φ-FEM: a finite element method on domains defined by level-sets

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ϕ-FEM: a finite element method on domains defined by level-sets

Michel Duprez, Alexei Lozinski

To cite this version:

Michel Duprez, Alexei Lozinski. ϕ-FEM: a finite element method on domains defined by level-sets.

SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, In press.

�hal-02521111�

φ-FEM: a finite element method on domains defined by level-sets

Michel Duprez ^∗and Alexei Lozinski ^† March 27, 2020

Abstract

We propose a new fictitious domain finite element method, well suited for elliptic problems posed in a domain given by a level-set function without requiring a mesh fitting the boundary. To impose the Dirichlet boundary conditions, we search the approximation to the solution as a product of a finite element function with the given level-set function, which is also approximated by finite elements. Unlike other recent fictitious domain-type methods (XFEM, CutFEM), our approach does not need any non- standard numerical integration (on cut mesh elements or on the actual boundary). We consider the Poisson equation discretized with piecewise polynomial Lagrange finite elements of any order and prove the optimal convergence of our method in the H¹-norm. Moreover, the discrete problem is proven to be well conditioned,i.e. the condition number of the associated finite element matrix is of the same order as that of a standard finite element method on a comparable conforming mesh. Numerical results confirm the optimal convergence in bothH¹andL²norms.

1 Introduction

We consider the Poisson-Dirichlet problem

−∆u=f on Ω, u= 0 on Γ, (1)

in a bounded domain Ω⊂R^d (d= 2,3) with smooth boundary Γ assuming that Ω and Γ are given by a level-set functionφ:

Ω :={φ <0}and Γ :={φ= 0}. (2)

Such a representation is a popular and useful tool to deal with problems with evolving surfaces or interfaces [17]. In the present article, the level-set function is supposed to be known onR^d, smooth, and to behave near Γ as the signed distance to Γ. We propose a finite element method for the problem above which is easy to implement, does not require a mesh fitted to Γ, and is guaranteed to converge optimally. Our basic idea is very simple: one cannot impose the Dirichlet boundary conditions in the usual manner since the boundary Γ is not resolved by the mesh, but one can search for the approximation touas a product of a finite element functionwh with the level-setφitself: such a product obviously vanishes on Γ. In order to make this idea work, some stabilization should be added to the scheme as outlined below and explained in detail in the next section. We coin our methodφ-FEM in accordance with the tradition of denoting the level-sets byφ.

More specifically, let us assume that Ω lies inside a simply shaped domain O (typically a box inR^d) and introduce a quasi-uniform simplicial meshT_h^O onO(the background mesh). LetTh be a submesh of T_h^O obtained by getting rid of mesh elements lying entirely outside Ω (the definition ofTh will be slightly

∗CEREMADE, Universit´e Paris-Dauphine & CNRS UMR 7534, Universit´e PSL, 75016 Paris, France.

mduprez@math.cnrs.fr

†Laboratoire de Math´ematiques de Besan¸con, UMR CNRS 6623, Universit´e Bourgogne Franche-Comt´e, 16, route de Gray, 25030 Besan¸con Cedex, France.alexei.lozinski@univ-fcomte.fr

changed afterwords). Denote by Ωh the domain covered by the mesh Th (so that typically Ωh is only slightly larger than Ω). Our starting point is the following formal observation: assuming that the right- hand sidef is actually well defined on Ωh, and the solutionucan be extended to Ωh so that−∆u=f on Ωh, we can introduce the new unknownw∈H¹(Ωh) such thatu=φwand the boundary condition on Γ is automatically satisfied. An integration by parts yields then

Z

Ω_h

∇(φw)· ∇(φv)− Z

∂Ω_h

∂

∂n(φw)φv= Z

Ω_h

f φv, ∀v∈H¹(Ω_h). (3) Given a finite element approximationφhtoφon the meshThand a finite element spaceVhonTh, one can then try to search for wh ∈Vh such that the equality in (3) with the subscriptsheverywhere is satisfied for all the test functionsvh ∈Vh and to reconstruct an approximate solution uh to (1) as φhwh. These considerations are very formal and, not surprisingly, such a method does not work as is. We shall show however that it becomes a valid scheme once a proper stabilization in the vein of the Ghost penalty [3] is added. The details on the stabilization and on the resulting finite element scheme are given in the next section.

Our method shares many features with other finite elements methods on non-matching meshes, such as XFEM [15, 14, 18, 11] or CutFEM [5, 6, 7, 4]. Unlike the present work, the integrals over Ω are kept in XFEM or CutFEM discretizations, which is cumbersome in practice since one needs to implement the integration on the boundary Γ and on parts of mesh elements cut by the boundary. The first attempt to alleviate this practical difficulty was done in [12] with method that does not require to perform the integration on the cut elements, but needs still the integration on Γ. In the present article, we fully avoid any non trivial numerical integration: all the integration inφ-FEM is performed on the whole mesh elements, and there are no integrals on Γ. We also note that an easily implementable version ofφ-FEM is here developed for Pk finite elements of any orderk≥1. This should be contrasted with the situation in CutFEM where some additional terms should be added in order to achieve the optimalPk accuracy if k >1, cf. [8]. An alternative approach avoiding non trivial quadrature is presented in a recent work on the shifted boundary method [13]. The optimal convergence with piecewise linear finite elements (k= 1) on a non-fitted mesh is achieved there by introducing a truncated Taylor expansion on the approximate boundary.

The article is structured as follows: ourφ-FEM method is presented in the next section. We also give there the assumptions on the level-setφand on the mesh, and announce our main result: thea priori error estimate forφ-FEM. We work with standard continuousP_k finite elements on a simplicial mesh and prove the optimal orderh^k for the error in theH¹norm and the (slightly) suboptimal orderh^k+1/2for the error in theL² norm.¹ The proofs of these estimates are the subject of Section 3. We contend ourselves with the error analysis pertinent to theh-refinement only, i.e. we do not attempt to track the dependence of the constants, appearing in our estimates, on the polynomial degree. In Section 4, we prove that the linear system produced by our method has the condition number of order 1/h², the same as that of a standard finite element method. Numerical illustrations are given in Section 5, including a test case covered by our theory that confirms the theoretical predictions and other test cases going slightly beyond the theoretical framework. Finally, conclusions and perspectives are presented in Section 6.

2 Definitions, assumptions, description of φ-FEM, and the main result

We recall that we work with a bounded domain Ω ⊂ O ⊂ R^d (d = 2,3) with boundary Γ given by a level-setφas in (2). We assume thatφis sufficiently smooth and behaves near Γ as the signed distance to Γ after an appropriate change of local coordinates. More specifically, we fix an integerk≥1 and introduce the following

1Our approach can also be realized usingQkfinite elements on a mesh consisting of rectangles/cubes. The convergence results and proofs can be straightforwardly passed over to this case.

Assumption 1. The boundary Γ can be covered by open sets Oi, i = 1, . . . , I and one can introduce on every Oi local coordinates ξ1, . . . , ξd with ξd = φ such that all the partial derivatives ∂^αξ/∂x^α and

∂^αx/∂ξ^α up to orderk+ 1 are bounded by someC0>0. Moreover, φis of class C^k+1 onO and|φ| ≥m onO \ ∪i=1,...,IOi with some m >0.

LetT_h^O be a quasi-uniform simplicial mesh onOof mesh sizeh, meaning thathT ≤handρ(T)≥βh for all simplexesT ∈ T_h^O with some mesh regularity parameterβ >0 (herehT = diam(T) andρ(T) is the radius of the largest ball inscribed inT). Consider, for an integerl≥1, the finite element space

V_h,O^(l) ={vh∈H¹(O) :vh|T ∈Pl(T)∀T ∈ T_h^O}

where P^l(T) stands for the space of polynomials ind variables of degree ≤l viewed as functions on T. Introduce an approximate level-setφh∈V_h,O^(l) by

φh:=I_h,O^(l) (φ) (4)

where I_h,O^(l) is the standard Lagrange interpolation operator on V_h,O^(l). We shall use this to approximate the physical domain Ω ={φ <0} with smooth boundary Γ ={φ= 0}by the domain {φh<0} with the piecewise polynomial boundary Γh={φh= 0}. We employφh rather thanφin our numerical method in order to simplify its implementation (all the integrals in the forthcoming finite element formulation will involve only the piecewise polynomials). This feature will also turn out to be crucial in our theoretical analysis.

We now introduce the computational meshThas the subset ofT_h^Ocomposed of the triangles/tetrahedrons having a non-empty intersection with the approximate domain{φh<0}. We denote the domain occupied byTh by Ω_h,i.e.

Th:={T ∈ T_h^O :T∩ {φh<0} 6=∅} and Ωh= (∪_T∈ThT)^o.

Note that we do not necessarily have Ω ⊂ Ωh. Indeed some mesh elements can be cut by the exact boundary {φ = 0} but not by the approximate one {φh = 0}. In such rare occasions, a mesh element containing a small portion of Ω will not be included intoTh.

Fix an integerk≥1 (the samekas in Assumption 1) and consider the finite element space V_h^(k)={vh∈H¹(Ωh) :vh|T ∈P^k(T)∀T ∈ Th}.

Theφ-FEM approximation to (1) is introduced as follows: findwh∈V_h^(k) such that:

ah(wh, vh) =lh(vh) for all vh∈V_h^(k), (5) where the bilinear forma_h and the linear forml_h are defined by

ah(w, v) :=

Z

Ω_h

∇(φhw)· ∇(φhv)− Z

∂Ω_h

∂

∂n(φhw)φhv+Gh(w, v) (6) and

lh(v) :=

Z

Ωh

f φhv+G^rhs_h (v), withGh andG^rhs_h standing for

Gh(w, v) :=σh X

E∈F_h^Γ

Z

E

∂

∂n(φhw) ∂

∂n(φhv)

+σh² X

T∈T_h^Γ

Z

T

∆(φhw)∆(φhv),

G^rhs_h (v) :=−σh² X

T∈T_h^Γ

Z

T

f∆(φhv),

whereσ >0 is anh-independent stabilization parameter,T_h^Γ⊂ Th contains the mesh elements cut by the approximate boundary Γh={φh= 0},i.e.

T_h^Γ={T ∈ Th:T∩Γh6=∅}, Ω^Γ_h:=

∪_T∈TΓ hTo

. (7)

andF_h^Γ collects the interior facets of the meshTheither cut by Γh or belonging to a cut mesh element F_h^Γ={E (an internal facet ofTh) such that∃T ∈ Th:T∩Γh6=∅andE∈∂T}.

The brackets inside the integral over E∈ F_h^Γ in the formula forGh stand for the jump over the facet E.

The first part inG_h actually coincides with the ghost penalty as introduced in [3] forP₁ finite elements.

We shall also need the following assumptions on the mesh Th, more specifically on the intersection of elements ofT_h with the approximate boundary Γ_h={φ_h= 0}.

Assumption 2. The approximate boundary Γ_h can be covered by element patches {Π_i}_i=1,...,NΠ having the following properties:

• Each patch Πi is a connected set composed of a mesh elementTi∈ Th\ T_h^Γ and some mesh elements cut by Γh. More precisely,Πi=Ti∪Π^Γ_i with Π^Γ_i ⊂ T_h^Γ containing at most M mesh elements;

• T_h^Γ =∪^N_i=1^ΠΠ^Γ_i;

• Π_i andΠ_j are disjoint if i6=j.

Assumption 2 is satisfied forhsmall enough, preventing strong oscillations of Γ on the length scaleh.

It can be reformulated by saying that each cut element T ∈ T_h^Γ can be connected to an uncut element T⁰ ∈ Th\ T_h^Γ by a path consisting of a small number of mesh elements adjacent to one another; see [12]

for a more detailed discussion and an illustration (Fig. 2).

In what follows, k · k_k,D (resp. | · |_k,D) denote the norm (resp. the semi-norm) in the Sobolev space H^k(D) with an integerk≥0 whereDcan be a domain inR^d or a (d−1)-dimensional manifold.

Theorem 2.1. Suppose that Assumptions 1 and 2 hold true, l ≥k, the mesh Th is quasi-uniform, and f ∈H^k(Ωh∪Ω). Let u∈H^k+2(Ω) be the solution to (1)andwh∈V_h^(k) be the solution to (5). Denoting uh:=φhwh, it holds

|u−uh|1,Ω∩Ω_h ≤Ch^kkfkk,Ω∪Ω_h (8)

with a constantC >0depending on theC0,m,M in Assumptions 1, 2, on the maximum of the derivatives of φof order up tok+ 1, on the mesh regularity, and on the polynomial degrees kandl, but independent of h,f, andu.

Moreover, supposingΩ⊂Ω_h

ku−u_hk_0,Ω≤Ch^k+1/2kfk_k,Ωh (9)

with a constantC >0 of the same type.

3 Proof of the a priori error estimate

The proof of Theorem 2.1 is preceded with auxiliary lemmas in Sections 3.1 and 3.2, followed by the proof of coercivity of the forma_h in Section 3.3.

3.1 A Hardy-type inequality

Lemma 3.1. We assume that the domainΩis given by the level-setφ, cf. (2), and satisfies Assumption 1. Then, for anyu∈H^k+1(O)vanishing on Γ,

u φ k,O

≤Ckuk_k+1,O

withC >0 depending only on the constants in Assumption 1.

Proof. The proof is decomposed into three steps:

Step 1. We start in the one dimensional setting and adapt the proof oF Hardy’s inequality from [16].

Letu:R→Rbe aC^∞ function with compact support such that u(0) = 0. Setw(x) =u(x)/xforx6= 0 andw(0) =u⁰(0). We shall prove thatwis a C^∞function onRand, for any integers≥0,

Z ∞

−∞

|w^(s)(x)|²dx ^1/2

≤C Z ∞

−∞

|u^(s+1)(x)|²dx ^1/2

(10) withC depending only ons.

Observe, for anyx >0,

w(x) = u(x) x = 1

x Z x

0

u⁰(t)dt= Z 1

0

u⁰(xt)dt.

Hence,

w^(s)(x) = Z 1

0

u^(s+1)(xt)t^sdt. (11)

It implies lim_x→0+w^(s)(x) = u^(s+1)(0)/(s+ 1). The same formula holds for the limit as x→0⁻. This means thatwis continuous (the special cases= 0), andw^(s)(0) exists for alls≥1.

We have now by (11) and the integral version of Minkowski’s inequality Z ∞

0

|w^(s)(x)|²dx 1/2

= Z ∞

0

Z 1 0

u^(s+1)(xt)t^sdt

2

dx

!^1/2

≤ Z 1

0

Z ∞ 0

|u^(s+1)(xt)|²dx ^1/2

t^sdt=C Z ∞

0

|u^(s+1)(x)|²dx ^1/2

withC=R1

0 t^s−1/2dt= 1/(s+ 1/2). Applying the same argument to negativex, we get (10).

Step 2. Let now u : R^d → R be a compactly supported C^∞ function vanishing at xd = 0 and set w=u/xd. We shall prove

|w|_k,Rd≤C|u|_k+1,Rd (12)

withC depending only onk.

To keep things simple, we give here the proof for the cased= 2 only (the cased= 3 is similar but would involve more complicated notations). Take any integerst, s≥0 witht+s=k, apply (10) to ^∂_∂x^twt

1

=_x¹

2

∂^tu

∂x^t₁

treated as a function ofx₂ (note that ^∂_∂x^tut 1

vanishes atx₂= 0) and then integrate with respect tox₁. This gives

∂^kw

∂x^t₁∂x^s_{2 0,}

R^d

≤C

∂^k+1u

∂x^t₁∂x^s+1₂ 0,R^d

.

Thus,

|w|²_k,Rd=

k

X

s=0

∂^kw

∂x^k−s₁ ∂x^s₂

2

0,R^d

≤C²

k

X

s=0

∂^k+1u

∂x^k−s₁ ∂x^s+1₂

2

0,R^d

≤C²|u|²_k+1,Rd

so that (12) is proved.

Step 3. Consider finally the domains Ω⊂ Oas announced in the statement of this Lemma, letube a C^∞function onOvanishing on Γ, and setw=u/φ. Assume first thatuis compactly supported inOl, one of the sets forming the cover of Γ as announced in Assumption 1. Recall the local coordinatedξ1, . . . , ξd

onOlwith ξd =φand denote by ˆu(resp. ˆw) the function u(resp. w) treated as a function of ξ1, . . . , ξd. Since ˆw= ˆu/ξd, (12) implieskwkˆ _k,Rd≤Ckukˆ _k+1,Rd. Passing from the coordinatesx1, . . . , xd toξ1, . . . , ξd

and backwards we conclude kwkk,O_l ≤Ckukk+1,O_l with a constant C that depends on the maximum of partial derivatives∂^αx/∂ξ^α up to orderkand that of∂^αξ/∂x^αup to orderk+ 1. Introducing a partition of unity subject to the cover{Ol}we can now easily provekwkk,O≤Ckukk+1,O noting that 1/φis of class C^k outside∪_l{O_l}. This estimate holds also true foru∈H^k+1(O) by density ofC^∞ inH^k+1.

3.2 Some technical lemmas

This Section regroups some technical results to be used later in the proofs of the coercivity ofah (Section 3.3) and the a priori error estimates (Sections 3.4 and 3.5). The most important contribution here is Lemma 3.3 which extends to finite elements of any degree a result from [12]. This lemma will be the keystone of the proof of coercivity by allowing us to handle the non necessarily positive terms on the cut elements. It shows indeed that the H¹ norm of a finite element function on Ω^Γ_h can be bounded by its norm on the whole computational domain Ω_h multiplied by a number strictly smaller than 1, modulo the addition of stabilization terms. We recall that our stabilization is strongly inspired by that of [3] but differs from it by some extra terms involving the Laplacian on mesh elements. The proof that such a stabilization is sufficient in Lemma 3.3 relies on a simple observation on polynomials, announced and proven in Lemma 3.2.

Lemma 3.2. LetT be a triangle/tetrahedron,E one of its sides andpa polynomial onT of degrees≥0 such that p= ^∂p_∂n = 0 onE and∆p= 0onT. Then p= 0 onT.

Proof. Let us consider only the 2D case (3D is similar). Without loss of generality, we can assume that E lies on the x-axis in (x, y) coordinates. Let p = P

pijxⁱy^j with i, j ≥ 0, i+j ≤ s as above. We shall prove by induction on m = 0,1, . . . , l that pim = 0, ∀i. Indeed, this is valid for m = 0,1 since p(x,0) =P

ipi0xⁱ= 0 and ^∂p_∂y(x,0) =P

ipi1xⁱ= 0. Now, ∆p= 0 implies for all indicesi, j≥0 (i+ 2)(i+ 1)pi+2,j+ (j+ 2)(j+ 1)pi,j+2= 0

so thatp_im= 0,∀iimpliesp_i,m+2= 0, ∀i.

Lemma 3.3. Under Assumption 2, for anyβ >0 ands∈N^∗ one can choose 0< α <1 depending only on the mesh regularity andssuch that, for each vh∈V_h^(s),

|vh|²_1,ΩΓ h

≤α|vh|²_1,Ω

h+βh X

E∈F_h^Γ

∂vh

∂n

2

0,E

+βh² X

T∈T_h^Γ

k∆vhk²_0,T. (13)

Proof. Choose anyβ >0, consider the decomposition of Ω^Γ_h in element patches{Πk}as in Assumption 2, and introduce

α:= max

Π_k,v_h6=0

|vh|²_1,ΠΓ k

−βhP

E∈F_k

∂v_h

∂n

2

0,E−βh²P

T⊂Π_kk∆vhk²_0,T

|vh|²_1,Π

k

, (14)

where the maximum is taken over all the possible configurations of a patch Π_k allowed by the mesh regularity and over all the piecewise polynomial functions on Π_k (polynomials of degree≤s). The subset F_k⊂ F_h^Γ gathers the edges internal to Π_k. Note that the quantity under the max sign in (14) is invariant under the scaling transformation x7→ hx and is homogeneous with respect to v_h. Recall also that the

patch Πk contains at most M elements. Thus, the maximum is indeed attained since it is taken over a bounded set in a finite dimensional space.

Clearly,α≤1. Supposingα= 1 would lead to a contradiction. Indeed, ifα= 1 then we can take Πk, vh yielding this maximum and suppose without loss of generality|vh|1,Π_k= 1. We observe then

|vh|²_1,Tk+βh X

E∈Fk

∂v_h

∂n

2

0,E

+βh² X

T⊂Πk

k∆vhk²_0,T = 0 since |vh|²_1,Π

k =|vh|²_1,T

k+|vh|²_1,ΠΓ k

. This implies vh = c = const on Tk, _∂vh

∂n

= 0 on allE ∈ Fk, and

∆vh = 0 on all T ⊂ Πk. Thus applying Lemma 3.2 to vh−c, we deduce that vh = c on Πk, which contradicts|vh|1,Π_k= 1.

This proves α <1. We have thus

|v_h|²_1,ΠΓ k

≤α|v_h|²_1,Π

k+βh X

E∈Fk

∂vh

∂n

2

0,E

+βh² X

T⊂Πk

k∆v_hk²_0,T

for allv_h∈V_hand all the admissible patches Π_k. Summing this over Π_k, k= 1, . . . , N_Π yields (13).

Lemma 3.4. For allvh∈V_h^(k), it holds

kφ_hv_hk_0,ΩΓ h

≤Ch|φ_hv_h|_1,ΩΓ h

, (15)

kφhvhk_0,Ω

h\Ω≤Ch|φhvh|_1,Ω

h , (16)

with a constantC >0 depending only on the regularity ofT_h andk.

Proof. It is easy to see that the supremum

C= sup

p_h6=0,T

kp_hk_0,T hT|ph|1,T

over all the polynomials ph ∈ Pk+l(T) vanishing at a point of T and all the simplexesT satisfying the regularity assumption hT/ρ(T) ≥ β is attained so that C is finite. Taking any T ∈ T_h^Γ and putting ph = φhvh, this implies kφhvhk_0,T ≤ ChT|φhvh|_1,T for any Vh ∈ V_h^(k). Summing over all T ∈ T_h^Γ concludes the proof of (15). Estimate (16) is proven similarly, adding, if necessary, neighbour elements to T ∈ T_h^Γ.

Lemma 3.5. For allvh∈V_h^(k)

X

E∈F_h^Γ

kφhvhk²_0,E ≤Ch|φhvh|²_1,Ωh (17) and

kφ_hv_hk²_0,∂Ω

h ≤Ch|φ_hv_h|²_1,Ω

h (18)

with a constantC >0 depending only on the regularity ofTh. Proof. LetE∈ F_h^Γ. Recall the well-known trace inequality

kvk²_0,E ≤C 1

hkvk²_0,T+h|v|²_1,T

(19) for eachv∈H¹(E). Summing this over allE∈ F_h^Γ gives

X

E∈F_h^Γ

kφhvhk²_0,E ≤C 1

hkφhvhk²_0,ΩΓ

h+h|φhvh|²_1,ΩΓ h

leading, in combination with (15), to (17). The proof of (18) is similar.

Lemma 3.6. Under Assumption 1, it holds for all v ∈H^s(Ωh) with integer 1≤s≤k+ 1, v vanishing onΩ,

kvk_0,Ω

h\Ω≤Ch^skvk_s,Ω

h\Ω. (20)

Proof. Consider the 2D case (d = 2). For simplicity, we can assume that v is C^∞ regular and pass to v ∈ H^s(Ω_h) by density. By Assumption 1, we can pass to the local coordinates ξ₁, ξ₂ on every set Ok

covering Γ assuming thatξ₁ varies between 0 andL and, for anyξ₁ fixed, ξ₂ varies on Ω_h\Ω from 0 to someb(ξ₁) with 0≤b(ξ₁)≤Ch. We observe using the bounds on the mapping (x₁, x₂)7→(ξ₁, ξ₂)

kvk²_0,(Ω

h\Ω)∩Ok≤C Z L

0

Z b(ξ₁) 0

v²(ξ1, ξ2)dξ2dξ1

(recall that ∂^αv

∂ξ^α₂(ξ1,0) = 0 forα= 0, . . . , s-1 andb≤Ch)

=C Z L

0

Z b(ξ₁) 0

Z ξ₂ 0

(ξ₂−t)^s−1 (s−1)!

∂^sv

∂ξ₂^s(ξ1, t)dt

!² dξ2dξ1

≤C Z L

0

h^2s Z b(ξ₁)

0

∂^sv

∂ξ^s₂(ξ1, t)

2

dtdξ1

≤Ch^2s|v|²_s,(Ω

h\Ω)∩Ok.

Summing over all neighbourhoods Ok gives (20). The proof in the 3D case is the same up to the change of notations.

3.3 Coercivity of the bilinear form ah

Lemma 3.7. Under Assumption 2, the bilinear formah is coercive onV_h^(k) with respect to the norm

|||vh|||_h:=q

|φhvh|²_1,Ω

h+Gh(vh, vh)

i.e. a_h(v_h, v_h)≥c|||vh|||²_h for all v_h ∈V_h^(k) withc > 0 depending only on the mesh regularity and on the constants in Assumption 2.

Proof. Letv_h∈V_h^(k) andB_h be the strip between Γ_hand∂Ω_h,i.e. B_h={φ_h>0} ∩Ω_h. Sinceφ_hv_h= 0 on Γ_h,

Z

∂Ω_h

∂(φ_hv_h)

∂n φ_hv_h= Z

∂B_h

∂(φ_hv_h)

∂n φ_hv_h

= X

T∈T_h^Γ

Z

∂(Bh∩T)

∂(φhvh)

∂n φhvh− X

T∈T_h^Γ

X

E∈F_h^cut(T)

Z

Bh∩E

∂(φhvh)

∂n φhvh,

whereT_h^Γ is defined in (7) andF_h^cut(T) regroups the facets of a mesh elementT cut by Γ_h. By divergence theorem,

Z

∂Ω_h

∂(φ_hv_h)

∂n φ_hv_h= Z

B_h

|∇(φhv_h)|²+ X

T∈T_h^Γ

Z

B_h∩T

∆(φ_hv_h)φ_hv_h

− X

E∈F_h^Γ

Z

E∩Bh

φhvh

∂(φ_hv_h)

∂n

.