F

h

≤ C

Let us now take any p _h ∈ Q _h . By the bounds on ψ _h proven above and by an inverse inequality we deduce on any triangle T ∈ T _h ⁱ

|p _h ψ h | _1,T ≤ Ch|p _h | _1,T +Ckp _h k _0,T ≤ Ck p _h k _0,T (36) A similar bound also holds on any cut triangle T ∈ T _h ^Γ . One cannot use a straight- forward inverse inequality in this case, since the width of the cut portion F _T , say ε, can be much smaller than h. However, the construction of ψ h implies in such a situation kψ _h k ∞,F

≤ Cε. Combining this with the inverse inequality |p _h | 1,F

≤

C

ε kp _h k 0,F

one arrives at (36). Summing this over all the triangles T ∈ T _h ⁱ yields

|p _h ψ _h | _1,F ≤ Ck p _h k _0,F . Finally, in order to bound p _h ψ _h in L ² (Γ ) we recall. Hence k p _h ψ h k _1,Γ ≤ k p _h ψ h k _{0,∂ F}

h

+C √

h| p _h ψ h | _1,B

h

≤ Chkp _h k _{0,∂ F}

h

+C √

hk p _h k 0,F

By scaling, kp _h k 0,E ≤ ^{√ C}

h kp _h k 0,T for any edge E ∈ ∂ F _h adjacent to a triangle T ∈ T _h ⁱ . The summation over all such edges yields kp _h k _{0,∂ F}

h

≤ ^{√ C}

h k p _h k _0,F

h

and

consequently kp _h ψ h k _1,Γ ≤ C √

hk p _h k 0,F . u t

Proposition 10. For any p _h ∈ Q _h there exists v _h ^p ∈ V _h such that

− Z

F

p _h divv _h ^p = k p _h k ² _0,F and |v _h ^p | 1,F + 1

√ h kv _h ^p k _0,Γ ≤ Ck p _h k 0,F (37) Proof. The continuous inf-sup condition implies that for all p _h ∈ Q _h there exists v _p ∈ (H ₀ ¹ (F)) ^d satisfying (18). Recalling the interpolation operator I _h ⁰ from Propo- sition 5, we observe

− Z

F

p _h div I _h ⁰ v _p = − Z

F

p _h divI _h ⁰ v _p = k p _h k ² _0,F

+ Z

F

p _h div(v _p − I _h ⁰ v _p )

= k p _h k ² _0,F

+ Z

∂ F

h

p _h n · v _p − Z

F

h

∇p _h · (v _p − I _h ⁰ v _p )

≥ kp _h k ² _0,F

h

−Ck p _h k _0,B

|v _p | _1,B

h

−Ch|p _h | _1,F

|v _p | _1,F

h

≥ kp _h k ² _0,F

−C|v _p | _1,F

kp _h k ² _0,B

h

+ h ² |p _h | ² _1,F

≥ 1

2 k p _h k ² _0,F

−Ckp _h k ²

0,B

−Ch ² |p _h | ² _1,F

(38) We have used Proposition 8, the interpolation estimate from Proposition 5, Young inequality and |v _p | ²

1,B

+ |v _p | ²

1,F

= |v _p | ² _1,F ≤Ckp _h k ² _0,F = C(kp _h k ²

0,F

+k p _h k ²

0,F

).