Recent studies demonstrated that 5-to-6-yearolds' abilities to detect and solve ambiguities in speech is a strong predictor of their later reading abilities in first-to-third grade. However, the origins of this relationship remain unclear. For some researchers, success in ambiguity detection is reflective of global language knowledge, while for others, it is reflective of cognitive control skills. In this study, we examined the contributions of oral language knowledge, SES, and cognitive control skills to preschoolers' ability to detect ambiguities. Later we will investigate the relationship among all these factors to early reading development. Our preliminary results show that 5-to-6-year-olds vary greatly in their ability to detect both lexical and structural ambiguities in speech. Moreover, regression analysis revealed that children's cognitive control skills is the strongest predictor of kindergarteners' ambiguity resolution abilities. This is the first study demonstrating that the ambiguity detection task designed to predict reading performance in children is also reflecting their cognitive control skills.

Why is ambiguity detection in kindergarten a predictor of reading abilities ?

Violette Bigot 1 ; John Trueswell 2 & Alex de Carvalho 1

BACKGROUND

DISCUSSION

RESULTS

Figure 2-A-B: Results of the Ambiguity Detection Task. The box-and-whiskers represents the average of the scores in (A) Ambiguous vs Unambiguous Trials on the left, and (B) in each type of Ambiguous sentence, on the right. Each gray dot represents the average for one participant in each condition. Within each plot, the white dashed line represents the mean of the distribution, and the solid black line in the middle represents the median. Children performed better in Unambiguous trials than Ambiguous trials (t(1,37)=12.37, p<.001***) and in Structural Ambiguities than Lexical Ambiguities (t(1,37)= 4.09, p<.001***). Within lexical ambiguities, participants performed better in detecting Noun-Noun Ambiguities than Noun-Verb Ambiguities (t(1,37)=5.13, p<.001***).

Table 2. Results of a multiple regression showing the relationship between ambiguity cost (z-scores of the difference in performance between Unambiguous minus Ambiguous trials) and performance (z-scores) on the other cognitive, linguistic and SES measures.

Participants: Thirty-eight French-learning preschoolers (Mage=67.6mo, from 61 to 75months) Procedure: Participants were tested in 3 different tasks and their SES was indexed by the number of years of maternal education after the end of high school.

Ambiguity detection task (adapted from [2]) :

o On each trial, the child heard a sentence and had to find the picture(s) that matched it.

§ 12 sentences contained a lexical ambiguity (6 with noun-noun homophones and 6 with noun-verb homophones) § 12 sentences contained structural ambiguities For the ambiguous sentences, amongst the four pictures presented, two depicted different meanings for the same ambiguous sentence. § 12 sentences were unambiguous (6 trials contained two pictures illustrating the same meaning for a given sentence, 6 trials had only one target picture)

o Participants earned 4 points if they spontaneously identified two pictures illustrating each of the possible meanings; 3 points if they answered correctly after one additional prompt ("Are there any two pictures that sound EXACTLY alike?"); 2 points for a correct answer after another prompt (the experimenter named each picture, played the sentence again and reasked the question). If after all these attempts children did not identify the two correct pictures, no points were awarded in the trial and the experimenter passed to the next trial. • Five-to-six-year-olds' abilities to detect and solve ambiguities in speech is a strong predictor of their later reading abilities in first-to-third grade [START_REF] Cairns | Detecting the Ambiguity of Sentences: Relationship to Early Reading Skill[END_REF][START_REF] Kamowski-Shakibai | Kindergarten children can be taught to detect lexical ambiguities[END_REF].

Cognitive control task

But why does this relationship exist ? • For some researchers, success in ambiguity detection is reflective of better language skills, which varies with socioeconomic status [START_REF] Huang | Developmental parsing and linguistic knowledge: Reexamining the role of cognitive control in the kindergarten path effect[END_REF] and is known to impact reading [START_REF] Lervåg | Unpicking the Developmental Relationship Between Oral Language Skills and Reading Comprehension: It's Simple, But Complex[END_REF]. • For others, kindergarteners' ability to deal with representational conflict in speech is better explained by their cognitive control skills [START_REF] Woodard | Taking your own path: Individual differences in executive function and language processing skills in child learners[END_REF], which also impacts literacy attainment [START_REF] Cartwright | Insights From Cognitive Neuroscience: The Importance of Executive Function for Early Reading Development and Education[END_REF].

• What is the contribution of oral language knowledge, SES, and cognitive control skills to children's ability to detect ambiguities ?

• And what is the relationship among all these factors to early reading development ?

METHODS AND MATERIALS

CONCLUSIONS REFERENCES

• From the age of 5, children are able to detect and successfully solve ambiguities in oral language. As a group, they successfully detected and solved both lexical and structural ambiguities. • However, at the individual level, these ambiguity detection skills vary greatly from child to child. • In accordance with the results of [START_REF] Woodard | Taking your own path: Individual differences in executive function and language processing skills in child learners[END_REF], we found that children's cognitive control skills measured via a Flanker task is the strongest predictor of preschoolers' ambiguity resolution skills.

• This is the first study demonstrating that the ambiguity detection task designed (by [START_REF] Cairns | Detecting the Ambiguity of Sentences: Relationship to Early Reading Skill[END_REF], [START_REF] Kamowski-Shakibai | Kindergarten children can be taught to detect lexical ambiguities[END_REF]) to predict reading performance in children is also reflecting their cognitive control skills.

• The next step is to test these same participants again once they learn to read (in two years) and evaluate the impact of all these factors in children's reading comprehension skills. We predict that any relationship between ambiguity detection and reading ability will be explained by cognitive control.

• The relationship between individual differences on all the measures with children's performance in the ambiguity detection task was investigated. • For the ambiguity detection task, we kept only the comparison with ambiguous and unambiguous sentences with two targets in our analyses.

• Our results (Fig2-A-B) show that 5-to-6-year-olds vary greatly in their ability to detect both lexical and structural ambiguities and their performance in the two type of ambiguities are strongly correlated. o Children had to "feed the hungry fish" in the middle of the screen by pressing a key at the left or the right side of the keyboard, depending on the direction in which the fish in the middle of the screen was swimming. When they saw empty fish bowls around the middle fish, they were told they should not press any button. o 36 trials (12 in each condition: congruent, incongruent and no-go, e.g, Fig. 2) o In the vocabulary task, bands composed of 6 images were presented (e.g., Fig. 3) and children were asked to point towards the image corresponding to the word they heard (e.g., "winter"). 15 words were presented.

RESEARCH QUESTIONS

o In the oral language comprehension task, the experimenter pronounced a sentence and presented 4 images to the child and asked her to show the image that best corresponded to the spoken sentence they heard (e.g., Fig 4). 10 sentences were presented. • Children's cognitive control skills (i.e., Flanker Switch Cost) is the strongest predictor of kindergarteners' ambiguity resolution abilities. This effect does not interact with the type of ambiguity tested (lexical vs structural).

Figure 1 .

 1 Figure 1. Examples of the trials used in the Ambiguity Detection Task.

Figure 3 .

 3 Figure 3. Relation between Ambiguity Detection Cost (residuals) and Flanker Switch Cost (residuals), following (5).

Figure 2 .

 2 Figure 2. Examples of the three conditions in theFlanker / No-go task from[START_REF] Woodard | Taking your own path: Individual differences in executive function and language processing skills in child learners[END_REF]

Figure 3 .

 3 Figure 3. Example of images presented for the vocabulary task (BSEDS): Band of images corresponding to the item "WINTER".

Figure 4 .

 4 Figure 4. Example of images presented for the oral language comprehension task (BSEDS): images corresponding to the item "the pencil is behind the box".

  Detection Scores by Ambiguity Type

Table 1 .

 1 Summary of all measures (unstandardized)

	M	SD	Range (low:high)
	22.3	5.05	9:30
	7.5	1.7	4:10
	4.5	2.9	0:11
	-.10	.25	-1:.08
	336	323	-169:1125
	.01	.06	-.11:.13
	184	228.5	-262:737
	-10.35	2.46	-12:0
	1.47	1.01	0:3.8
	3.49	.42	2.67:4

Assessing the relationship between Age, SES and Cognitive Control on children's ability to detect and solve ambiguities in speech

	Fixed effects	β	SE	t-value	p-value
	Intercept	-0.01064	0.16301	-0.065	0.9484
	Age	-0.14696	0.18258	-0.805	0.4270
	Flanker Congruency Cost (Incongruent minus Congruent) -0.12228	0.18044	-0.678	0.5030
	Flanker Switch Cost (Switch minus No Switch)	0.61461	0.27424	2.241	0.0323*
	Overall_Language (Vocabulary and Sentence Comp)	0.24483	0.21128	1.159	0.2554
	SES (nb of years of maternal education)	-0.16856	0.18274	-0.922	0.3634
	No-Go Cost (Omission errors minus Hits)	0.04380	0.18018	0.243	0.8095


Rodolfo Araya

Franz Chouly

Residual a posteriori error estimation for frictional contact with Nitsche method

Keywords: unilateral contact, Tresca friction, Coulomb friction, elasticity, Lagrange finite elements, Nitsche method, residual a posteriori error estimates AMS Subject Classification: 65N12, 65N15, 65N30, 74M10, 74M15, 74M20

We consider frictional contact problems in small strain elasticity discretized with finite elements and Nitsche method. Both bilateral and unilateral contact problems are taken into account, as well as both Tresca and Coulomb models for the friction. We derive residual a posteriori error estimates for each friction model, following [Chouly et al, IMA J. Numer. Anal. (38) 2018, pp. 921-954]. For the incomplete variant of Nitsche, we prove an upper bound for the dual norm of the residual, for Tresca and Coulomb friction, without any extra regularity and without a saturation assumption. Numerical experiments allow to assess the accuracy of the estimates and their interest for adaptive meshing in different situations.

Introduction

For a wide range of systems in structural mechanics, it is crucial to consider account contact and friction between rigid or elastic bodies. Among numerous applications, let us mention foundations in civil engineering, metal forming processes, crash-tests of cars, design of car tires (see, e.g., [START_REF] Wriggers | Computational Contact Mechanics[END_REF]). Contact and friction conditions are usually formulated with a set of inequalities and non-linear equations on the boundary of each body, with corresponding unknowns that are displacements, velocities and surface stresses. Basically, contact conditions allow to enforce non-penetration on the whole candidate contact surface, and the actual contact surface is not known in advance. A friction law may be taken into account additionally, and various models exist that correspond to different surface properties, the most popular one being Coulomb's friction (see, e.g., [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF] and references therein).

Frictional contact problems can be formulated weakly within the framework of variational inequalities (see, e.g., [START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF][START_REF] Fichera | Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambigue condizioni al contorno[END_REF][START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF]). Those are the very basis of most existing Finite Element Methods (FEM), see, e.g., [START_REF] Glowinski | Augmented Lagrangian and operator-splitting methods in nonlinear mechanics[END_REF][START_REF] Han | Quasistatic contact problems in viscoelasticity and viscoplasticity[END_REF][START_REF] Haslinger | Numerical methods for unilateral problems in solid mechanics[END_REF][START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF][START_REF] Laursen | Computational contact and impact mechanics[END_REF][START_REF] Wohlmuth | Variationally consistent discretization schemes and numerical algorithms for contact problems[END_REF][START_REF] Wriggers | Computational Contact Mechanics[END_REF]. For numerical computations with the FEM, various techniques have been devised to enforce contact and friction conditions at the discrete level, and the foremost are penalty methods (see, e.g., [START_REF] Chernov | A priori error estimates for hp penalty BEM for contact problems in elasticity[END_REF][START_REF] Chouly | On convergence of the penalty method for unilateral contact problems[END_REF][START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF][START_REF] Kikuchi | Penalty-finite-element approximation of a class of unilateral problems in linear elasticity[END_REF][START_REF] Oden | Finite element methods for constrained problems in elasticity[END_REF][START_REF] Oden | Interior penalty methods for finite element approximations of the Signorini problem in elastostatics[END_REF]) and mixed methods (see, e.g., [START_REF] Belgacem | Hybrid finite element methods for the Signorini problem[END_REF][START_REF] Haslinger | Numerical methods for unilateral problems in solid mechanics[END_REF][START_REF] Hild | Numerical implementation of two nonconforming finite element methods for unilateral contact[END_REF][START_REF] Hüeber | An optimal a priori error estimate for nonlinear multibody contact problems[END_REF][START_REF] Laborde | Fixed point strategies for elastostatic frictional contact problems[END_REF][START_REF] Wohlmuth | Variationally consistent discretization schemes and numerical algorithms for contact problems[END_REF]).

Nitsche's method has been considered recently to discretize contact and friction conditions. The Nitsche method orginally proposed in [START_REF] Nitsche | Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind[END_REF] aims at treating the boundary or interface conditions in a weak sense, with appropriate consistent terms that involve only the primal variables. It differs in this aspect from standard penalization techniques which are generally non-consistent [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF]. Moreover, no additional unknown (Lagrange multiplier) is needed and, therefore, no discrete inf-sup condition must be fulfilled, contrarily to mixed methods (see, e.g., [START_REF] Haslinger | Numerical methods for unilateral problems in solid mechanics[END_REF][START_REF] Wohlmuth | Variationally consistent discretization schemes and numerical algorithms for contact problems[END_REF]). Most of the applications of Nitsche's method during the last two decades involved linear conditions on the boundary of a domain or at the interface between sub-domains: see, e.g,. [START_REF] Stenberg | On some techniques for approximating boundary conditions in the finite element method[END_REF] for the Dirichlet problem, [START_REF] Becker | A finite element method for domain decomposition with non-matching grids[END_REF] for domain decomposition with non-matching meshes and [START_REF] Hansbo | Nitsche's method for interface problems in computational mechanics[END_REF] for a global review. In some previous works [START_REF] Hansbo | A finite element method for the simulation of strong and weak discontinuities in solid mechanics[END_REF][START_REF] Heintz | Stabilized Lagrange multiplier methods for bilateral elastic contact with friction[END_REF] it has been adapted for bilateral (persistent) contact, which still corresponds to linear boundary conditions on the contact zone. We remark furthermore that an algorithm for unilateral contact which makes use of Nitsche's method in its original form is presented and implemented in [START_REF] Hansbo | A finite element method for the simulation of strong and weak discontinuities in solid mechanics[END_REF], An extension to large strain bilateral contact has been performed in [START_REF] Wriggers | A formulation for frictionless contact problems using a weak form introduced by Nitsche[END_REF]. In [START_REF] Chouly | A Nitsche-based method for unilateral contact problems: numerical analysis[END_REF][START_REF] Chouly | Symmetric and non-symmetric variants of Nitsche's method for contact problems in elasticity: theory and numerical experiments[END_REF] a new Nitsche-based FEM was proposed and analyzed for Signorini's problem, where a linear elastic body is in frictionless contact with a rigid foundation. Conversely to bilateral (persistent) contact, Signorini's problem involves non-linear boundary conditions associated to unilateral contact, with an unknown actual contact region.

For this Nitsche-based FEM, optimal convergence in the H 1 (Ω)-norm of order O(h s-1 ) has been proved, provided the solution has a regularity H s (Ω), 3/2 < s ≤ 1 + k (k = 1, 2 is the polynomial degree of the Lagrange finite elements). To this purpose there is no need of additional assumption on the contact/friction zone.

Moreover the Nitsche-based FEM encompasses symmetric and nonsymmetric variants depending upon a parameter called θ. The symmetric case of [START_REF] Chouly | A Nitsche-based method for unilateral contact problems: numerical analysis[END_REF] is recovered when θ = 1. When θ = 1 positivity of the contact term in the Nitsche variational formulation is generally lost. Nevertheless some other advantages are recovered, mostly from the numerical viewpoint. Namely, one of the variants (θ = 0) involves a reduced quantity of terms, which makes it easier to implement and to extend to contact problems involving non-linear elasticity. In addition, this nonsymmetric variant θ = 0 performs better in the sense that it requires less Newton iterations to converge, for a wider range of the Nitsche parameter, than the variant θ = 1, see [START_REF] Renard | Generalized Newton's methods for the approximation and resolution of frictional contact problems in elasticity[END_REF]. Concerning the skew-symmetric variant θ = -1, the well-posedness of the discrete formulation and the optimal convergence are preserved irrespectively of the value of the Nitsche parameter. Lately, various extensions of the method proposed in [START_REF] Chouly | A Nitsche-based method for unilateral contact problems: numerical analysis[END_REF][START_REF] Chouly | Symmetric and non-symmetric variants of Nitsche's method for contact problems in elasticity: theory and numerical experiments[END_REF] have been carried out (see for instance [START_REF] Chouly | An overview of recent results on Nitsche's method for contact problems[END_REF] or [START_REF] Hu | Skew-symmetric Nitsche's formulation in isogeometric analysis: Dirichlet and symmetry conditions, patch coupling and frictionless contact[END_REF] for overviews, as well as [START_REF] Burman | Augmented Lagrangian finite element methods for contact problems[END_REF] for the link between Nitsche and the augmented Lagrangian). Particularly, an extension to Tresca's friction has been studied in [START_REF] Chouly | An adaptation of Nitsche's method to the Tresca friction problem[END_REF]. Optimal convergence in H 1 (Ω)-norm has been established as well, without any assumption other than usual Sobolev regularity. An extension to (static) Coulomb's friction has been formulated in [START_REF] Renard | Generalized Newton's methods for the approximation and resolution of frictional contact problems in elasticity[END_REF] and tested numerically using a generalized Newton algorithm. Later on, the existence of solutions for Coulomb friction has been studied in [START_REF] Chouly | Nitsche-based finite element method for contact with Coulomb friction[END_REF][START_REF]Nitsche method for contact with Coulomb friction: existence results for the static and dynamic finite element formulations[END_REF], and application to fracture mechanics has been considered in [START_REF] Beaude | Mixed and Nitsche's discretizations of Coulomb frictional contact-mechanics for mixed dimensional poromechanical models[END_REF][START_REF]Mixed and Nitsche's discretizations of frictional contact-mechanics in fractured porous media[END_REF]. The case of contact between two elastic bodies is addressed in [START_REF] Chouly | An unbiased Nitsche's approximation of the frictional contact between two elastic structures[END_REF][START_REF] Fabre | A fictitious domain method for frictionless contact problems in elasticity using Nitsche's method[END_REF][START_REF] Mlika | An unbiased nitsches formulation of large deformation frictional contact and self-contact[END_REF]. In [START_REF] Fabre | A fictitious domain method for frictionless contact problems in elasticity using Nitsche's method[END_REF] Nitsche's method is combined with a cut-FEM / fictitious domain discretization, in the small deformations framework. In [START_REF] Chouly | An unbiased Nitsche's approximation of the frictional contact between two elastic structures[END_REF][START_REF] Mlika | An unbiased nitsches formulation of large deformation frictional contact and self-contact[END_REF] an unbiased variant implements the frictional contact between two elastic bodies without making any difference between master and slave contact surfaces. The contact condition is the same on each surface. This is an advantage for treatment of self-contact or multi-body contact. Residual-based a posteriori error estimates are presented in [START_REF] Chouly | Residual-based a posteriori error estimation for contact problems approximated by Nitsche's method[END_REF]. Upper and lower bounds are proved under a saturation assumption, for frictionless contact and Tresca friction, and the performance of the error estimates is investigated numerically for frictionless contact.

In this paper we focus on different friction models, namely Tresca friction and Coulomb friction, discretized with Lagrange FEM and Nitsche's method. We consider both unilateral and bilateral contact settings. For these discrete friction problems, we provide residual based a posteriori error estimates. Moreover, for θ = 0, and following some ideas of [START_REF] Di Pietro | A posteriori error estimates via equilibrated stress reconstructions for contact problems approximated by Nitsche's method[END_REF], we prove an upper bound for the dual norm of the corresponding residual, without any extra regularity assumption or saturation assumption, and valid for both Tresca and Coulomb friction. Numerical experiments allow to assess the performance of the error estimates. For Tresca friction discretized with Nitsche-FEM, a residual based a posteriori error estimator has already been suggested in [START_REF] Chouly | Residual-based a posteriori error estimation for contact problems approximated by Nitsche's method[END_REF] (see the Appendix page 951). Its reliability, under a saturation assumption, and its efficiency have been established. Nevertheless, it has never been implemented and assessed numerically. For Coulomb friction with Nitsche, no a posteriori error estimates exist to the best of our knowledge.

To put our work in perspective, let us mention that, in a recent contribution [START_REF] Gustafsson | Stabilized finite elements for Tresca friction problem[END_REF], another kind of residual a posteriori error estimate has been designed for a stabilized mixed method close to the symmetric variant of Nitsche's method. Notably, thanks to additional terms, the authors of [START_REF] Gustafsson | Stabilized finite elements for Tresca friction problem[END_REF] manage to prove an upper bound for this estimator without any saturation assumption. For frictionless (Signorini) contact, such a technique has already been presented in [START_REF] Gustafsson | On Nitsche's method for elastic contact problems[END_REF] (in addition to an a priori error bound under minimal regularity assumption). In the same manner, an upper bound for an a posteriori error estimator based on equilibrated fluxes has been proven, without saturation assumption and for the scalar Signorini problem [START_REF] Capatina | Local flux reconstruction for a frictionless unilateral contact problem[END_REF]. Still for an equilibrated fluxes estimator and frictionless contact in elasticity, an upper bound has been established with a saturation assumption in [START_REF] Di Pietro | A posteriori error estimates via equilibrated stress reconstructions for contact problems approximated by Nitsche's method[END_REF]. As well, for Tresca friction discretized with the Virtual Element Method (VEM), a new a posteriori estimate has been designed in [START_REF] Deng | A posteriori error estimates of virtual element method for a simplified friction problem[END_REF]. For Coulomb friction, the upper and lower bounds for residual a posteriori error estimates are difficult to establish, see for instance [START_REF] Hild | Residual error estimators for Coulomb friction[END_REF] and references therein for mixed methods. See also [START_REF] Hüeber | Equilibration techniques for solving contact problems with Coulomb friction[END_REF] for an estimator for Coulomb friction based on equilibrated fluxes. In fact, very few error estimators with mathematical justification of robustness or efficiency have been proposed for Coulomb friction.

This paper is outlined as follows. In Section 2 we introduce a frictional contact problem and its Nitsche finite element approximation. Section 3 details the residual a posteriori error estimate. Section 4 presents an upper bound for the dual norm of the residual and the incomplete variant of Nitsche. Section 5 reports different numerical experiments with uniform and adaptive refinement, for contact with Tresca and Coulomb friction, in order to assess the performance of the estimator in practice.

Let us introduce some useful notations. In what follows, bold letters like u, v, indicate vector or tensor valued quantities, while the capital ones (e.g., V, K . . .) represent functional sets or operators involving vector fields.

As usual, we denote by H s (•), s ∈ R, the Sobolev spaces of real-valued functions, and H s (•; R d ), s ∈ R, d ∈ N * the Sobolev spaces of vector-valued functions in R d (see [START_REF] Adams | Sobolev spaces[END_REF]). For D a domain, the standard scalar product (resp. norm) of H s (D; R d ) is denoted by (•, •)s,D (resp. • s,D ) and we keep the same notation for all the values of d. The letter C stands for a generic constant, independent of the discretization parameters.

Setting and Nitsche-FEM

In this section we first present the frictional contact problems under consideration and then their approximation with Nitsche-FEM.

Frictional contact

We consider an elastic body whose reference configuration is represented by the domain Ω in R d with d = 2 or d = 3. Small strain assumption is made, as well as plane strain when d = 2. The boundary ∂Ω of Ω is polygonal or polyhedral and we partition ∂Ω in three nonoverlapping parts ΓD, ΓN and the (potential) contact/friction boundary ΓC , with meas(ΓD) > 0 and meas(ΓC ) > 0. The contact/friction boundary is supposed to be a straight line segment when d = 2 or a planar polygon when d = 3 to simplify. The normal unit outward vector on ∂Ω is denoted n. The body is clamped on ΓD for the sake of simplicity. It is subjected to volume forces f ∈ L 2 (Ω; R d ) and to surface loads fN ∈ L 2 (ΓN ; R d ).

The contact problem under consideration consists in finding the displacement field u : Ω → R d verifying the equations and conditions (1)-( 2)-(3):

div σ(u) + f = 0 in Ω, σ(u) = C ε(u) in Ω, u = 0 on ΓD, σ(u)n = fN on ΓN , (1)
where σ = (σij), 1 ≤ i, j ≤ d, stands for the stress tensor field and div denotes the divergence operator of tensor valued functions. The notation ε(v) = (∇v + ∇v T )/2 represents the linearized strain tensor field and C is the fourth order symmetric elasticity tensor having the usual uniform ellipticity and boundedness property. For any displacement field v and for any density of surface forces σ(v)n defined on ∂Ω we adopt the following decomposition into normal and tangential components, v = vnn + vt and σ(v)n = σn(v)n + σt(v).

where vn := v • n and vt := v -vnn (the same applies for the vector σ(v)n). The contact conditions on ΓC are formulated as:

σn = -[un -σn]C (2)
where [•]C is a projection operator onto a convex subset of R. Two special cases will be considered:

1. Unilateral contact conditions, for which [•]C is the positive part operator:

[x] C = 1 2 (x + |x|) for x ∈ R. Then condition (2) is equivalent to the Signorini conditions un ≤ 0, σn(u) ≤ 0, σn(u) un = 0.
2. Bilateral contact conditions, for which [•]C is the identity: [START_REF] Alart | A generalized Newton method for contact problems with friction[END_REF] is the generalized Dirichlet condition:

[x] C = x for x ∈ R. Then condition
un = 0.
The friction conditions on ΓC are given by

   |σt(u)| ≤ S(u) if ut = 0, (i) σt(u) = -S(u) ut |ut| otherwise, (ii) (3)
where | • | stands for the euclidean norm in R d-1 . The above formulation encompasses both Tresca and Coulomb friction models. Indeed, for Tresca, we set S(u) = sT where sT ∈ L 2 (ΓC ), s ≥ 0 is a given threshold. For Coulomb friction, we set S(u) = -F σn(u), where F ≥ 0 is the friction coefficient. Note that conditions (3)-(i) and ( 3)-(ii) imply that |σt(u)| ≤ S(u) in all cases, and that if |σt(u)| < S(u), we must have ut = 0.

For Coulomb friction, conditions (3) mean that, at a given contact point on ΓC , sliding can not occur while the magnitude of the tangential stress |σt(u)| is strictly below the threshold -F σn(u). When the threshold -F σn(u) is reached, sliding may happen, in a direction opposite to σt(u) (see, e.g., [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF]Chapter 10]). Remark that this static form of Coulomb friction is an adaptation of the quasi-static (or dynamic) Coulomb's law, in which the tangential velocity ut plays the same role as the displacement ut (see e.g., [START_REF] Ballard | Steady sliding frictional contact problems in linear elasticity[END_REF][START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF][START_REF] Eck | Unilateral contact problems[END_REF]). A formulation such as (3) is obtained for instance when the quasi-static Coulomb's law is discretized with a time-marching scheme (see, e.g., [START_REF] Wohlmuth | Variationally consistent discretization schemes and numerical algorithms for contact problems[END_REF]).

Remark 2.1. In the Tresca friction model (S(u) = sT in (3)), it is assumed that the amplitude of the normal friction threshold is known (i.e., F |σn(u)| = sT , see, e.g., [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF]Section 10.3]). The introduction of the Tresca friction model is rather motivated by its mathematical simplicity than by physical reasons, though it can be of use in special situations. For instance when a thin belt/tape is pressed against an elastic body, with a known pressure (see, e.g., [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF]Chapter 10]), or for persistent contact between solids with high intensity of contact pressures, such as it may occur in earth sciences (see, e.g., [START_REF] Pipping | Variational formulation of rate-and state-dependent friction problems[END_REF]). Moreover, the Tresca friction model can be useful, for instance, when Coulomb friction is approximated iteratively (see, e.g., [START_REF] Haslinger | Approximation of the Signorini problem with friction by a mixed finite element method[END_REF] and [51, Section 9, Theorem 7]).

We introduce the Hilbert space V and the set K of admissible displacements which satisfy the condition on the contact zone ΓC :

V := v ∈ H 1 (Ω; R d ) : v = 0 on ΓD , K := {v ∈ V : [vn]C = 0 on ΓC } . Define a(u, v) := Ω σ(u) : ε(v) dΩ, L(v) := Ω f • v dΩ + Γ N fN • v dΓ, j(u; v) := Γ C S(u)|vt| dΓ,
for any u and v in V, and u regular enough.

The weak formulation of Problem ( 1)-( 2)-( 3) as a (quasi-)variational inequality is (see, e.g., [START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF]):

Find u ∈ K (regular enough) such that: a(u, v -u) + j(u; v) -j(u; u) ≥ L(v -u), ∀ v ∈ K. (4)
In the case of Tresca friction (S(u) = s), the above weak formulation is a variational inequality of the second kind that admits a unique solution (see, e.g., [32, Theorem 5.1, Remark 5.2, p.69]). Contact with Coulomb friction in elastostatics remains a difficult problem, with still some open issues in its mathematical analysis, both for the continuous and the discrete problems. In the continuous case, there is indeed no complete characterization of existence and uniqueness when the friction coefficient is varied (see, e.g., [START_REF] Eck | Existence results for the static contact problem with Coulomb friction[END_REF][START_REF] Eck | Unilateral contact problems[END_REF][START_REF] Necas | On the solution of the variational inequality to the Signorini problem with small friction[END_REF] for existence results when the friction coefficient is small). Moreover it can be proved that uniqueness is lost in some configurations and multiple solutions can be obtained, see, e.g., [START_REF] Ballard | Steady sliding frictional contact problems in linear elasticity[END_REF][START_REF]Non unique slipping in the Coulomb friction model in two dimensional linear elasticity[END_REF].

The Nitsche-based finite element method

Let V h ⊂ V be a family of finite dimensional vector spaces (see [START_REF] Brenner | The mathematical theory of finite element methods[END_REF][START_REF] Ciarlet | The finite element method for elliptic problems[END_REF][START_REF] Ern | Theory and practice of finite elements[END_REF]) indexed by h coming from a family T h of triangulations of the domain Ω (h = max T ∈T h hT where hT is the diameter of T ). We suppose that the family of triangulations is regular, i.e., there exists σ > 0 such that ∀T ∈ T h , hT /ρT ≤ σ where ρT denotes the radius of the inscribed ball in T . Furthermore, we suppose that this family is conformal to the subdivision of the boundary into ΓD, ΓN and ΓC (i.e., a face of an element T ∈ T h is not allowed to have simultaneous non-empty intersection with more than one part of the subdivision). We choose a standard Lagrange finite element method of degree k with k = 1 or k = 2, i.e.:

V h := v h ∈ C 0 (Ω; R d ) : v h | T ∈ P k (T ; R d ), ∀T ∈ T h , v h = 0 on ΓD . (5)
However, the analysis would be similar for any C 0 -conforming finite element method. Moreover, for any α ∈ R + , we introduce the notation [•]α for the orthogonal projection onto B(0, α) ⊂ R d-1 , where B(0, α) is the closed ball centered at the origin 0 and of radius α. This operation can be defined analytically, for x ∈ R d-1 by:

[x]α = x if |x| ≤ α, α x |x| otherwise.
The next result has been pointed out earlier in [START_REF] Alart | A generalized Newton method for contact problems with friction[END_REF] (see as well [START_REF] Chouly | An adaptation of Nitsche's method to the Tresca friction problem[END_REF][START_REF] Chouly | A Nitsche-based method for unilateral contact problems: numerical analysis[END_REF][START_REF]Nitsche method for contact with Coulomb friction: existence results for the static and dynamic finite element formulations[END_REF] for detailed formal proofs). Proposition 2.2. Let γ be a positive function defined on ΓC . The contact with friction conditions (2)-( 3) can be reformulated as follows:

σn(u) = -[γ un -σn(u)] C , (6)
σt(u) = -[γ ut -σt(u)] S(u) . (7)
Remember that for Tresca friction, we have S(u) = sT where sT ∈ L 2 (ΓC ), s ≥ 0 is a given threshold. For Coulomb friction, we can use [START_REF] Becker | A finite element method for domain decomposition with non-matching grids[END_REF] to have two different expressions of the solution-dependent threshold:

S(u) = -F σn(u) = F [γ un -σn(u)]
C . Let now θ ∈ R be a fixed parameter and γ a positive function defined on ΓC . The Nitsche method for frictional contact is derived as follows [START_REF] Chouly | An overview of recent results on Nitsche's method for contact problems[END_REF][START_REF] Hu | Skew-symmetric Nitsche's formulation in isogeometric analysis: Dirichlet and symmetry conditions, patch coupling and frictionless contact[END_REF]. Let u be the solution to the frictional contact problem in its strong form (1)-( 2)-( 3). We assume that u is sufficiently regular so that all the following calculations make sense. From the Green formula and equation (1), we get for every v ∈ K:

a(u, v) - Γ C σn(u) vn dΓ - Γ C σt(u) • vt dΓ = L(v). Note that vn = 1 γ (γvn -θσn(v)) + θ γ σn(v), vt = 1 γ (γvt -θσt(v)) + θ γ σt(v).
So:

a(u, v) - Γ C θ γ σn(u) σn(v) dΓ - Γ C 1 γ σn(u) (γvn -θσn(v)) dΓ - Γ C θ γ σt(u) • σt(v) dΓ - Γ C 1 γ σt(u) • (γvt -θσt(v)) dΓ = L(v).
Finally, using conditions ( 6) and ( 7), we obtain:

a(u, v) - Γ C θ γ σn(u) σn(v) dΓ + Γ C 1 γ [γ un -σn(u)] C (γvn -θσn(v)) dΓ - Γ C θ γ σt(u) • σt(v) dΓ + Γ C 1 γ [γ ut -σt(u)] S(u) • (γvt -θσt(v)) dΓ = L(v). (8)
Formula ( 8) is the starting point of the Nitsche-based formulation. We remark that it may have no sense at the continuous level if u lacks of regularity (the only assumption u ∈ V is not sufficient to justify the above calculations). Nevertheless we consider in what follows that γ is a positive piecewise constant function on the contact and friction interface ΓC which satisfies

γ|T ∩Γ C = γ0 hT , (9)
for every T that has a non-empty intersection of dimension d -1 with ΓC , and where γ0 is a positive given constant (the Nitsche parameter). Note that the value of γ on element intersections has no influence. Let us introduce the discrete linear operators P n θ,γ :

V h → L 2 (ΓC ) v h → γv h n -θσn(v h ) and P t θ,γ : V h → (L 2 (ΓC )) d-1 v h → γv h t -θσt(v h )
.

Define as well the bilinear form:

A θγ (u h , v h ) := a(u h , v h ) - Γ C θ γ σ(u h )n • σ(v h )n dΓ.
The Nitsche-based method for frictional contact then reads:

             Find u h ∈ V h such that: A θγ (u h , v h ) + Γ C 1 γ [P n 1,γ (u h )] C P n θ,γ (v h ) dΓ + Γ C 1 γ [P t 1,γ (u h )] S h (u h ) • P t θ,γ (v h ) dΓ = L(v h ), ∀ v h ∈ V h . ( 10
)
For Tresca friction we take simply:

S h (u h ) = sT .
For Coulomb friction we preferably take:

S h (u h ) = F P n 1,γ (u h ) C .
For unilateral contact, it ensures that the threshold for the projection operator remains nonnegative. The parameter θ can be set to some particular values, namely:

1. for θ = 1 we recover a symmetric method.

2. for θ = 0 we recover a simple method close to penalty and to the augmented Lagrangian.

3. for θ = -1 the skew-symmetric method is well-posed irrespectively of the value of γ0 > 0.

Existence and uniqueness of solutions to the discrete problem [START_REF] Burman | Augmented Lagrangian finite element methods for contact problems[END_REF] has been studied in [START_REF] Chouly | An adaptation of Nitsche's method to the Tresca friction problem[END_REF] for Tresca friction: in this case well-posedness is ensured provided that γ0 is large enough (γ0 > 0 for θ = 1). For Coulomb friction, a fixed point argument allows to assess the existence of solutions to [START_REF] Burman | Augmented Lagrangian finite element methods for contact problems[END_REF] for γ0 large enough, and every value of the friction coefficient. Uniqueness can be recovered provided a restrictive condition, for friction small enough, and dependent of the mesh size, see [START_REF]Nitsche method for contact with Coulomb friction: existence results for the static and dynamic finite element formulations[END_REF] for details. In [START_REF] Chouly | An adaptation of Nitsche's method to the Tresca friction problem[END_REF] an optimal a priori error bound for Tresca friction in the natural norm has been derived, for 2D and 3D problems, and for linear and quadratic FEM.

3 Residual-based a posteriori error estimate An explicit residual-based a posteriori error estimate can be derived for Problem [START_REF] Burman | Augmented Lagrangian finite element methods for contact problems[END_REF], that has been proposed in [START_REF] Chouly | Residual-based a posteriori error estimation for contact problems approximated by Nitsche's method[END_REF] for frictionless contact and Tresca friction. We extend it here to the Coulomb friction model. This a posteriori error estimate has its origin in [START_REF] Becker | A finite element method for domain decomposition with non-matching grids[END_REF] (see also, e.g., [START_REF] Verfürth | A review of a posteriori error estimation techniques for elasticity problems[END_REF] for linear elasticity). We introduce standard notations for this purpose:

• We define E h the set of edges/faces of the triangulation and define

E int h := {E ∈ E h : E ⊂ Ω}
:= ET ∩ E int h , E N T := ET ∩ E N h , E C T := ET ∩ E C h .
• For an edge/face E of an element T , introduce νT,E the unit outward normal vector to T along E.

Furthermore, for each edge/face E, we fix one of the two normal vectors and denote it by νE. The jump of some vector valued function v across an edge/face

E ∈ E int h at a point y ∈ E is defined as v E (y) := lim α→0 + v(y + ανE) -v(y -ανE).
• Let ωT be the union of all elements having a nonempty intersection with T . Similarly for a node x and an edge/face E, let ωx := ∪T :x∈T T and ωE := ∪ x∈E ωx.

• fT (resp. fN,E) is a computable quantity that approximates f on the element T ∈ T h (resp. fN on the edge E ∈ E N h ). The a posteriori error estimator is defined as follows.

Definition 3.1. For the unified formulation (10) the local error estimators ηT and the the global estimator η are defined by

ηT := 4 i=1 η 2 iT 1/2 , η1T := hT div σ(u h ) + fT 0,T , η2T := h 1/2 T   E∈E int T ∪E N T JE,n(u h ) 2 0,E   1/2 , η3T := h 1/2 T   E∈E C T σt(u h )+ P t 1,γ (u h ) S h (u h ) 2 0,E   1/2 , η4T := h 1/2 T   E∈E C T σn(u h )+ P n 1,γ (u h ) C 2 0,E   1/2 , η :=   T ∈T h η 2 T   1/2
, where JE,n(u h ) means the constraint jump of u h in the normal direction, i.e.,

JE,n(u

h ) := σ(u h )νE E , ∀E ∈ E int h , σ(u h )νE -fN,E, ∀E ∈ E N h . (11)
The local and global approximation terms are given by

ζT :=   h 2 T T ⊂ω T f -f T 2 0,T + hE E⊂E N T fN -fN,E 2 0,E   1/2 , ζ :=   T ∈T h ζ 2 T   1/2 .
4 An upper bound for the residual

In [START_REF] Chouly | Residual-based a posteriori error estimation for contact problems approximated by Nitsche's method[END_REF] a global upper bound, under a saturation assumption, and for regularity H s (Ω), s > 3/2, of the solution u, have been proven, for frictionless contact and also for Tresca friction. Local optimal lower bounds have been established too.

We can try here to follow the idea of a recent work from Di Pietro et al [START_REF] Di Pietro | A posteriori error estimates via equilibrated stress reconstructions for contact problems approximated by Nitsche's method[END_REF] and to derive an upper bound for the dual norm of the residual without any saturation assumption and with solely a regularity H 1 (Ω) for u. This can be done for 2D and 3D problems, linear and quadratic finite elements, and above all for both friction models: Tresca friction and Coulomb friction. As in [START_REF] Di Pietro | A posteriori error estimates via equilibrated stress reconstructions for contact problems approximated by Nitsche's method[END_REF], we focus on the incomplete version of Nitsche and set θ = 0.

The residual R can be viewed as an application

R : V h → V
where V is the topological dual of V. This application is defined, for w h ∈ V h as:

     R(w h ), v :=L(v) -a(w h , v) - Γ C [P n 1,γ (w h )] C vn dΓ - Γ C [P t 1,γ (w h )] S h (u h ) • vt dΓ, ∀ v ∈ V. (12)
We see here why it is not convenient to define a residual for θ = 0: in this situation the Nitsche boundary terms may have no meaning in L 2 (ΓC ), since we suppose only v ∈ H 1 (Ω; R d ). The dual norm of the residual is then defined as

R(w h ) * := sup v∈V R(w h ), v v 1,Ω .
The following result holds:

Proposition 4.1. Let us suppose that θ = 0, γ0 > 0 large enough and let u h ∈ V h ⊂ V be a solution to Nitsche formulation [START_REF] Burman | Augmented Lagrangian finite element methods for contact problems[END_REF]. Then the residual defined by [START_REF] Chernov | A priori error estimates for hp penalty BEM for contact problems in elasticity[END_REF] is bounded by the a posteriori error estimate and the approximation terms of Definition 3.1 as follows

R(u h ) * ≤ C (η + ζ)
where the constant C > 0 is independent of h.

Proof. Let u h ∈ V h ⊂ V be a solution to Nitsche formulation [START_REF] Burman | Augmented Lagrangian finite element methods for contact problems[END_REF], let v ∈ V be a test function, and let us write first

R(u h ), v = L(v) -a(u h , v) - Γ C [P n 1,γ (u h )] C vn dΓ - Γ C [P t 1,γ (u h )] S h (u h ) • vt dΓ.
Since u h solves (10), then there holds: R(u h ), v h = 0, for every test function v h ∈ V h . So there holds also

R(u h ), v = R(u h ), v -R(u h ), v h = R(u h ), v -v h = L(v -v h ) -a(u h , v -v h ) - Γ C [P n 1,γ (u h )] C (vn -v h n ) dΓ - Γ C [P t 1,γ (u h )] S h (u h ) • (vt -v h t ) dΓ.
Now we can transform the above expression by integrating by parts on each triangle T , using the definition of JE,n(u h ) and splitting up the integrals on ΓC into normal and tangential components. As a result, we get:

R(u h ), v = T ∈T h T (div σ(u h ) + f ) • (v -v h ) dΓ - E∈E C h E [P n 1,γ (u h )] C + σn(u h ) (vn -v h n ) dΓ - E∈E C h E [P t 1,γ (u h )] S h (u h ) + σt(u h ) • (vt -v h t ) dΓ - E∈E int h ∪E N h E JE,n(u h ) • (v -v h ) dΓ + E∈E N h E (fN -fN,E) • (v -v h ) dΓ. ( 13
)
We now need to estimate each term of this right-hand side. For that purpose, we take

v h = SZ h (v), (14)
where SZ h is the quasi-interpolation operator of Scott & Zhang, see [START_REF] Scott | Finite element interpolation of nonsmooth functions satisfying boundary conditions[END_REF] and [START_REF] Ern | Theory and practice of finite elements[END_REF].

We start with the integral term on simplices T . Cauchy-Schwarz's inequality implies

T ∈T h T (div σ(u h ) + f ) • (v -v h ) dΓ ≤ T ∈T h div σ(u h ) + f 0,T v -v h 0,T ,
and it suffices to estimate vv h 0,T for any simplex T . From the definition of v h and the approximation properties of SZ h , we get:

v -v h 0,T = v -SZ h v 0,T ≤ ChT v 1,ω T . As a consequence Ω (div σ(u h ) + f ) • (v -v h ) dΓ ≤ C(η + ζ) v 1,Ω.
For the interior and Neumann boundary terms in [START_REF] Chouly | An adaptation of Nitsche's method to the Tresca friction problem[END_REF], the application of Cauchy-Schwarz's inequality leads to

E∈E int h ∪E N h E JE,n(u h ) • (v -v h ) dΓ ≤ E∈E int h ∪E N h JE,n(u h ) 0,E v -v h 0,E .
Using once again the approximation properties of SZ h , we get:

v -v h 0,E = v -SZ h v 0,E h 1/2 E v 1,ω E ,
and we deduce that

E∈E int h ∪E N h E JE,n(u h ) • (v -v h ) dΓ ≤ Cη v 1,Ω.

Moreover we can bound

E∈E N h E (fN -fN,E) • (v -v h ) dΓ ≤ Cζ v 1,Ω.
Let us focus now on the friction term:

E∈E C h E [P t 1,γ (u h )] S h (u h ) + σt(u h ) • (vt -v h t ) dΓ ≤ E∈E C h [P t 1,γ (u h )] S h (u h ) + σt(u h ) 0,E v -v h 0,E .
Still we use the quasi-interpolation properties of SZ h and then we bound

E∈E C h E [P t 1,γ (u h )] S h (u h ) + σt(u h ) • (vt -v h t ) dΓ ≤ Cη v 1,Ω.
We do exactly the same for the contact term:

E∈E C h E [P n 1,γ (u h )] C + σn(u h ) • (vn -v h n ) dΓ ≤ Cη v 1,Ω.
We combine all the previous results and get finally

R(u h ), v ≤ C(η + ζ) v 1,Ω.
By definition of the dual norm, we conclude that

R(u h ) * ≤ C(η + ζ).
and this ends the proof.

Remark 4.2. In the frictionless case, one can recover from the above result an upper bound on the natural norm of the discretization error ( u -u h 1,Ω), under a saturation assumption and with enough regularity on the continuous solution u, see [START_REF] Di Pietro | A posteriori error estimates via equilibrated stress reconstructions for contact problems approximated by Nitsche's method[END_REF]. This is also the case for Tresca friction, and one then recovers the upper bound stated in [START_REF] Chouly | Residual-based a posteriori error estimation for contact problems approximated by Nitsche's method[END_REF], in the case θ = 0. For Coulomb friction, the obtention of an upper bound on the natural norm of the discretization error remains an open issue.

Numerical experiments

Contact and friction problems can be solved by using a generalized (semi-smooth) Newton method, which means that Problem ( 10) is derived with respect to u h to obtain the tangent system. The term "generalized Newton's method" comes from the fact that the operators such as [•]

C and [•] S(u) are not Gâteaux-differentiable at some specific points. However, no special treatment is considered. If a point of non-differentiability is encountered, the tangent system corresponding to one of the two alternatives (x < 0 or x > 0 for [•]

C ) is chosen arbitrarily. Note that, for frictionless contact, the situation where the solution is non-differentiable at an integration point is very rare and corresponds to what is called a "grazing contact" (both un = 0 and σn = 0). Problem ( 10) is implemented within the open source finite element library FEniCS [START_REF] Logg | Automated solution of differential equations by the finite element method: The FEniCS book[END_REF]. We took advantage of the capabilities of FEniCS in terms of automatic symbolic differentiation: the weak form [START_REF] Burman | Augmented Lagrangian finite element methods for contact problems[END_REF] is written directly, and we did not need to specify explicitely the tangent problem.

First we present a first numerical test with Tresca friction and a manufactured solution, from [START_REF] Chouly | A hybrid high-order discretization combined with Nitsche's method for contact and Tresca friction in small strain elasticity[END_REF]. Another test, still for Tresca, is from a classical benchmark, originally presented in [START_REF] Bostan | A posteriori error analysis for finite element solutions of a frictional contact problem[END_REF]. The last numerical experiment is about Coulomb friction and comes from [START_REF] Lleras | Modélisation, analyse et simulation de problèmes de contact en mécanique des solides et des fluides[END_REF].

A manufactured solution

We first carry out some tests on the problem with a manufactured solution, presented in [START_REF] Chouly | A hybrid high-order discretization combined with Nitsche's method for contact and Tresca friction in small strain elasticity[END_REF]. In this example we consider Ω := (0, 1) × (0, 1), ΓC := (0, 1) × {0}, ΓD := ∂Ω \ ΓC . The Lamé coefficients are λ = 1000 and µ = 2. The rhs f and the Dirichlet boundary condition uD are such that the exact solution is given by: u := (u1, u2), where u1(x, y)

:= 1 + 1 1 + λ x e x+y , u2(x, y) := -1 + 1 1 + λ y e x+y .
The friction thresold s, defined on ΓC , is given by

s(x) := µ 1 + 1 1 + λ x e x .
Note that this setting is slightly different from Section 2, since there is no Neumann boundary, and a nonhomogeneous Dirichlet boundary condition. Lagrange finite elements of order P k are used (k = 1, 2). Nitsche's parameter is set as γ0 = λ when θ = -1 and θ = 0. For the symmetric variant θ = 1, it is set as γ0 = 4λ for k = 1 and γ0 = 6λ for k = 2.

Uniform refinement

Problem ( 10) is solved on a sequence of 7 structured meshes, with uniform refinement: the mesh size is divided by 2 each time. This allows to check the convergence properties of the error estimate.

For the skew-symmetric variant (θ = -1), results are presented in Table 1, Table 2, Table 3 and Table 4. For Table 1 (order k = 1) and Table 3 (order k = 2), we detail the error in the L 2 -norm, H 1 semi-norm and norm and the value of the residual error estimate η. The empirical convergence rates are denoted by r2, rs1, r1 and rη, for the L 2 -norm, the H 1 semi-norm, the H 1 norm and η, respectively. The efficiency index is denoted by E and defined, as usual, as:

E := η u -u h 1,Ω .
As it can be seen in Table 1 for order k = 1, the errors in L 2 and H 1 norm are reduced with a rate higher than expected. This slight superconvergence phenomenon is unexplained. The a posteriori error estimate η decreases with the expected rate of approximately 1. The efficiency index E is large.

For order k = 2 we can observe from Table 3 that similar conclusions hold than for k = 1, still with a slight superconvergence for the a priori error rates. Note however, that the convergence rate of the residual estimator η is slightly below the theoretical rate of 2. The efficiency index is still large.

For Table 2 (order k = 1) and Table 4 (order k = 2), we report the value of the different components of η and their convergence rates. As in [START_REF] Chouly | Residual-based a posteriori error estimation for contact problems approximated by Nitsche's method[END_REF], we observe that the components η1 and η2 (residual and jumps on the edges) that are, roughly speaking, related to the elasticity equations, are predominent, and converge with the expected rate of approximately 1, for linear finite elements. The components related to Tresca friction and to unilateral contact, respectively η3 and η4, are of smaller magnitude, and converge faster, with a rate around 1.5. We have no explanation for this phenomenon. Remark also that the estimator related to Tresca friction is even smaller than the estimator for contact. For order k = 2, we can draw similar conclusions from Table 4. Moreover we remark that for η1 and η2 the convergence order is below 2, whereas the components η3 and η4 for friction and contact converge much faster, with rates around 2.5. Remark also that the component related to Tresca friction is smaller than the one for contact. These results are in agreement with the previous ones for frictionless contact, reported in [START_REF] Chouly | Residual-based a posteriori error estimation for contact problems approximated by Nitsche's method[END_REF]. Table 4: Problem with a manufactured solution. Components of the estimator η and their respective convergence order. Polynomial order k = 2. Nitsche parameters γ0 = λ and θ = -1. Table 6: Problem with a manufactured solution. Components of the estimator η and their respective convergence order. Polynomial order k = 1. Nitsche parameters γ0 = λ and θ = 0. Table 8: Problem with a manufactured solution. Components of the estimator η and their respective convergence order. Polynomial order k = 2. Nitsche parameters γ0 = λ and θ = 0. Table 12: Problem with a manufactured solution. Components of the estimator η and their respective convergence order. Polynomial order k = 2. Nitsche parameters γ0 = 6λ and θ = 1.

h u -u h 0,Ω r 2 |u -u h | 1,Ω r s1 u -u h 1,Ω r
h u -u h 0,Ω r 2 |u -u h | 1,Ω r s1 u -u h 1,Ω r 1 η reta E 0.
h u -u h 0,Ω r 2 |u -u h | 1,Ω r s1 u -u h 1,Ω r 1 η reta E 0.
h u -u h 0,Ω r 2 |u -u h | 1,Ω r s1 u -u h 1,Ω r 1 η reta E 0.

Adaptive refinement

We now provide numerical results when mesh adaptation driven by the local components of η is carried out. Figure 1 depicts the initial mesh and the final adapted mesh. We check that the refinement occurs in the expected region at the top right corner of the square. On Figure 2 we observe that the final adapted solution is undistinguishable from the exact solution, contrarily to the solution on the initial mesh.

A test case from Bostan & Han

We reproduce here the test case described in Bostan & Han [START_REF] Bostan | A posteriori error analysis for finite element solutions of a frictional contact problem[END_REF] for Tresca friction with stick and slip transition.

In this example, we set Ω := (0, 8) × (0, 4), ΓD := (0, 8) × {4}, ΓN := {0} × (0, 4) ∪ {8} × (0, 4). The contact boundary is located at the bottom: ΓC :=]0, 8[×{0}. The Lamé coefficients are λ := 576.9 and µ := 384.6. The source term is f := 0. The Dirichlet and Neumann boundary conditions are given, respectively, by: uD := 0, and σ(u)n = g, with g = (400, 0) T .

The friction thresold s, defined on ΓC , is constant, and given by s(x) := 150.

Note that in this case we do not dispose of an analytical solution. Figure 3 depicts the initial and the final adapted meshes. Figure 4 depicts the final solution and the deformed adapted mesh. As expected, refinement occurs at corners, and, more interestingly, near the transition point between slip and stick. The results are in good agreement with [START_REF] Bostan | A posteriori error analysis for finite element solutions of a frictional contact problem[END_REF].

Coulomb friction

Finally, we reproduce the test case from [START_REF] Hild | Residual error estimators for Coulomb friction[END_REF][START_REF] Lleras | Modélisation, analyse et simulation de problèmes de contact en mécanique des solides et des fluides[END_REF] to test the behavior of the residual error estimate η in the case of Coulomb friction. In this example, we set Ω := (0, 1) × (0, 1) with ΓD := {0} × (0, 1), ΓC := {1} × (0, 1) and ΓN := (0, 1) × ({0} ∪ {1}). The Young modulus and Poisson ratio are respectively EY := 10 6 and ν := 0.3. The rhs source term is given by f := (0, -76518). The Dirichlet and Neumann boundary conditions are given, respectively, by: uD := 0, and σ(u)n = 0

The friction coefficient F , defined on ΓC , is F := 0.2. Note that in this case we do not dispose of an analytical solution yet we will use as reference a P1 solution computed on a highly uniform refined mesh of 800 • 000 triangles. The computed solution, obtained with the symmetric Nitsche method and linear Lagrange finite elements is depicted Figure 5, Figure 6 and Figure 7. The results are in good agreement with those of [START_REF] Lleras | Modélisation, analyse et simulation de problèmes de contact en mécanique des solides et des fluides[END_REF] obtained with a mixed method. Table 13 presents the a priori error in L 2 and H 1 norms with their respective convergence rates, for the skew-symmetric variant (θ = -1). The computed rates are slightly suboptimal (around 1.6 in the L 2 norm and 0.7 in the H 1 norm, in comparison with the expected optimal rates of 2 and 1, respectively). Note that, for Coulomb friction, the best theoretical a priori error bounds are in O(h 1 2 ) [START_REF] Hild | An error estimate for the Signorini problem with Coulomb friction approximated by finite elements[END_REF], for mixed methods, but it is not known still if these theoretical rates can be improved or not. For Nitsche method, there is no theoretical a priori error estimate available for the moment. The limitation in the convergence rate may come from the limited regularity of the solution. For comparison purpose, rates of 1.12 and 0.95 were obtained in [START_REF] Chouly | Nitsche-based finite element method for contact with Coulomb friction[END_REF], but for a different test case with a solution expected to be smoother.

The different components of the estimator are reported Table 14. Suboptimal rates can be observed, and a global convergence rate of approximately 0.7 can be deduced for the overall estimator as well as for the components η2 associated with jumps on internal and Neumann edges. Notably the components η3 and η4 associated with friction and contact have a low, suboptimal rate of 0.5. This is in contrast with previous results for Tresca friction, were superconvergence was observed. Moreover, in this situation the relative magnitude of the components associated with contact and friction is high. The convergence rates are close to those of the a priori error in the H 1 norm, and the efficiency remains poor. Remark finally that the rate of 0.7 is in agreement with the rate reported in [START_REF] Hild | Residual error estimators for Coulomb friction[END_REF] and [START_REF] Lleras | Modélisation, analyse et simulation de problèmes de contact en mécanique des solides et des fluides[END_REF] for a mixed method and another error estimator. It seems to indicate a limited regularity of the continuous solution.

Results for the incomplete variant θ = 0 are reported Table 15 and Table 16. Conclusions similar as in the previous case θ = -1 can be drawn, except that better overall convergence rates can be observed, with, for instance, a rate slightly above 0.8 for the H 1 -norm. Surprisingly, the components of the estimator η3 and η4 associated with friction and contact converge with rates much better than in the case θ = -1, with even a superconvergence for η3. Results for the symmetric variant θ = 1 are reported Table 17 and Table 18, and are very similar to those obtained for the skew-symmetric variants. Finally, we carried out tests for larger values of the friction coefficient F = 0.4, 0.6, 0.8. We do not report the details but the same conclusions hold as for the case F = 0.2. Particularly, the difference of convergence rates for the components η3 and η4, for the specific value θ = 0 in comparison to the variants θ = -1, 1 is persisting. Table 18: Components of the error estimator η for a uniform refinement and their respective convergence order. Polynomial order k = 1. Nitsche parameters γ0 = 10λ and θ = 1.

h u -u h 0,Ω r 2 |u -u h | 1,Ω r s1 u -u h 1,

Figure 1 :

 1 Figure 1: Problem with a manufactured solution. Mesh refinement. Initial mesh (left) and final adapted mesh (right). Polynomial order k = 1 and symmetric variant (θ = 1).

Figure 2 :

 2 Figure 2: Problem with a manufactured solution. Solution for the initial mesh (left), for final adapted mesh (center), and exact solution (right). Polynomial order k = 1 and symmetric variant (θ = 1).

Figure 3 :

 3 Figure 3: Test from Bostan & Han. Initial mesh (left) and final adapted mesh (right). Polynomial order k = 1. Skew-symmetric Nitsche method (θ = -1).

Figure 4 :

 4 Figure 4: Test from Bostan & Han. Solution obtained using the final adapted mesh (left) and deformed final adapted mesh (right). Polynomial order k = 1. Skew-symmetric Nitsche method (θ = -1).

Figure 5 :

 5 Figure 5: Coulomb friction. Magnitude of the computed solution. Polynomial order k = 1 and symmetric Nitsche (θ = 1).

Figure 6 :

 6 Figure 6: Coulomb friction. Deformed mesh obtained using the computed solution. Polynomial order k = 1 and symmetric Nitsche (θ = 1).

Figure 7 :

 7 Figure 7: Coulomb friction. Normal displacement (blue) and tangential displacement (red) on the contact boundary ΓC . Polynomial order k = 1 and symmetric Nitsche (θ = 1).

  as the set of interior edges/faces of T h . We denote by E N For an element T , we denote by ET the set of edges/faces of T and according to the above notation, we set E int T

h := {E ∈ E h : E ⊂ ΓN } the set of Neumann edges/faces and similarly E C h := {E ∈ E h : E ⊂ ΓC } is the set of contact edges/faces. •

Table 1 :

 1 Problem with a manufactured solution. Errors for a uniform refinement. Polynomial degree k = 1. Nitsche parameters γ0 = λ and θ = -1.

	1	η	reta	E

Table 2 :

 2 Problem with a manufactured solution. Components of the estimator η and their respective convergence order. Polynomial degree k = 1. Nitsche parameters γ0 = λ and θ = -1.

	h	u -u h 0,Ω	r 2	|u -u h | 1,Ω	r s1	u -u h 1,Ω	r 1	η	reta	E
	0.314	0.009256	-	0.190779	-	0.191004	-	122.42	-	640.9
	0.157	0.001242	2.89	0.055794	1.77	0.055808	1.77	26.89	2.18	481.8
	0.078	0.000162	2.94	0.016672	1.74	0.016673	1.74	7.54	1.83	452.3
	0.039	0.000021	2.97	0.004534	1.87	0.004534	1.87	2.33	1.69	515.2
	0.019	0.000002	3.18	0.001023	2.14	0.001023	2.14	0.70	1.72	690.7
	0.009	0.000000	3.41	0.000198	2.36	0.000198	2.36	0.19	1.83 1000.4
	0.004	0.000000	3.50	0.000036	2.45	0.000036	2.45	0.05	1.91 1452.6

Table 3 :

 3 Problem with a manufactured solution. Errors for a uniform refinement. Polynomial order k = 2. Nitsche parameters γ0 = λ and θ = -1.

	h	η 1	r 1	η 2	r 2	η 3	r 3	η 4	r 4	η	rη
	0.314 116.60	-	37.28	-	0.055925	-	1.533677	-	122.42	-
	0.157	25.37	2.19	8.88	2.06 0.020291 1.46 0.406656 1.91	26.89	2.18
	0.078	7.20	1.81	2.24	1.98 0.003910 2.37 0.086384 2.23	7.54	1.83
	0.039	2.21	1.70	0.74	1.59 0.000838 2.22 0.015668 2.46	2.33	1.69
	0.019	0.65	1.75	0.25	1.52 0.000146 2.52 0.002847 2.46	0.70	1.72
	0.009	0.18	1.85	0.07	1.70 0.000021 2.77 0.000526 2.43	0.19	1.83
	0.004	0.04	1.93	0.02	1.85 0.000003 2.79 0.000096 2.45	0.05	1.91

Table 5 :

 5 Problem with a manufactured solution. Errors for a uniform refinement. Polynomial order k = 1. Nitsche parameters γ0 = λ and θ = 0.

	h	η 1	r 1	η 2	r 2	η 3	r 3	η 4	r 4	η	rη
	0.314 1352.65	-	1825.32	-	1.763916	-	115.923480	-	2274.84	-
	0.157	680.34	0.99	997.98	0.87 0.732652 1.26	47.342641	1.29 1208.74 0.91
	0.078	340.44	0.99	535.70	0.89 0.163662 2.16	18.499760	1.35	634.99	0.92
	0.039	170.25	0.99	287.83	0.89 0.049425 1.72	7.065624	1.38	334.49	0.92
	0.019	85.13	0.99	152.22	0.91 0.020210 1.29	2.609322	1.43	174.43	0.93
	0.009	42.56	0.99	78.59	0.95 0.006700 1.59	0.948872	1.45	89.38	0.96
	0.004	21.28	0.99	39.88	0.97 0.001919 1.80	0.342196	1.47	45.21	0.98

Table 7 :

 7 Problem with a manufactured solution. Errors for a uniform refinement. Polynomial order k = 2. Nitsche parameters γ0 = λ and θ = 0.

	314	0.009667	-	0.191831	-	0.192074	-	124.93	-	650.47
	0.157	0.001273	2.92	0.055813	1.78	0.055827	1.78	26.98	2.21	483.38
	0.078	0.000165	2.94	0.016654	1.74	0.016655	1.74	7.55	1.83	453.35
	0.039	0.000021	2.96	0.004530	1.87	0.004530	1.87	2.33	1.69	515.95
	0.019	0.000002	3.16	0.001023	2.14	0.001023	2.14	0.70	1.72	691.21
	0.009	0.000000	3.35	0.000198	2.36	0.000198	2.36	0.19	1.83 1000.83
	0.004	0.000000	3.38	0.000036	2.45	0.000036	2.45	0.05	1.91 1452.91
	h	η 1	r 1	η 2	r 2	η 3	r 3	η 4		r 4	η	rη
	0.314 118.99	-	36.93	-	0.070056	-	9.276328	-	124.93	-
	0.157	25.51	2.22	8.69	2.08 0.021140 1.72 1.248022 2.89	26.98	2.21
	0.078	7.21	1.82	2.22	1.96 0.003731 2.50 0.218437 2.51	7.55	1.83
	0.039	2.21	1.70	0.74	1.58 0.000799 2.22 0.042261 2.36	2.33	1.69
	0.019	0.65	1.75	0.25	1.51 0.000138 2.52 0.008389 2.33	0.70	1.72
	0.009	0.18	1.85	0.07	1.70 0.000020 2.76 0.001601 2.38	0.19	1.83
	0.004	0.04	1.93	0.02	1.85 0.000003 2.78 0.000294 2.44	0.05	1.91

Table 9 :

 9 Problem with a manufactured solution. Errors for a uniform refinement. Polynomial order k = 1. Nitsche parameters γ0 = 4λ and θ = 1.

	h	η 1	r 1	η 2	r 2	η 3	r 3	η 4	r 4	η	rη
	0.314 1349.43	-	1856.37	-	1.862072	-	228.896586	-	2306.40	-
	0.157	679.94	0.98 1001.31 0.89 0.682431 1.44	86.362286	1.40 1213.42 0.92
	0.078	340.39	0.99	535.91	0.90 0.126684 2.42	32.581834	1.40	635.71	0.93
	0.039	170.25	0.99	287.81	0.89 0.037613 1.75	11.864399	1.45	334.61	0.92
	0.019	85.13	0.99	152.20	0.91 0.014332 1.39	4.211949	1.49	174.44	0.93
	0.009	42.56	0.99	78.58	0.95 0.004429 1.69	1.493011	1.49	89.38	0.96
	0.004	21.28	0.99	39.88	0.97 0.001218 1.86	0.530160	1.49	45.21	0.98

Table 10 :

 10 Problem with a manufactured solution. Components of the estimator η and their respective convergence order. Polynomial order k = 1. Nitsche parameters γ0 = 4λ and θ = 1.

	h	u -u h 0,Ω	r 2	|u -u h | 1,Ω	r s1	u -u h 1,Ω	r 1	η	reta	E
	0.314	0.024923	-	0.278289	-	0.279403	-	153.61	-	549.79
	0.157	0.005953	2.06	0.080538	1.78	0.080757	1.79	28.79	2.41 356.56
	0.078	0.001476	2.01	0.021749	1.88	0.021799	1.88	7.74	1.89 355.44
	0.039	0.000370	1.99	0.005696	1.93	0.005708	1.93	2.37	1.70 415.87
	0.019	0.000093	1.99	0.001339	2.08	0.001342	2.08	0.71	1.73 531.84
	0.009	0.000023	1.99	0.000293	2.19	0.000294	2.18	0.19	1.84 677.35
	0.004	0.000006	2.00	0.000065	2.17	0.000065	2.17	0.05	1.92 806.11

Table 11 :

 11 Problem with a manufactured solution. Errors for a uniform refinement. Polynomial order k = 2. Nitsche parameters γ0 = 6λ and θ = 1.

	h	η 1	r 1	η 2	r 2	η 3	r 3	η 4	r 4	η	rη
	0.314 141.37	-	51.50	-	0.133297	-	30.925213	-	153.61	-
	0.157	26.60	2.40 10.30 2.32 0.026320 2.34	3.915698	2.98	28.79	2.41
	0.078	7.32	1.86	2.43	2.07 0.005198 2.34	0.687062	2.51	7.74	1.89
	0.039	2.23	1.70	0.77	1.66 0.001231 2.07	0.161259	2.09	2.37	1.70
	0.019	0.66	1.75	0.26	1.55 0.000237 2.37	0.039925	2.01	0.71	1.73
	0.009	0.18	1.86	0.07	1.72 0.000036 2.73	0.008839	2.17	0.19	1.84
	0.004	0.04	1.93	0.02	1.86 0.000005 2.90	0.001740	2.34	0.05	1.92

Table 13 :

 13 Coulomb friction. Errors for a uniform refinement and their respective convergence orders. Polynomial order k = 1. Nitsche parameters γ0 = 10λ and θ = -1.

	Ω	r 1

Table 14 :

 14 Coulomb friction. Components of the error estimator η for a uniform refinement and their respective convergence order. Polynomial order k = 1. Nitsche parameters γ0 = 10λ and θ = -1.

	h	u -u h 0,Ω	r 2	|u -u h | 1,Ω	r s1	u -u h 1,Ω	r 1
	0.353	0.010154	-	0.028469	-	0.030225	-
	0.176	0.004236	1.26	0.013869	1.03	0.014502	1.05
	0.088	0.001490	1.50	0.006825	1.02	0.006986	1.05
	0.044	0.000523	1.51	0.003757	0.86	0.003793	0.88
	0.022	0.000177	1.56	0.002153	0.80	0.002160	0.81
	0.011	0.000059	1.58	0.001239	0.79	0.001240	0.80
	0.005	0.000017	1.75	0.000688	0.84	0.000689	0.84

Table 15 :

 15 Coulomb friction. Errors for a uniform refinement and their respective convergence order. Polynomial order k = 1. Nitsche parameters γ0 = 10λ and θ = 0.

	h	η 1	r 1	η 2	r 2	η 3	r 3	η 4	r 4	η	r
	0.353 27053.19	-	60948.11	-	6567.94	-	10762.72	-	67863.99	-
	0.176 13526.59 1.00 40792.63 0.57 2755.17 1.25	2498.26	2.10 43137.45 0.65
	0.088	6763.29	1.00 24828.46 0.71 1063.29 1.37	629.45	1.98 25762.79 0.74
	0.044	3381.64	1.00 14611.70 0.76	385.29	1.46	334.65	0.91 15006.59 0.77
	0.022	1690.82	1.00	8537.63	0.77	149.61	1.36	247.77	0.43	8708.26	0.78
	0.011	845.41	1.00	5016.40	0.76	55.82	1.42	138.35	0.84	5089.32	0.77
	0.005	422.70	1.00	2970.76	0.75	22.24	1.32	66.85	1.04	3001.51	0.76

Table 16 :

 16 Components of the error estimator η for a uniform refinement and their respective convergence order. Polynomial order k = 1. Nitsche parameters γ0 = 10λ and θ = 0.

	h	u -u h 0,Ω	r 2	|u -u h | 1,Ω	r s1	u -u h 1,Ω	r 1
	0.353	0.010008	-	0.028979	-	0.030658	-
	0.176	0.004328	1.20	0.015067	0.94	0.015677	0.96
	0.088	0.001600	1.43	0.007955	0.92	0.008114	0.95
	0.044	0.000577	1.47	0.004657	0.77	0.004693	0.78
	0.022	0.000202	1.51	0.002856	0.70	0.002863	0.71
	0.011	0.000068	1.57	0.001767	0.69	0.001769	0.69
	0.005	0.000018	1.90	0.001065	0.73	0.001066	0.73

Table 17 :

 17 Coulomb friction. Errors for a uniform refinement and their respective convergence order. Polynomial order k = 1. Nitsche parameters γ0 = 10λ and θ = 1.

	h	η 1	r 1	η 2	r 2	η 3	r 3	η 4	r 4	η	r
	0.353 27053.19	-	63388.12	-	8399.56	-	15577.33	-	71155.71	-
	0.176 13526.59 1.00 42236.24 0.58 3446.07 1.28 10013.60 0.63 45596.23	0.64
	0.088	6763.29	1.00 25992.38 0.70 1557.03 1.14	6491.10	0.62 27674.99	0.72
	0.044	3381.64	1.00 15545.69 0.74	788.78	0.98	4487.18	0.53 16548.75	0.74
	0.022	1690.82	1.00	9299.40	0.74	484.59	0.70	3170.96	0.50	9981.36	0.72
	0.011	845.41	1.00	5640.33	0.72	324.71	0.57	2244.53	0.49	6137.70	0.70
	0.005	422.70	1.00	3480.77	0.69	225.48	0.52	1588.37	0.49	3855.93	0.677

Acknowledgements R.A. was partially supported by ANID-Chile through the projects: Centro de Modelamiento Matemático (FB210005) of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal, and Fondecyt Regular No 1211649 F.C. work was partially supported by the I-Site BFC project NAANoD and the EIPHI Graduate School (contract ANR-17-EURE-0002). F.C. is grateful of the Center for Mathematical Modeling grant FB20005.