A VISCO-ELASTO-PLASTIC EVOLUTION MODEL WITH REGULARIZED FRACTURE

DOI:10.1051/cocv/2015005 www.esaim-cocv.org

A VISCO-ELASTO-PLASTIC EVOLUTION MODEL WITH REGULARIZED FRACTURE

Luk´ aˇ s Jakabˇ cin

Abstract. We study a model for visco-elasto-plastic deformation with fracture, in which fracture is approximatedviaa diffuse interface model. We show that a discretized (in time) quasistatic evolution, converges to a solution of the continuous (in time) evolution, proving existence of a solution to our model.

Mathematics Subject Classification. 49J40, 49J45, 74C10, 74R20.

Received June 25, 2014.

Published online October 6, 2015.

1. Introduction

This paper deals with a visco-elasto-plastic model with regularized fracture. The model predicting fracture is based on Griffith’s criterion [13] that crack path and crack growth are determined by the competition between the elastic energy and the energy dissipated to produce a crack. The variational approach to fracture mechanics and a mathematical model have been developed by Francfort and Marigo [12], based on this idea. This approach has been then adapted by Dal Maso and Toader [9] to the study of fracture problems in elasto-plastic materials with cracks in the case of planar small strain elasto-plasticity where the fracture is represented by the compact crack set Γ ⊂Ωverifying an irrevesibility condition.

In Larsen, Ortner, Suli [16] existence and convergence results are proved for a regularized model of dynamic brittle fracture based on the Ambrosio−Tortorelli approximation. This model couples an elastodynamic equation with regularized fracture. Babadjian, Francfort, Mora [3], and Babadjian, Mora [2] study the approximation of dynamic and quasistatic evolution problems in elasto-plasticityviaviscosity regularization.

The goal of this paper is to propose a model that takes into consideration three dissipative terms: plastic flow, fracture and viscous dissipation. This is motivated by the modeling of the Earth crust considered as a visco-elasto-plastic solid in which cracks are allowed to propagate. This hypothesis is qualitatevely supported by analogue 2D-experiments of Peltzer and Tapponnier [17] that show faults propagation in a layer of plasticine.

We study from mathematical point of view a model in R² of visco-elasto-plastic material that may account for the behaviour observed in the plasticine experiments, but our results extend to any dimension. The main objective is to understand by which mechanisms energy can be dissipated in such model.

Keywords and phrases.Plasticity, regularized fracture, viscous dissipation.

1 Laboratoire Jean Kuntzmann, 51 rue des Math´ematiques, Campus de Saint Martin d’H`eres, BP 53, 38041 Grenoble cedex 09, France.lukas.jakabcin@imag.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2015

A VISCO-ELASTO-PLASTIC EVOLUTION MODEL WITH REGULARIZED FRACTURE

In this model the fracture is obtainedviaAmbrosio−Tortorelli regularization. We only consider fracturevia a diffuse interface model. In other words, the geometry of possible cracks is captured by a functionvwith values between 0 and 1,v= 1 in the healthy parts that do not contain cracks. The length of the cracks, a quantity that contributes to the total energy, is approximatedviaa functional introduced by Ambrosio and Tortorelli [1]. In other words, the continuous model is obtained coupling visco-elasto-plastic behaviour with regularized fracture evolution of the model of Larsen, Ortner and Suli [16].

A convenience of a such model is the fact, that it can be studied numerically thanks to the presence of regularized fracture. For more details, see [6–8] for the numerical studies in elastic case and [4,15] for the numerical studies of our models in the case of traction and plasticine experiments.

In this paper, we propose the mathematical analysis of a such model via a semi-discrete time procedure.

We approximate a continuous time evolution, via semi-discrete time evolutions obtained solving incremental minimum problems. We then prove an existence result to the continuous model when a time discretization parameter converges to zero. The main difficulty is pass to the limit in the discrete plastic flow rule and discrete crack propagation condition. For this reason, we prove particularly a strong compactness result for elastic strain (Prop. 3.12). As in [2], we prove that the discrete-time elastic strain e⁺_h converges strongly in L²(0, T_f, L²(Ω,M^2×2_sym)), but the presence ofv in our model requires the analysis and control of some additional terms associated tov.

The paper is organised as follows. After a short introduction, Section 1 is devoted to the definitions, math- ematical and mechanical settings necessary to the description of our model. The main result is then presented in Section 3. Firstly, we prove the existence of solutions for discrete minimum problem in Proposition3.2and Proposition3.4. Then, we study the convergence of these approximate evolutions as the time steph→0. The main result of this paper is presented as follows: There exists at least one limit evolution (u, v, e, p) that satisfies initial and irreversibility conditions, the equilibrium equation, plastic flow rule and crack propagation condition (Thm. 3.1).

2. Formulation of the model 2.1. Preliminaries and mathematical setting

Throughout the paper,Ωis a bounded connected open set inR²with Lipschitz boundary∂Ω=∂Ω_D∪∂Ω_N where ∂Ω_D, ∂Ω_N are disjoint open sets in∂Ω and H¹(∂Ω_D)>0. GivenT_f >0, we denote by L^p((0, T_f), X), W^k,p((0, T_f), X), the Lebesgue and Sobolev spaces involving time (see [11], p. 285), where X is a Banach space.

We will usually write u(t) :=u(., t) foru∈W^k,p((0, T_f), X).

The set of symmetric 2×2 matrices is denoted by M^2×2_sym. For ξ, ζ ∈ M^2×2_sym we define the scalar product between matricesζ:ξ:=

ijζ_ijξ_ij, and the associated matrix norm by|ξ|:=√ξ:ξ. Let A be the fourth order tensor of Lam´e coefficients. We assume that for some constants 0< α₁≤ α₂<∞, they satisfy the ellipticity conditions

∀e∈M^2×2_sym, α₁|e|²≤Ae:e≤α₂|e|². Fore∈M^2×2_sym andx∈Ω, we define |e|²_A(x):=A(x)e:eande²_A:=

Ω|e|²_Adx.

We recall that the mechanical unknowns of our model are the displacement field u:Ω×[0, T_f] →R², the elastic straine:Ω×[0, T_f]→M^2×2_sym, the plastic strain p:Ω×[0, T_f]→M^2×2_sym. We assumeuand ∇uremain small. So that the relation bewteen the deformation tensorE and the displacement field is given by

Eu:= 1

2(∇u+∇u^T).

We also assume thatEudecomposes as an elastic part and a plastic part Eu=e+p.

L. JAKABˇCIN

We also define the set of kinematically admissible fields by

A_adm :={(u, e, p)∈H¹(Ω,R²)×L²(Ω,M^2×2_sym)×L²(Ω,M^2×2_sym) : Eu = e+p a.e.in Ω, u= 0 a.e. on ∂Ω_D}.

We denote H_D¹ := {u ∈ H¹(Ω,R²); u = 0 on∂Ω_D}. For a fixed constant τ > 0, we define K:={q∈M^2×2_sym;|q| ≤τ}andH :M^2×2_sym →[0,∞] the support function ofKby

H(p) := sup

θ∈K θ:p=τ|p|. Forη >0, the elastic energy is defined as

Eel : L²(Ω,M^2×2_sym)×H¹(Ω,R)→R (e, v)−→ Eel(e, v) =1

2

Ω

v²+η

Ae:edx.

In the following, we will define an evolution as a limit of time discretizations with a time steph. In fact,pandp₀ represent the plastic deformation at 2 consecutive time steps, so that p−p₀

h ∼p. In the same way,˙ uand u₀ represent displacement field at 2 consecutive time steps, so that u−u₀

h ∼u. The plastic dissipated energy is˙ defined, by

Ep : L²(Ω,M^2×2_sym)×L²(Ω,M^2×2_sym)→R (p, p₀)−→ Ep(p, p₀) =

Ω

H(p−p₀) dx.

Givenβ >0, the viscoelastic dissipated energy is defined by Eve : H¹(Ω,R²)×H¹(Ω,R²)→R (u, u₀)−→ Eve(u, u₀) = β

2h

Ω

(Eu−Eu₀) : (Eu−Eu₀) dx.

The Griffith surface energy is approximated by the phase-field surface energy E_S : H¹(Ω,R)→R

v −→ ES(v) =

Ω

ε|∇v|²dx+

Ω

(1−v)² 4ε dx.

It is shown in [5] that in the elastic anti-plane case, where the displacement reduces to a scalar and Eureduces to∇u, the Ambrosio−Tortorelli functional

Eε(∇u, v) =Eel(∇u, v) +ES(v), Γ-converges, as 0< η→0, to the Griffith energyG, where

G(u) :=1 2

Ω

A|∇u|²dx+H¹(S(u)).

Here, S(u) denotes the discontinuity set of u, and H¹ is the 1-dimensional Hausdorff measure. In the case of n-dimensional elasticity the Γ-convergence result for Ambrosio−Tortorelli approximation has been proved recently by Iurlano [14].

A VISCO-ELASTO-PLASTIC EVOLUTION MODEL WITH REGULARIZED FRACTURE

For f ∈ C¹(0, T_f, L²(Ω,R²)) and g ∈ C¹(0, T_f, L²(∂Ω_N,R²)), the external forces at time t ∈ [0, T_f] with T_f >0 are collected into a functionall(t)∈(H_D¹)^∗, where (H_D¹)^∗ denotes the dual ofH_D¹:

l(t), ϕ:=

Ω

f(t).ϕdx+

∂ΩN

g(t).ϕds ∀ϕ∈H_D¹.

In the following framework, we approximate the continuous-time visco-elasto-plastic evolutionviadiscrete-time evolutions obtained by solving incremental variational problems. GivenT_f > 0 and a positive integer N_f, at each discrete timet_i=ih, i= 1, . . . , N_f, with h= _N^Tf

f, let us assume that the approximate visco-elasto-plastic evolution (u(t_i−1), v(t_i−1), e(t_i−1), p(t_i−1)) is known att_i−1. We then define (u(t_i), v(t_i), e(t_i), p(t_i)) as follows:

(u(t_i), e(t_i), p(t_i)) is defined at time t_i as a minimizer of a deformation energy:

Edef(u, v(t_i−1), e, p) =Eel(e, v(t_i−1)) +Eve(u, u(t_i−1)) +Ep(p, p(t_i−1))− l(t_i), u,

withv(t_i−1) fixed and among all (u, e, p) triplets satisfying the kinematic admissibility condition. Thenv(t_i) is determined as a minimizer of the following variational problem:

v(t_i) := argmin

v≤v(ti−1)Eel(e(t_i), v) +ES(v).

In the following section, we describe a continuous time evolution of the proposed model.

2.2. The visco-elasto-plastic evolution with regularized fracture

We assume that the stress σ = aAe+βEu˙ is the sum of two terms. The first term represents the stress associated to elastic deformation. It is affected by fracture via the factor a(x, t). The second term represents the effect of viscous deformation. Let u ∈ W^1,∞(0, T_f, H¹(Ω;R²)), v ∈ W^1,∞(0, T_f, H¹(Ω;R)), e∈W^1,∞(0, T_f, L²(Ω;M^2×2_sym)),p∈W^1,∞(0, T_f, L²(Ω;M^2×2_sym)). We call

(u, v, e, p) :Ω×[0, T_f]−→R²×R×M^2×2_sym×M^2×2_sym a continuous evolution if it satisfies the following properties:

• (H1) Initial conditions: (u(0), v(0), e(0), p(0)) = (u₀, v₀, e₀, p₀) with

(u₀, e₀, p₀) ∈ A_adm, and v₀ ∈ H¹(Ω) with v₀ = 1 on ∂Ω_D and 0 ≤ v₀ ≤ 1 a.e. in Ω. We suppose also (v²₀+η)|Ae₀| ≤τ.

• (H2) for everyt∈[0, T_f],v(t) = 1 on∂Ω_D, and 0≤v(t)≤1 a.e. in Ω,

• (H3) Irreversibility: for a.e.t∈[0, T_f], ˙v(t)≤0 a.e. inΩ,

• (H4) Kinematic compatibility: for everyt∈[0, T_f], v(t) = 1 on∂Ω_D,

Eu(t) =e(t) +p(t) a.e.inΩ and u(t) = 0 a.e. on ∂Ω_D.

• (H5) Equilibrium condition: for a.e.t∈[0, T_f],

−div(σ(t)) =f(t), a.e. in Ω, σ(t).n=g(t), a.e. on ∂Ω_N, whereσ(t) =a(t)Ae(t) +βEu(t) and˙ a(t) = (v(t))²+η.

• (H6) Plastic flow rule: for a.e.t∈[0, T_f],

a(t)Ae(t)∈∂H( ˙p(t)) fora.e. x∈Ω.

• (H7) Crack propagation condition: for a.e.t∈[0, T_f], Eel(e(t), v(t)) +ES(v(t)) = inf

v∈H¹(Ω), v=1 on∂ΩD,v≤v(t)Eel(e(t), v) +ES(v).

L. JAKABˇCIN

3. Existence result

The main result of the paper is the existence result for the visco-elasto-plastic model with fracture.

Theorem 3.1. Let β >0, τ >0,ε >0,η >0. We suppose that v₀ satisfies the crack propagation condition (H7) with e₀ i.e.

Eel(e₀, v₀) +ES(v₀) = inf

v∈H¹(Ω), v=1on∂ΩD,v≤v0Eel(e₀, v) +ES(v).

Then, there exists at least one solution

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

u∈W^1,∞(0, T_f, H¹(Ω;R²)), v∈W^1,∞(0, T_f, H¹(Ω;R)), e∈W^1,∞(0, T_f, L²(Ω;M^2×2_sym)), p∈W^1,∞(0, T_f, L²(Ω;M^2×2_sym)), satisfying (H1)−(H7).

3.1. Time discretization

The proof of Theorem 3.1is based on a time discretization procedure. We consider a partition of the time interval [0, T_f] intoN_f sub-intervals of equal length h:

0 =t⁰_h< t¹_h< . . . < tⁿ_h=nh < . . . < t^N_h^f =T_f,with h= T_f

N_f =tⁿ_h−tⁿ⁻¹_h →0.

We setv_h⁰=v₀,u⁰_h=u₀,e⁰_h=e₀,p⁰_h=p₀. Note that in the whole text,C >0 denotes a generic constant which is independent of the discretization parameters. Forn= 1, . . . , N_f, N_f ≥2, we construct (uⁿ_h, v_hⁿ, eⁿ_h, pⁿ_h) using an alternate minimization procedure. We define the deformation energy

Edef(z, v_hⁿ⁻¹, ξ, q) = 1 2

Ω

aⁿ⁻¹_h Aξ:ξdx+ 1

2hβEz−Euⁿ⁻¹_h ²_L2

+τ

Ω

|q−pⁿ⁻¹_h |dx− l(tⁿ_h), z

=Eel(v_hⁿ⁻¹, ξ) +Eve(z, uⁿ⁻¹_h ) +Ep(q, pⁿ⁻¹_h )− l(tⁿ_h), z,

withaⁿ⁻¹_h := [vⁿ⁻¹_h ]²+η,β >0,τ >0. SinceEdef(., vⁿ⁻¹_h , ., .) is strictly convex, coercive andA_adm is a convex closed set, we have

Proposition 3.2. Suppose that (uⁿ⁻¹_h , eⁿ⁻¹_h , pⁿ⁻¹_h )∈ A_adm. There exists unique minimizer to the variational problem

(z,ξ,q)∈Aminadm

Edef(z, v_hⁿ⁻¹, ξ, q). (3.1)

We now define (uⁿ_h, eⁿ_h, pⁿ_h) as a solution of (3.1) and we derive the Euler−Lagrange equation satisfied by this solution.

Proposition 3.3. Let (uⁿ_h, eⁿ_h, pⁿ_h)∈A_adm be a solution to (3.1) with

σⁿ_h :=aⁿ⁻¹_h Aeⁿ_h+βEuⁿ_h−Euⁿ⁻¹_h

h =aⁿ⁻¹_h Aeⁿ_h+βEδuⁿ_h. (3.2)

A VISCO-ELASTO-PLASTIC EVOLUTION MODEL WITH REGULARIZED FRACTURE

Then, for all n∈ {1, . . . , Nf}:

⎧⎪

⎨

⎪⎩

−div(σⁿ_h) =f(tⁿ_h), a.e. in Ω, σⁿ_h.n=g(tⁿ_h), a.e on ∂Ω_N, aⁿ⁻¹_h Aeⁿ_h∈∂H(pⁿ_h−pⁿ⁻¹_h ) a.e. in Ω.

Proof. Let (z, ξ, q)∈A_adm, then (uⁿ_h+sz, eⁿ_h+sξ, pⁿ_h+sq)∈A_adm is an admissible triplet for every 0< s <1.

We have

Edef(uⁿ_h, vⁿ⁻¹_h , eⁿ_h, pⁿ_h)≤ Edef(uⁿ_h+sz, v_hⁿ⁻¹, eⁿ_h+sξ, pⁿ_h+sq), hence

0≤s

Ω

aⁿ⁻¹_h Aeⁿ_h:ξdx+s

Ω

βEuⁿ_h−Euⁿ⁻¹_h

h : (ξ+q) dx +τ

Ω

|pⁿ_h+sq−pⁿ⁻¹_h | − |pⁿ_h−pⁿ⁻¹_h |dx−sl(tⁿ_h), z+o(s).

LetΨ(s) :=τ

Ω|pⁿ_h+sq−pⁿ⁻¹_h |dx. Using the convexity ofΨ we haveΨ(s)−Ψ(0)≤s(Ψ(1)−Ψ(0)). Dividing this inequality by s and letting s tend to zero implies that

Ω

aⁿ⁻¹_h Aeⁿ_h:ξdx+β

Ω

Euⁿ_h−Euⁿ⁻¹_h

h : (ξ+q) dx +τ

Ω

|pⁿ_h−pⁿ⁻¹_h +q| − |pⁿ_h−pⁿ⁻¹_h |dx

l(tⁿ_h), z. (3.3)

Testing (3.3) with (z, ξ, q) =±(φ, Eφ,0) for anyφ∈C_c^∞(Ω,R²), we obtain

Ω

σ_hⁿ:E(φ) dx=l(tⁿ_h), φ (3.4)

and −div(σⁿ_h) =f(tⁿ_h) a.e. inΩ. Furher, picking φ∈C^∞(Ω,R²), with φ= 0 on ∂Ω_D in ±(φ, Eφ,0) as a test function for (3.3) and integrating (3.4) by parts, we also obtain thatσⁿ_h.n=g(tⁿ_h) a.e. on∂Ω_N. Testing (3.3) with (0,−q+pⁿ_h−pⁿ⁻¹_h , q−pⁿ_h+pⁿ⁻¹_h ) for anyq∈L²(Ω,M^2×2_sym), we have

τ

Ω

|q|dx≥τ

Ω

|pⁿ_h−pⁿ⁻¹_h |dx+

Ω

aⁿ⁻¹_h Aeⁿ_h: (q−(pⁿ_h−pⁿ⁻¹_h )) dx, so that

τ|q| ≥τ|pⁿ_h(x)−pⁿ⁻¹_h (x)|+aⁿ⁻¹_h (x)Aeⁿ_h(x) : (q−(pⁿ_h(x)−pⁿ⁻¹_h (x)))

(3.5) for allq∈M^2×2_sym and for a.e. x∈Ω, which by definition of the subdifferential implies that

aⁿ⁻¹_h Aeⁿ_h∈∂H(pⁿ_h−pⁿ⁻¹_h ) a.e.inΩ. (3.6) Proposition 3.4. For given eⁿ_h∈L²(Ω,M^2×2_sym)there exists an unique minimizerv_hⁿ to

v_hⁿ:= argmin

v∈H¹(Ω), v=1on∂ΩD, v≤vⁿ⁻¹_h

{Eel(eⁿ_h, v) +ES(v)}. (3.7)

L. JAKABˇCIN

Additionally, for alln∈ {1, . . . , Nf},v_hⁿ satisfies the following variational inequality:

2ε

Ω

∇vⁿ_h∇(vⁿ_h−ϕ) dx+

Ω

vⁿ_hAeⁿ_h:eⁿ_h(v_hⁿ−ϕ) dx + (2ε)⁻¹

Ω

(vⁿ_h−1)(v_hⁿ−ϕ) dx≤0, (3.8) for anyϕ∈H¹(Ω),ϕ= 1on∂Ω_D andϕ≤v_hⁿ⁻¹. Furthermore,vⁿ_h satisfies the comparison principle0≤v_hⁿ≤ v_hⁿ⁻¹ a.e. inΩ.

Proof. The existence and uniqueness of the solution of (3.7) follow from the strict convexity and coercivity of the functional Eel(eⁿ_h, .) +ES(.), and since {v ∈H¹(Ω), v= 1 on∂Ω_D, v≤vⁿ⁻¹_h } is a closed convex set. Letϕ an admissible function for (3.7), thenψ=vⁿ_h+t(ϕ−v_hⁿ) with 0< t <1 is an admissible function for (3.7). In fact,ψ∈H¹(Ω),ψ= 1 on∂Ω_D and

ψ≤v_hⁿ+t(v_hⁿ⁻¹−vⁿ_h) =vⁿ_h(1−t) +tvⁿ⁻¹_h

≤v_hⁿ⁻¹(1−t) +tvⁿ⁻¹_h =vⁿ⁻¹_h . By definition ofv_hⁿ,Eel(eⁿ_h, vⁿ_h) +ES(v_hⁿ)≤ Eel(eⁿ_h, ψ) +ES(ψ). We obtain:

2ε

Ω

∇vⁿ_h∇(vⁿ_h−ϕ) dx+

Ω

v_hⁿAeⁿ_h :eⁿ_h(v_hⁿ−ϕ) dx+ (2ε)⁻¹

Ω

(vⁿ_h−1)(vⁿ_h−ϕ) dx≤0. (3.9) Testing (3.9) withϕ= max(0, v_hⁿ) gives

2ε

{v_hⁿ≤0}∇v_hⁿ∇vⁿ_hdx+

{v_hⁿ≤0}

(v_hⁿ)²Aeⁿ_h:eⁿ_hdx+ (2ε)⁻¹

{vⁿ_h≤0}

(vⁿ_h−1)v_hⁿdx≤0, (3.10) so thatv_hⁿ= 0 a.e. on{v_hⁿ≤0}. It follows thatvⁿ_h0 a.e. inΩ.

Remark 3.5. Testing (3.9) withϕ=v_hⁿ⁻¹ and withϕ= 2vⁿ_h−vⁿ⁻¹_h we derive the equality 2ε

Ω

∇v_hⁿ∇(vⁿ_h−vⁿ⁻¹_h ) dx+

Ω

vⁿ_hAeⁿ_h:eⁿ_h(v_hⁿ−v_hⁿ⁻¹) dx+ (2ε)⁻¹

Ω

(v_hⁿ−1)(v_hⁿ−vⁿ⁻¹_h ) dx= 0. (3.11)

3.2. A priori estimates

We define for alln1, δuⁿ_h:= uⁿ_h−uⁿ⁻¹_h

h , δvⁿ_h:= v_hⁿ−vⁿ⁻¹_h

h , δeⁿ_h:= eⁿ_h−eⁿ⁻¹_h

h , δpⁿ_h:= pⁿ_h−pⁿ⁻¹_h

h ·

Proposition 3.6. There exists a constant C >0 independent of handnsuch that

{1,..Nmaxf}(δuⁿ_hH¹,δv_hⁿ_H¹,δeⁿ_hL²,δpⁿ_hL²)≤C. (3.12) For the proof of Proposition3.6we need following lemma.

Lemma 3.7. For all n 1, we have aⁿ⁻¹_h Aeⁿ_h ∈ K a.e. in Ω. Additionally, there exists a constant C > 0, independent ofh andnsuch that

Aeⁿ_hL^∞≤C. (3.13)

A VISCO-ELASTO-PLASTIC EVOLUTION MODEL WITH REGULARIZED FRACTURE

Proof. Testing (3.5) with q =aⁿ⁻¹_h Aeⁿ_h(x) +pⁿ_h(x)−pⁿ⁻¹_h (x) leads to |aⁿ⁻¹_h Aeⁿ_h(x)| ≤τ for a.e.x∈Ω. Since

η≤aⁿ⁻¹_h (x) for allx∈Ω, we haveAeⁿ_hL^∞≤C.

Lemma 3.8. For all q∈K={q^∗∈M^2×2_sym;|q^∗| ≤τ} andn1, we have

aⁿ⁻¹_h Aeⁿ_h:δpⁿ_hq:δpⁿ_h, a.e. inΩ. (3.14) Proof. By convex duality

aⁿ⁻¹_h Aeⁿ_h ∈∂H(pⁿ_h−pⁿ⁻¹_h )⇐⇒pⁿ_h−pⁿ⁻¹_h ∈∂H^∗(aⁿ⁻¹_h Aeⁿ_h)

whereH^∗(q^∗) = 1_K(q^∗) denotes the convex conjugate of H. Sinceaⁿ⁻¹_h Aeⁿ_h ∈K a.e.inΩ, we have for allq∈K

aⁿ⁻¹_h Aeⁿ_h:δpⁿ_hq:δpⁿ_h a.e. inΩ.

Lemma 3.9. For all n∈ {1, . . . , N_f}, there exists a constant C >0 independent of handnsuch that

δvⁿ_h²_L2+∇δvⁿ_h²_L2≤Cδeⁿ_h²_L2. (3.15) Proof. We proceed as in the proof of Lemma 3.4 in [16]. We write inequality (3.9) at timetⁿ⁻¹_h :

2ε

Ω

∇vⁿ⁻¹_h ∇(v_hⁿ⁻¹−ϕ) dx+

Ω

v_hⁿ⁻¹Aeⁿ⁻¹_h :eⁿ⁻¹_h (v_hⁿ⁻¹−ϕ) dx + (2ε)⁻¹

Ω

(v_hⁿ⁻¹−1)(vⁿ⁻¹_h −ϕ) dx≤0 (3.16)

for anyϕ≤v_hⁿ⁻²,ϕ∈H¹(Ω),ϕ= 1 on∂Ω_D. We chooseϕ=vⁿ_h ≤vⁿ⁻²_h and divide (3.16) byh:

2ε

Ω

∇v_hⁿ⁻¹∇δv_hⁿdx+

Ω

v_hⁿ⁻¹Aeⁿ⁻¹_h :eⁿ⁻¹_h δvⁿ_hdx + (2ε)⁻¹

Ω

(v_hⁿ⁻¹−1)δv_hⁿdx0. (3.17)

We divide (3.11) byhand substract (3.17). Then we use the equalitya²−b² = (a−b)²+ 2(a−b)b, the fact thatδvⁿ_h ≤0, and the Cauchy−Schwarz inequality to obtain

δv_hⁿ|eⁿ_h|A²_L2+ (2ε)⁻¹δvⁿ_h²_L2+ (2ε)∇δv_hⁿ²_L2

≤ 1 h

Ω

(|eⁿ⁻¹_h |²_A− |eⁿ_h|²_A)vⁿ⁻¹_h δv_hⁿdx

= 1 h

Ω

|(eⁿ⁻¹_h −eⁿ_h)|²_Av_hⁿ⁻¹δvⁿ_hdx +2

h

Ω

A(eⁿ⁻¹_h −eⁿ_h) :eⁿ_hv_hⁿ⁻¹δvⁿ_hdx

≤ 2 h

Ω

(|eⁿ_h|A|δvⁿ_h|) (|v_hⁿ⁻¹||(eⁿ⁻¹_h −eⁿ_h)|A) dx

≤2(δv_hⁿ)|eⁿ_h|AL² vⁿ⁻¹_h |δeⁿ_h|AL² . (3.18)

The result follows from the Young inequalityab≤a²/2 +b²/2.

We now prove Proposition3.6.

L. JAKABˇCIN

Proof. Testing (3.3) with (z, ξ, q) =±(hδuⁿ_h, E(hδuⁿ_h),0) whereE(hδuⁿ_h) =h(δeⁿ_h+δpⁿ_h), we obtain

Ω

aⁿ⁻¹_h Aeⁿ_h: (eⁿ_h−eⁿ⁻¹_h ) dx+

Ω

aⁿ⁻¹_h Aeⁿ_h : (pⁿ_h−pⁿ⁻¹_h ) dx+hβEδuⁿ_h²_L2

= l(tⁿ_h), uⁿ_h−uⁿ⁻¹_h ,

Ω

aⁿ⁻¹_h Aeⁿ_h: (pⁿ_h−pⁿ⁻¹_h ) dx=l(tⁿ_h), uⁿ_h−uⁿ⁻¹_h −hβEδuⁿ_h²_L2

−

Ω

aⁿ⁻¹_h Aeⁿ_h: (eⁿ_h−eⁿ⁻¹_h ) dx. (3.19) Testing (3.3) with (0, hδpⁿ_h,−hδpⁿ_h) yields

Ω

aⁿ⁻¹_h Aeⁿ_h: (pⁿ_h−pⁿ⁻¹_h ) dxτ

Ω

|pⁿ_h−pⁿ⁻¹_h |dx. (3.20) Combining (3.19) and (3.20), we get

Ω

aⁿ⁻¹_h Aeⁿ_h : (eⁿ_h−eⁿ⁻¹_h ) dx+hβEδuⁿ_h ²_L2 +τ

Ω

|pⁿ_h−pⁿ⁻¹_h |dx

l(tⁿ_h), uⁿ_h−uⁿ⁻¹_h . (3.21)

The first term on the left-hand side can be analysed in a similar way as in [16]:

Ω

aⁿ⁻¹_h Aeⁿ_h: (eⁿ_h−eⁿ⁻¹_h ) dx=E_el(eⁿ_h, vⁿ_h)− E_el(eⁿ⁻¹_h , vⁿ⁻¹_h ) (3.22) + 1

2h²(aⁿ⁻¹_h )^1/2|δeⁿ_h|A²_L2−1 2

Ω

(aⁿ_h−aⁿ⁻¹_h )|eⁿ_h|²_Adx.

Further, we observe that

aⁿ_h−aⁿ⁻¹_h = (v_hⁿ)²−(v_hⁿ⁻¹)²=h(v_hⁿ+v_hⁿ⁻¹)δv_hⁿ= 2hv_hⁿδvⁿ_h−h²|δv_hⁿ|². Thanks to (3.11), we obtain

−1 2

Ω

(aⁿ_h−aⁿ⁻¹_h )|eⁿ_h|²_Adx= (2ε)⁻¹

Ω

(v_hⁿ−1)(v_hⁿ−vⁿ⁻¹_h ) dx +2ε

Ω

∇v_hⁿ(∇v_hⁿ− ∇v_hⁿ⁻¹) dx+1

2h²(δvⁿ_h)|eⁿ_h|A²_L2, and rewritingv_hⁿ−vⁿ⁻¹_h = (v_hⁿ−1)−(v_hⁿ⁻¹−1), yields

− 1 2

Ω

(aⁿ_h−aⁿ⁻¹_h )|eⁿ_h|²_Adx=ES(v_hⁿ)− ES(vⁿ⁻¹_h ) (3.23) +h²

(4ε)⁻¹δvⁿ_h²_L2+ε∇δv_hⁿ²_L2

+1

2h²δvⁿ_h|eⁿ_h|A²_L2. Summing (3.21) for 1≤n≤N and using (3.22) and (3.23) we obtain

Eel(e^N_h, v^N_h) +ES(v_h^N) + N n=1

hβEδuⁿ_h²_L2+ N n=1

h

Ω

|δpⁿ_h|dx

≤ N n=1

l(tⁿ_h), uⁿ_h−uⁿ⁻¹_h +Eel(e₀, v₀) +ES(v₀),