DOI:10.1051/m2an/2012064 www.esaim-m2an.org
QUADRATIC FINITE ELEMENTS WITH NON-MATCHING GRIDS FOR THE UNILATERAL BOUNDARY CONTACT
S. Auliac
1, Z. Belhachmi
2, F. Ben Belgacem
3,4and F. Hecht
5Abstract. We analyze a numerical model for the Signorini unilateral contact, based on the mortar method, in the quadratic finite element context. The mortar frame enables one to use non-matching grids and brings facilities in the mesh generation of different components of a complex system. The convergence rates we state here are similar to those already obtained for the Signorini problem when discretized on conforming meshes. The matching for the unilateral contact driven by mortars preserves then the proper accuracy of the quadratic finite elements. This approach has already been used and proved to be reliable for the unilateral contact problems even for large deformations. We provide however some numerical examples to support the theoretical predictions.
Mathematics Subject Classification. 35J85, 65N30, 74M15.
Received May 11, 2012. Revised September 24, 2012.
Published online June 17, 2013.
1. Introduction
High-order finite elements are more and more relevant for complex systems since they allow for large compu- tations with increased accuracy. Engineers of unilateral contact may be interested in as an afforded response to capture small structures in their numerical simulations. Letting apart the linear finite element to which a wide literature is devoted. We refer to the books [22,31,44] as classical references, then to [12] for earliera priorierror estimates and to [24,29,35] for more recent studies which improve former results in the literature. Conducting the convergence analysis of the finite element approximation applied to unilateral problems turns out to be a pain in the neck, presumably because they are expressed by variational inequalities. Unlike the linear variational equations for which a general theory of the finite element methods has been developed, specific tools should be used for the unilateral contact inequality according to the degree of the finite elements. Here, we are interested in the Signorini boundary contact. Some of the mixed finite elements that are performing tools to efficiently handle the numerical locking caused by the nearly incompressible materials in elasticity, are based on quadratic elements (see [11]). One may think for example of Taylor-Hood elements already used for the unilateral contact (see [6,28,40]). These all are as many reasons why it is worth to study quadratic finite elements for the unilateral
Keywords and phrases.Unilateral contact conditions, quadratic finite elements, non-matching grids, mortar matching.
1 LJLL, C.N.R.S. Universit´e Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France.
2 LMIA, EA CNRS, Universit´e deq Haute Alsace, Rue des Fr`eres Lumi`ere, 68096 Mulhouse, France.
3 LMAC, EA 2222, Universit´e de Technologie de Compi`egne, BP 20529, 60205 Compi`egne Cedex, France.
4 I2M (UMR CNRS 5295), Site ENSCBP, 16 Avenue Pey Berland, 33607 Pessac Cedex, France.
5 LJLL, C.N.R.S. Universit´e Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France.
Article published by EDP Sciences c EDP Sciences, SMAI 2013
contact. We are involved in non-matching meshes and in the use of the mortar method. Deep attention has been successfully devoted to a single mesh context in Signorini’s problem or in the case of compatible grids that is the meshes within each component of the system to handle numerically coincide at the interfaces. Detailed studies with optimal convergence estimates are available for those finite element methods with matching grids (see [4]). When the meshes are constructed independently, resulting in dissimilar grids that do not match at interfaces, a substantial implementation and validation work have been achieved (see [20,21,30,34,37,38]) either in unilateral or bilateral contact with or without friction. We recommend in particular the interesting survey provided in [33]. In the numerical analysis chapter, only few attempts have been realized for the unilateral configurations (see [27,30]). We pursue here a study of some numerical modelings of unilateral contact in the frame of the mortar-matching for finite elements with degree two, applied to the primal variational Signorini Inequality. A literature exists on the linear mortar finite element method for the Signorini problem. We refer to [3,5,25,26,42] without being exhaustive. Users are recommended not to enforce the unilateral conditions on the discrete solution before running the mortar projection. Such a mortar matching, first introduced in [8], allows to express the unilateral inequalities, in the discrete level, on traces built on the same grid and avoids by then possible instabilities along interfaces. It is nowadays widely known that writing unilateral inequalities directly on the approximated solution at the interfaces as done in the classical node-to-segment treatments, generates low numerical accuracy. We refer to [21,32,43] for a worthy discussion about the efficiency of the mortar method in interface contact mechanics.
Our willing is to focus only on the mortaring and technical issues related to, for conciseness. We exclude therefore any other features which are doubtless as important in unilateral contact and that are either already successfully handled or under consideration in many research teams. We pay all our attention to the mortar projections along interfaces of dissimilar grids, to the way the unilateral contact is enforced at the discrete level and finally to show how the combination of both yields optimal convergence results, owing to Falk’s Lemma.
The outline of the paper is as follows. Section 2 is a presentation of some models where unilateral contact is involved. We describe briefly the static problem of the unilateral contact between two elastic bodies, the solution to be computed is the displacement field. We recall also the unilateral conditions in the case of the Laplace equation. The regularity issue for the solution of the Signorini-Laplace problem is discussed at the end of the section. In Section3, we introduce the mortar framework for the quadratic finite elements and the unilateral conditions are then expressed at the nodes of the interfaces. After recalling a suitable version of Falk’s Lemma, we dedicate Section4to the numerical analysis of the mortar quadratic finite element method for the Signorini- Laplace problem. The convergence rates we exhibit in our main result, Theorem 4.1, are similar to the case where matching grids are used. As a result, using incompatible grids in the simulation of unilateral contact does not degrade the accuracy when mortar-matching drives the communication between different meshes. The final section is dedicated to some numerical experiences to check the theoretical predictions. Lastly, in Section5, we describe and comment some experiences realized inFreefem++to assess the theoretical predictions proved here.
Some notations.LetC(Ω) be the space of real valued continuous function on a given domainΩ. The Lebesgue space of square integrable functionsL2(Ω) is endowed with the natural inner product (·,·)L2(Ω); the associated norm is · L2(Ω). The Sobolev spaceH1(Ω) involves all the functions that are inL2(Ω) so as their first order derivatives. It is provided with the norm · H1(Ω)and the semi-norm is denoted by| · |H1(Ω). For any portion of the boundaryΥ ⊂∂Ω, the spaceH01(Ω, Υ) contains all the functions ofH1(Ω) that vanish onΥ. We recall that, for anyθ∈]0,1[, the Sobolev spaceHθ(Υ) can be obtained by a Hilbertian interpolation betweenH1(Υ) andL2(Υ) andH−θ(Υ) is the dual space (Hθ(Υ)). Notice finally that any functionv∈H01(Ω, ∂Ω\Υ) whose Laplacian is inL2(Ω) has a normal derivative (∂nv)|Υ that belongs toH−1/2(Υ). We refer to [1] for a detailed study of the fractional Sobolev spaces.
2. Examples of unilateral conditions
A large number of thermal, mechanical or electrical devices can be modeled by unilateral boundary conditions.
We refer the reader to the book by Duvaut and Lions [17] for various examples. In fact the two examples we describe here are picked up from that reference.
ΓC
Ω Ωs
i
Figure 1. Two solids are in unilateral contact alongΓC. Inter-penetration is not authorized.
2.1. Unilateral contact between two solids
Let us consider two elastic solids occupying the domains Ωs and Ωi in their initial unconstrained configu- rations. They are assumed in contact with each other along a common part of their boundaries (∂Ωs∩∂Ωi).
These boundaries ∂Ωs and ∂Ωi are supposed regular and the unit external normal is denoted bynwhen no confusion is feared or ni and ns for the corresponding bodies. Each of∂Ω (=i or s) is divided into three portionsΓD, ΓN etΓC (see Fig.1). The solidΩis fixed alongΓD and is subjected to a traction or compression forceg(∈L2(ΓN)) onΓN and to a volume forcef(∈L2(Ω)), which is most often its own weight. Both solids share a common zoneΓC=ΓCs =ΓCi, candidate to an effective contact.
The unilateral contact problem consists in :finding a displacement fields u= (u)= (u|Ωi,u|Ωs)satisfying the boundary value problem
−divσ(u) =f in Ω, (2.1)
u= 0 on ΓD, (2.2)
σ(u)n=g on ΓN. (2.3)
The bold symboldivdenotes the divergence operator of a tensor field and is defined asdivσ= ∂σkr
∂xr
k
·The bodies are made of Hook type material whose constitutive laws are given byσ(u) =A(x) (u), whereA(x) is a fourth order symmetric and elliptic tensor. Frictionless unilateral contact conditions enforced onΓC allow to close the system
(σini)·ni= (σsns)·ns=σn, (2.4) [u·n]≤0, σn≤0, σn[u·n] = 0, (2.5)
σti=σts= 0. (2.6)
The notation [u·n] = (ui·ni+us·ns) stands for the jump of the normal displacements contact acrossΓC. Condition (2.4) expresses the Newton action and reaction principle, (2.6) indicates that the contact occurs without friction. Most often, the modeling of the contact condition is formulated using a gap functionζdefined on ΓC, so that instead of [u·n] ≤0 and of the saturation condition σn[u·n] = 0 we have [u·n] ≤ζ and σn([u·n]−ζ) = 0 on the contact zone ΓC (see [17]). As the whole subsequent analysis can be extended straightforwardly to the case whereζdoes not vanish, we choose, only for conciseness,ζ= 0. In what follows we will writeσforσ(u). The analysis is limited to infinitesimal deformations (small perturbations) and therefore
to the case of linear elasticity where the strain tensor produced by the displacementv is (v) =1
2(∇v+ (∇v)T).
In order to derive the variational formulation of the unilateral contact problem (2.1)–(2.6) we need the spaces X(Ω) =H10(Ωi, ΓDi)×H10(Ωs, ΓDs).
This space is endowed with the broken semi-norm:∀v∈X(Ω),
|v|∗,H1 =
(vi)2L2(Ωi)+ (vs)2L2(Ωs)
1/2
.
which is, because of the Dirichlet condition and Korn’s inequality, a norm equivalent to the broken norm.∗,H1. Then, the appropriate closed convex setK(Ω) of admissible displacements is defined by
K(Ω) =
v= (vi,vs)∈X(Ω), ([v·n])|ΓC ≤0
.
The weak formulation results in a variational inequality that reads as:findu∈K(Ω)such that
a(u,v−u)≥F(v−u), ∀v∈K(Ω). (2.7) In (2.7), we set:∀u,v∈X(Ω)
a(u,v) =
Ωi
Ai(x) (ui) : (vi) dx+
Ωs
As(x) (us) : (vs) dx, F(v) =
Ωi
fi·vidx+
Γgi
gi·vi dΓ+
Ωs
fs·vsdx+
Γgs
gs·vsdΓ,
The existence and uniqueness of u ∈ K(Ω) which solves problem (3.5) result from Stampacchia’s Theorem (see [22,31]).
2.2. The Signorini–Laplace problem
Consider the domain Ω as the union of two non-intersecting components Ωs and Ωi. Assume that both sub-domains share a common boundaryΓC. Then, denote byΓNs andΓNi the remaining parts of the boundaries ofΩs andΩi, respectively. We setΓN =ΓNs ∪ΓNi and Γ=ΓC∪ΓN for=i, s. Finally, let nC (nif there is no risk of confusion) be the unit normal toΓC oriented fromΩi towardΩs.
Assume now that g = (gs, gi)∈ L2(ΓN) and f ∈L2(Ω). The unilateral contact problem we interested in consists in findingp= (pi, ps), satisfying the following equations:
p−div(a∇p) =f in Ω, =i, s, (2.8)
(a∂n)p=g on ΓN, =i, s. (2.9) The coefficient a= (ai, as) lies inL∞(Ω). It is such that 0< a∗ ≤a(x)≤a∗ <∞,∀x∈Ω. To complete the model, unilateral conditions should be provided alongΓC. They read as
(ai∂n)pi= (as∂n)ps, (2.10)
[p] =pi−ps≥0, (ai∂n)pi≥0, (ai∂n)pi[p] = 0. (2.11) We aim at a variational framework fitted to the problem (2.8)–(2.11). It is provided by the broken space
H∗1(Ω) =H1(Ωi)×H1(Ωs).
The broken norm is denoted by · ∗,H1. Taking into account the unilateral contact condition onΓC in the weak formulation is ensured by incorporating it in the closed convex cone
K(Ω) =
q∈H∗1(Ω), ([q])|ΓC = (qi−qs)|ΓC ≥0
.
The primal variational principle applied to problem (2.8)–(2.11) produces the variational inequality consisting in:findingp∈K(Ω)such that:
a(p, q−p)≥F(q−p), ∀q∈K(Ω). (2.12) In (2.12), we have set
a(p, q) =
Ω
pqdx+
Ωi
ai(∇p)(∇q) dx+
Ωs
as(∇p)(∇q) dx, (2.13)
F(v) =
Ω
f qdx+g , q1/2,Γ. (2.14)
Applying here again Stampacchia’s theorem (see [22]), we state that the variational inequality (2.12) is well posed and has only one solution inp∈K(Ω) that depends continuously on the data (f, g).
Remark 2.1. In the variational formulation, the mathematical sense given to conditions (2.11) is as follows (ai∂n)p , μ1/2,ΓC ≥0, ∀μ∈H1/2(ΓC), μ≥0,
(ai∂n)pi , [p]1/2,ΓC = 0.
Roughly, the first formula says (ai∂n)pi ≥ 0 on ΓC while the second expresses the saturation condition (ai∂n)pi[p] = 0 onΓC.
Remark 2.2. Variational inequality (2.12) can be viewed as the optimality condition of a quadratic minimiza- tion problem on the closed convexK(Ω). Hence, the solutionp∈K(Ω) is the one that satisfies
1
2a(p, p)−F(p) = min
q∈K(Ω)
1
2a(q, q)−F(q). (2.15)
2.3. About singularities
The numerical analysis of any finite element method applied the unilateral contact problem (2.12) requires the knowledge of the regularity of the solutionp= (pi, ps). Since the work by Moussaoui and Khodja (see [36]), it is admitted that the unilateral condition may generate some singular behavior at the neighborhood ofΓC. We assumeΓC is the union of regular portions connected to each other through some corners. Some singularities are generated by the angular interfaces and some others are specifically created by the contact condition. We will assume thata= 1, this is the most easy case to chack out for the regularity of the Signorini solution.
Let us first look at the portions of the unilateral boundary that are straight-lines. The unilateral contact on ΓC creates some singularities caused by the points where the contact changes from binding6 (ps = pi) to non-binding (pi > ps). Let m be one of those points. Considering the global pressure p = (pi, ps) at the neighborhood of m, we have the transmission conditions (pi, ∂npi) = (ps, ∂nps) along the binding portion of ΓC. As a result, pis harmonic at the neighborhood of that binding boundary. Things happen as if we were involved in a Poisson equation around m and the non-binding part of ΓC acts like an insulating crack. The shape of the singularities affecting both (ps, pi) are
Ski(r, θ) =αk ϕ(r)rk+1/2 cos(k+ 1/2)θ, 0≤θ≤π, Sks(r, θ) =αk ϕ(r)rk+1/2 cos(k+ 1/2)θ, π≤θ≤2π.
6We adopt here the terminology from the solid mechanics.
The polar coordinates (r, θ) are used with origin m and ϕis a cut-off function around m. The effective con- tact takes place for θ = π and the pressure p is discontinuous at θ = 0(2π). Writing the unilateral condi- tions (2.10)–(2.11) at both sides ofm
Ski(r,0)− Sks(r,2π)>0, ∂nSki(r, π) =∂nSks(r, π)≥0
producesαk(1−(−1)k)>0 and (−1)k+1αk ≥0. This yields thatk is necessarily odd and αk >0. Then only the oddk-singularities are involved in the expansion ofp, produces that
p(r, θ) =preg(r, θ) +
k
S2k+1(r, θ).
We expect therefore that p ∈ Hσ(Vm) for any σ < 52 and Vm is a an open set containing m. Furthermore, if the first singularities (S1) are switched off then the next ones (S3) command the regularity of (pi, ps), at least at the neighborhood of ΓC and one obtains that p ∈ Hσ(Vm) for any σ < 92· In fact, the point to be fixed once for all is that the possible values of the limit Sobolev regularity index increases by jumps of length two, it equals to 5/2,9/2,· · ·. Now, let us focus on the corners. At a vertexm with an aperture of the angle ω ∈]0,2π[7, three situations are possible. Either, effective contact holds along both edges of the sector, and the global solution (pi, ps) is blind to the interface and then non singularities arise. The second situation is where effective contact takes place along one edge while on the other we have no contact. This configuration is a reminiscence to the upper case of a straight-line. The binding edge has no role to play in the shape of the singularities arising around m. Only singularities like those discussed for a straight-line enter the asymptotic expansions of (pi, ps). It is noticeable that these first two possibilities for corners do not degrade the regularity of the solution (pi, ps), already limited by the changing of the contact conditions along a straight-line. The last case consists in the absence of contact on both sides (pi < ps). The homogeneous normal derivative of both (pi, ps) allows to compute the admissible singularities,
Ski(r, θ) =ηk+αk ϕ(r)rkπω |logr| coskπ
ω θ, 0≤θ≤ω, Sks(r, θ) =βk ϕ(r)r2π−ωkπ |logr| cos kπ
2π−ω(2π−θ) ω≤θ≤2π.
where ηk >0 is sufficiently large to ensure that pi > ps within the support ofϕ (at the neighborhood of the corner m). Moreover, the logarithm is there forSki only when kπω is an integer, idem for Sks and 2πkπ−ω· The regularity of the first singular functions (S1i,S1s) will fully decide of the smoothness of (pi, ps). This predicts that pi∈Hσ(VΓC) for anyσ <1 +πω andps∈Hσ(Vm) for anyσ <1 +2ππ−ω·Anyway, we obtainp∈H3/2(VΓC).
Would these singularities be switched off, there would be a jump on the regularity exponents, since 5/2< σ <
1 + min(2ωπ,2π2−πω) and so on.
3. The quadratic mortar finite element method
We pursue the numerical simulation of the unilateral boundary conditions by non-matching finite element grids. The main advantage is the possible generation of meshes well adapted to the local features of the do- main components and to the physical parameters. As said earlier, the convergence rate of the finite element approximation depends on the regularity of the solutionp. According the previous regularity discussion it may reasonably happen that p belongs to a more regular space than H3/2 or even than H5/2, at least near ΓC. Therefore, the approximation of the variational inequality (2.12) by affine finite elements fails to fully account for that regularity ofp(at least at the vicinity ofΓC). We are indeed limited byH2. Quadratic finite elements for the Signorini problem have been analyzed in [4,27]. Many factors command the accuracy of the discretization
7ωis the angle withinΩi. The angle inΩs is then (2π−ω).
such as the smoothness of ΓC, the regularity off and g, the wayΓC is approximated by a given mesh . . . , and the unilateral singularities alongΓC. Our scrutiny work is exclusively oriented toward the effect of this last point, and we do not pay attention to the other factors. They are all classical and have been considered in many works (see [9,15]). Assume then that the domainΩis polygonal so that it can be exactly covered by rectilinear finite elements and thatΓC is a union of straight-lines.
Towards the construction of the discrete space, we recall some fundamental notions. Let h= (hi, hs) be a given pair of real positive numbers that will decay towards zero. With eachΩ, we associate a regular family of triangulationsTh, with triangular elements, denoted byκ, whose diameter does not exceedh. The unilateral boundary ΓC inherits two independent meshes Th,C( =i, s) from Th. The mesh Th,C on ΓC is the set of all the edges of κ∈ Th on the contact zone. The nodes of Th,C are (xm)1≤m≤m∗, (tm =]xm,xm+1[)1≤m≤m∗ are the edges and the middle nodes oftmisxm+1/2. Denote byP2 the space of the polynomials onκwith a global degree≤2. With anyκ, we associate the finite setΞκof the vertices ofκ, so that (κ,P2, Ξκ) is a finite element of Lagrange type. We defineΞ=
κ∈ThΞκ.The finite element space inΩ is then Xh(Ω) =
qh ∈C(Ω), ∀κ∈ Th, (qh)|κ∈ P2
,
and Xh(Ω) =Xh(Ωi)×Xh(Ωs). The unilateral condition (2.11) in a discrete level should be driven carefully.
Comparing directlyqhi andqsh, built on different meshes, has to be avoided since it is known to create numerical troubles. It is rather recommended to realize it betweenqihand some projection ofqhson the opposite meshThi,C. We therefore need the traces space
Wh(ΓC) =
ϕh= (qh)|ΓC, qh ∈Xh(Ω)
,
which will be the mortar space for a fixed . Given thatΓC is a closed curve there is no need to differentiate the Lagrange Multipliers space and the mortar space as currently made (see [8]). The mortar projection π is defined onWh(ΓC) as follows. Letψ∈L2(ΓC), we haveπψ∈Wh(ΓC) and
ΓC
(ψ−πψ)χhdΓ = 0 ∀χh∈Wh(ΓC). (3.1)
The operatorπ is an orthogonal projection with respect to the norm ofL2(ΓC) and is therefore expected to satisfy an optimal error estimate with respect to the norm of that space. It is also mandatory that it yields an optimal estimate for the norm ofH1/2(ΓC). We hence add a mild assumption on the meshTh,C introduced in [16]: two arbitrary edgestmand tm satisfy
|tm|
|tm| ≤ηα|m−m|, (1≤m, m≤m∗),
where 1≤α <9 and η does not depend onhs.Th,C is then called an (M)-mesh8. As a result, the following stability holds:∀ψ∈H1/2(ΓC),
πψH1/2(ΓC)≤CψH1/2(ΓC). (3.2) The proof may be issued from the results in [16,39] and an interpolation artifice of linear operators in Hilbert spaces. Moreover,π satisfies some approximation results. Let μ ∈ [0,3[, the following approximation holds:
∀ψ∈Hμ(ΓC),
ψ−πψH−1/2(ΓC)+hψ−πψH1/2(ΓC)≤C(h)μ+1/2ψHμ(ΓC). (3.3) We proceed with the construction of the discrete convex cone. We have then to decide which side ofΓC plays the role of the mortar, let us say Ωi, the same ΓC as a part of ∂Ωs is hence the non-mortar. The purpose
8(M)-meshes authorizes the adaptativity contrary to the quasi-uniform meshes.
Figure 2.A test functionsψh. Observe that its affine at both extreme edges.
consists in enforcing the non-negativity on the values of (πs(qhi)−qhs) at the vertices ofΓC coming from theΩs side. The discrete closed convex cone is hence defined to be
Kh(Ω) =
qh= (qsh, qih)∈Xh(Ω), [qh]h(x) = (πs(qih)−qhs)(x)≥0, ∀x∈Ξs∩ΓC
.
The overall tools useful for the discrete variational inequality are available. The Ritz-Galerkin approximation reads as follows:findph∈Kh(Ω)such that
a(ph, qh−ph)≥F(qh−ph), ∀qh∈Kh(Ω). (3.4) The setKh(Ω) is obviously an external approximation ofK(Ω),i.e. Kh(Ω)⊂K(Ω), the mortar discretization is of course non-conforming. Checking that the discrete problem (3.4) has only one solution ph ∈Kh(Ω) is an easy matter from Stampacchia’s Theorem.
Remark 3.1. It is possible to splitΓCinto more than one mortar(resp. non-mortar), each edge ofΩi(resp.Ωs) may be considered as mortar(resp. non-mortar)by itself. The matching projection has to be particularized to the non mortars, whose union formsΓCas part ofΩs. Each of these projections should be defined in the specific mortaring way (see [8,41]). The test functionsψhinvolved in the definition of [8] has to be reshaped as depicted in Figure2. Nevertheless, for the issue under examination, the theory for the mortar method may be conducted similarly. We do not therefore consider this case in details.
4. Convergence of the Mortar method
The analysis of the accuracy of the approximation, as said earlier, will be specifically dedicated to the effect of unilateral condition, excluding the other factors such as the regularity of the data that have been widely investigated in the literature. Henceforth,f andg are taken regular. Moreover, for seek of simplicity we fix the parametera to unity, that is ai =as = 1. The forthcoming results are extended to the general case provided thatais smooth enough.
Relying on the discussion in Section 2.3, we will pay a particular attention to the case where the regularity exponent of the solution p is either 3/2 or 5/2. The proofs for σ = 3/2 can be reproduced, with a minor adaptation, when the regularity exponentσ lies in [1,3/2[ and those developed for σ = 5/2 can be extended to the case where σ ∈]2,5/2[ following the same lines. In the subsequent the generic constant C depends on the solution p= (pi, ps). Our main result in this paper, that we state here and prove in several steps, is the following.
Theorem 4.1. Let p= (pi, ps)∈K(Ω) be the solution of the variational inequality (2.12). Assume that p ∈ H3/2(Ω). Then we have that
p−ph∗,H1 ≤C (hi)1/2+ (hs)1/2 . Assume thatp∈H5/2(Ω). Then we have that
p−ph∗,H1 ≤C (hi)3/2+ (hs)3/2 . The constantC is dependent on pHσ(Ω).
Remark 4.2. The analysis of the intermediary caseσ∈]3/2,2] requires some more work. The important fact to point at is the necessity here of a technical assumption on the effective contact region that seems to be firstly introduced in [12]. The effective contact portion ofΓC is supposed to be the union of a finite number of segments. In addition, some Sobolev and Sobolev Murray embeddings combined to finite element approximation results Lebesgue spacesLr(r >2) are needed in the proof.
Remark 4.3. Notice that the issue of optimal convergence rates for higher regularity that isσ∈]5/2,3] is still open. This has nothing to do with the mortaring process. Such results are also missing for the Signorini problem where only a single mesh is involved.
4.1. Falk’s Lemma in the Mortar context
Deriving an estimate of the error (p−ph) with respect to the energy norm relies on a basic tool obtained from the Falk Lemma (see [5,19]).
Lemma 4.4. Let p∈K(Ω)be the solution of the variational inequality (2.12)andph∈Kh(Ω)be the solution of the discrete variational inequality (3.4). Then we have that
p−ph2∗,H1 ≤C
qh∈infKh(Ω)
p−qh2∗,H1+∂npi , [qh]1/2,ΓC
+ inf
q∈K(Ω)∂npi , [q−ph]1/2,ΓC
. (4.1) Remark 4.5. The last infimum is the consistency error, it is the “variational crime” and is due to the noncon- formity of the approximation.
4.2. The best approximation error
We pursue an optimal bound of the first infimum in the abstract Falk’s estimate (4.1). The following holds.
Proposition 4.6. Assume that p∈Hσ(Ω)withσ= 3/2 or5/2. Then we have that
qh∈infKh(Ω)
p−qh2∗,H1+∂npi , [qh]1/2,ΓC
≤C (hi)σ−1+ (hs)σ−12
. (4.2)
Proof. Observe first that [p] = (pi−ps) ∈ Hσ−1/2(ΓC). Using the lifting theorem (see [1]) we may find r ∈ Hσ(Ωs) such thatr|ΓC = [p]≥0 verifying the stability
rHσ(Ωs)≤C piHσ(Ωi)+psHσ(Ωs)
. (4.3)
Setting (˜pi,p˜s) = (pi, ps+r)∈Hσ(Ωi)×Hσ(Ωs), it is clear that [˜p] = 0 on ΓC and thus ˜p∈H1(Ωi∪Ωs).
We are therefore in the bilateral contact context. Proceeding as in [5] (see also [8]) we are able to build (˜qhi,q˜sh)∈Xh(Ωi)×Xh(Ωs) satisfying the mortar matching that is [˜qh]h= 0. Moreover, thanks to (3.2) there holds that
p−q˜h∗,H1 ≤C (hi)σ−1+ (hs)σ−1
. (4.4)
Now, letrh∈Xh(Ωs) be the Lagrange interpolant ofrand defineqh as follows
qh = (qih, qsh) = (˜qih,q˜sh−rh). (4.5) We check easily that, for allx∈Ξs∩ΓC, we have
[qh]h(x) =rh(x) =r(x) = [p](x)≥0.
As a result, we have thatqh∈Kh(Ω) and the following bound holds
p−qh∗,H1 ≤ p−q˜h∗,H1+r−rhH1(Ωs)≤Cp−q˜h∗,H1+C(hs)σ−1rHσ(Ωs)
Using (4.4) together with (4.3) yields
p−q˜h∗,H1 ≤C (hi)σ−1+ (hs)σ−1 .
The first part of the infimum (4.2) is bounded in an optimal way. There remains to estimate∂npi , [qh]1/2,ΓC
which is pretty longer. Furthermore, different technicalities are required according to the regularity exponent σ. We choose to provide the proofs into two separate lemmas that come just after. Once these are stated, the
proof of the proposition will be completed.
Lemma 4.7. Assume that p∈H3/2(Ω)and qh given as in (4.5). Then we have that ∂npi , [qh]1/2,ΓC ≤C (hi)1/2+ (hs)1/22
. Proof. From the saturation (∂npi)[p] = 0 onΓC, we derive that
∂npi, [qh]1/2,ΓC =∂npi, [˜qh]1/2,ΓC +∂npi , rh1/2,ΓC
=∂npi, [˜qh]1/2,ΓC +∂npi ,(rh−r)1/2,ΓC. (4.6) The first term is handled in a standard way (see [5,8]). There remains to bound the second one. Thatpi ∈ H3/2(Ωi) with (−Δ)pi∈L2(Ωi) results in (∂npi)∈L2(ΓC) (see [1]). We can then write that
∂npi ,(rh−r)1/2,ΓC =
ΓC
(∂npi)(rh−r) dΓ ≤ ∂npiL2(ΓC)r−rhL2(ΓC)
≤C(hs)∂npiL2(ΓC)rH1(ΓC)≤C(hs)piH3/2(Ωi)rH3/2(Ωs).
Using (4.3) withσ= 3/2, completes the proof.
Lemma 4.8. Assume that p∈H5/2(Ω)and qh given as in (4.5). Then we have that
∂npi , [qh]1/2,ΓC ≤C (hi)3/2+ (hs)3/22 .
Proof. The proof is a slightly more complicated than in the previous case. Again, the point is to bound the second term of (4.6). Notice that, due to the regularity ofp, we obtain
ΓC
(∂npi)(rh−r) dΓ =
ΓC
(∂npi)(rh−r) dΓ
=
m∗
i=1
tm
(∂npi)(r−rh) dΓ ≤
m∗
i=1
∂npiL2(tm)r−rhL2(tm).
Some indices in the sum may be removed. Are kept there only the indices m ∈ M˜ for which (∂npi)|ΓC ⊂ H1(ΓC)⊂C(ΓC) vanishes at least once in tm. Otherwise if (∂npi)|tm >0 then r|tm = [p]|tm = 0. This gives (rh)|tm = 0 and the associated integral is zero. Then, the inequality reduces to
ΓC
(∂npi)(rh−r) dΓ ≤C
m∈M˜
∂npiL2(tm)(hs)2|r|H2(tm).
Applying now Lemma 8.1 of [4] to ∂npi∈H1(ΓC), we obtain
ΓC
(∂npi)(rh−r) dΓ ≤
m∈M˜
C(hi)|∂npi|H1(tm)(hs)2|r|H2(tm)
≤C(hi)(hs)2
m∈M˜
|∂npi|2H1(tm)
1/2 m∈M˜
|r|2H2(tm)
1/2
≤C(hi)(hs)2|∂npi|H1(ΓC)|r|H2(ΓC).
The proof is achieved due to the estimate (4.3) withσ= 5/2.
