Many boundary value problems with non-homogeneous boundary data may be cast in the abstract variational form: find u ∈ X such that γu = g ∈ M and

SPACES ON LOCALLY REFINED MESHES

MARK AINSWORTH, JOHNNY GUZM ´ AN, AND FRANCISCO–JAVIER SAYAS

Abstract. The existence of uniformly bounded discrete extension operators is established for conforming Raviart-Thomas and N´ edelec discretisations of H(div) and H(curl) on locally refined partitions of a polyhedral domain into tetrahedra.

1. Introduction

Many boundary value problems with non-homogeneous boundary data may be cast in the abstract variational form: find u ∈ X such that γu = g ∈ M and

(1.1) B(u, v) = F (v) ∀v ∈ X 0

where X is a Hilbert space over a domain Ω, M is a Hilbert space over the boundary Γ of Ω, and γ : X → M is a trace operator with X 0 = {v ∈ X : γv = 0}. We assume that the bilinear and linear forms B : X × X → R and F : X → R satisfy suitable conditions for the problem to be well-posed;

specific examples will be given later.

A Galerkin finite element approximation u h ≈ u is obtained by selecting a finite dimensional subspace X h ⊂ X, setting M h = γX h , constructing a suitable approximation g h ≈ g of the non- homogeneous boundary data, and seeking u h ∈ X h such that γu h = g h ∈ M h and

(1.2) B(u h , v h ) = F (v h ) ∀v h ∈ X h0 = {v h ∈ X h : γv h = 0}.

The discrete problem (1.2) is well-posed provided that there exists a positive constant β > 0 such that

(1.3) sup

v

∈X

,kv

k

=1

B(w h , v h ) ≥ βkw h k X for all w h ∈ X h .

The issue of the accuracy of the resulting approximation is usually addressed by reference to the following classical C´ea type estimate

(1.4) ku − u h k X ≤ 

1 + κ β

 inf

w

∈X

:γw

=g

ku − w h k X ,

where κ is the continuity constant of the bilinear form B. It is clear that the accuracy depends on both the choice of finite dimensional subspace X h ⊂ X and on the choice g h ≈ g of the approximate Dirichlet boundary condition. Nevertheless, the bound (1.4) is somewhat unsatisfactory. In particu- lar, whilst X h ⊂ X and g h ≈ g can essentially be chosen independently of one another, the influence of each choice on the accuracy of the resulting finite element approximation is obscured through the

Date: June 11, 2014.

1991 Mathematics Subject Classification. 76M10,65N30,65N12.

Key words and phrases. finite elements, Stokes, conforming, divergence-free.

Partial support for MA under AFOSR contract FA9550-12-1-0399 is gratefully acknowledged. FJS was partially funded by the NSF grant DMS 1216356.

1

requirement that the choice of comparator w h ∈ X h is constrained to satisfy the boundary condition γw h = g h .

Under what conditions is it possible to obtain an error estimate of the form

(1.5) ku − u h k X ≤ C( inf

v

∈X

ku − v h k X + kg − g h k M ),

in which the individual contributions to the error corresponding to the choice of X h ⊂ X and g h ≈ g are isolated? Suppose that there exists a uniformly bounded discrete extension operator L h : M h → X h such that

(1.6) (P 1) γL h µ h = µ h ∀µ h ∈ M h ; (P 2) kL h µ h k X ≤ C L kµ h k M ∀µ h ∈ M h .

Let v h ∈ X h be arbitrary, and set w h = v h − L h (γv h − g h ) ∈ X h , so that γw h = g h on Γ and u h − w h ∈ X 0h . With this choice, estimate (1.4) then gives

(1.7) ku − u h k X ≤ 

1 + κ β

 ku − v h + L h (γv h − g h )k X .

With the aid of the triangle inequality and (P 2), the right hand side in the above estimate may be bounded by ku − v h k X + C L kγv h − g h k M . The second term in this expression can be bounded by inserting 0 = γu − g, applying the triangle inequality and using the continuity of the trace operator γ : X → M to obtain kγv h − g h k M ≤ Ckv h − uk X + kg − g h k M . Combining the above estimates, we conclude that (1.5) holds whenever there exists a discrete extension operator satisfying (P 1)-(P 2).

Interestingly, the existence of an operator satisfying (P 1) − (P 2) is also necessary for a bound of the form (1.5) to hold [9, 14].

The existence of uniformly bounded discrete extension operators satisfying (P 1)-(P 2) is important in many areas of numerical analysis including the construction of domain decomposition precondi- tioners [17]. The main purpose of the current work is to establish the existence of discrete extension operators satisfying (P 1)-(P 2) in the case where the discrete spaces are taken to be conforming discretisations of H(div) and H(curl), i.e. Raviart-Thomas and N´edelec spaces, on a possibly non- quasi-uniform partitioning of a polyhedral domain into tetrahedra.

Various results concerning stable extension operators are interspersed in the literature. There is a number of results available in the literature concerning discrete extensions from the boundary to the interior on a single isolated element [8], but the fact that the norms on the trace spaces are not additive means that one cannot prove the results for collections of elements by simply summing contributions from individual elements. The case of Raviart-Thomas elements on a two-dimensional domain appears in [3] applied to the analysis of weakly imposed essential boundary conditions for the mixed Laplacian for the case of quasi-uniform triangulations, and was subsequently extended [10] to cover meshes that are quasi-uniform in a neighborhood of the boundary. Subsequently, the case of general non-quasi-uniform meshes was covered in [11] although, unfortunately, the arguments used seem to be limited to the two-dimensional setting.

The approach employed in the present work for the treatment of Raviart-Thomas elements is

similar to the idea used in [3, 10] without, however, requiring quasi-uniformity of the mesh. The key

to relaxing the conditions on the mesh is to develop local regularity estimates along with discrete

norm equivalences valid on locally refined meshes [1], and to show that the operator defined in [10] is,

in fact, uniformly bounded on general shape regular meshes. The treatment of the N´ed´elec spaces is

different again. The idea is to first split the discrete trace using a discrete Hodge decomposition and

to then use our extension result for Raviart-Thomas elements to handle one component of the split-

ting, with the remaining component treated using an idea adapted from [11]. The resulting extension

yields a divergence-free field which therefore belongs Brezzi-Douglas-Marini space. Consequently,

our results for the Raviart-Thomas case (i.e. N´ed´elec spaces in the three dimensional case [12])

extend to the three dimensional counterpart of the Brezzi-Douglas-Marini finite element [5, 13].

The plan of the paper is as follows. The main results are stated in Section 2. Proofs are given in Sections 3 for the Raviart-Thomas elements, and in Section 4 for the N´ed´elec elements. The local regularity estimates needed in Section 3 are given in the Appendix. Basic results on the spaces H ¹ (Ω), H(div, Ω), and H(curl, Ω) will be assumed throughout. The symbol . will be used as follows:

for two quantities a h and b h depending on the triangulations (see below), we write a h . b h , whenever there exists C > 0 independent of h, such that a h ≤ Cb h . The quantity C will be allowed to depend on: the polynomial degree, the shape-regularity of the triangulation, and the domain.

2. Main results

Let Ω ⊂ R ³ be a connected polyhedral Lipschitz domain. The unit outward pointing normal vector field on Γ := ∂Ω will be denoted by n. Let now T h be a shape regular simplicial triangulation of Ω which, however, need not be quasi-uniform. For each K ∈ T h we let h K denote the diameter of K. We then consider the spaces of Raviart-Thomas and N´ed´elec finite elements:

V h := {q ∈ H(div, Ω) : q| K ∈ [P ^k (K)] ³ + P ^k (K)x, for all K ∈ T h }, N h := {q ∈ H(curl, Ω) : q| K ∈ [P ^k (K)] ³ + [P ^k (K)] ³ × x, for all K ∈ T h },

where P k (S) is the space of polynomials of degree k or less defined on S. On the boundary Γ, we consider the induced triangulation,

Γ h = {∂Ω ∩ ∂K : for all K ∈ T h }.

and two spaces

M h := {q · n : q ∈ V h } = {v : v| F ∈ P ^k (F ) for all F ∈ Γ h }

R h := {q × n : q ∈ N h } = {r ∈ H(div Γ , Γ) : r| F ∈ [P ^k (F )] ² + P ^k (F )x t for all F ∈ Γ h }.

In the last space x t := x − (x · n)n is the tangential position vector, and div Γ is the tangential divergence operator. We also consider M _h ⁰ to be the subset of M h consisting of elements whose average value vanishes.

Let S be a d-dimensional domain, then the fractional Sobolev norms on S are defined as follows:

for non-negative integer k and 0 < s < 1, we define

kvk ² _H

(S) = kvk ² _H

(S) + |v| ² _H

(S)

where

|v| ² _H

(S) = X

|α|=k

Z

S

Z

S

|∂ ^α v(x) − ∂ ^α v(y)| ²

|x − y| ^d+2s dxdy

is the Slobodetskij seminorm and k · k H

(S) is the usual Sobolev norm. For negative s, H ^−s (S) is the dual space of H ₀ ^s (S), the closure in H ^s (S) of the set of smooth compactly supported functions.

In particular, in the case of the closed surface Γ, we can write for functions v ∈ L ² (Γ):

kvk _H

(Γ) = sup

w∈C

(Γ),kwk

=1

Z

Γ

vw

As noted before, the operator V h 3 v 7→ v · n ∈ M h is surjective. The next result shows that there is a right-inverse of this operator that is bounded as an operator H ^−1/2 (Γ) → H(div; Ω), uniformly in the mesh size. Using the result [11, Theorem 5.1] it is enough to establish the uniform extension for data in M _h ⁰ .

Theorem 2.1. There exists a constant C depending only on the shape regularity of T h and on Ω such that for any g h ∈ M _h ⁰ there exists σ h ∈ V h with the following properties:

(a) σ h · n = g h on Γ,

(b) kσ h k H(div;Ω) . kg h k _H

_(∂Ω) , (c) div σ h = 0 in Ω.

The second result concerns the N´ed´elec space, asserting the existence of a uniformly bounded right-inverse of the operator N h 3 w h 7→ w h × n ∈ R h .

Theorem 2.2. There exists a constant C depending only on the shape regularity of T h and on Ω such that for any r h ∈ R h there exists w h ∈ N h with the following properties:

(a) w h × n = r h on Γ,

(b) kw h k H(curl;Ω) . kr h k _H

_(Γ) + kdiv Γ r h k _H

_(Γ) .

3. Discrete Extension Operators for Raviart-Thomas Finite Element Spaces We first recall some properties of the Raviart-Thomas projection. Let v ∈ H ^1/2+s (Ω) for some s > 0. Then we define Πv ∈ V h satisfying

Z

F

(Πv · n)w = Z

F

(v · n)w ∀w ∈ P ^k (F ) ∀F ∈ E h , Z

K

Πv · w = Z

K

v · w ∀w ∈ [P ^k−1 (K)] ³ ∀K ∈ T h

(see [4, Example 2.5.3]). Here E h is the set of all faces of the triangulation. The following classical result can be found in [4, Propositions 2.5.1, 2.5.2].

Proposition 3.1. For every v ∈ (H ^1/2+s (Ω)) ³ one has (a) div Πv = P div v,

(b) kΠv − vk L

(K) . h ^1/2+s _K kvk _H

(K) for all K ∈ T h ,

where P is the L ² (Ω)-orthogonal projection onto the space of piecewise P ^k (K) functions.

The following inverse inequality will play a key role in our analysis.

Lemma 3.2. For any g ∈ M h we have X

F ∈Γ

h F kg h k ² _L

(F ) . kg h k ² _H

(Γ)

Proof. This result is a consequence of basic estimates given in [1]. Let {ϕ i,F : i = 1, . . . , dimP ^k (F )}

be a basis for P ^k (F ) built by pushing forward the Lagrange basis on the reference element. Then we decompose g h = P

F

P

i g i,F ϕ i,F , and estimate X

F ∈Γ

h F kg h k ² _L

(F ) . X

F

X

i

h ² _F |g i,F | ² kϕ i,F k ² _L

(F )

. X

F

X

i

h ³ _F |g i,F | ² (simple computation)

. X

F

X

i

kg i,F ϕ i,F k ² _H

(Γ) (by [1, Theorem 4.8]) .kg h k ² _H

(Γ) , (by [1, Lemma 5.4])

which finishes the proof. 

We also need elliptic regularity results; see for example [7] for the case g = 0. Consider the Poisson problem with Neumann boundary conditions

−4u = f on Ω

(3.1a)

∇u · n = g on Γ, (3.1b)

under the assumption that R

Γ g + R

Ω f = 0 and R

Ω u = 0. Then, there exist C > 0 and s ∈ (0, 1/2) such that

(3.2) kuk _H

(Ω) ≤ C kfk _H

(Ω) + kgk H

(Γ)
.

We can localize this regularity result to obtain:

Theorem 3.3. Suppose that f ≡ 0 in (3.1) and let σ = ∇u. Then for each K ∈ T h we have kσk _H

(K) ≤ C(h ^−1/2−s _K kσk _L

_(D

₎ + kgk H

(∂D

∩Γ) + h ^−s _K kgk _L

_(∂D

_∩Γ) ), where

D K := ∪{K ⁰ ∈ T h : K ∩ K ⁰ 6= ∅}, is the collection of tetrahedra sharing one or more vertices with K.

The proof of this result is contained in Appendix A.

Proof of Theorem 2.1. Let u satisfy

−4u = 0 on Ω,

∇u · n = g h on Γ,

and set σ = ∇u ∈ (H ^1/2+s (Ω)) ³ (see (3.2)). Note that div σ = 0. We define σ h = Πσ and we note that Theorem 2.1 (a) holds, and that by Proposition 3.1(a), div σ h = 0. Therefore, using elliptic regularity, Proposition 3.1(b) and Lemma 3.2,

kσ h k H(div;Ω) =kσ h k L

(Ω) ≤ kσk L

(Ω) + kσ − σ h k L

(Ω)

.kg h k H

(Γ) + kσ − σ h k L

(Ω)

.kg h k _H

(Γ) + ( X

K∈T

h ^2(1/2+s) _K kσk ² _H

(K) ) ^1/2 .kg h k _H

_(Γ)

+ X

K∈T

kσk ² _L

(D

) + h ^2(1/2+s) _K kg h k ² _H

(∂D

∩Γ) + h K kg h k ² _L

(∂D

∩Γ)

! 1/2

The shape regularity of the elements means that X

K∈T

kσk ² _L

(D

) . kσk ² _L

(Ω) .

Also, if we let D F to be the macro-element surrounding F (triangles sharing a vertex with F ), we have that

X

K∈T

h ^2(1/2+s) _K kg h k ² _H

(∂D

∩Γ) . X

F ∈Γ

h ^2(1/2+s) _F kg h k ² _H

(D

)

. X

F ∈Γ

h F kg h k ² _L

(D

) . X

F ∈Γ

h F kg h k ² _L

(F ) ,

where we have used a standard local inverse estimate for piecewise polynomial functions. Applying Lemma 3.2 completes the proof.



As an application we can get an error estimate for the Laplacian in mixed form with Neumann boundary conditions. In this case X = H(div; Ω) × L ² (Ω) and the trace space is M = H ^−1/2 (∂Ω) with trace operator γ(σ, u) = σ · n. The bilinear form B and the linear form F are given by

B((σ, u), (η, w)) :=

Z

Ω

(σ · η − u div η + w div σ)

F ((η, w)) :=

Z

Ω

fw for a given f ∈ L ² .

Of course, the finite element space will be X h = V h × U h where U h = {w : Ω → R : w| K ∈ P ^k (K) ∀K ∈ T h }.

In view of Theorem 2.1 and the introductory discussion, we have the following error estimates for Raviart-Thomas elements.

Corollary 3.4. Let X = H(div; Ω)×L ² (Ω) , M = H ^−1/2 (∂Ω) and X h = V h ×U h . Let (σ, u) satisfy (1.1) and (σ h , u h ) satisfy (1.2) then the following holds

kσ − σ h k H (div;Ω) + ku − u h k L

(Ω) . inf

v ∈V

kσ − vk H (div;Ω) + inf

w∈U

ku − wk L

(Ω) + kg − g h k _H

(Γ) . Finally, we note that other applications of the existence of an extension operator like L h are given in the analysis of a variety of discretization methods for the Stokes-Darcy problem [10, 11].

4. Discrete Extension Operators for N´ ed´ elec Finite Element Spaces

The proof of Theorem 2.2 relies on a Helmholtz-Hodge type decomposition of R h , two liftings (one for Lagrange finite elements and the one provided by Theorem 2.1), and local estimates in the space of Lagrange finite elements on the boundary

P h := {φ h ∈ C(Γ) : φ h | F ∈ P k+1 (F ) for all F ∈ Γ h }.

We begin with some technical results:

Lemma 4.1. If r h ∈ R h satisfies div Γ r h = 0, then r h = curl Γ φ h with φ h ∈ P h .

Proof. By definition of R h , there exists w h ∈ N h such that w h × n = r h . Note now that q h := ∇ × w h ∈ V h , q h · n = div Γ (w h × n) = 0.

Therefore, by the exactness of the discrete de-Rham complex with essential boundary conditions (see for example [2]), there exists w ⁰ _h ∈ N h such that ∇ × w ⁰ _h = q h and w ⁰ _h × n = 0. We then consider the difference w h − w ⁰ _h and note, again, by the exactness of the discrete de-Rham complex, there exists u h in the finite element space

(4.1) W h := {u h ∈ C(Ω) : u h | K ∈ P k+1 (K) for all K ∈ T h },

satisfying ∇u h = w h − w ⁰ _h . Take φ h = u h | Γ . The result follows since w h × n = ∇u h × n =

curl Γ φ h . 

Lemma 4.2. It holds

kφ h k _H

(Γ) . kcurl Γ φ h k _H

(Γ) for all φ h ∈ P h .

Proof. Let {φ j : j = 1, . . . , N} be a basis of P h , and θ j := h j curlφ j , where h j is the diameter of any of the elements contained in the support of φ j . We decompose φ h = P

j∈N α j φ j and use [1, Formula (5.1)] and [1, Theorem 4.8 (1)] to bound

(4.2) kφ h k ² _H

(Γ) . X

j∈N

α ² _j kφ j k ² _H

(Γ) ≈ X

j∈N

α ² _j h ² _j . On the other hand, we can use a basis {ψ ` } of M h and write

θ j = X

`∈S

c ` ψ ` , |c ` | . 1, #S j ≈ 1, and then use [1, Theorem 4.8 (1)] to show that

(4.3) h ² _j ≈ h ⁻² _j X

`∈S

kψ ` k ² _H

(Γ) ≈ h ⁻² _j kθ j k ² _H

(Γ) = kcurl Γ φ j k ² _H

(Γ) .

By equations (4.2) and (4.3) it follows that kφ h k ² _H

(Γ) . X

j∈N

α ² _j kcurl Γ φ j k ² _H

(Γ) ,

and the result is then a consequence of [1, Formula (5.13)].  Lemma 4.3. For all v h ∈ V h with div v h = 0, there exists w h ∈ N h such that ∇ × w h = v h and kw h k L

(Ω) . kv h k L

(Ω) .

Proof. By [6] uniformly bounded projections Π h : H(curl, Ω) → N h and Q h : H(div, Ω) → V h exist such that ∇ × Π h w = Q h (∇ × w) for all w ∈ H(curl, Ω). On the other hand the curl operator is surjective from H(curl, Ω) to {v ∈ H(div, Ω) : divv = 0} and has therefore a bounded right-inverse.

We then apply this right-inverse to v h to obtain w ∈ H(curl, Ω) and define w h = Π h w. Then

∇ × w h = Q h (∇ × w) = Q h v h = v h and the result is proved.  Proof of Theorem 2.2. Let r h ∈ R h . Then div Γ r h ∈ M _h ⁰ and by Theorem 2.1 we can find v h ∈ V h

such that

div v h = 0, v h · n = div Γ r h , and

kv h k H(div,Ω) . kdiv r h k _H

_(Γ) . We then use Lemma 4.3 to obtain w h ∈ N h such that ∇ × w h = v h , and (4.4) kw h k H(curl,Ω) . kv h k H(div,Ω) . kdiv r h k _H

_(Γ) . Consider now the function m h := r h − w h × n ∈ R h and note that

km h k _H

_(Γ) . kr h k _H

_(Γ) + kdiv Γ r h k _H

_(Γ)

by (4.4) and the continuity of the tangential trace operator from H(curl, Ω) to H ^−1/2 (Γ) ³ . Addi- tionally

div Γ m h = div Γ r h − (∇ × w h ) · n = 0,

by construction of w h . We then apply Lemma 4.1 to find φ h ∈ P h such that r h − w h × n = curl Γ φ h

and use Lemma 4.2 to bound

(4.5) kφ h k _H

(Γ) . kr h − w h × nk _H

(Γ) . kr h k _H

(Γ) + kdiv Γ r h k _H

(Γ) .

We then take u h in the finite element space W h (see (4.1)) such that u h | Γ = φ h and

(4.6) ku h k _H

_(Ω) . kφ h k _H

(Γ) .

This can be accomplished by first taking u ∈ H ¹ (Ω) whose trace is φ h and satisfying kuk H

(Ω) . kφ h k _H

_(Γ) and then applying the Scott-Zhang interpolation operator [16] to u. The desired lifting of r h is the function w h + ∇u h ∈ N h . The bound

kw h + ∇u h k H(curl,Ω) ≤ kw h k H(curl,Ω) + k∇u h k L

(Ω) . kr h k _H

_(Γ) + kdiv Γ r h k _H

_(Γ) is a direct consequence of (4.4), (4.5), and (4.6). The fact that it is a lifting follows from

(w h + ∇u h ) × n = w h × n + curl Γ φ h = w h × n + m h = r h .

This finishes the proof. 

As an application to Theorem 2.2 we consider problem (1.1) with X = H(curl; Ω), M = H ^−1/2 (div Γ ; ∂Ω) :=

{µ ∈ H ^−1/2 (∂Ω) : div Γ µ ∈ H ^−1/2 (∂Ω)}. The finite element space are the N´ed´elec elements X h = N h . The bilinear form B and linear form F are as follows

B(u, v) = Z

Ω

(∇ × u) · (∇ × v) + u · v, F (v) =

Z

Ω

f · v.

Theorem 2.2 and the Introductory discussion now gives the following error estimates for N´ed´elec elements.

Corollary 4.4. Let X = H(curl; Ω) , M = H ^−1/2 (div Γ ; ∂Ω) and X h = N h . Let u satisfy (1.1) and u h ) satisfy (1.2) then the following holds

ku − u h k H(curl;Ω) . inf

v ∈N

ku − vk H(curl;Ω) + kg − g h || _H

_(div

_;Γ) . Appendix A. Proof of Theorem 3.3

For each K ∈ T h we can find a cut–off function ω = ω K ∈ C ^∞ (D K ) with the following properties:

ω ≡ 1 in K, (A.1a)

ω ≡ 0 in Ω \ D K , (A.1b)

kD ^s ωk L

(D

) . h ^−s _K for s = 0, 1, 2.

(A.1c) Note that

(A.2) kuk _H

_(K) ≤ kωuk _H

_(Ω) ≤ C(k − 4(ωu)k _H

_(Ω) + k∇(ωu) · nk H

(Γ) ), by (3.2).

Let us first deal with elements K such that D K does not contain a face in ∂Ω. In this case

∇(ωu) · n ≡ 0 on ∂Ω. To bound the first term we let v ∈ H ^1/2−s (Ω), and define m(v) = _|D ¹

_| R

D

v.

Then, we have

− Z

Ω

4(ωu)v = − Z

Ω

4(ωu)(v − m(v)) − Z

Ω

4(ωu)m(v) = − Z

Ω

(2∇ω · ∇u + u4ω)(v − m(v)) where we used that 4u = 0 and that ∇(ωu) · n ≡ 0 on ∂Ω. Therefore

− Z

Ω

4(ωu)v ≤ (k∇ω · ∇uk L

(D

) + ku4ωk L

(D

) )kv − m(v)k L

(D

)

. (h ^−1/2−s _T k∇uk L

(D

) + h ^−3/2−s _K kuk L

(D

) )kvk _H

(D

) ,

where we have used (A.1c), the Poincar´e inequality and an interpolation argument. Taking the supremum over v we have

k − 4(ωu)k _H

(Ω) ≤ (h ^−1/2−s _K k∇uk _L

_(D

₎ + h ^−3/2−s _K kuk _L

_(D

₎ ).

Hence, in the case D K does not contain a face in ∂Ω we have

kuk _H

(K) . (h ^−1/2−s _K k∇uk L

(D

) + h ^−3/2−s _K kuk L

(D

) ) If we replace u with m(u), and note that

ku − m(u)k L

(D

) . h K k∇uk L

(D

)

we get

k∇uk _H

(K) . h ^−1/2−s _K k∇uk L

(D

) .

Next we consider the case when ∂D K contains one or more faces on Γ. To bound the first term in the right of (A.2) we get

    Z

Ω

4(ωu)v    

≤ (2k∇ω · ∇uk L

(D

) + ku4ωk L

(D

) )kvk L

(D

)

. (h ^−1/2−s _K k∇uk L

(D

) + h ^−3/2−s _K kuk L

(D

) )h ^1/2−s _K kvk _H

_(D

₎ ,

where we have used (A.1c) and the fact that v vanishes on at least one face of ∂D K , which allows us to use an inequality in the form

(A.3) kvk L

(D

) . h ^1/2−s _K kvk _H

_(D

₎ . Therefore, as above we have that

k − 4(ωu)k _H

(Ω) . h ^−1/2−s _K k∇uk L

(D

) + h ^−3/2−s _K kuk L

(D

) . To bound the second term on the right of (A.2) we first use the product rule and get

k∇(ωu) · nk H

(∂Ω) ≤ ku∇ω · nk H

(∂Ω) + kωgk H

(Γ)

. h ^−1/2−s _K ku∇ωk L

(D

) + h ^1/2−s _K k∇(u∇ω)k L

(D

) + kωgk H

(Γ)

. h ^−3/2−s _K kuk L

(D

) + h ^−1/2−s _K k∇uk L

(D

) + kωgk H

(Γ) ,

after using a localized version of the trace theorem and (A.1c). By a simple interpolation argument and (A.1c), we have

kωgk H

(Γ) . kgk H

(∂D

∩Γ) + h ^−s _K kgk L

(∂D

∩Γ) . Combining the above inequalities we have

kuk _H

_(K) . (h ^−1/2−s _K k∇uk L

(D

) + h ^−3/2−s _K kuk L

(D

) ) +(kgk H

(D

∩Γ) + h ^−s _K kgk L

(D

∩Γ) ).

If we apply the above argument to u − m(u), we obtain our result.

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E-mail address: mark [email protected]

Divison of Applied Mathematics, Brown University, Providence, RI E-mail address: johnny [email protected]

Division of Applied Mathematics, Brown University, Providence, RI E-mail address: [email protected]

Department of Mathematical Sciences, University of Delaware, Newark,DE