T Chaumont-Frelet

A SIMPLE EQUILIBRATION PROCEDURE LEADING TO POLYNOMIAL-DEGREE-ROBUST A POSTERIORI ERROR ESTIMATORS FOR THE CURL-CURL PROBLEM

Keywords: A posteriori error estimates, Electromagnetics, Finite element methods, High order methods

We introduce two a posteriori error estimators for Nédélec finite element discretizations of the curl-curl problem. These estimators pertain to a new Prager-Synge identity and an associated equilibration procedure. They are reliable and efficient, and the error estimates are polynomial-degree-robust. In addition, when the domain is convex, the reliability constants are fully computable. The proposed error estimators are also cheap and easy to implement, as they are computed by solving divergence-constrained minimization problems over edge patches. Numerical examples highlight our key findings, and show that both estimators are suited to drive adaptive refinement algorithms. Besides, these examples seem to indicate that guaranteed upper bounds can be achieved even in non-convex domains.

Introduction

Given a domain Ω, a partition {Γ D , Γ N } of its boundary, and a divergence free field J : Ω → R 3 , the curl-curl problem consists in finding A : Ω → R 3 such that

∇ × ∇ × A = J , ∇ • A = 0,
in Ω, (1.1a)

A × n = o, on Γ D , (1.1b) (∇ × A) × n = o, A • n = 0 on Γ N , (1.1c)
together with a finite set of suitable orthogonality conditions to filter out harmonic forms when the topology is non-trivial [START_REF] Fernandes | Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions[END_REF]. This problem is the central model for magnetostatic applications and is the basis of Maxwell's equations, which are instrumental in the modeling of electromagnetic phenomena [START_REF] Griffiths | Introduction to Eelectrodynamics[END_REF].

In general geometries, numerical schemes are required to approximate the solution to (1.1). Here, we consider finite element methods [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF][START_REF] Monk | Finite element methods for Maxwell's equations[END_REF][START_REF] Nédélec | Mixed finite elements in R 3[END_REF], and focus on a posteriori error estimation for Nédélec elements. This topic is already largely covered in the literature [START_REF] Beck | Residual based a posteriori error estimators for eddy current computation[END_REF][START_REF] Braess | Equilibrated residual error estimators for edge elements[END_REF][START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF][START_REF] Chaumont-Frelet | p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. application to a posteriori analysis of the curl-curl problem[END_REF][START_REF] Gedicke | An equilibrated a posteriori error estimator for arbitrary-order Nédédec elements for magnetostatic problems[END_REF][START_REF] Nicaise | A posteriori error estimation for heterogeneous Maxwell equations on isotropic and anisotropic meshes[END_REF][START_REF] Schöberl | A posteriori error estimates for Maxwell equations[END_REF], motivated by the variety of important applications as well as the mathematical challenges involved.

A posteriori error estimators for the curl-curl problem where first proposed in [START_REF] Beck | Residual based a posteriori error estimators for eddy current computation[END_REF] for convex domains (see [5, Assumption 2] for details). The estimator in [START_REF] Beck | Residual based a posteriori error estimators for eddy current computation[END_REF] is of residual type, and has been generalized to arbitrary polyhedral Lipschitz domains in [START_REF] Nicaise | A posteriori error estimation for heterogeneous Maxwell equations on isotropic and anisotropic meshes[END_REF][START_REF] Schöberl | A posteriori error estimates for Maxwell equations[END_REF]. While this approach provides reliable and efficient estimators, it still suffers from two drawbacks, namely: (i) the constants appearing in the reliability estimates are not computable in practice, and (ii) the constants in the efficiency estimates deteriorate as the polynomial degree p is increased. Notice that this is not specific to the curl-curl problem, and residual-based estimators exhibit the same downsides even in the simpler context of scalar elliptic problems [START_REF] Ainsworth | A posteriori error estimation in finite element analysis[END_REF][START_REF] Melenk | On residual-based a posteriori estmiation in hp-FEM[END_REF].

In this work, we focus on so-called "equilibrated" error estimators, that have the ability to provide (i) guaranteed upper bounds free of unknown constants and (ii) polynomial-degree-robust (or simply, p-robust) efficiency constants. The concept of equilibrated flux can be traced back to the seminal work of Prager and Synge [START_REF] Prager | Approximations in elasticity based on the concept of function space[END_REF], where the authors establish that equilibrated fluxes can be employed to provide guaranteed error upper bounds. For scalar elliptic problems, several constructions of equilibrated fluxes have then been proposed, leading to practically usable error estimators, see e.g. [START_REF] Ainsworth | A posteriori error estimation in finite element analysis[END_REF][START_REF] Ladevèze | Error estimate procedure in the finite method and applications[END_REF][START_REF] Luce | A local a posteriori error estimator based on equilibrated fluxes[END_REF]. Here, we focus on the approach initially proposed in [START_REF] Destuynder | Explicit error bounds in a conforming finite element method[END_REF] and later extended in [START_REF] Ern | Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations[END_REF]. It relies on a partition of unity via finite element shape functions and local mixed finite element problems, and lead to p-robust estimates [START_REF] Braess | Equilibrated residual error estimates are p-robust[END_REF][START_REF]Stable broken H 1 and H(div) polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions[END_REF].

For the curl-curl problem, the construction of equilibrated fluxes turns out to be a much more arduous task than for scalar elliptic equations. A procedure for the lowest-order Nédélec elements have been introduced early [START_REF] Braess | Equilibrated residual error estimators for edge elements[END_REF], but its generalization to arbitrary orders has only been proposed recently [START_REF] Gedicke | An equilibrated a posteriori error estimator for arbitrary-order Nédédec elements for magnetostatic problems[END_REF]. Actually, to the best of the author's knowledge, they are currently only two constructions of equilibrated estimator that lead to p-robust estimates for the curl-curl problem, namely (i) the construction in [START_REF] Gedicke | A polynomial-degree-robust a posteriori error estimator for Nédélec discretizations of magnetostatic problems[END_REF] which is based on [START_REF] Gedicke | An equilibrated a posteriori error estimator for arbitrary-order Nédédec elements for magnetostatic problems[END_REF], and (ii) the approach in [START_REF] Chaumont-Frelet | p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. application to a posteriori analysis of the curl-curl problem[END_REF] which employs a partition of unity in the spirit of [START_REF] Destuynder | Explicit error bounds in a conforming finite element method[END_REF][START_REF] Ern | Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations[END_REF]. While the estimators in [START_REF] Chaumont-Frelet | p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. application to a posteriori analysis of the curl-curl problem[END_REF] and [START_REF] Gedicke | A polynomial-degree-robust a posteriori error estimator for Nédélec discretizations of magnetostatic problems[END_REF] provide constant-free reliability estimates and p-robust efficiency constants, there are not fully satisfactory as there construction is complicated. Indeed, they hinge on over-constrained minimization problems, and require several passes through the mesh. In addition, the patches involved in the efficiency estimates are rather large.

Another approach called "broken patchwise equilibration" has been proposed in [START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF], where the authors introduce a p-robust a posteriori error estimator for the curlcurl problem that do not rely on equilibration. The key assets of this estimator is that it is cheap and straightforward to compute (especially as compared to [START_REF] Chaumont-Frelet | p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. application to a posteriori analysis of the curl-curl problem[END_REF][START_REF] Gedicke | A polynomial-degree-robust a posteriori error estimator for Nédélec discretizations of magnetostatic problems[END_REF]), and that the efficiency estimates are established on tight edge patches. On the other hand, because the reliability estimate does not pertain to a Prager-Synge identity, the reliability bound contains constants that are either unavailable, or cumbersome to compute in practice.

The equilibrated estimators of [START_REF] Braess | Equilibrated residual error estimators for edge elements[END_REF][START_REF] Chaumont-Frelet | p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. application to a posteriori analysis of the curl-curl problem[END_REF][START_REF] Gedicke | An equilibrated a posteriori error estimator for arbitrary-order Nédédec elements for magnetostatic problems[END_REF][START_REF] Gedicke | A polynomial-degree-robust a posteriori error estimator for Nédélec discretizations of magnetostatic problems[END_REF] are based on the following Prager-Synge identity: if H ∈ H ΓN (curl, Ω) is any "equilibrated field" satisfying ∇×H = J , then

(1.2) (∇ × (A -A h ), ∇ × v) Ω = (H -∇ × A h , ∇ × v) Ω ,
for any field A h , v ∈ H ΓD (curl, Ω) (the notations are rigorously introduced in Section 2 below), leading to the estimate

(1.3) ∇ × (A -A h ) Ω ≤ H -∇ × A h Ω .
This motivates the construction of equilibrated fields H with prescribed curl. Unfortunately, as previously mentioned, the construction of such fields is rather involved [START_REF] Chaumont-Frelet | p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. application to a posteriori analysis of the curl-curl problem[END_REF][START_REF] Gedicke | An equilibrated a posteriori error estimator for arbitrary-order Nédédec elements for magnetostatic problems[END_REF][START_REF] Gedicke | A polynomial-degree-robust a posteriori error estimator for Nédélec discretizations of magnetostatic problems[END_REF].

Here, we introduce an alternative Prager-Synge identity that leads to a much simpler equilibration procedure. It relies on the observation that whenever v ∈ H 1 (Ω), we have ∇ × (vu k ) = ∇v × u k , where {u k } 3 k=1 is the canonical basis of R 3 . Simple manipulations then show the following identity: if, for each k ∈ {1, 2, 3},

S k ∈ H ΓN (div, Ω) satisfies ∇ • S k = J k , we have (1.4) (∇ × (A -A h ), ∇ × v) Ω = - 3 k=1 (S k + u k × ∇ × A h , ∇v k ) Ω for all A h ∈ H ΓD (curl, Ω) and v ∈ H 1 ΓD (Ω)
. Interestingly, as opposed to (1.2), identity (1.4) only constrains the divergence of the equilibrated fields, simplifying the equilibration process. On the other hand, the downside of (1.4) is that it requires the gradient of the test function v ∈ H 1 ΓD (Ω) instead of its curl in (1.2). As usual [START_REF] Nicaise | A posteriori error estimation for heterogeneous Maxwell equations on isotropic and anisotropic meshes[END_REF][START_REF] Schöberl | A posteriori error estimates for Maxwell equations[END_REF], this is remedied by employing "regular decompositions", leading to the estimate

(1.5) ∇ × (A -A h ) Ω ≤ C L,Ω 3 k=1 S k + u k × ∇ × A h Ω , 1/2
, where the constant C L,Ω is one in convex domains (with Γ D = ∂Ω or Γ N = ∂Ω), but is not practically computable in general (see Section 2.7).

In this work, we carefully establish the new Prager-Synge identity introduced in (1.4), and we use it to design and analyze two novel a posteriori error estimators for curl-curl problem (1.1). We show that they are reliable and efficient with probust constants. We also provide a set of numerical experiments showing that both estimators are suitable to drive adative mesh refinements processes. Our estimators share some similarities with the broken patchwise estimator introduced in [START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF] and the equilibrated estimators based on (1.2) provided in [START_REF] Chaumont-Frelet | p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. application to a posteriori analysis of the curl-curl problem[END_REF][START_REF] Gedicke | A polynomial-degree-robust a posteriori error estimator for Nédélec discretizations of magnetostatic problems[END_REF], which we now describe.

As in [START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF], our construction relies on a vectorial partition of unity via edge functions (see Section 2.8), which leads to a localization on tight edge patches. In contrast to [START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF] however, the proposed approach has two main assets. Indeed, (i) it uses divergence-constrained minimization problems which are easier to implement and cheaper to solve than the curl-constrained problems employed in [START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF]. Besides, (ii) the present approach hinges on an equilibration principle, which leads to nicer upper bounds than in [START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF]. The two approaches can be seen as dual to one another in some sense, as we elaborate in Remark 3.4.

As compared to the usual equilibration procedures of [START_REF] Braess | Equilibrated residual error estimators for edge elements[END_REF][START_REF] Chaumont-Frelet | p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. application to a posteriori analysis of the curl-curl problem[END_REF][START_REF] Gedicke | An equilibrated a posteriori error estimator for arbitrary-order Nédédec elements for magnetostatic problems[END_REF][START_REF] Gedicke | A polynomial-degree-robust a posteriori error estimator for Nédélec discretizations of magnetostatic problems[END_REF] based on (1.2), the main drawback of the present approach is that apart from convex domains, the upper bound contains the unknown constant C L,Ω . While this appears as a major downside, numerical experiments suggest that the constant C L,Ω might be spurious, and that taking C L,Ω = 1 provides guaranteed upper bounds, even in non-convex domains. We also elaborate on why the constant C L,Ω may not be necessary in Remark 5.2.

The remaining of this work is organized as follows. In Section 2, we describe the setting, and recall preliminary results. Section 3 presents the key abstract arguments to localize the construction of the estimators. We introduce our two estimators in Sections 4 and 5 where we establish that they are reliable and efficient with p-robust constants. Section 6 illustrates the key theoretical findings with a set of numerical examples, and we present concluding remarks in Section 7.

Setting

2.1. Domain. We consider a Lipschitz polyhedral domain Ω ⊂ R 3 . The boundary of Ω is split into two disjoint (relatively open) parts Γ D and Γ N in such a way that ∂Ω = Γ D ∪ Γ N . We assume that both Γ D and Γ N are polygonal. Notice that we do not assume that Ω is simply connected, nor that Γ D or Γ N are connected.

H Γ (curl, ω) := w ∈ H(div, ω); (w, ∇ × p) -(∇ × v, p) = 0 ∀p ∈ H 1 Γ c (ω) with Γ c := ∂ω \ Γ.
We refer the reader to [START_REF] Adams | Sobolev spaces[END_REF][START_REF] Fernandes | Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions[END_REF][START_REF] Girault | Finite element methods for Navier-Stokes equations: theory and algorithms[END_REF] for an in-depth presentation of the above spaces.

For

m ≥ 2, H m (ω) is the space of functions v ∈ L 2 (ω) such that ∂ α v ∈ L 2 (ω) for all |α| ≤ m. If U is a collection of open sets ω, then H m (U ) contains those functions v ∈ L 2 ( U ) such that v| ω ∈ H m (ω)
for all ω ∈ U . We also employ the notations H m (ω) := (H m (ω))

3 and H m (U ) := (H m (U )) 3 . The usual seminorm of H m (U ) is denoted by

|v| 2 H m (U ) := ω∈U |α|=m ∂ α v 2 ω ∀v ∈ H m (ω),
and we set

|v| 2 H m (U ) := 3 k=1 |v k | 2 H m (U ) when v ∈ H m (U ).
We also employ the notation

Λ ΓD (Ω) := {v ∈ H ΓD (Ω); ∇ × v = o} ,
for functions with vanishing curl. When Ω is simply connected and Γ D is connected, we simply have Λ ΓD (Ω) = ∇ H 1 ΓD (Ω) . In the general case however,

Λ ΓD (Ω) = ∇ H 1 ΓD (Ω) + H ΓD (Ω)
where H ΓD (Ω) is a finite dimensional "cohomology" space. The dimension of H ΓD (Ω) depends on the topology of Ω and Γ D , and its structure is explicitly known [START_REF] Fernandes | Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions[END_REF]. We write Λ ⊥ ΓD (Ω) for the orthogonal complement of Λ ΓD (Ω) in L 2 (Ω).

Model problem.

Assuming that J ∈ L 2 (Ω), the weak form of (1.1) consists in finding (A, l) ∈ H ΓD (curl, Ω) × Λ ΓD (Ω) such that

(2.1) (∇ × A, ∇ × v) Ω + (l, v) Ω = (J , v) Ω (A, w) Ω = 0 for all (v, w) ∈ H ΓD (curl, Ω) × Λ ΓD (Ω). Problem (2.1
) is a standard saddle-point problem, and admits a unique solution [START_REF] Brezzi | On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers[END_REF]. We will assume throughout this work that J ∈ Λ ⊥ ΓD (Ω), so that actually, l = o, and (2.2)

(∇ × A, ∇ × v) Ω = (J , v) Ω ∀v ∈ H ΓD (curl, Ω).
2.4. Computational mesh and edge patches. The domain Ω is partitioned into a mesh T h of (open) tetrahedral elements K. We assume that the mesh is conforming in the sense of [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF], meaning that the intersection K -∩ K + of two distincts elements K ± ∈ T h is either empty, or it is a full face, edge or vertex of both K -and K + . The set vertices, edges and faces of the mesh are respectively denoted by V h , E h and F h . Classically, we assume that the mesh is conforming with the partition of the boundary, i.e., that for all faces F ∈ F h such that F ⊂ ∂Ω, either F ⊂ Γ D or F ⊂ Γ N . For each K ∈ T h , the quantity κ K := h K /ρ K (where, as usual, h K is the diameter of K and ρ K is the diameter of the largest ball contained in K) is the shape-regularity parameter of the element K.

Consider an edge ∈ E h . The mesh patch T h gathers those elements K ∈ T h having as an edge. Besides, we denote by ω the open domain covering the elements K ∈ T h , and h is the diameter of ω . We also associate with an arbitrary, but fixed, unit tangent vector τ := (b -a)/|b -a|, where a, b ∈ V h are the two vertices of . We then introduce the edge function

(2.3) ψ := |b -a|(ψ a ∇ψ b -ψ b ∇ψ a )
where

ψ a , ψ b ∈ P 1 (T h ) ∩ H 1 (Ω) are the usual hat functions satisfying ψ a (c) = δ a,c and ψ b (c) = δ b,c for all c ∈ V h . Notice that ψ | • τ = δ , for all ∈ E h .
We also denote by κ := min K∈T h κ K the shape-regularity paremeter of the edge patch.

2.5. Finite element spaces. For all integer q ≥ 0 and elements K ∈ T h , P q (K) stands for the space of polynomials from K to R of degree less than or equal to q, and P q (K) := (P q (K)) 3 . As usual [START_REF] Nédélec | Mixed finite elements in R 3[END_REF][START_REF] Raviart | A mixed finite element method for 2nd order elliptic problems[END_REF], the Raviart-Thomas and Nédélec polynomial spaces are respectively defined by RT q (K) := xP q (K) + P q (K), N q (K) := x × P q (K) + P q (K).

If T ⊂ T h and ω is the domain covering the elements of T , then P q (T ) gathers the functions u : L 2 (ω) → R such that u| K ∈ P q (K) for all K ∈ T h . Notice that there are no "build in" compatibility conditions in this space. We also define P q (T ), RT q (T ) and N q (T ) in a similar fashion.

If v ∈ L 2 (ω), we denote by π T ,q (v) ∈ P q (T ) its L 2 (ω)-projection over P q (T ), which is uniquely defined by

(v -π T ,q (v), w h ) ω = 0 ∀w h ∈ P q (T ),
and we have the estimate

(2.4) v -π T ,q (v) ω ≤ h T π q+1 |v| H q+1 (T )
with h T := max K∈T h K , whenever v ∈ H q+1 (T ). We also employ the symbol π T ,q for the L 2 (ω)-projection over P q (T ). Notice that since K ∈ T h is convex, (2.4) can be obtained by repeated applications of the Poincaré inequality established in [START_REF] Payne | An optimal Poincaré inequality for convex domains[END_REF] to v k , 1 ≤ k ≤ 3, and its derivatives ∂ α v k for |α| ≤ q.

2.6. Discrete solution. Throughout this work, we consider a discrete function

A h ∈ H ΓD (curl, Ω) such that (2.5) A h | ω ∈ N p (T h )
for some p ≥ 0 for ∈ E h and

(2.6) (∇ × A h , ∇ × ψ ) Ω = (J , ψ ) Ω for all edge ∈ E h such that ⊂ Γ D .
Observe that in particular, (2.6) typically holds true if A h is obtained from a Galerkin approximation to (2.1), since the edge functions involved in (2.6) belong to the lowest-order Nédélec space. On the other hand, (2.5) is satisfied if A h belongs to a hp-adaptive Nédélec finite element space, and we can simply take p = p for all ∈ E h in the particular case of a uniform polynomial degree p ≥ 0.

2.7. Regular decomposition. For all v ∈ H ΓD (curl, Ω), there exists a function

w ∈ H 1 ΓD (Ω) such that ∇ × w = ∇ × v and (2.7) ∇w Ω ≤ C L,Ω ∇ × v Ω .
In addition, if Ω is convex and either Γ D or Γ N is empty, we can take C L,Ω = 1. We refer the reader to [START_REF] Costabel | Singularities of Maxwell interface problems[END_REF][START_REF] Girault | Finite element methods for Navier-Stokes equations: theory and algorithms[END_REF][START_REF] Hiptmair | Discrete regular decompositions of tetrahedral discrete 1-forms[END_REF] for a proof of these results.

2.8. Partition of unity via edge functions. For all ∈ E h , supp ψ = ω , and we have

(2.8) ψ L ∞ (ω ) + h ∇ × ψ L ∞ (ω ) ≤ C(κ ) ∀ ∈ E h ,
where C(κ ) is a constant that only depends on the shape regularity parameter κ of the edge patch. In addition, the identity (2.9)

w = ∈E h (w • τ )ψ
holds true for all w ∈ L 2 (Ω). We refer the reader to [10, Section 5.3] for a proof of these facts.

Local functional spaces and Poincaré inequalities. Consider an edge

∈ E h . If ⊂ Γ D , H 1 (ω ) is the subset of H 1 (ω ) of functions with vanishing mean value and H 0 (div, ω ) := H ∂ω (div, ω ). If on the other hand, ⊂ Γ D , H 1 (ω ) := H 1 Γ (ω ) and H 0 (div, ω ) := H Γ c (div, ω ) where Γ := ∂ω ∩ Γ D and Γ c := ∂ω \ Γ D .
For all edges ∈ E h , the constant (2.10)

C P, := h -1 sup w∈H 1 (ω ) ∇w ω =1
w ω is finite and only depends on κ . Since every cell K ∈ T h is convex [START_REF] Payne | An optimal Poincaré inequality for convex domains[END_REF], the following elementwise Poincaré inequality holds true:

(2.11) v -π K,0 v K ≤ h K π ∇v K ∀v ∈ H 1 (K),
where π K,0 denotes the L 2 (K)-projection of v over P 0 (K).

Localization of the residual functional

In this section, we start by presenting abstract arguments later used to localize the computation of the estimator. Specifically, we introduce a residual functional R ∈ (H ΓD (curl, Ω)) by setting

(3.1) R, v := (J , v) Ω -(∇ × A h , ∇ × v) Ω for all v ∈ H ΓD (curl, Ω).
Observe that because of (2.6), we have

(3.2) R, ψ = 0 ∀ ∈ E h ; ⊂ Γ D .
We will employ two key norms for the residual functional. On the one hand the usual dual norm reads

(3.3) R ,Ω := sup v∈HΓ D (curl,Ω) ∇×v =1 R, v .
On the other hand, we also introduce a localized norm

(3.4) R , := sup w∈H 1 (ω ) ∇w ω =1 R, wψ for each ∈ E h . Notice that as R, v = (∇ × (A -A h ), ∇ × v) Ω for all v ∈ H ΓD (Ω), we have (3.5) R ,Ω = ∇ × (A -A h ) Ω ,
so that the dual norm of R is the quantity we actually want to estimate. Directly estimating R ,Ω , however, would lead to global (and expensive) computations.

Henceforth, the goal of this section is to establish equivalence results between the dual norm R ,Ω and the squared sum of the localized norms R , , which are more suitable to tackle numerically. We start by establishing an upper bound.

Theorem 3.1 (Continous reliability). We have

(3.6) R ,Ω ≤ √ 6C L,Ω ∈E h R 2 , 1/2 . Proof. Let v ∈ H ΓD (curl, Ω) with ∇ × v Ω = 1. Recalling the discussion in Section 2.7, there exists w ∈ H 1 ΓD (Ω) with ∇ × v = ∇ × w and (3.7) ∇w Ω ≤ C L,Ω ∇ × v Ω = C L,Ω .
Then, we use (2.9) and (3.2) to show that

R, v = R, w = ∈E h R, (w • τ )ψ = ∈E h R, (w • τ -ϑ )ψ with ϑ := 0 if ⊂ Γ D , 1 |ω | ω w • τ otherwise.
We then observe that w •τ -ϑ ∈ H 1 (ω ), so that recalling the definition of R , in (3.4), we have

R, v = ∈E h R , ∇(w • τ ) ω ≤ ∈E h R , ∇w ω ≤ √ 6 ∈E h R 2 , 1/2
∇w Ω , and (3.6) follows from (3.7).

Next, we provide a lower bound.

Theorem 3.2 (Continuous efficiency). For all edges ∈ E h , we have

(3.8) R , ≤ C cont, ∇ × (A -A h ) ω where C cont, := ψ L ∞ (ω ) + C P, h ∇ × ψ L ∞ (ω ) only depends on κ .
Proof. First, we observe that there exists w ∈ H 1 (ω ) with ∇w ω = 1 such that

R , = R, w ψ = (∇ × (A -A h ), ∇ × (w ψ )) Ω ≤ ∇ × (A -A h ) ω ∇ × (w ψ ) ω .
Then, (3.8) follows since

∇ × (w ψ ) ω ≤ ∇ × ψ L ∞ (ω ) w ω + ψ L ∞ (ω ) ∇w ω ≤ ψ L ∞ (ω ) + C P, h ∇ × ψ L ∞ (ω ) .
The fact that C cont, only depends on κ is a direct consequence of (2.8) and (2.10).

Finally, we provide an alternative expression for R , by employing a duality argument. This dual expression corresponds to a minimization problem and is the basis of the estimators we propose in this work. Theorem 3.3 (Dual characterization). For all ∈ E h , the equality

(3.9) R , = σ + ψ × ∇ × A h ω holds true with (3.10) σ := arg min v∈H0(div,ω ) ∇•v=ψ •J-∇×ψ •∇×A h v + ψ × ∇ × A h ω .
Proof. Fix ∈ E h . We introduce a Riesz representant r defined as the unique element of H 1 (ω ) such that (3.11) (∇r , ∇w) ω = R, wψ

for all w ∈ H 1 (ω ). Since R, wψ = (J , wψ ) Ω -(∇ × A h , ∇ × (wψ )) Ω = (J , wψ ) ω -(∇ × A h , ∇w × ψ + w∇ × ψ ) ω = (J • ψ -∇ × A h • ∇ × ψ , w) ω -(ψ × ∇ × A h , ∇w) ω ,
one easily sees that

(∇r + ψ × ∇ × A h , ∇w) ω = (ψ • J -∇ × ψ • ∇ × A h , w) ω
for all w ∈ H 1 (ω ). Then, since (3.10) is the mixed formulation of (3.11), we have -σ = ∇r + ψ × ∇A h , and (3.9) follows.

Notice that because of (2.6), for ∈ E h , we do have

(3.12) (ψ • J -∇ × ψ • ∇ × A h , 1) ω = (J , ψ ) Ω -(∇ × A h , ∇ × ψ ) Ω = 0
whenever ψ ⊂ Γ D , ensuring the well-posedness of (3.10).

Remark 3.4 (Broken patchwise equilibration). The broken patchwise equilibration procedure in [START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF] hinges on related localized norms of the residual functional, namely:

R †, := sup v∈H0(curl,ω ) ∇×v =1 R, v , ∈ E h ,
where H 0 (curl, ω ) is defined as H 0 (div, ω ) in Section 2.9. Since vψ

∈ H 0 (curl, ω ) with ∇ × (vψ ) ω ≤ C cont, ∇v ω when v ∈ H 1 (ω ), we have R , ≤ C cont, R †, .
On the other hand, we have from [10, Lemma 5.5] that

R †, ≤ ∇ × (A -A h ) ω ,
where, in comparison to (3.8), the constant C cont, is omitted. As a result, the constant C cont, is moved from the reliability estimate to the efficiency estimate here as compared to [START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF].

An edge-based a posteriori estimator

This section introduces a first a posteriori error estimator that is attached to the edges of the mesh. It is simply defined by mimicking the definition of σ in (3.10) at the discrete level. As a result, for all edges ∈ E h , we fix a polynomial degree q ≥ p + 1, and we set (4.1) σ ,h := arg min

v h ∈RT q (T h )∩H0(div,ω ) ∇•v h =πq (ψ •J-∇×ψ •∇×A h ) v h + ψ × ∇ × A h ω .
as well as

(4.2) η := σ ,h + ψ × ∇ × A h ω .
Notice that (4.1) indeed provides a sound definition for σ ,h since the compatibility condition in (3.12) holds whenever required.

Recalling the results of Section 3, it is clear that η will make a good estimator if the discrete minimizer σ ,h is sufficiently close to the continuous minimizer σ for each ∈ E h . This is classical in the analysis of equilibrated estimator [START_REF] Braess | Equilibrated residual error estimates are p-robust[END_REF][START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF][START_REF]Stable broken H 1 and H(div) polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions[END_REF] and the corresponding result is often called "stable discrete minimization". The result we actually need is stable discrete minimization in the H(div) Sobolev space over an edge patch T h . For the sake of shortness, we skip the proof of this result, as it can be easily obtained by combining the proof of stable discrete minimization in H(div) over a vertex patch [START_REF] Braess | Equilibrated residual error estimates are p-robust[END_REF][START_REF]Stable broken H 1 and H(div) polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions[END_REF], and the proof of stable discrete minimization in H(curl) over an edge patch [START_REF] Chaumont-Frelet | Vohralík, Polynomial-degree-robust H(curl)-stability of discrete minimization in a tetrahedron[END_REF][START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF]. Proposition 4.1 (Stable discrete minimization). Consider an edge ∈ E h and a polynomial degree q ≥ 0. Let r ∈ P q (T h ) and g ∈ RT q (T h ), and if ⊂ Γ D , assume that (r, 1) ω = 1. Then, we have

(4.3) min v h ∈RT q (T h )∩H0(div,ω ) ∇•v h =r v h + g ω ≤ C st, min v∈H0(div,ω ) ∇•v=r v + g ω
where the constant C st, only depends on κ .

Remark that since the minimization set in the left-hand side of (4.3) is contained in the minimization of the right-hand side, the reverse inequality (4.4) min

v∈H0(div,ω ) ∇•v=r v + g ω ≤ min v h ∈RT q (T h )∩H0(div,ω ) ∇•v h =r v h + g ω trivially holds true.
Let us further comment that since both sides of (4.3) vanish if and only if g ∈ RT q (T h ) ∩ H(div, ω ) with ∇ • g = r, by linearity, it is clear that the inequality should hold for some constant C st, . It is also clear, based on standard scaling arguments and Piola mappings, that C st, is independent of the mesh size. The non-trivial part is two show that C st, does not depend on q.

Before establishing our key reliability and efficiency results, we need to cope with the fact that the divergence constraints in the continuous and discrete minimiaztion problems (3.10) and (4.1) defining σ and σ ,h are different when J is not a polynomial. Classically, this is done by introducing an oscillation term. Notice that since q ≥ p + 1, the estimate in (4.7) shows that osc converges to zero faster than the error, justifying the "oscillation" terminology (recall from (2.8) that ψ L ∞ (ω ) 1). Lemma 4.2 (Data oscillation). For all ∈ E h , we have

(4.5) R , - min v∈H0(div,ω ) ∇•v=πq (ψ •J-∇×ψ •∇×A h ) v + ψ × ∇ × A h ω ≤ osc with (4.6) osc := C P, h ψ • J -π T h ,q (ψ • J ) ω .
In addition, if J ∈ H q +1 (T h ), we have

(4.7) osc ≤ C P, ψ L ∞ (ω ) π q h q +1 |J | H q +1 (T h ) .
Proof. Let ∈ E h . Following the proof of Theorem 3.3, we know that R , = ∇r ω where r is the unique element of H 1 (ω ) such that

(∇r , ∇w) ω = (ψ • J -∇ × ψ • ∇ × A h , w) ω -(ψ × ∇ × A h , ∇w) ω
for all w ∈ H 1 (ω ). Similarly, we have min

v∈H0(div,ω ) ∇•v=πq (ψ •J-∇×ψ •∇×A h ) v + ψ × ∇ × A h ω = ∇r ω where r is the unique element of H 1 (ω ) such that (∇r , ∇w) ω = (π q (ψ • J -∇ × ψ • ∇ × A h ), w) ω -(ψ × ∇ × A h , ∇w) ω for all w ∈ H 1 (ω ). Since ∇ × ψ • ∇ × A h ∈ P q (T h ), it follows that (∇(r -r ), ∇w) ω = (ψ • J -π T h ,q (ψ • J ), w) ω ≤ C P, h ψ • J -π q (ψ • J ) ω ∇w ω .
for all w ∈ H 1 (ω ). Picking w = r -r , we have

∇(r -r ) ω ≤ C P, h ψ • J -π q (ψ • J ) ω ,
and (4.5) follows from the reverse triangle inequality.

Finally, since ψ ∈ P 1 (T h ), (4.7) follows from

ψ • J -π T h ,q (ψ • J ) ω ≤ ψ • J -ψ • (π T h ,q -1 (J )) ω ≤ ψ L ∞ (ω ) J -π T h ,q -1 (J ) ω ≤ ψ L ∞ (ω ) h π q |J | H q (ω ) ,
where we have used (2.4) together with the fact that h K ≤ h for all K ∈ T h .

We are now ready to state the main result of this section, which is a direct consequence of the three theorems of Section 3, identity (3.5), as well as estimates (4.3), (4.4) and (4.5).

Theorem 4.3 (Edge-based error estimator). The estimates

(4.8) ∇ × (A -A h ) Ω ≤ √ 6C L,Ω ∈E h (η 2 + osc 2 ) 1/2 and (4.9) η ≤ C st, C cont, ∇ × (A -A h ) ω + C cont, osc ∀ ∈ E h hold true.
Remark 4.4 (An interpretation of what η measures). Assume that J ∈ RT p (T h ) and that η = 0 for an edge ∈ E h . Then, we have

ψ •J -∇×ψ •∇×A h = ∇•(ψ ×∇×A h ) = -∇×ψ •∇×A h +ψ •∇×∇×A h , meaning that ψ • (∇ × ∇ × A h -J ) = 0.
We then see that A h locally satisfies the curl-curl problem strongly in ω , at least in the τ direction.

Remark 4.5 (Alternate construction by sequential sweeps). We can decrease the computational cost for constructing σ ,h by following the approach presented in [10, Theorem 3.2]. In this case, instead of solving the patchwise minimization in (4.1), we construct an alternative local contribution σ ,h by sweeping through the edge patch. In this approach, for each K ∈ T h , σ ,h | K is defined through an elementwise minimization problem in K, similar to (4.1).

Remark 4.6 (p-adaptive reconstruction).

Let ∈ E h . In our presentation, we employed the assumption that A h ∈ N p (T h ). It is possible to further take advantage of the fact that A h ∈ N p K (K) for all K ∈ T h for some 0 ≤ p K ≤ p . Indeed, instead of seeking σ ,h in the space RT q (T h ) (recall that q ≥ p + 1), we can instead require that σ ,h ∈ RT p K +1 (K) for all K ∈ T h . In doing so, it is still true that (4.3) holds true, but with a constant C st, that may depend on the distribution {p K } K∈T h . Indeed, it is not clear wether constant C st, is polynomial-degree-robust in this case as, in particular, the proof techniques employed in [START_REF] Braess | Equilibrated residual error estimates are p-robust[END_REF][START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF][START_REF]Stable broken H 1 and H(div) polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions[END_REF] fail in this case.

Prager-Synge type estimates

In this section, we elaborate a second error estimator. It hinges on a new Prager-Synge type identity and a corresponding discrete equilibration procedure. We actually show that the local contributions σ ,h previously introduced to build the edge-based estimator can be recombined to provide equilibrated fields.

We begin with our new Prager-Synge type identity.

Theorem 5.1 (Prager-Synge identity). Assume that, for 1 ≤ k ≤ 3, S k ∈ H ΓN (div, Ω), then we have

(5.1) (∇ × (A -A h ), ∇ × v) Ω = 3 k=1 (J k -∇ • S k , v k ) Ω -(S k + u k × ∇ × A h , ∇v k ) Ω for all v ∈ H 1 ΓD (Ω). In addition, if (5.2) (∇ • S k -J k , r h ) Ω = 0 ∀r h ∈ P 0 (T h ),
then,

(5.3) ∇ × (A -A h ) Ω ≤ C L,Ω K∈T h 3 k=1 h K π J k -∇ • S k K + S k + u k × ∇ × A h K 2 1/2 . Proof. Let v ∈ H 1 ΓD (Ω). We have (5.4) (∇ × (A -A h ), ∇ × v) Ω = (J , v) Ω -(∇ × A h , ∇ × v) Ω .
On the one hand, we have

(5.5) (J , v) Ω = 3 k=1 (J k , v k ) Ω = 3 k=1 {(J k -∇ • S k , v k ) Ω -(S k , ∇v k ) Ω }
On the other hand, it holds that

∇ × v = d k=1 ∇ × (v k u k ) = d k=1 ∇v k × u k , leading to (∇ × A h , ∇ × v) Ω = d k=1 (∇ × A h , ∇v k × u k ) Ω (5.6) = d k=1 (u k × ∇ × A h , ∇v k ) Ω .
Identity (5.1) then easily follows from (5.4), (5.5) and (5.6). Finally, estimate (5.3) is a direct consequence of (5.1), orthogonality property (5.2) together with elemenwise Poincaré inequality (2.11) and the discussion of Section 2.7, since

(J k -∇ • S k , v k ) K = (J k -∇ • S k , v k -π K,0 v k ) K ≤ h K π J -∇ • S k K ∇v k K .
Remark 5.2 (The constant C L,Ω ). The constant C L,Ω is the price we pay for working with a test function in H 1 ΓD (Ω) instead of H ΓD (curl, Ω). Specifically, starting from a general function v ∈ H ΓD (curl, Ω), we introduce a function w ∈ H 1 ΓD (Ω) such that ∇×v = ∇×w. When doing this operation, there is no reason to think that ∇w Ω ≤ ∇ × v Ω in the general case, hence the need for the constant C L,Ω . However, for our purposes, we do not need to impose that ∇ × v = ∇ × w. Indeed, we simply need that

(∇ × (A -A h ), ∇ × v) Ω = (∇ × (A -A h ), ∇ × w),
which is a much less demanding condition. Although the author is not currently aware of a way to take advantage of this idea, it hints toward the fact that a sharper estimate without the constant C L,Ω may be achieved. We further point out that this observation is not limited to the particular estimators considered in this work, but to general a posteriori estimators using regular decompositions [START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF][START_REF] Nicaise | A posteriori error estimation for heterogeneous Maxwell equations on isotropic and anisotropic meshes[END_REF][START_REF] Schöberl | A posteriori error estimates for Maxwell equations[END_REF]. A similar remark also holds true for the upper bounds of Theorems 3.1 and 4.3.

Remark 5.3 (Improved oscillation term).

Under the additional assumption that (∇ • S k -J k , r h ) Ω = 0 for all r h ∈ P q (T h ) for some q ≥ 0, it is possible to improve the factor h K /π in the oscillation term to Ch K /q (see, e.g., [START_REF] Babuška | The h-p version of the finite element method with quasiuniform meshes[END_REF]Lemma 4.1]). For the sake of simplicity though, we only focus on the simpler version, in particular because the constant appearing in the improved version is not easily computable in practice.

The next step is to provide a discrete construction of fields S k

h satisfying the requirements of of Theorem 5.1. As previously advertised, this is easily achieved by recombining the local contributions σ ,h introduced at (4.1). We thus set, for 1 ≤ k ≤ 3, the field (5.7)

S k h := ∈E h (τ • u k )σ ,h . Lemma 5.4 (Discrete equilibration). For 1 ≤ k ≤ 3, we have S k h ∈ H ΓN (div, Ω) with (5.8) (∇ • S k h -J k , r h ) Ω = 0 ∀r h ∈ P 0 (T h ). Proof. Let k ∈ {1, 2, 3}. That S k h ∈ H ΓN (div, Ω
) is a direct consequence of the fact that σ ,h ∈ H 0 (div, ω ) for all ∈ E h . Then, consider an element K ∈ T h and r h ∈ P 0 (K). Recalling (4.1), we see that

(∇ • S K h , r h ) K = ∈E h (τ • u k )(∇ • σ ,h , r h ) K = ∈E h (τ • u k ) (π q (ψ • J -∇ × ψ • ∇ × A h ) , r h ) K = ∈E h (τ • u k ) (ψ • J -∇ × ψ • ∇ × A h , r h ) K .
By linearity, and recalling (2.9), we have

(∇ • S k h , r h ) K = (u k • J -∇ × u k • ∇ × A h , r h ) K = (J k , r h ) K , leading to (5.8).
Having introduced the equilibrated fields S k h , we simply define our estimator with the elementwise contributions in the right-hand side of (5.3). We thus set (5.9)

η k K := u k × ∇ × A h + S k h K
for each element K ∈ T h and k ∈ {1, 2, 3}. As we establish below, this estimator is reliable and efficient.

Theorem 5.5 (Equilibrated estimator). The following upper bound

(5.10) ∇ × (A -A h ) Ω ≤ C L,Ω K∈T h 3 k=1 η k K + osc k K 2 1/2
holds true with

(5.11) osc k K := h K π ∇ • S k h -J k K ∀K ∈ T h , 1 ≤ k ≤ 3.
In addition, we have the lower bounds (5.12)

3 k=1 (η k K ) 2 ≤ 6 ∈E K (C st, C cont, ∇ × (A -A h ) ω + C cont, osc ) 2 for all K ∈ T h .
Proof. On the one hand, (5.10) is a direct consequence of Theorem 5.1 and Lemma 5.4. On the other hand, we have

S k h + u k × ∇ × A h K = ∈E K (τ • u k )σ ,h + ∈E K (u k • τ )ψ × ∇ × A h K ≤ ∈E K |u k • τ | • σ ,h + ψ × ∇ × A h K ≤ ∈E K |u k • τ | • σ ,h + ψ × ∇ × A h ω = ∈E K |u k • τ |η ,
and (5.12) follows from (4.9) using Cauchy-Schwarz inequality

3 k=1 (η k K ) 2 = 3 k=1 S k h + u k × ∇ × A h 2 K ≤ 3 k=1 ∈E K |u k • τ |η 2 ≤ 6 ∈E K 3 k=1 |u k • τ | 2 η 2
and observing that

3 k=1 |u k • τ | 2 = |τ | 2 = 1.

Numerical examples

6.1. Settings. This section presents a set numerical examples. We first describe the general setting.

6.1.1. Estimators. For the sake of shortness, we disregard all the oscillations terms. That way, we can simply set

η 2 K := 3 k=1 (η k K ) 2 ,
for all K ∈ T h and

η 2 edge := 6 ∈E h η 2 , η 2 cell := K∈T h η 2 K .
6.1.2. Discrete solution. The discrete solution A h is computed using Nédélec elements of uniform degree p ≥ 0 through the usual Galerkin formulation. For the sake of simplicity, we only consider cases where Ω is simply connected and Γ D = ∂Ω, so that Λ ΓD (Ω) = ∇ H 1 ΓD (Ω) . This leads to the definition of A h has the unique element of

N p (T h ) ∩ H ΓD (curl, Ω) such that (∇ × A h , ∇ × v h ) Ω = (J , v h ) Ω (A h , ∇q h ) Ω = 0 for all v h ∈ N p (T h ) ∩ H ΓD (curl, Ω) and q h ∈ P p+1 (T h ) ∩ H 1 ΓD (Ω).
Remark in particular that A h satisfies the assumptions of Section 2.6 with p = p for all ∈ E h . The estimator is then computed with the lowest polynomial degree possible, namely q = q := p + 1 for all ∈ E h . 6.1.3. Error evaluation. In the first two examples, the analytic solution A to the problem is available, so that we can readily compute the true error err Ω := ∇ × (A -A h ) Ω for comparison purposes. In the last experiment however, we do not have access to A, and if A h ∈ N p (T h ) ∩ H ΓD (Ω) we assess the error using the quantity err

Ω := ∇ × ( A h -A h ) Ω , where A h ∈ N p+2 (T h ) ∩ H ΓD (Ω)
is computed using the same mesh than A h , but with a higher polynomial degree. 6.1.4. Mesh generation. The meshes we employ are generated with with gmsh [START_REF] Geuzaine | Gmsh: A 3-D finite element mesh generator with built-in preand post-processing facilities[END_REF] and mmg3D [START_REF] Dobrzynski | D: User guide[END_REF]. Starting from a file "geom.geo" describing the geometry of the experiment, we generate a first tetrahedral mesh using the command "gmsh -3 geom.geo -format mesh". This generates a mesh stored in "geom.mesh" that is then passed to mmg3D for further refinements. Specifically, when we talk about a "mesh of size h max " when it has been generated using the command "mmg3D -in geom.mesh -out mesh.mesh -hmax h max ", which generates two files: "mesh.mesh" describing the mesh, and "mesh.sol" describing the mesh size around each vertex.

We also employ mmg3D for iterative mesh refinements. In this case, based on finite element computation with the mesh described by the files "mesh.mesh" and "mesh.sol" we create a new mesh the following way:

(1) Find an ordering ι : {1, . . . , E h } → E h such that j → η ι(j) is decreasing.

(2) Select the smallest integer m such that

m j=1 η 2 ι(j) ≤ θ ∈E h η 2 , θ := 0.1.
(3) Mark all the vertices associated with the edges ι(j), j ∈ {1, . . . , m}.

(4) Generate a new file "refinement.sol" where the mesh size associated with all the marked vertices is divided by two. (5) Generate the refined mesh with the command "mmg3D -in mesh.mesh -sol refinement.sol -out refined mesh.mesh -hgrad 10.". We also employ the equilibrated estimator {η K } K∈T h instead of {η } ∈E h . In this case, we follow the same procedure with edges replaced by cells. 6.1.5. Quantities of interest and legends. Throughout this section, we will focus on 5 quantities of interest, namely: the error err Ω , the summed estimators η edge and η cell , and the effectivity indices eff edge := η edge / err Ω and eff cell := η cell / err Ω . These quantities will be represented on two kinds of figures. On the one hand, we will display the error, and summed estimator on "Error and estimators" plot with the following legend: We first consider a "h-convergence" example where we fix the polynomial degree p, and consider a sequence of uniform meshes generated by mmg3D with h max := 1, 1/2, 1/4, 1/16 and 1/32. Figure 6.1 presents the results. As can be seen from the left-panel, the expected convergence rate in O(h p+1 ) is achieved. The right-panel illustrates that both estimators are reliable and efficient, since the efficiency indices are grater that one and independent of h.

Next, we perform a "p-convergence" study where h max is fixed, but p increases from 0 to 6. The expected exponential convergence rate is observed on the leftpanel of Figure 6.2. The right-panel of Figure 6.2 shows that both estimators are reliable and efficient. This experiment further highlights the p-robustness of the estimator, since the effectivity indices are indeed independent of p.

We further note that the effectivity index of the equilibrated estimator is fairly close to one. Notice that the fact that the effectivity is greater than one is ensured by Theorem 5.5, since the domain is convex in this example, and we can take C L,Ω = 1 in (5.10). For the edge-based estimator, the effectivity index is close to, and always greater than, √ 6. This may hint that the factor √ 6 in (4.8) is spurious, although the author is not aware of a way to suppress it from a theoretical stand point.

6.3.

Edge singularity in L-type domains. Given an angle φ ∈ (0, 2π), we consider a domain of the form Ω := L × (0, 1), where Following [START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF][START_REF] Chaumont-Frelet | p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. application to a posteriori analysis of the curl-curl problem[END_REF][START_REF] Gedicke | An equilibrated a posteriori error estimator for arbitrary-order Nédédec elements for magnetostatic problems[END_REF], we consider a solution that is singular around the edge {(0, 0)} × (0, 1). Specifically, it reads A := (0, 0, s), where (6.1) s(r, θ, x 3 ) := χ(r)r α sin(αθ) with α := π/(2π -φ) and χ is a cutoff function employed to satisfy boundary conditions as in [START_REF]Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem[END_REF][START_REF] Chaumont-Frelet | p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. application to a posteriori analysis of the curl-curl problem[END_REF]. For each angle φ, we perform an adaptive mesh refinement as described above. In all cases, we start with a mesh generated with h max := 2 and perform three different refinements for polynomial degrees p = 0, 1 and 2.

L := {x = r(cos(θ), sin(θ)) ∈ R 2 ; |x 1 |, |x 2 | < 1, 0 < θ < 2π -φ}.
When φ = 3π/4, the adaptive process is driven by the cell estimator {η K } K∈T h and the results are reported on Figure 6.4. As can be seen on the left-panel, the optimal convergence rates are observed. Notice that for p = 2, the convergence rate is not N -(p+1)/3 dofs = N dofs because we are using isotropic elements, see [3, Section 4.2.3] for more details. Similar to the cube experiment, the effectivity indices of the edge-based and equilibrated estimators are respectively close to √ 6 and 1. We use the edge estimator {η } ∈E h to drive the refinements when φ = π/2. The results are similar to the previous case, and are reported on Figure 6.5. Figure 6.6 illustrates the last case where φ = π/8, and the adaptive process is driven by the cell estimator. The results are again similar, except that the use of isotropic elements also reduces the optimal convergence rate when p = 1, as expected [START_REF] Apel | Anisotropic finite elements: local estimates and applications[END_REF].

Interestingly, we observe that the effectivity indices eff edge and eff cell respectively stays above √ 6 and 1 (apart from the first iterations when φ = π/2, which is due to data oscillation). Notice that our theory does not cover this estimate, as it requires the constant C L,Ω , which is not 1 in this case. The fact that the effictivity indices remains independent of the angle φ hints that the C L,Ω may not be compulsory to obtain guaranteed estimates. 6.4. Corner singularity in the Fichera cube. In this last experiment, we consider the Fichera domain Ω := (-1, 1) 3 \ (0, 1) 3 . We select the right-hand side J := (1, 1, 0). Starting from an initial mesh generated with h max := 2, we consider to adaptive refinement processes. We first set p = 0 and employ the edge-based estimator to drive the adaptive process and then, we use the cell-based estimator with p = 1. Figure 6.7 presents the results. As for the L-type domain example, we obtain the optimal convergence rates, and the estimators seem to provide guaranteed upper bounds even when C L,Ω is omitted.

Conclusion

We propose two a posteriori error estimators that are motivated by a novel Prager-Synge identity for the curl-curl problem. Both estimators are polynomialdegree-robust, and rely on divergence-constrained minimization problems over edge patches. When the domain is convex, these estimators also provide guaranteed and fully computable upper bounds. In the general case however, the reliability estimate involves a constant C L,Ω related to regular decompositions of H(curl) fields that is not easily computable in practice.

We present a set of numerical examples which illustrates the key theoretical findings: both estimators are reliable and efficient with p-robust constants, and provide guaranteed upper bounds in convex domains. In addition, when the domain is not convex, these numerical tests suggest that the estimators can still be employed without the constant C L,Ω to provide a guaranteed upper bound. While we are not able to prove this result, we provide some theoretical reasons why it may be the case. We also employ both estimators to drive adaptive mesh refinements, and obtain optimal convergence rates in domains featuring re-entrant edges and corners.

In practice, it seems that the cell-based equilibrated estimator should be preferred, as it provides improved results as compared to the edge-based estimator, only at the price of a moderate additional complexity in the implementation. From a theoretical viewpoint however, the edge-based estimator may be preferable since the associated efficiency estimates involve smaller mesh patches, which may be of interest to design adaptive refinement algorithms that provably converge with optimal rates.

edge eff cell 6 . 2 .

 62 err Ω η edge η cellOn the other hand, "Effectivity indices" figures have the following legend:eff Smooth solution in a cube. We start by considering the unit cube Ω := (0, 1) 3 together with the load term J := (3π) 2   cos(πx 1 ) sin(πx 2 ) sin(πx 3 ) -sin(πx 1 ) cos(πx 2 ) sin(πx 3 ) 0One readily sees that ∇ • J = 0 and that the associated solution isA :=   cos(πx 1) sin(πx 2 ) sin(πx 3 ) -sin(πx 1 ) cos(πx 2 ) sin(πx 3 ) 0   .

Figure 6 . 1 .

 61 Figure 6.1. Smooth solution example: h-convergence

Figure 6 . 2 .

 62 Figure 6.2. Smooth solution example: p-convergence

( 8 Figure 6 . 3 .

 863 Figure 6.3. L-type domains

10 2 10 3 10

 10

Figure 6 . 4 .

 64 Figure 6.4. L-type experiment with φ = 3π/4.

10 2 10 3 10

 10

Figure 6 . 5 .

 65 Figure 6.5. L-type experiment with φ = π/2.

10 2 10 3 10 Figure 6 . 6 .

 1066 Figure 6.6. L-type experiment with φ = π/8.

Figure 6 . 7 .

 67 Figure 6.7. Fichera domain example

2.2. Functional spaces. If ω ⊂ Ω is an open set, L 2 (ω) and L 2 (ω) are the spaces of scalar and vector-valued square integrable functions defined on ω. The usual inner products and norms of these spaces are denoted by (•, •) ω and • ω . H 1 (ω), H(div, ω) and H(curl, ω) are the usual Sobolev spaces respectively containing square-integrable functions with square-integrable gradient, divergence, and curl. We also employ the notationH 1 (ω) := H 1 (ω) 3 . If Γ ⊂ ∂ω is a relatively open subset, H 1 Γ (ω)is the set of functions of H 1 (ω) with vanishing trace on Γ, and we also noteH 1 Γ (ω) := H 1 Γ (ω) 3 . Similarly, we set H Γ (div, ω) := v ∈ H(div, ω); (v, ∇q) + (∇ • v, q) = 0 ∀q ∈ H 1 Γ c (ω) and


Devadas Bhat Panemangalore

Dr Rajashekhara Shabadi
email: rajashekhara.shabadi@univ-lille1.fr

David Tingaud

Matthieu Touzin

Biocompatible silica-based magnesium composites

Keywords: magnesium, biocompatibility, silica, spark plasma sintering

In this study, an in-situ formed Mg2Si-Mg composite is investigated to understand the formation of Mg2Si and its influence in mechanical properties and corrosion behavior. A starting blend of 2.5 wt.% SiO2 nanoparticles and magnesium powders was used to elaborate in-situ composite through mechanical blending followed by spark plasma sintering technique.

High temperature X-ray diffraction measurements and differential scanning calorimetry results confirmed the formation of Mg2Si phase. The addition of 2.5 wt.% SiO2 enhanced hardness while the stiffness values remained the same. The results of potentiodynamic polarization tests revealed an improved corrosion resistance of Mg reinforced with SiO2 nanoparticles. Thus, this work is an attempt to understand the fabrication (dissociation of silica into Mg2Si in presence of Mg) and corrosion properties of silica reinforced magnesium composites with those of pure magnesium targeting biomedical applications.

Introduction

High strength and corrosion resistant materials like titanium, Ti-6Al-4V and SS316L are being largely used for fixation of bone and as implant materials in orthopedic applications.

However, major shortcomings of these materials are their high elastic moduli which lead to stress-shielding effect [START_REF] Nagels | Stress shielding and bone resorption in shoulder arthroplasty[END_REF]. Moreover they are non-degradable [START_REF] Gupta | Insight into Designing Biocompatible Magnesium Alloys and Composites[END_REF] and, hence, require postoperative measures, which come at an additional expense and pain that causes a lot of inconvenience to the patients. Therefore, it is important to develop next generation implant materials that resemble the mechanical properties of bones that counter the 'stress shielding effect' and also are bio-degradable.

Over the past few years, research on biodegradable metals like magnesium, its alloys and composites has gained lot of momentum [START_REF] Farraro | Revolutionizing orthopaedic biomaterials: The potential of biodegradable and bioresorbable magnesium-based materials for functional tissue engineering[END_REF] as magnesium has its elastic modulus close to that of cortical bone [START_REF] Gu | A review on magnesium alloys as biodegradable materials[END_REF] and is an essential mineral for a human metabolism. More importantly, it is biodegradable, i.e., gets absorbed after fixing the bone and its regeneration [START_REF] Zhang | In vivo evaluation of biodegradable magnesium alloy bone implant in the first 6 months implantation[END_REF]. However, a major concern is its poor corrosion behavior under conducive environments [START_REF] Song | Recent progress in corrosion and protection of magnesium alloys[END_REF]. To improve their bio-corrosion behavior and to make it bio-compatible with controlled degradability, two metallurgical approaches can be used: 1. alloying and 2. synthesized composites. In the literature, alloying magnesium involve binary [START_REF] Zhang | Research on an Mg-Zn alloy as a degradable biomaterial[END_REF], ternary [START_REF] Gu | In vitro degradation performance and biological response of a Mg-Zn-Zr alloy[END_REF] or quaternary additions [START_REF] Liao | In vitro response of chondrocytes to a biodegradable Mg-Nd-Zn-Zr alloy[END_REF] to pure magnesium in order to reduce hydrogen gas formation and maintain the surface and chemical integrity of implant materials. However, the major shortcomings of these alloys are maintaining the purity, homogeneity and composition to prevent excessive degradation. Discussion on the alloys is not considered here as it falls out of the scope of the present work. We limit further discussions only to composites and in particularly, silica-based Mg matrix composites.

Magnesium based composites with reinforcements including several oxides, alumina [START_REF] Kang | MgF2-coated porous magnesium/alumina scaffolds with improved strength, corrosion resistance, and biological performance for biomedical applications[END_REF], zirconia [START_REF] Amiri | Electrophoretic deposition of nano-zirconia coating on AZ91D magnesium alloy for bio-corrosion control purposes[END_REF], titania [START_REF] Amaravathy | Bioactive HA/TiO2 coating on magnesium alloy for biomedical applications[END_REF], silica [START_REF] Bakhsheshi-Rad | Structure, corrosion behavior, and antibacterial properties of nanosilica/graphene oxide coating on biodegradable magnesium alloy for biomedical applications[END_REF] have been studied, either as a reinforcing material or in the form of coatings. Alumina particle reinforcements on magnesium scaffolds were studied by Kang et al [START_REF] Kang | MgF2-coated porous magnesium/alumina scaffolds with improved strength, corrosion resistance, and biological performance for biomedical applications[END_REF] for biomedical applications. A 5 wt.% addition led to a slight increase in corrosion as compared to pure magnesium in SBF solution and hence a coating of MgF2 was introduced that reduced the corrosion rate drastically. Nano-zirconia coating using electrophoretic deposition on AZ91D magnesium alloy for biomedical applications were done by Amiri et al. [START_REF] Amiri | Electrophoretic deposition of nano-zirconia coating on AZ91D magnesium alloy for bio-corrosion control purposes[END_REF]. Although AZ91D shows better corrosion resistance, it is not widely accepted as a biocompatible material as a 9 wt.% aluminum in this alloy could induce risk for Alzheimer's disease [START_REF] Ferreira | Aluminum as a risk factor for Alzheimer's disease[END_REF]. Coatings of titania on magnesium have been studied in detail [START_REF] Amaravathy | Bioactive HA/TiO2 coating on magnesium alloy for biomedical applications[END_REF] but a very limited research is available on its usage as a reinforcement material for pure Mg for biomedical applications. Nano silica was selected in this study as a reinforcement because of its high biocompatibility [START_REF] Beck | Bioactive silica-based nanoparticles stimulate bone-forming osteoblasts, suppress bone-resorbing osteoclasts, and enhance bone mineral density in vivo[END_REF] and in moderate dosages, it shows no signs of toxicity in the human body fluids [START_REF] Yildirimer | Toxicology and clinical potential of nanoparticles[END_REF]. It is bioactive and biodegradable material that aids in bone repair and regeneration [START_REF] Wang | Bioactive and biodegradable silica biomaterial for bone regeneration[END_REF]. Hence, in the present study we are concentrating on the influence of silica on the dissociation capacity into Mg2Si in the presence of Mg to become Mg2Si-Mg composites, during in-situ synthesis, and also because there is very little research available on the nano-silica Mg composites. Mesoporous silica nanoparticles which are hollow and biodegradable are known to enhance antitumor efficiency [START_REF] Kong | Biodegradable hollow mesoporous silica nanoparticles for regulating tumor microenvironment and enhancing antitumor efficiency[END_REF]. Using tetraethyl orthosilicate, Kursawe et al. [START_REF] Kursawe | Biodegradable Silica Fibers from Sols[END_REF] synthesized biodegradable silica fibers for reinforcement of medical implants using sol-gel processing. Bio-silica, which is the main mineral component of sponge silicatein aids in treatment of human bone diseases and dysfunctions [START_REF] Wang | Bio-silica and biopolyphosphate: Applications in biomedicine (bone formation)[END_REF]. Also, bio-silica is known to stimulate the formation of hydroxyapatite by mineralizing cells [START_REF] Natalio | Bioengineering of the silica-polymerizing enzyme silicatein-α for a targeted application to hydroxyapatite[END_REF]. Silica nanoparticles are also known to be multifunctional as they possess the potential of imaging and gene delivery [START_REF] Bhakta | Multifunctional silica nanoparticles with potentials of imaging and gene delivery[END_REF]. There have been several ongoing researches on reinforcing silica with magnesium for the enhancement of mechanical properties. Parande et al. used low volume fraction of 10-20 nm silica nanoparticulates to enhance the hardness, compression and damping response of magnesium [START_REF] Parande | Enhancing the hardness / compression / damping response of magnesium by reinforcing with biocompatible silica nanoparticulates[END_REF].

Lu et al. studied the formation of Mg2Si via mechanical alloying of elemental Mg and Si powders [START_REF] Lu | Formation of nanocrystalline Mg2Si and Mg2Si dispersion strengthened Mg-Al alloy by mechanical alloying[END_REF]. Wang et al. synthesized nanocrystalline Mg2Si using ball milling of elemental powders and found enhanced fracture toughness compared to pure magnesium [START_REF] Wang | Fabrication and mechanical properties of bulk nanocrystalline intermetallic Mg2Si[END_REF]. Kondoh et al. also adopted powder metallurgy route in solid-state synthesis of Mg2Si intermetallic compounds but using a technique called repeated plastic working with compaction and extrusion using pure elemental powders [START_REF] Kondoh | In-situ Synthesis of Mg2Si Intermetallics via Powder Metallurgy Process[END_REF]. Sun et al. studied the synthesis kinetics during the formation of Mg-Mg2Si composite and concluded that at temperatures below 580 o C, the elemental powders did not react completely to form Mg2Si [START_REF] Sun | Kinetic Analysis of Solid-State Formation of Mg2Si by Powder Metallurgy Process[END_REF]. In-house scraps of the wrought optical silica glass fiber were milled with AZ31 by Kondoh et al. to fabricate composites by consolidation, heating and hot-extrusion [START_REF] Kondoh | New process to fabricate magnesium composites using SiO2 glass scraps[END_REF]. However, they couldn't be used for bio-medical applications as they did contain lot of impurities as they were manufactured from scrap materials. Using hot-pressing technique, Myalska et al. synthesized in-situ Mg metal matrix composites based on hydrophilic fumed silica nanoparticles of 5 nm size and they comprised of Mg2Si and MgO dispersoids [START_REF] Olszówka-Myalska | Microstructure of in-situ Mg Metal Matrix Composites Based on Silica Nanoparticles[END_REF]. Gupta et al. [START_REF] Gupta | Electrochemical Behavior and Localized Corrosion Associated with Mg2Si Particles in Al and Mg Alloys[END_REF] studied the electrochemical behavior of Mg2Si particles in AS31 magnesium alloy and AA6360-T6 and AA7075-T651 aluminum alloys in both acidic and alkaline medium. The intermetallic Mg2Si was found to be active at low pH and passive at high pH conditions. Also the Mg2Si in Al matrix was found to serve as a local anode, whereas in Mg matrix, the Mg2Si remained intact while the surrounding matrix gets corroded. Ben-Hamu et al. found improved corrosion behavior of wrought Mg-Zn-Mn alloy with the presence of polygonal shape Mg2Si intermetallics [START_REF] Ben-Hamu | The role of Mg2Si on the corrosion behavior of wrought Mg-Zn-Mn alloy[END_REF][START_REF] Ben-Hamu | The role of Si and Ca on new wrought Mg-Zn-Mn based alloy[END_REF]. Also, as reported by Kondoh et al. the corrosion resistance of Mg2Si is better than that of stainless steels [START_REF] Kondoh | New process to fabricate magnesium composites using SiO2 glass scraps[END_REF]. This conclusion was arrived based on salt spray tests according to JIS Z 2731 standard conducted on intermetallic bulk Mg2Si and conventional 304 stainless steel. At room temperature, corrosion phenomenon was not observed after 1000h of 5% salt spray for Mg2Si whereas the corrosion initiated after 300-600h for stainless steel. The corrosion potential for Mg2Si in aluminum alloys is more positive than that of pure Mg as studied by Birbilis et.al [START_REF] Birbilis | Electrochemical Characteristics of Intermetallic Phases in Aluminum Alloys[END_REF]. This intermetallic Mg2Si has a high melting point and hardness [START_REF] Suveen | Magnesium technology[END_REF][START_REF] Niu | Formation of magnesium silicide by mechanical alloying[END_REF][START_REF] Michel | Effect of temperature on hardness and indentation cracking of fused silica[END_REF].

In the present work, it is expected that Mg dissociates SiO2 into Mg2Si and MgO at the sintering temperatures using Spark Plasma Sintering technique (SPS) there by forming an insitu composite of Mg2Si +Mg as per the following reaction [START_REF] Olszówka-Myalska | Microstructure of in-situ Mg Metal Matrix Composites Based on Silica Nanoparticles[END_REF]:

With the fine blend of nano-silica into the Mg matrix, it is expected to have uniformly distributed Mg2Si in the Mg matrix. In this article, we summarize our observations on the synthesis, mechanical properties and general corrosion behavior targeting biomedical applications.

Materials and Methods

Materials

Pure magnesium (Mg) powder of 99.8% purity with an average particles size of 44 µm, and silica (SiO2) nano powders of 99.9% purity with a size range of 10-20 nm were used as the base material and reinforcement respectively.

Synthesis

Pure Mg and 2.5 wt.% SiO2 reinforced magnesium nanocomposites were synthesized using mechanical mixing/crushing and sintered using SPS. SPS experiments have been carried out using a Dr. Sinter 515S Syntex (Fuji Electronic Industrial Co., Japan) belonging to the "Plateforme CNRS de Frittage Ile de France" (ICMPE, Thiais, France). Pre-weighed amounts of pure Mg powder and SiO2 nano-powders were carefully blended for 1 h using a mortar and pestle. The blended powder mixtures were first pre-compacted under a pressure of 100 MPa, and then sintered using SPS at a temperature of 585 o C under a pressure of 60 MPa for 10 min in the form of cylindrical pellets of 10 mm in diameter and 3 mm thickness.

Density measurements

The density of two sintered samples was measured using a Helium gas pycnometer. 20

measurements have been taken on each sample to get a good statistical evaluation.

Optical and scanning electron micrographs

To conduct microstructural characterizations, specimens were polished up to 4000 grit paper and then surfaces were given a finishing with diamond paste up to 1/4 µm. Optical micrographs were taken on the samples using ZEISS Axio Observer inverted microscope after mechanical polishing and etching with acetic-picral. Macrographs of the corroded surfaces were recorded using a ZEISS Stemi 2000C stereo microscope. A scanning electron microscope (JEOL 7800F) equipped with an energy dispersive spectrometer (EDS) was used to study the microstructure, and the precipitates of Mg2Si in the processed samples.

Microhardness and elastic modulus

Micro-indentation experiments were carried out on flat and polished specimens at room temperature using a CSM 2-107 Tester using a pyramidal Vickers indenter. The indentation loads from 0.1 to 25 N and the loading rates between 0.5 N/min and 300 N/min were available on the instrument range. For this experiment, 0.5 N load and a loading rate of 1 N/min was applied. The indenter was maintained at the maximum load during 5 s. The tests were conducted for 10 repeatable readings to ensure reproducibility. Oliver-Pharr method was applied to the Load Vs Displacement curves to determine the microhardness and the Elastic modulus.

Corrosion behavior

Electrochemical measurements were done using Voltalab instrument. The standard threeelectrode setup was used: i.e. saturated calomel electrode as reference, platinum as counter and the test sample mounted in a resin as working electrodes. Potentiodynamic polarization tests using Phosphate Buffer Saline (PBS) solution at controlled temperature of 37 o C was used and the scan rate adopted was 1 mV/s ranging from -2.5 V to -0.5 V. The composition of PBS solution is given in [START_REF]Phosphate buffered saline tablet | Sigma-Aldrich[END_REF].

Minimum of three trials for each sample were conducted. The samples before corrosion were polished using SiC abrasive papers to 4000 grit and then washed with ethanol. SEM coupled with energy dispersive spectrometry was used to record the microstructure of corroded samples.

Results and discussion

SPS sintered samples were tested for their physical properties, mechanical properties and their corrosion behavior. In the following sections, we summarize different observations of those properties: glass powder mixture [START_REF] Kondoh | New process to fabricate magnesium composites using SiO2 glass scraps[END_REF]. The DSC curve in their studies showed an exothermic peak at 457 o C. In the current study an exothermic peak is observed in all the repeated measurements.

This peak is attributed to an in-situ reaction of magnesium with silica that has taken place to form Mg2Si but a peak-split could be seen. It is considered that they are not two peaks but one peak split into two as there is no available literature for the existence of two peaks that could correspond to two different reactions for Mg-SiO2 composite. There is only one reaction that takes place and it is the formation of Mg2Si. Since nano-silica is used in this case, the reaction takes place slightly earlier, at 410 o C and it commences upto 460 o C. The first peak is attributed to Mg2Si formation due to deformation of the particles done before DSC. The second peak is attributed to the solid state reaction of residual Mg with residual SiO2 to form Mg2Si at a slightly higher temperature. Mechanical milling induced sub-processes were studied before on the formation of nanocrystalline Mg2Si [START_REF] Wang | The effect of mechanical milling on the formation of nanocrystalline Mg2Si through solid-state reaction[END_REF]. As the peaks for Mg2Si in HT-XRD appear after 470 o C in Figure 2, it can be deduced that the reaction product in DSC after the experiment is Mg2Si. Slightly higher temperature 457 o C in the results obtained by Kondoh et.al [START_REF] Kondoh | New process to fabricate magnesium composites using SiO2 glass scraps[END_REF] could be due to the fact that, it was an alloy AZ31 instead of pure magnesium and also because SiO2 was a glass powder mixture in their study. DSC analysis done by Myalska et.al on a powder mixture of granulated Mg and nanosized silica showed an exothermic peak at 515 o C. An increase in the temperature could be because the initial powders were not mixed properly [START_REF] Olszówka-Myalska | Microstructure of in-situ Mg Metal Matrix Composites Based on Silica Nanoparticles[END_REF]. Also a 1:1 powder mixture was considered that delayed the onset of phase formation. By adopting mechanical alloying process, Lu et.al studied the formation of nanocrystalline Mg2Si using elemental Mg and Si powders [START_REF] Lu | Formation of nanocrystalline Mg2Si and Mg2Si dispersion strengthened Mg-Al alloy by mechanical alloying[END_REF]. They observed an exothermic peak at 190 o C after 10h of mechanical alloying and the formation of Mg2Si took place at the same temperature for increased milling time. Because of the attrition between the powders, this process introduced lot of defects and resulted in decreasing the activation energy for diffusion.

Density

Using SPS technique, highly dense Mg and Mg-SiO2 composites were synthesized. The density measurements are shown in Table 1. The experimental densities are always higher than the theoretical predicted density for pure magnesium (1.7 g/cc). This is due to the presence of MgO in the starting powders and also due to the formation of MgO during sintering process. Mg-2.5 wt.% SiO2 exhibits maximum experimental density value due to the incorporation of SiO2, which is only approx. 1% greater than that of pure Mg.

The higher experimental density values for both samples are attributed to the presence of low

MgO content (about 2-3 vol.%) present on all the sintered samples. MgO has a density of 3.58 g/cm 3 and its presence has certainly increased the overall density of the sintered samples.

Further, a marginal increase in the density value of the composite is due to the presence of dense Mg2Si particles in the sample. The density of Mg2Si is 1.99 g/cm 3 , which is slightly higher than that of pure magnesium.

Optical Microscopy

The optical observations of sintered samples etched using acetic-picral and their corresponding grain size histograms calculated for 500 grains are shown in Figure 4.

Mechanical Characterizations

Table 1 shows the microhardness results for pure Mg and Mg-2.5wt.% SiO2 after sintering.

Mg-2.5wt.% SiO2 sintered composite has an increased microhardness value than the pure Mg material. This could be due to the presence of Mg2Si. Mg2Si being an intermetallic compound with high hardness, low density and a high melting temperature [START_REF] Niu | Formation of magnesium silicide by mechanical alloying[END_REF]. Hard nano-silica [START_REF] Michel | Effect of temperature on hardness and indentation cracking of fused silica[END_REF] particles distributed uniformly due to grinding and sintering led to a dispersion strengthened Mg2Si-Mg composite with the formation of intermetallic Mg2Si that has a good interface with Mg matrix. Hence the hardness of SiO2 reinforced composite is higher than that of pure magnesium.

Hardness and elastic modulus values are obtained from micro indentation hardness tester using Oliver-Pharr method with 500 mN of constant load and a dwell time of 5s. Hardness values show a 39 percent increase for Mg-2.5 wt.% SiO2 composite as to pure Mg.

The resulting load-displacement curves are shown in Figure 6. Using these curves, reduced elastic modulus was calculated. These values are in the range of 45-48 GPa, which is in close comparison to the elastic modulus of cortical bone which is 10-20 GPa [START_REF] Gupta | Insight into Designing Biocompatible Magnesium Alloys and Composites[END_REF].

Figure 6: Load-displacement curves for sintered magnesium and it's composite Apart from corrosion resistance, another important property of a biomaterial targeting orthopedic applications is elastic modulus. Conventional implants made of titanium and stainless steels have elastic modulus in the order of 100-200 GPa [START_REF] Gupta | Insight into Designing Biocompatible Magnesium Alloys and Composites[END_REF]. Because of the difference in the stiffness of an implant and the bone, a phenomenon called stress shielding occurs. This leads to decrease in bone mass, which is known as bone resorption. Also implant materials made of magnesium is biodegradable, which rules out the need for its removal during second surgery. Thus the synthesized in-situ Mg2Si-Mg composite with its properties similar to that of natural bone has the potential to replace commercially available expensive implants. The composite does not show increased elastic moduli with the addition of SiO2 or the in-situ precipitation of Mg2Si. This is rather beneficial as the elastic modulus of bone is similar to that of pure Mg.

Potentiodynamic Polarization Test

Figure 7 shows the potentiodynamic polarization curves plotted for pure Mg and Mg-2.5 wt.% SiO2. Immediately after immersion in the PBS solution, the voltage sweep was carried out to measure the current for freshly prepared specimens. Mg-2.5% SiO2 shows greater corrosion potential and lower corrosion current density compared with pure magnesium. The polarization parameters are tabulated in Table 1. As seen in the table, the corrosion current for the silica reinforced composite is two orders of magnitude lower than for pure magnesium.

Also the corrosion rate calculated using tafel slope plot is 0.2 mm/y in comparison with 35.2 mm/y for pure magnesium. This shows the improvement in the corrosion behavior with the reinforcement of nano silica. Figure 8 shows the macrograph of the corroded surfaces of pure Mg and Mg-2.5 wt.% SiO2

after potentiodynamic polarization test. The degradation layer with a certain thickness that appears as whitish gray spread across the complete surface of pure magnesium. This layer is fragile and it could easily be chipped off. On the other hand in the case of Mg-2.5 wt.% SiO2, the entire surface was not corroded. Corrosion took place in certain areas that are well connected, leaving aside some areas uncorroded. To study the corrosion morphology in detail, SEM analysis of corroded samples after potentiodynamic polarization studies was carried out and their morphology is shown in Figure 9. In the case of pure magnesium, as seen in Figure 9a, preferential oxidation of grain boundaries was observed along with the grain interiors oxidizing uniformly, the corrosive products were deposited everywhere in the sample. These products do not have a definitive morphology and they are irregular. In the case of Mg-2.5 wt. % SiO2, these products mostly form globular shape and they are not as dispersed as that of pure magnesium. conducted for a long time [START_REF] Kondoh | New process to fabricate magnesium composites using SiO2 glass scraps[END_REF]. Ben Hamu et al. introduced different amounts of Si to Mg-6Zn-0.5Mn alloy to study their corrosion behavior [START_REF] Ben-Hamu | The role of Mg2Si on the corrosion behavior of wrought Mg-Zn-Mn alloy[END_REF]. When the Si content is below 1 wt.% (Chinese script type morphology of Mg2Si), the corrosion rate was found to be higher. But when the volume fraction of silicon is higher (polygonal type morphology of Mg2Si), the corrosion rate was low. Mg2Si is resistant to corrosion [START_REF] Birbilis | Electrochemical Characteristics of Intermetallic Phases in Aluminum Alloys[END_REF] as compared to pure Mg although, it was proven in aluminum alloys. Birbilis et al. studied several intermetallics on AA7075 and it was found that Mg2Si did not show any breakdown potential and it corroded freely above their Ecorr [START_REF] Birbilis | Electrochemical Characteristics of Intermetallic Phases in Aluminum Alloys[END_REF]. They conducted electrochemical testing of aluminum alloys via microcell method and in NaCl solution the corrosion potential of intermetallic Mg2Si was found to be more positive than pure Mg [START_REF] Birbilis | Electrochemical Characteristics of Intermetallic Phases in Aluminum Alloys[END_REF]. For further clear understanding of the role of intermetallic Mg2Si, it is necessary to conduct microcapillary electrochemical measurements.

In a study pursued by Andreatta et al. [START_REF] Andreatta | Volta potential of second phase particles in extruded AZ80 magnesium alloy[END_REF] on the corrosion behavior of the intermetallics in a magnesium alloy, they found that among several intermetallics, Mg2Si showed the weakest cathodic behavior relative to the matrix. The difference in the electrochemical potential between Mg2Si and Mg or MgO and Mg leads to the formation of galvanic couples between these phases and the matrix. These phases are distributed uniformly throughout the matrix.

The localized corrosion was not observed for the synthesized Mg2Si-Mg composite. The degradation took place uniformly.

With the addition of SiO2 and the formation of Mg2Si, the quantity of Mg decreased. Pure Mg undergoes rapid corrosion due to the Cl -ion from the solution that gets trapped in the surface film which is pseudo-passive. It is possible that the presence of these phases in the film reduces the porosity of the film structure and prevents Cl -ions from attacking pure Mg and enhances the corrosion resistance of Mg-Mg2Si composite.

Hence, the addition of silica nanoparticulates is beneficial in synthesizing a bionanocomposite that can delay the degradation process as an implant in a human body.

Conclusions

Magnesium-silica based composites were prepared by powder metallurgy for in-situ synthesis of Mg2Si. The existing works on micro-silica addition to Mg and its alloys consists of Mg2Si particles in Chinese script type or polygonal with particle sizes in the order of 10-60 μm [START_REF] Ben-Hamu | The role of Mg2Si on the corrosion behavior of wrought Mg-Zn-Mn alloy[END_REF].

Another study of incorporating micro SiO2 to pure Mg consisted of agglomeration due to increased size of Mg2Si phase. This decreases the mechanical properties of the composite material [START_REF] Aydin | Microstructure and hardness of Mg-based composites reinforced with Mg2Si particles[END_REF]. Also the density of the composites gets lowered if the particle size of the reinforcement is beyond a certain limit.

Hence we used nanometer silica, to not only have a uniform distribution of Mg2Si phase but also to ensure the complete conversion into Mg2Si due to an increased reactivity of nanosilica. This enhances the mechanical properties, density and the corrosion resistance. Also SiO2 nanoparticles have been extensively used for biomedical applications such as drugdelivery agents as it has greater surface properties. As this work is associated with developing Mg composites targeting biomedical applications, usage of nano-silica was reasonable.

After determining the structure, mechanical and corrosion behavior of these materials, the following conclusions were attained.

1. Dense sintered pure magnesium and silica reinforced magnesium composites were synthesized using mechanical alloying and SPS technique.

2. The elastic modulus of sintered composites did not change significantly as compared to pure magnesium.

3. A slight increase in the microhardness of the composites implies that low weight percent silica reinforcement offers resistance to localized deformation.

4. The corrosion behavior measured using potentiodynamic polarization test in PBS solution at 37 o C showed a significant improvement for silica reinforced composites.

Thus, these nanocomposites could be a viable candidate as a biomaterial for implants and fixation devices and need further evaluation with series of cell culture studies.

Acknowledgements

One of the authors Devadas Bhat Panemangalore thanks University of Lille for supporting his graduate program and providing facilities for carrying out the experiment. The authors also

2. 3 .

 3 Microstructural Characterizations 2.3.1. X-ray Diffraction at room temperature and in-situ high-temperature measurements X-ray Diffractometer (Philips) using cobalt source with a voltage of 40 kV and tube current of 20 mA was used to characterize the phases formed after SPS. A 20 o -120 o range of diffraction angle measurement was considered using a scan rate of 0.01 o /s. For high temperature X-Ray Diffraction (HT-XRD) measurements, Bruker D8 Advance instrument with copper source was used. The generator voltage was set to 40 kV and the current was set to 40 mA. A 5 o C/min heating rate was used and a step size of 0.5s to record the scans sweeping from 20 o -100 o in N2 atmosphere. 2.3.2. Differential Scanning Calorimetry A Differential Scanning Calorimeter (DSC) from NETSZCH, model 404C was used. The DSC response was recorded under argon flow of 60 cc/min with samples weighing approximately 10 mg. In all experiments, the samples were heated at 10 o C/min from ambient temperature 25 o C to 500 o C. The tests were conducted for two repeatable readings.

3. 1 .

 1 Microstructural characterization 3.1.1. X-ray diffraction at room temperature The XRD patterns of the sintered samples are shown in Figure 1. For both samples, the pyramidal peak (1 0 1̅ 1̅ ) at 42.8 o is the most intense. It is followed by the basal peaks (0002) at 40.3 o and the prismatic (1 0 1̅ 0) at 37.6 o . Therefore, addition of 2.5 wt.% SiO2 did not affect the texture of pure magnesium. For sintered Mg-2.5 wt.% SiO2 diffractogram, Mg2Si peaks are observed, as identified by their diffraction peaks at 28.2 o and 46.8 o corresponding to (111) and (002) diffraction planes respectively. The presence of MgO peaks at 50.4 o and 88 o could be seen for both the samples.

Figure 1 :

 1 Figure 1: X-ray diffractogram of sintered samples

Figure 2 :

 2 Figure 2: HT-XRD of Mg-2.5wt.% SiO2 blended powders

Figure 3 :

 3 Figure 3: DSC thermograms of pure Mg and blended Mg-2.5 wt.% SiO2 powders

Figure 4 :

 4 Figure 4: Optical micrographs and histograms for Mg and Mg-2.5 wt.% SiO2 sintered samples

Figure 5 :

 5 Figure 5: EDS maps for Mg, O and Si and its corresponding BSE image for Mg-2.5 wt.% SiO2 composite

Figure 7 :

 7 Figure 7: Potentiodynamic Polarization Curves for sintered pure Mg and Mg-2.5 wt.% SiO2

Figure 8 :

 8 Figure 8: Macrograph of the corroded surfaces of pure Mg and Mg-2.5 wt.% SiO2 after potentiodynamic polarization test

Figure 9 :

 9 Figure 9: SEM images of corroded surfaces of Mg and Mg-2.5 wt.% SiO2 sintered samplesA detailed EDS analysis is presented in Table2. For pure magnesium corroded sample, the

Table 1 :

 1 Measurements related to density, hardness and corrosion

	Material	Density	Hardness	Corrosion rate in PBS solution at 37 o C

Table 2 :

 2 EDS analysis of Mg and Mg-2.5 wt.% SiO2 composites. The values are in weight percent

	Material Spot	Mg	O	Si	P	Cl
	Mg	A	28.5 49.6		19.5 2.4
		B	30.1 38.2		28.6 3
		C	26.3 52.3		20.6 0.9
		D	25.6 54.5		18.5 1.5

In-situ precipitated Mg2Si with its uniform distribution has improved the general corrosion behavior significantly. It is possible that these precipitates act as a barrier to further corrosion by stabilizing the hydroxide film formed on it's surface. Also, it minimized the α-Mg content subjected to corrosion by introducing Mg2Si precipitates. As reported by

Kondoh et al.,

according to Japanese International Standard JIS Z 2371, the corrosion process was initiated in a conventional stainless steel over Mg2Si intermetallic during a salt spray test at 35 o C

acknowledge Conseil Regional du Nord-Pas de Calais and the European Regional Development Fund (ERDF) for the support of the XRD, SEM national facility in Lille (France).