# Variational and Numerical Analysis for Frictional Contact Problem with Normal Compliance in Thermo-Electro-Viscoelasticity

Authors: Mustapha Bouallala; El Hassan Essoufi; Mohamed Alaoui
Journal: International Journal of Differential Equations
 (2019)

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## Abstract

In this paper, we consider a mathematical model of a contact problem in thermo-electro-viscoelasticity with the normal compliance conditions and Tresca’s friction law. We present a variational formulation of the problem, and we prove the existence and uniqueness of the weak solution. We also study the numerical approach using spatially semidiscrete and fully discrete finite element schemes with Euler’s backward scheme. Finally, we derive error estimates on the approximate solutions.


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## Body

## 1. Introduction

In the recent years, piezoelectric contact problems have been of great interest to modern engineering. General models of electroelastic characteristics of piezoelectric materials can be found in [1, 2]. The problems of piezo-viscoelastic materials have been studied with different contact conditions within linearized elasticity in [3–5] and within nonlinear viscoelasticity in [6–8]. The modeling of these problems does not take into account the thermic effect. Mindlin [9] was the first to propose the thermo-piezoelectric model. The mathematical model which describes the frictional contact between a thermo-piezoelectric body and a conductive foundation is already addressed in the static case in [10, 11].Sofonea et al. considered in [12] the modeling of quasistatic viscoelastic problem with normal compliance friction and damage; they proved the existence and uniqueness of the weak solution, and they derived error estimates on the approximate solutions.In the article [13], we find the recent result of a new quasistatic mathematical model which describes the chosen thermo-electro-viscoelastic body behavior and the contact by Signorini condition with nonfrictional and nonconductive foundation; also, the variational formulation of this problem is derived and its unique weak solvability is established.In this paper, we consider a quasi-static contact problem with Tresca’s friction between a thermo-electro viscoelastic body and an electrically and thermally conductive rigid foundation. The novelty in this model, which can be considered as the generalization of the model presented in [13], lies in the use of the penalized normal compliance contact condition: (1)σνuν−g=−1εuν−g+,ε>0.This means that we allow a weak interpenetration between the body and the foundation. On the contact zone, we consider the following regularized electrical and thermal conditions:(2)D⋅ν=ψuν−gϕLφ−φF,∂q∂ν=kcuν−gϕLθ−θF,which describe both the thermal and electrical conductivities of the foundation. This leads to nonlinear coupling between the mechanical displacement and thermal and electrical fields and hence more complexities on the model.Since the friction conditions are inequalities, we derive a quasivariational formulation of this problem and we prove the existence and uniqueness of the weak solution based on arguments variational inequalities, Galerkin method, compactness method, and Banach fixed point theorem. We derive error estimates for the numerical approximations based on discrete schemes.The paper is structured as follows. In Section2, we present the model of equilibrium process of the thermo-electro-viscoelastic body in frictional contact with a conductive rigid foundation and we introduce the notations and assumptions on the problem data. We also derive the variational formulation of the problem. We state the main results concerning the existence and the uniqueness of a weak solution. We present a spatially semidiscrete scheme and a fully discrete scheme to approximate the contact problem. We then use the finite element method to discretize the domain Ω and Euler’s forward scheme to discretize the time derivatives. Finally, the proofs are established in Section 3.


## 2. Formulation and Main Results

### 2.1. Problem Setting

We consider a body of a piezoelectric material which occupies the domainΩ⊂ℝdd=2,3 in the reference configuration which will be supposedly bounded with a smooth boundary ∂Ω=Γ. This boundary is divided into three open disjoint parts ΓD, ΓN, and ΓC on one hand and a partition of ΓD∪ΓN into two open parts Γa and Γb on the other hand, such that measΓD>0 and measΓa>0. Let 0;T be the time interval, where T>0.The body is submitted to the action of body forces of densityf0, a volume electric charge of density q0, and a heat source of constant strength q1. It is also submitted to mechanical, electrical, and thermal constants on the boundary. Indeed, the body is assumed to be clamped in ΓD, and therefore, the displacement field vanishes there. Moreover, we assume that a density of traction forces, denoted by f2, acts on the boundary part ΓN. We also assume that the electrical potential vanishes on Γa, and a surface electrical charge of density q2 is prescribed on Γb. We consider that the temperature θ0 is prescribed on the surface ΓD∪ΓN.In the reference configuration, the body may come in contact overΓC with an electrically thermally conductive foundation. Assume that its potential and temperature are maintained at φF and θF. The contact is frictional, and there may be electrical charges and heat transfer on the contact surface. The normalized gap between ΓC and the rigid foundation is denoted by g.In the following sections, we useSd to denote the space of second-order symmetric tensors on ℝd while “.” and . will represent the inner product and the Euclidean norm on Sd and ℝd, that is(3)u⋅v=uivi,v=v⋅v1/2,∀u,v∈ℝd,σ⋅τ=σijτij,τ=τ⋅τ1/2,∀σ,τ∈Sd.We denoteu:Ω×0;T⟶ℝd as the displacement field, σ:Ω⟶Sd and σ=σij the stress tensor, θ:Ω×0;T⟶ℝd the temperature, q:Ω⟶ℝd and q=qi the heat flux vector, and D:Ω⟶ℝd and D=Di the electric displacement field. We also denote Eφ=Eiφ as the electric vector field, where φ:Ω×0;T⟶ℝ is the electric potential. Moreover, let εu=εiju denote the linearized strain tensor given by εiju=1/2ui,j+uj,i, and “Div” and “div” denote the divergence operators for tensor and vector valued functions, respectively, i.e., Divσ=σij,j and divξ=ξj,j. We shall adopt the usual notation for normal and tangential components of displacement vector and stress: υn=υ⋅n,υτ=υ−υnn,σn=σn⋅n, and στ=σn−σnn, where n denotes the outward normal vector on Γ.ProblemP. Find a displacement field u:Ω×0,T⟶ℝd, an electric potential φ:Ω×0,T⟶ℝ, and a temperature field θ:Ω×0,T⟶ℝ such that (4)σ=ℑεu−ε∗Eφ−θℳ+Cεu˙inΩ×0,T,(5)D=ℰεu+βEφ−θ−θ∗PinΩ×0,T,(6)Divσ+f0=0,inΩ×0,T,(7)divD=q0,inΩ×0,T,(8)θ˙+divq=q1,inΩ×0,T,(9)u=0,onΓD×0,T,(10)σν=f2,onΓN×0,T,(11)φ=0,onΓa×0,T,(12)D⋅ν=qbonΓb×0,T,(13)θ=0,onΓD∪ΓN×0,T,(14)u0,x=u0,inΩ,(15)θ0,x=θ0,inΩ,(16)σνuν−g=−1εuν−g+,ε>0,onΓC×0,T,(17)στ≤S,onΓC×0,T,(18)στ<S⟹u˙τ=0,onΓC×0,T,(19)στ=S⟹∃λ≠0such thatστ=−λu˙τonΓC×0,T,(20)D⋅ν=ψuν−gϕLφ−φF,onΓC×0,T,(21)∂q∂ν=kcuν−gϕLθ−θF,onΓC×0,T.Here, the equations (4) and (5) represent the thermo-electro-viscoelastic constitutive law of the material in which σ=σij denotes the stress tensor, εu is the linearized strain tensor, Eφ is the electric field. ℑ=fijkl, ℰ=eijk, ℳ=mij, β=βij, P=pi, and C=cijkl are, respectively, elastic, piezoelectric, thermal expansion, electric permittivity, pyroelectric tensor, and (fourth-order) viscosity tensor. ℰ∗ is the transpose of ℰ given by(22)ℰ∗=eijk∗,eijk∗=ekij,(23)ℰσυ=σℰ∗υ,∀σ∈Sd,υ∈ℝd.The constantθ∗ represents the reference temperature.Fourier’s law of heat conduction is given by(24)q=−K∇θ,inΩ×0,T,where K=kij denotes the thermal conductivity tensor.Equations (6)–(8) represent the equilibrium equations for the stress. Relations (9) and (10), (11) and (12), and (13) represent the mechanical, the electrical, and the thermal boundary conditions. The unilateral boundary condition (16) represents the normal compliance condition and (17)–(19) represent Tresca’s friction law in which S is the coefficient of friction.Following [14], the contact conditions (20) and (21) on ΓC are obtained as follows:When there is no contact at a point on the surface, there is no free electrical charges on the surface and no thermal transfer; that is(25)uν<g⟹D⋅ν=0,q⋅ν=0.If the contact holds, i.e.,uν≥g, the normal component of the electric displacement field or the free charge (resp., thermal transfer) is assumed to be proportional to the difference between the potential of foundation and the body’s surface potential (resp., to the difference between the temperature of foundation and the body’s surface temperature). Thus,(26)uν≥g⟹D⋅ν=kφφ−φF,q⋅ν=kθθ−θF.We combine (25) and (26) to obtain(27)D⋅ν=kφχ0,+∞uν−gφ−φF,q⋅ν=kθχ0,+∞uν−gθ−θF,where χ0,+∞ is the characteristic function of the interval 0,+∞ defined by(28)χ0,+∞s=0,ifs<0,1,ifs≥0.Equation (27) represents the regularization electrical contact condition and the heat flux condition on ΓC, where(29)ϕLs=−L,ifs<−L,s,if−L≤s≤L,L,ifs>L,ψr=0,ifr<0,cδr,if0≤r≤1δ,c,ifr>1δ,and where c=kφ,kθ, and L is a large positive constant, δ>0 is a small parameter, kc:r⟶kcr is supposed to be zero for r<0 and positive, otherwise nondecreasing and Lipschitz continuous.Remark 1.
We note that whenψ≡0, equality (20) becomes(30)D⋅ν=0onΓC×0,T,which models the case when the foundation is a perfect electric insulator.
Similarly, in equality (21), we have(31)q⋅ν=0onΓC×0,T.


### 2.2. Weak Formulation and Uniqueness Result

To obtain a variational formulation of ProblemP, we need additional notations and need to recall some definitions in the sequel.We use the following functional Hilbert spaces:(32)L2Ω=L2Ωd,H1Ω=H1Ωd,ℋ=σ∈Sd:σ=σij,σij=σji∈L2Ω,(33)W=D=Di∈H1Ω:Di∈L2Ω,divD∈L2Ω,endowed with the canonical inner product given by(34)u,vL2Ω=∫Ωuividx,σ,τℋ=∫Ωσiτidx,u,vH1Ω=u,vL2Ω+εu,εvℋ,D⋅E=D⋅EL2Ω+divD⋅divEL2Ω,and the associated norms .L2Ω, .H1Ω, .ℋ, and .W, respectively.LetV, W, and Q be the closed subspaces of H1Ω given by(35)V=v∈H1Ω:v=0onΓD,W=ξ∈H1Ω:ξ=0onΓa,Q=η∈H1Ω:η=0onΓD∪ΓN,and the set of admissible displacements(36)Vad=v∈V:vν−g≤0onΓC.It is known thatV, W, and Q are real Hilbert spaces with the inner products u,vV=εu,εvℋ, φ,ξW=∇φ,∇ξL2Ω, and θ,ηQ=∇θ,∇ηL2Ω, respectively.Moreover, the associated normvV=εvℋ is equivalent on V to the usual norm .H1Ω and ξW=∇ξL2Ω and ηQ=∇ηL2Ω are equivalent on W and Q, respectively, with the usual norms .H1Ω.By Sobolev’s trace theorem, there exists three positive constantsCs1, Cs2, and Cs3 depending on Ω, ΓC, ΓN, ΓD, Γa, and Γb:(37)vL2Γd≤Cs1vV,∀v∈V,(38)ξL2Γc≤Cs2ξW,∀ξ∈W,(39)ηL2Γc≤Cs3ηQ,∀η∈Q.SincemeasΓD>0 and Korn’s inequality hold(40)εvℋ≥CKvH1Ω,∀v∈V,where CK in a nonnegative constant depending only on Ω and ΓD. Notice also that since measΓa>0, the following Friedrichs–Poincaré inequalities hold:(41)∇ξW≥CF1ξW,∀ξ∈W,(42)∇ηL2Ω≥CF2ηQ,∀η∈Q,where CF1 and CF2 are the positive constants which depend only on Ω, Γa, ΓD, and ΓN.For a real Banach spaceX and 1≤p≤∞, we consider the Banach spaces C0,T;X and C10,T;X of continuous and continuously differentiable functions from 0,T to X with the norms(43)uC0,T;X=supt∈0,TutX,uC10,T;X=supt∈0,TutX+supt∈0,Tu˙tX.To simplify the writing, we denote bya:V×V⟶ℝ, b:W×W⟶ℝ, c:V×V⟶ℝ, and d:Q×Q⟶ℝ the following bilinear and symmetric applications:(44)au,v≔ℑεu,εvℋ,bφ,ξ≔β∇φ,∇ξL2Ω,cu,v≔Cεu,εvℋ,dθ,η≔K∇θ,∇ηL2Ω,and by e:V×W⟶ℝ, m:Q×V⟶ℝ, and p:Q×W⟶ℝ, the following bilinear applications:(45)ev,ξ≔ℰεv,∇ξL2Ω=ℰ∗∇ξ,εvV,mθ,v≔ℳθ,εvQ,pθ,ξ≔P∇θ,∇ξL2Ω.In the study of mechanical ProblemP, we make the following assumptions:HP1. The elasticity operator ℑ:Ω×Sd⟶Sd, the electric permittivity tensor β=βij:⟶Ω×ℝd⟶ℝd, the viscosity tensor c:Ω×Sd⟶Sd, and the thermal conductivity tensor K=kij:Ω×ℝd⟶ℝd satisfy the usual properties of symmetry, boundedness, and ellipticity:(46)fijkl=fjikl=flkij∈L∞Ω,βij=βji∈L∞Ω,cijkl=cjikl=clkij∈L∞Ω,kij=kji∈L∞Ω,and there exists thatmℑ,mβ,mc,mK>0 such that(47)fijklxξkξl≥mℑξ2,∀ξ∈Sd,∀x∈Ω,cijklxξkξl≥mcξ2,∀ξ∈Sd,∀x∈Ω,βijζiζj≥mβζ2,kijζiζj≥mKζ2,∀ζ∈ℝd.HP2. The piezoelectric tensor Ε=eijk:Ω×Sd⟶ℝ, the thermal expansion tensor ℳ=mij:Ω×ℝ⟶ℝ, and the pyroelectric tensor P=pi:Ω⟶ℝd satisfy(48)eijk=eikj∈L∞Ω,mij=mji∈L∞Ω,pi∈L∞Ω,and there exist the positive constantsMℑ,Mβ,Mc,MK,Mℰ,Mℳ, and MP such that(49)au,v≤MℑuVvV,bφ,ξ≤MβφWξW,cu,v≤McuVvV,dθ,η≤MKθQηQ,ev,ξ≤MΕvVξW,mθ,v≤MℳθQvV,pθ,ξ≤MPθQξW.HP3. The surface electrical conductivity ψ:ΓC×ℝ⟶ℝ+ and the thermal conductance kc:ΓC×ℝ⟶ℝ+ satisfy the following hypothesis for π=ψ,kc:∃Mπ>0 such that πx,u≤Mπ,∀u∈ℝ, a.e., x∈ΓC and x⟶πx,u is measurable on ΓC for all u∈ℝ and is zero for all u≤0.The functionu⟶πx,u is a Lipschitz function on ℝ for all x∈Γ3. πx,u1−πx,u2≤Lπu1−u2,∀u1,u2∈ℝ, where Lπ is a positive constant.HP4. The forces, the traction, the volume, the surfaces charge densities, and the strength of the heat source are as follows:(50)f0∈C0,T;L2Ωd,f2∈C0,T;L2ΓNd,qb∈L20,T;L2Γb,q1∈L20,T;L2Ω,q0∈L20,T;L2Ω.The potential and the temperature satisfy(51)φF∈L20,T;L2ΓC,θF∈L20,T;L2ΓC.The initial conditions, the friction-bounded function, and the gap function satisfy(52)u0∈K,θ0∈Q,S∈L∞ΓC,S≥0,g∈L2ΓC,g≥0.Using Riesz’s representation theorem, we definef:0,T⟶V, qe:0,T⟶W, and qth:0,T⟶Q by the following:(53)ft,vV=∫Ωf0t⋅vdx+∫ΓNf2t⋅vda,∀v∈V,(54)qet,ξW=∫Ωq0t⋅ξdx−∫Γbqbt⋅ξda,∀ξ∈V,(55)qtht,ηQ=∫Ωq1t·ηdx,∀η∈Q.We define the mappingsj:V⟶ℝ, ϱ:V×V⟶ℝ, ℓ:V×W2⟶ℝ, and χ:V×Q2⟶ℝ, a.e., t∈0,T by (56)jv=∫ΓCSvτda,∀v∈V,(57)ϱu,v=∫ΓCuν+−gvνda=uν+−g,vνΓC,∀u,v∈V,(58)ℓut,φt,ξ=∫ΓCψuνt−gϕLφt−φFξda,∀u∈V,∀φ,ξ∈W,(59)χut,θt,η=∫ΓCkcuνt−gϕLθt−θFηda,∀u∈V,∀θ,η∈Q,respectively.Now, by a standard variational technique, it is straightforward to see that ifu,φ,θ satisfies the conditions (4)–(21), a.e. t∈0;T, then(60)σt,εv−εu˙tℋ+1εϱut,v−u˙t+jv−ju˙t≥ft,v−u˙tV,∀v∈Vad,(61)Dt,∇ξL2Ω=ℓut,φt,ξ−qet,ξW,∀ξ∈W,(62)qt,∇ηL2Ω=θ˙t,ηQ+χut,θt,η−qtht,ηQ,∀η∈Q.We assume that the initial conditionsu0 and θ0 satisfy the following compatibility condition: there exists φ0∈W such that(63)bφ0,ξ−eu0,ξ−pθ0,ξ+ℓu0,φ0,ξ=qe0,ξW,∀ξ∈W.This nonlinear problem, has a unique solutionφ0, by using the fixed point theorem.Using all of these assumptions, notations, andE=−∇φ, we obtain the following variational formulation of the Problem P.ProblemPV: find a displacement field u:0;T⟶Vad, an electric potential φ:0;T⟶W, and a temperature field θ:0;T⟶Q, a.e., t∈0,T such that(64)cu˙t,v−u˙t+aut,v−u˙t+ev−u˙t,φt−mθt,v−u˙t+1ϵϱut,v−u˙t+jv−ju˙t≥ft,v−u˙tV,∀v∈Vad,(65)bφt,ξ−eut,ξ−pθt,ξ+ℓut,φt,ξ=qet,ξW,∀ξ∈W,(66)dθt,η+θ˙t,ηQ+χut,θt,η=qtht,ηQ,∀η∈Q,(67)u0,x=u0x,θ0,x=θ0x.We present now the existence and the uniqueness of solution to ProblemPV.Theorem 1.
Assume that the assumptionsHP1−HP4, (37)–(42), for ε>0,and the conditions(68)mβ>Mψc12,mK<c2Mkcc2+LkkLc02,hold. Then, Problem PV has a unique solution as follows:(69)u∈C10,T;V,φ∈L20,T;W,θ∈L20,T;Q.


### 2.3. Spatially Semidiscrete Approximation

In this paragraph, we consider a semidiscrete approximation of the ProblemPV by discretizing the spatial domain, using the finite element method. Let Th be a regular finite element partition of the domain Ω compatible with the boundary partition Γ=Γ¯C∪Γ¯D∪Γ¯N. We then define a finite element space Vh⊂VandVadh=Vad∩Vh, for the approximates of the displacement field u, Wh⊂W for the electric potential φ, and Qh⊂Q for the temperature θ defined by(70)Vh=vh∈CΩ¯d;vTrh∈ℙ1Trd∀Tr∈Th;vh=0onΓ¯C,Wh=ξh∈CΩ¯;ξTrh∈ℙ1Tr∀Tr∈Th;ξh=onΓ¯a,Qh=ηh∈CΩ¯;ηTrh∈ℙ1Tr∀Tr∈Th;ηh=0onΓ¯D∪Γ¯N.A spatially semidiscrete scheme can be formed as the following problem:ProblemPVh.Find uh:0;T⟶Vh, φh:0;T⟶Wh, and θh:0;T⟶Qh, a.e., t∈0,T such that for vh∈Vadh, ξh∈Wh, and ηh∈Qh(71)cu˙ht,vh−u˙ht+auht,vh−u˙ht+evh−u˙ht,φht−mθht,vh−u˙h+1εϱuht,vh−u˙ht+jvh−ju˙ht≥ft,vh−u˙ht,(72)bφht,ξh−euht,ξh−pθht,ξh+ℓuht,φht,ξh=qet,ξh,(73)dθht,ηh+θ˙ht,ηh+χuht,ηht,ηh=qtht,ηh,(74)uh0=u0h,φh0=φ0h,θh0=θ0h.Here,u0h∈Vh, φ0h∈Wh, and θ0h∈Qh are appropriate approximations of u0, φ0, and θ0, respectively.Using the same argument presented in Section 2, it can be shown that ProblemPVh has a unique solution uh∈C10,T;Vh, φh∈L20,T;Wh, and θh∈L20,T;Qh.In this paragraph, we are interested in obtaining estimates for the errorsu−uh, φ−φh, and θ−θh.Using the initial value condition, we have(75)ut=∫0twsds+u0,θt=∫0tθ˙sds+θ0,(76)uht=∫0tu˙hsds+u0h,θht=∫0tθ˙hsds+θ0h.Theorem 2.
Assume that the assumptions stated in Theorem 1 are hold, forε>0. Then, under the conditions(77)u0−u0hV⟶0,θ0−θ0hV⟶0,ash⟶0,the semi-discrete solution of PVh converges as follows:(78)u−uhL20,T;V⟶0,φ−φhL20,T;W⟶0,θ−θhL20,T;Q⟶0,ash⟶0.


### 2.4. Fully Discrete Approximation

In this paragraph, we consider a fully discrete approximation of ProblemPV. We use the finite element spaces Vh, Wh, and Qh introduced in Section 2.3. We introduce a partition of the time interval 0;T:0=t0<t1<…<tN=T. We denote the step size Δt=kn=tn−tn−1 for n=1,2,…,N and let k=maxnkn be the maximal step size. For a sequence vnn=0N, we denote δw=wn−wn−1/kn.The fully discrete approximation method is based on the backward Euler scheme, and it has the following form.ProblemPVnhk. Find a displacement field unhkn=0N⊂Vadh, an electric potential φnhkn=0N⊂Wh, and a temperature field θnhkn=0N⊂Qh for all vh∈Vh,ξh∈Wh,ηh∈Qh, and n=1,…,N such that(79)cwnhk,vh−wnhk+aunhk,vh−wnhk+evh−wnhk,φn−1hk−mθn−1hk,vh−wnhk+1εϱunhk,vh−wnhk+jvh−jwnhk≥fn,vh−wnhkV,(80)bφnhk,ξh−eunhk,ξh−pθnhk,ξh+ℓunhk,φnhk,ξh=qen,ξhW,(81)dθnhk,ηh+δθnhk,ηh+χunhk,θnhk,ηh=qthn,ηhQ,(82)u0hk=u0h,φ0hk=φ0h,θ0hk=θ0h.Remark 2.
The choice ofθn−1h and φn−1h instead of θnh and φnh is motivated by the fixed point method in the proof of the existence and uniqueness. Otherwise, we may get another different condition for the uniqueness of the solution of fixed iteration problem (79)–(82). In addition, this choice will be helpful for the application of discrete Grönwall’s lemma in the next.To simplify again the notation, we introduce the velocity(83)wnhk=δunhk,n=1,…,N,unhk=∑j=1nkjwjhk+u0h,θnhk=∑j=1nkjδθjhk+θ0h,n≥1.This problem has a unique solution, and the proof is similar to that used in Theorem 1.We now derive the following convergence result.Theorem 3.
Assuming that the initial valuesu0h∈Vh, φ0h∈Wh, and θ0h∈Qh are chosen in such a way that(84)u0−u0hV⟶0,φ0−φ0hW⟶0,θ0−θ0hQ⟶0,ash⟶0,under the condition stated in Theorem 1 and for ε>0, the fully discrete solution converges, i.e.,(85)max1≤n≤Nun−unhkV+wn−wnhkV+φn−φnhkW+θn−θnhkQ⟶0,ash,k⟶0.


## 2.1. Problem Setting

We consider a body of a piezoelectric material which occupies the domainΩ⊂ℝdd=2,3 in the reference configuration which will be supposedly bounded with a smooth boundary ∂Ω=Γ. This boundary is divided into three open disjoint parts ΓD, ΓN, and ΓC on one hand and a partition of ΓD∪ΓN into two open parts Γa and Γb on the other hand, such that measΓD>0 and measΓa>0. Let 0;T be the time interval, where T>0.The body is submitted to the action of body forces of densityf0, a volume electric charge of density q0, and a heat source of constant strength q1. It is also submitted to mechanical, electrical, and thermal constants on the boundary. Indeed, the body is assumed to be clamped in ΓD, and therefore, the displacement field vanishes there. Moreover, we assume that a density of traction forces, denoted by f2, acts on the boundary part ΓN. We also assume that the electrical potential vanishes on Γa, and a surface electrical charge of density q2 is prescribed on Γb. We consider that the temperature θ0 is prescribed on the surface ΓD∪ΓN.In the reference configuration, the body may come in contact overΓC with an electrically thermally conductive foundation. Assume that its potential and temperature are maintained at φF and θF. The contact is frictional, and there may be electrical charges and heat transfer on the contact surface. The normalized gap between ΓC and the rigid foundation is denoted by g.In the following sections, we useSd to denote the space of second-order symmetric tensors on ℝd while “.” and . will represent the inner product and the Euclidean norm on Sd and ℝd, that is(3)u⋅v=uivi,v=v⋅v1/2,∀u,v∈ℝd,σ⋅τ=σijτij,τ=τ⋅τ1/2,∀σ,τ∈Sd.We denoteu:Ω×0;T⟶ℝd as the displacement field, σ:Ω⟶Sd and σ=σij the stress tensor, θ:Ω×0;T⟶ℝd the temperature, q:Ω⟶ℝd and q=qi the heat flux vector, and D:Ω⟶ℝd and D=Di the electric displacement field. We also denote Eφ=Eiφ as the electric vector field, where φ:Ω×0;T⟶ℝ is the electric potential. Moreover, let εu=εiju denote the linearized strain tensor given by εiju=1/2ui,j+uj,i, and “Div” and “div” denote the divergence operators for tensor and vector valued functions, respectively, i.e., Divσ=σij,j and divξ=ξj,j. We shall adopt the usual notation for normal and tangential components of displacement vector and stress: υn=υ⋅n,υτ=υ−υnn,σn=σn⋅n, and στ=σn−σnn, where n denotes the outward normal vector on Γ.ProblemP. Find a displacement field u:Ω×0,T⟶ℝd, an electric potential φ:Ω×0,T⟶ℝ, and a temperature field θ:Ω×0,T⟶ℝ such that (4)σ=ℑεu−ε∗Eφ−θℳ+Cεu˙inΩ×0,T,(5)D=ℰεu+βEφ−θ−θ∗PinΩ×0,T,(6)Divσ+f0=0,inΩ×0,T,(7)divD=q0,inΩ×0,T,(8)θ˙+divq=q1,inΩ×0,T,(9)u=0,onΓD×0,T,(10)σν=f2,onΓN×0,T,(11)φ=0,onΓa×0,T,(12)D⋅ν=qbonΓb×0,T,(13)θ=0,onΓD∪ΓN×0,T,(14)u0,x=u0,inΩ,(15)θ0,x=θ0,inΩ,(16)σνuν−g=−1εuν−g+,ε>0,onΓC×0,T,(17)στ≤S,onΓC×0,T,(18)στ<S⟹u˙τ=0,onΓC×0,T,(19)στ=S⟹∃λ≠0such thatστ=−λu˙τonΓC×0,T,(20)D⋅ν=ψuν−gϕLφ−φF,onΓC×0,T,(21)∂q∂ν=kcuν−gϕLθ−θF,onΓC×0,T.Here, the equations (4) and (5) represent the thermo-electro-viscoelastic constitutive law of the material in which σ=σij denotes the stress tensor, εu is the linearized strain tensor, Eφ is the electric field. ℑ=fijkl, ℰ=eijk, ℳ=mij, β=βij, P=pi, and C=cijkl are, respectively, elastic, piezoelectric, thermal expansion, electric permittivity, pyroelectric tensor, and (fourth-order) viscosity tensor. ℰ∗ is the transpose of ℰ given by(22)ℰ∗=eijk∗,eijk∗=ekij,(23)ℰσυ=σℰ∗υ,∀σ∈Sd,υ∈ℝd.The constantθ∗ represents the reference temperature.Fourier’s law of heat conduction is given by(24)q=−K∇θ,inΩ×0,T,where K=kij denotes the thermal conductivity tensor.Equations (6)–(8) represent the equilibrium equations for the stress. Relations (9) and (10), (11) and (12), and (13) represent the mechanical, the electrical, and the thermal boundary conditions. The unilateral boundary condition (16) represents the normal compliance condition and (17)–(19) represent Tresca’s friction law in which S is the coefficient of friction.Following [14], the contact conditions (20) and (21) on ΓC are obtained as follows:When there is no contact at a point on the surface, there is no free electrical charges on the surface and no thermal transfer; that is(25)uν<g⟹D⋅ν=0,q⋅ν=0.If the contact holds, i.e.,uν≥g, the normal component of the electric displacement field or the free charge (resp., thermal transfer) is assumed to be proportional to the difference between the potential of foundation and the body’s surface potential (resp., to the difference between the temperature of foundation and the body’s surface temperature). Thus,(26)uν≥g⟹D⋅ν=kφφ−φF,q⋅ν=kθθ−θF.We combine (25) and (26) to obtain(27)D⋅ν=kφχ0,+∞uν−gφ−φF,q⋅ν=kθχ0,+∞uν−gθ−θF,where χ0,+∞ is the characteristic function of the interval 0,+∞ defined by(28)χ0,+∞s=0,ifs<0,1,ifs≥0.Equation (27) represents the regularization electrical contact condition and the heat flux condition on ΓC, where(29)ϕLs=−L,ifs<−L,s,if−L≤s≤L,L,ifs>L,ψr=0,ifr<0,cδr,if0≤r≤1δ,c,ifr>1δ,and where c=kφ,kθ, and L is a large positive constant, δ>0 is a small parameter, kc:r⟶kcr is supposed to be zero for r<0 and positive, otherwise nondecreasing and Lipschitz continuous.Remark 1.
We note that whenψ≡0, equality (20) becomes(30)D⋅ν=0onΓC×0,T,which models the case when the foundation is a perfect electric insulator.
Similarly, in equality (21), we have(31)q⋅ν=0onΓC×0,T.


## 2.2. Weak Formulation and Uniqueness Result

To obtain a variational formulation of ProblemP, we need additional notations and need to recall some definitions in the sequel.We use the following functional Hilbert spaces:(32)L2Ω=L2Ωd,H1Ω=H1Ωd,ℋ=σ∈Sd:σ=σij,σij=σji∈L2Ω,(33)W=D=Di∈H1Ω:Di∈L2Ω,divD∈L2Ω,endowed with the canonical inner product given by(34)u,vL2Ω=∫Ωuividx,σ,τℋ=∫Ωσiτidx,u,vH1Ω=u,vL2Ω+εu,εvℋ,D⋅E=D⋅EL2Ω+divD⋅divEL2Ω,and the associated norms .L2Ω, .H1Ω, .ℋ, and .W, respectively.LetV, W, and Q be the closed subspaces of H1Ω given by(35)V=v∈H1Ω:v=0onΓD,W=ξ∈H1Ω:ξ=0onΓa,Q=η∈H1Ω:η=0onΓD∪ΓN,and the set of admissible displacements(36)Vad=v∈V:vν−g≤0onΓC.It is known thatV, W, and Q are real Hilbert spaces with the inner products u,vV=εu,εvℋ, φ,ξW=∇φ,∇ξL2Ω, and θ,ηQ=∇θ,∇ηL2Ω, respectively.Moreover, the associated normvV=εvℋ is equivalent on V to the usual norm .H1Ω and ξW=∇ξL2Ω and ηQ=∇ηL2Ω are equivalent on W and Q, respectively, with the usual norms .H1Ω.By Sobolev’s trace theorem, there exists three positive constantsCs1, Cs2, and Cs3 depending on Ω, ΓC, ΓN, ΓD, Γa, and Γb:(37)vL2Γd≤Cs1vV,∀v∈V,(38)ξL2Γc≤Cs2ξW,∀ξ∈W,(39)ηL2Γc≤Cs3ηQ,∀η∈Q.SincemeasΓD>0 and Korn’s inequality hold(40)εvℋ≥CKvH1Ω,∀v∈V,where CK in a nonnegative constant depending only on Ω and ΓD. Notice also that since measΓa>0, the following Friedrichs–Poincaré inequalities hold:(41)∇ξW≥CF1ξW,∀ξ∈W,(42)∇ηL2Ω≥CF2ηQ,∀η∈Q,where CF1 and CF2 are the positive constants which depend only on Ω, Γa, ΓD, and ΓN.For a real Banach spaceX and 1≤p≤∞, we consider the Banach spaces C0,T;X and C10,T;X of continuous and continuously differentiable functions from 0,T to X with the norms(43)uC0,T;X=supt∈0,TutX,uC10,T;X=supt∈0,TutX+supt∈0,Tu˙tX.To simplify the writing, we denote bya:V×V⟶ℝ, b:W×W⟶ℝ, c:V×V⟶ℝ, and d:Q×Q⟶ℝ the following bilinear and symmetric applications:(44)au,v≔ℑεu,εvℋ,bφ,ξ≔β∇φ,∇ξL2Ω,cu,v≔Cεu,εvℋ,dθ,η≔K∇θ,∇ηL2Ω,and by e:V×W⟶ℝ, m:Q×V⟶ℝ, and p:Q×W⟶ℝ, the following bilinear applications:(45)ev,ξ≔ℰεv,∇ξL2Ω=ℰ∗∇ξ,εvV,mθ,v≔ℳθ,εvQ,pθ,ξ≔P∇θ,∇ξL2Ω.In the study of mechanical ProblemP, we make the following assumptions:HP1. The elasticity operator ℑ:Ω×Sd⟶Sd, the electric permittivity tensor β=βij:⟶Ω×ℝd⟶ℝd, the viscosity tensor c:Ω×Sd⟶Sd, and the thermal conductivity tensor K=kij:Ω×ℝd⟶ℝd satisfy the usual properties of symmetry, boundedness, and ellipticity:(46)fijkl=fjikl=flkij∈L∞Ω,βij=βji∈L∞Ω,cijkl=cjikl=clkij∈L∞Ω,kij=kji∈L∞Ω,and there exists thatmℑ,mβ,mc,mK>0 such that(47)fijklxξkξl≥mℑξ2,∀ξ∈Sd,∀x∈Ω,cijklxξkξl≥mcξ2,∀ξ∈Sd,∀x∈Ω,βijζiζj≥mβζ2,kijζiζj≥mKζ2,∀ζ∈ℝd.HP2. The piezoelectric tensor Ε=eijk:Ω×Sd⟶ℝ, the thermal expansion tensor ℳ=mij:Ω×ℝ⟶ℝ, and the pyroelectric tensor P=pi:Ω⟶ℝd satisfy(48)eijk=eikj∈L∞Ω,mij=mji∈L∞Ω,pi∈L∞Ω,and there exist the positive constantsMℑ,Mβ,Mc,MK,Mℰ,Mℳ, and MP such that(49)au,v≤MℑuVvV,bφ,ξ≤MβφWξW,cu,v≤McuVvV,dθ,η≤MKθQηQ,ev,ξ≤MΕvVξW,mθ,v≤MℳθQvV,pθ,ξ≤MPθQξW.HP3. The surface electrical conductivity ψ:ΓC×ℝ⟶ℝ+ and the thermal conductance kc:ΓC×ℝ⟶ℝ+ satisfy the following hypothesis for π=ψ,kc:∃Mπ>0 such that πx,u≤Mπ,∀u∈ℝ, a.e., x∈ΓC and x⟶πx,u is measurable on ΓC for all u∈ℝ and is zero for all u≤0.The functionu⟶πx,u is a Lipschitz function on ℝ for all x∈Γ3. πx,u1−πx,u2≤Lπu1−u2,∀u1,u2∈ℝ, where Lπ is a positive constant.HP4. The forces, the traction, the volume, the surfaces charge densities, and the strength of the heat source are as follows:(50)f0∈C0,T;L2Ωd,f2∈C0,T;L2ΓNd,qb∈L20,T;L2Γb,q1∈L20,T;L2Ω,q0∈L20,T;L2Ω.The potential and the temperature satisfy(51)φF∈L20,T;L2ΓC,θF∈L20,T;L2ΓC.The initial conditions, the friction-bounded function, and the gap function satisfy(52)u0∈K,θ0∈Q,S∈L∞ΓC,S≥0,g∈L2ΓC,g≥0.Using Riesz’s representation theorem, we definef:0,T⟶V, qe:0,T⟶W, and qth:0,T⟶Q by the following:(53)ft,vV=∫Ωf0t⋅vdx+∫ΓNf2t⋅vda,∀v∈V,(54)qet,ξW=∫Ωq0t⋅ξdx−∫Γbqbt⋅ξda,∀ξ∈V,(55)qtht,ηQ=∫Ωq1t·ηdx,∀η∈Q.We define the mappingsj:V⟶ℝ, ϱ:V×V⟶ℝ, ℓ:V×W2⟶ℝ, and χ:V×Q2⟶ℝ, a.e., t∈0,T by (56)jv=∫ΓCSvτda,∀v∈V,(57)ϱu,v=∫ΓCuν+−gvνda=uν+−g,vνΓC,∀u,v∈V,(58)ℓut,φt,ξ=∫ΓCψuνt−gϕLφt−φFξda,∀u∈V,∀φ,ξ∈W,(59)χut,θt,η=∫ΓCkcuνt−gϕLθt−θFηda,∀u∈V,∀θ,η∈Q,respectively.Now, by a standard variational technique, it is straightforward to see that ifu,φ,θ satisfies the conditions (4)–(21), a.e. t∈0;T, then(60)σt,εv−εu˙tℋ+1εϱut,v−u˙t+jv−ju˙t≥ft,v−u˙tV,∀v∈Vad,(61)Dt,∇ξL2Ω=ℓut,φt,ξ−qet,ξW,∀ξ∈W,(62)qt,∇ηL2Ω=θ˙t,ηQ+χut,θt,η−qtht,ηQ,∀η∈Q.We assume that the initial conditionsu0 and θ0 satisfy the following compatibility condition: there exists φ0∈W such that(63)bφ0,ξ−eu0,ξ−pθ0,ξ+ℓu0,φ0,ξ=qe0,ξW,∀ξ∈W.This nonlinear problem, has a unique solutionφ0, by using the fixed point theorem.Using all of these assumptions, notations, andE=−∇φ, we obtain the following variational formulation of the Problem P.ProblemPV: find a displacement field u:0;T⟶Vad, an electric potential φ:0;T⟶W, and a temperature field θ:0;T⟶Q, a.e., t∈0,T such that(64)cu˙t,v−u˙t+aut,v−u˙t+ev−u˙t,φt−mθt,v−u˙t+1ϵϱut,v−u˙t+jv−ju˙t≥ft,v−u˙tV,∀v∈Vad,(65)bφt,ξ−eut,ξ−pθt,ξ+ℓut,φt,ξ=qet,ξW,∀ξ∈W,(66)dθt,η+θ˙t,ηQ+χut,θt,η=qtht,ηQ,∀η∈Q,(67)u0,x=u0x,θ0,x=θ0x.We present now the existence and the uniqueness of solution to ProblemPV.Theorem 1.
Assume that the assumptionsHP1−HP4, (37)–(42), for ε>0,and the conditions(68)mβ>Mψc12,mK<c2Mkcc2+LkkLc02,hold. Then, Problem PV has a unique solution as follows:(69)u∈C10,T;V,φ∈L20,T;W,θ∈L20,T;Q.


## 2.3. Spatially Semidiscrete Approximation

In this paragraph, we consider a semidiscrete approximation of the ProblemPV by discretizing the spatial domain, using the finite element method. Let Th be a regular finite element partition of the domain Ω compatible with the boundary partition Γ=Γ¯C∪Γ¯D∪Γ¯N. We then define a finite element space Vh⊂VandVadh=Vad∩Vh, for the approximates of the displacement field u, Wh⊂W for the electric potential φ, and Qh⊂Q for the temperature θ defined by(70)Vh=vh∈CΩ¯d;vTrh∈ℙ1Trd∀Tr∈Th;vh=0onΓ¯C,Wh=ξh∈CΩ¯;ξTrh∈ℙ1Tr∀Tr∈Th;ξh=onΓ¯a,Qh=ηh∈CΩ¯;ηTrh∈ℙ1Tr∀Tr∈Th;ηh=0onΓ¯D∪Γ¯N.A spatially semidiscrete scheme can be formed as the following problem:ProblemPVh.Find uh:0;T⟶Vh, φh:0;T⟶Wh, and θh:0;T⟶Qh, a.e., t∈0,T such that for vh∈Vadh, ξh∈Wh, and ηh∈Qh(71)cu˙ht,vh−u˙ht+auht,vh−u˙ht+evh−u˙ht,φht−mθht,vh−u˙h+1εϱuht,vh−u˙ht+jvh−ju˙ht≥ft,vh−u˙ht,(72)bφht,ξh−euht,ξh−pθht,ξh+ℓuht,φht,ξh=qet,ξh,(73)dθht,ηh+θ˙ht,ηh+χuht,ηht,ηh=qtht,ηh,(74)uh0=u0h,φh0=φ0h,θh0=θ0h.Here,u0h∈Vh, φ0h∈Wh, and θ0h∈Qh are appropriate approximations of u0, φ0, and θ0, respectively.Using the same argument presented in Section 2, it can be shown that ProblemPVh has a unique solution uh∈C10,T;Vh, φh∈L20,T;Wh, and θh∈L20,T;Qh.In this paragraph, we are interested in obtaining estimates for the errorsu−uh, φ−φh, and θ−θh.Using the initial value condition, we have(75)ut=∫0twsds+u0,θt=∫0tθ˙sds+θ0,(76)uht=∫0tu˙hsds+u0h,θht=∫0tθ˙hsds+θ0h.Theorem 2.
Assume that the assumptions stated in Theorem 1 are hold, forε>0. Then, under the conditions(77)u0−u0hV⟶0,θ0−θ0hV⟶0,ash⟶0,the semi-discrete solution of PVh converges as follows:(78)u−uhL20,T;V⟶0,φ−φhL20,T;W⟶0,θ−θhL20,T;Q⟶0,ash⟶0.


## 2.4. Fully Discrete Approximation

In this paragraph, we consider a fully discrete approximation of ProblemPV. We use the finite element spaces Vh, Wh, and Qh introduced in Section 2.3. We introduce a partition of the time interval 0;T:0=t0<t1<…<tN=T. We denote the step size Δt=kn=tn−tn−1 for n=1,2,…,N and let k=maxnkn be the maximal step size. For a sequence vnn=0N, we denote δw=wn−wn−1/kn.The fully discrete approximation method is based on the backward Euler scheme, and it has the following form.ProblemPVnhk. Find a displacement field unhkn=0N⊂Vadh, an electric potential φnhkn=0N⊂Wh, and a temperature field θnhkn=0N⊂Qh for all vh∈Vh,ξh∈Wh,ηh∈Qh, and n=1,…,N such that(79)cwnhk,vh−wnhk+aunhk,vh−wnhk+evh−wnhk,φn−1hk−mθn−1hk,vh−wnhk+1εϱunhk,vh−wnhk+jvh−jwnhk≥fn,vh−wnhkV,(80)bφnhk,ξh−eunhk,ξh−pθnhk,ξh+ℓunhk,φnhk,ξh=qen,ξhW,(81)dθnhk,ηh+δθnhk,ηh+χunhk,θnhk,ηh=qthn,ηhQ,(82)u0hk=u0h,φ0hk=φ0h,θ0hk=θ0h.Remark 2.
The choice ofθn−1h and φn−1h instead of θnh and φnh is motivated by the fixed point method in the proof of the existence and uniqueness. Otherwise, we may get another different condition for the uniqueness of the solution of fixed iteration problem (79)–(82). In addition, this choice will be helpful for the application of discrete Grönwall’s lemma in the next.To simplify again the notation, we introduce the velocity(83)wnhk=δunhk,n=1,…,N,unhk=∑j=1nkjwjhk+u0h,θnhk=∑j=1nkjδθjhk+θ0h,n≥1.This problem has a unique solution, and the proof is similar to that used in Theorem 1.We now derive the following convergence result.Theorem 3.
Assuming that the initial valuesu0h∈Vh, φ0h∈Wh, and θ0h∈Qh are chosen in such a way that(84)u0−u0hV⟶0,φ0−φ0hW⟶0,θ0−θ0hQ⟶0,ash⟶0,under the condition stated in Theorem 1 and for ε>0, the fully discrete solution converges, i.e.,(85)max1≤n≤Nun−unhkV+wn−wnhkV+φn−φnhkW+θn−θnhkQ⟶0,ash,k⟶0.


## 3. Proof of Main Results

In this section, we prove the theorems presented in the previous section.


### 3.1. Proof of Theorem 1

The proof of Theorem 1 is based on fixed point argument, Galerkin method, and compactness method, similar to those used in [14, 15] but with a different choice of the operators.We turn now the following existence and uniqueness result.Letℱ∈C0,T;V given by(86)ℱt,v−u˙ℱtV=ev−u˙ℱt,φℱt−mθℱt,v−u˙ℱt.In the first step, we consider the intermediate ProblemPVdf.ProblemPVdf. Find uℱ∈Vad for, a.e., t∈0,T such that(87)cu˙ℱt,v−u˙ℱt+auℱt,v−u˙ℱt+ℱt,v−u˙ℱtV+1εϱuℱt,v−u˙t+jv−ju˙ℱt≥ft,v−u˙ℱt,uℱ0=u0,∀v∈V.We have the following result forPVdf.Lemma 1.
For allv∈Vad and for, a.e., t∈0,T, the Problem PVdf has a unique solution uℱ∈C10,T;V.Proof.
Using Riesz’s representation theorem, we define the operatorA:V⟶V and the element fℱt∈V by(88)fℱt,vV=ft,vV−ℱt,vV,(89)Auℱt,v=auℱt,v+1εϱuℱt,v.
Then, ProblemPVdf can be written in the following form:(90)cu˙ℱt,v−u˙ℱt+Auℱt,v−u˙ℱt+jv−ju˙ℱt≥fℱt,v−u˙ℱt,uℱ0=u0.
For allu,v∈V, there exists a constant c>0 which depends only on Mℑ, Cs1, and ϵ such that(91)Au,v≤cuVvV.
The assumptionHP4 and ℱ∈C0,T;V imply that fℱ∈C0,T;V and by HP1−HP2, the operator c is continuous and coercive.
We use now the result presented in pp. 61–65 in [16], and we conclude Problem PVdf has a unique solution uℱ∈C10,T;V.
Next, we use the displacement fielduℱ obtained in the first step and we consider the following lemma proved in [15].Lemma 2.
(a)
For allη∈Q and, a.e., t∈0,T, the problem
(92)
d
θ
ℱ
t
,
η
+
θ
˙
ℱ
t
,
η
Q
+
χ
u
ℱ
t
,
θ
ℱ
t
,
η
=
q
th
t
,
η
Q
,
∀
η
∈
Q
,
θ
ℱ
0
=
θ
0
,
has a unique solutionθℱ∈L20,T;Q.
(b)
For allξ∈W and for, a.e., t∈0,T, the problem
(93)
b
φ
ℱ
t
,
ξ
−
e
u
ℱ
t
,
ξ
−
p
θ
ℱ
t
,
ξ
+
ℓ
u
ℱ
t
,
φ
ℱ
t
,
ξ
=
q
e
t
,
ξ
W
,
φ
ℱ
0
=
φ
0
,has a unique solution φℱ∈L20,T;W.In the last step, forℱ∈L20,T;V, φℱ and θℱ are the functions obtained in Lemma 2 and we consider the operator ℒ:C0,T;V⟶C0,T;V defined by(94)ℒℱt,vV=ev,φℱt−mθℱt,v,for all v∈V and for, a.e., t∈0,T.For the operatorℒ, we have the following result obtained in [15].Lemma 3.
There exists a uniqueℱ˜∈C0,T;V such that ℒℱ˜=ℱ˜.We now turn to a proof of Theorem 1.Existence. Let ℱ˜∈C0,T;V be the fixed point of the operator ℒ and us denote x˜=u˜ℱ,φ˜ℱ,θ˜ℱ the solution of variational Problem PVℱ, for ℱ˜=ℱ; the definition of ℒ and Problem PVℱ prove that x˜ is a solution of Problem PV.Uniqueness. The uniqueness of the solution follows from the uniqueness of the fixed point of the operator ℒ.


### 3.2. Proof of Theorem 2

To prove Theorem 2, we need the following resultLemma 4.
Assume thatHP1−HP3. Then, we have the estimate as follows:(95)ut−uhtL20,T;V+φt−φhtL20,T;W+θt−θhtL20,T;Q≤c∗infvh∈L20,T;Vhξh∈L20,T;Whηh∈L20,T;Qhu˙t−vhL20,T;V+φt−ξhL20,T;W+θt−ηhL20,T;Q+St,vh,u˙tL20,T1/2+u0−u0hV+θ0−θ0hQ,for a positive constant c∗.Proof.
Takev=u˙ht in (64) and add the inequality to (71), we have(96)cu˙t−u˙ht,u˙t−u˙ht≤cu˙ht,vh−u˙t+aut,u˙ht−u˙t+auht,vh−u˙ht+eu˙ht−u˙t,φt+evh−u˙ht,φht−mθt,u˙ht−u˙t−mθht,vh−u˙ht+1εϱut,u˙ht−u˙t+ϱuht,vh−u˙ht+jvh−ju˙ht+ju˙ht−ju˙t−f,vh−u˙t−f,u˙h−u˙t.
Then,(97)cu˙t−u˙ht,u˙t−u˙ht≤St,vh,u˙t+SΦh+cu˙ht−u˙t,vh−u˙t+auht−ut,vh−u˙ht+evh−u˙ht,φht−φt−mθht−θt,vh−u˙ht,where(98)St,vh,u˙t=cu˙t,vh−u˙t+aut,vh−u˙t+evh−u˙t,φt−mθt,vh−u˙t+jvh−eu˙t−f,vh−u˙t,(99)Sϱh=1εϱut,u˙ht−u˙t+ϱuht,vh−u˙ht−ϱut,vh−u˙t.
We takeξ=ξh in (65) and η=ηh in (66) and by subtracting to (72) and to (73), respectively, we deduce that(100)bφt−φht,φt−φht=Sℓh+bφt−φht,φt−ξh−eut−uht,φt−ξh−eut−uht,φht−φt−pθt−θht,φt−ξh−pθt−θht,φht−φt,(101)dθt−θht,θt−θht=Sχh+dθt−θht,θt−ηh−θ˙t−θ˙ht,ηh−θt−θ˙t−θ˙ht,θt−θht,where(102)Sℓh=−ℓut,φt,ξh−φt−ℓut,φt,φt−φht+ℓuht,φht,ξh−φt+ℓuht,φht,φt−φht, (103)Sχh=χuht,θht,ηh−θt+χuht,θht,θt−θht−χut,θt,ηh−θt−χut,θt,θt−θht.
We now add (87), (100), and (101); we obtain(104)cu˙t−u˙ht,u˙t−u˙ht+bφt−φht,φt−φht+dθt−θht,θt−θht≤St,vh,u˙t+Sϱh+Sℓ+Sχ+cu˙ht−u˙t,vh−u˙t+auht−ut,vh−u˙ht+evh−u˙ht,φht−φt−mθht−θt,vh−u˙ht,bφt−φht,φt−ξh−eut−uht,φt−ξh−eut−uht,φht−φt−pθt−θht,φt−ξh−pθt−θht,φht−φt+dθt−θht,θt−ηh−θ˙t−θ˙ht,ηh−θt−θ˙t−θ˙ht,θt−θht.
FromHP1 and the previous inequality, it follows that(105)mcu˙t−u˙htV2+mβφt−φhtW2+mKθt−θhtQ2≤St,vh,u˙t+Sϱh+Sℓh+Sχh+Sh,where(106)Sh=cu˙ht−u˙t,vh−wt+auht−ut,vh−u˙ht+evh−u˙ht,φht−φt−mθht−θt,vh−u˙htbφt−φht,φt−ξh−eut−uht,φt−ξh−eut−uht,φht−φt−pθt−θht,φt−ξh−pθt−θht,φht−φt+dθt−θht,θt−ηh−θ˙t−θ˙ht,ηh−θt−θ˙t−θ˙ht,θt−θht.
Let us estimate each of the terms in (105), HP1−HP3, and(107)xy≤βx2+14βy2,∀β>0,allowing us to obtain(108)Sh≤αhu˙t−u˙htV2+ut−uhtV2+φt−φhtW2+θt−θhtQ2+θ˙t−θ˙htQ2+u˙ht−vhV2+φht−ξhW2+θt−ηhQ2.
Using the assumptionsHP3, (38), and (58), we find(109)Sℓh≤−ℓut,φt,ξh−φt−ℓut,φt,φt−φht+ℓuht,φht,ξh−φt+ℓuht,φht,φt−φht≤∫ΓCψutν−gϕLφt−φFξ−φtda+∫ΓCψutν−gϕLφt−φFφ−φhtda≤∫ΓCψutν−gϕLφt−φFξ−φtda+∫ΓCψutν−gϕLφt−φFφ−φhtda+∫ΓCψuhtν−gϕLφht−φFξh−φtda+∫ΓCψuhtν−gϕLφht−φFφ−φhtda≤Mψ∫ΓCφt−φht2da+LLψ∫ΓCφt−φhtuνt−uνhtda≤MψCs1φt−φhtW2+LLψCs1Cs2ut−uhtVφt−φhtW+MψCs2φt−φhtWφt−ξhL2ΓC+LLψc0ut−uhtVφt−ξhL2ΓC≤αℓφt−φhtW2+ut−uhtV2+φt−ξhW.
Similarly, we find(110)Sχh≤αχθt−θhtW2+ut−uhtV2+θt−ηhQ2.
By combining the inequality(111)x+−y+≤x−y,with (107), we observe that(112)Sϱh≤cϱut−uhtV2+u˙t−u˙htV2+vh−u˙tV2.
Thus, by (106) and (108)–(110), we have the inequality(113)u˙t−u˙htV2+φt−φhtW2+θt−θhtQ2≤c1∗ut−uhtV2+θ˙t−θ˙htQ2+Rt;vh,u˙t+u˙t−vhV2+φt−ξhW2+θt−ηhQ2.
Then, by (75) and (76), we have(114)ut−uht=∫0tu˙s−u˙hsds+u0−u0h,θt−θht=∫0tθ˙s−θ˙hsds+θ0−θ0h,and so(115)ut−uhtV2≤c2∗∫0tu˙s−u˙hsV2ds+u0−u0hV2,θt−θhtQ2≤c3∗∫0tθ˙s−θ˙hsQ2ds+θ0−θ0hQ2.
Consequently, from the previous inequalities and Grönwall's inequality in (113), we find (95).Proof of Theorem 2.
To estimate the error provided by the approximation of the finite element spaceVh,Wh,andQh,we useΠhu,Πhφ,andΠhθ, the standard finite element interpolation operator of u,φandθ, respectively. We then have the interpolation error estimate [16](116)u˙−Πhu˙V≤chu˙L20,T;V,φ−ΠhφW≤chφL20,T;W,θ−ΠhθQ≤chθL20,T;Q.
We bound now the termS.;vh.u˙..Using the properties of c, a, e, m, f, and (98), there exists positive constant c depending on Mc, Mℑ, Mℰ, Mℳ, fL20,T;V, uL20,T;V, φL20,T;V, and θL20,T;V such that(117)St,vh,u˙t≤cvht−u˙tV+jvht−ju˙t.
Taking (116) in (95), we have(118)u−uhL20,T;V+φ−φhL20,T;W+θ−θhL20,T;Q≤c+u˙−Πhu˙L20,T;Vφ−ΠhφL20,T;W+θ−ΠhθL20,T;Q+u˙−Πhu˙L20,T;V1/2+jΠhu˙−ju˙L20,T;V1/2+u0−u0hV+θ0−θ0hQ.
From (116), the constant c is independent of h, u, φ, and θ. This implies(119)u˙−Πhu˙V⟶0,φ−ΠhφW⟶0θ−ΠhθQash⟶0.
FromHP4 and (53), it follows that,(120)jΠhu˙−ju˙L20,T;V⟶0.
Finally, we find the result (78).


### 3.3. Proof of Theorem 3

Add (80) with vh=vnh and (64) with v=wnhk at t=tn, we find(121)cwnhk,vnh−wnhk+cwn,wnhk−wn+aunhk,vnh−wnhk+aun,wnhk−wn+evnh−wnhk,φn−1hk+ewnhk−wn,φn−mθn−1hk,vnh−wnhk−mθn,wnhk−wn+1εϱunhk,vnh−wnhk+ϱun,wnhk−wn+jvnh−jwnhk+jwnhk−jwn≥fn,vnh−wnhk+fn,wnhk−wn.This equality,(122)cwn−wnhk,wn−wnhk=cwnhk,wnhk−vnh+cwn,wn−vnh−cwnhk,wn−vnh+cwn,vnh−wnkh,allows us to obtain(123)cwn−wnhk,wn−wnhk≤Rvnh,wn+Sϱhk+cwnhk−wn,vnh−wn+aunhk−un,vnh−wnhk+evnh−wnhk,φn−1hk−φn−mθn−1hk−θn,vnh−wnhk,where(124)Snvnh,wn=cwn,vnh−wn+aun,vnh−wn+evnh−wn,φn−mθn,vnh−wn+jvnh−jwn−fn,vnh−wn, (125)Sϱhk=1εϱunhk,vnh−wnhk+ϱun,wnhk−wn−ϱun,vnh−wn.For alln≥1, we subtract (65) from (81) at t=tn; we obtain that(126)bφn−φnhk,ξh−eun−unhk,ξh−pθn−θnhk,ξh+ℓun,φn,ξh−ℓunhk,φnhk,ξh=0.We replaceξh by φn−φnhk and φn−ξh in (126); we find that(127)bφn−φnhk,φn−φnhk≤Sℓhk+bφn−φnhk,φn−ξh+eun−unhk,ξh−φnhk+pθn−θnhk,ξh−φnhk,where(128)Sℓhk=ℓun,φn,φn−ξh−ℓunhk,φnhk,φn−ξh−ℓun,φn,φn−φnhk+ℓunhk,φnhk,φn−φnhk.For alln≥1 and similar to (127) and (128), we have(129)dθn−θnhk,θn−θnhk≤Sχhk+dθn−θnhk,θn−ηh+δθn,θn−ηh−δθn,θn−θnhk−δθnhk,θn−ηh+δθnhk,θn−θnhk,where δθn=1/knθn−θn−1 and δθnhk=1/knθnhk−θn−1hk, and(130)Sχhk=χun,θn,θn−ηh−χunhk,θnhk,θn−ηh+χunhk,θnhk,θn−θnhk−χun,θn,θn−θnhk.Adding (124), (128), and (130), we find(131)cwn−wnhk,wn−wnhk+bφn−φnhk,φn−φnhk+dθn−θnhk,θn−θnhk≤Snvnh,wn+Sϱhk+Sℓhk+Sχhk+cwnhk−wn,vnh−wn+aunhk−un,vnh−wnhk+evnh−wnhk,φn−1hk−φn−mθn−1hk−θn,vnh−wnhk+bφn−φnhk,φn−ξh+dθn−θnhk,θn−ηh+eun−unhk,ξh−φnhk+pθn−θnhk,ξh−φnhk+δθn,θn−ηh−δθn,θn−θnhk−δθnhk,θn−ηh+δθnhk,θn−θnhk.The ellipticity of operatorsc, b, d, and the previous inequality follows that(132)mCwn−wnhkV2+mβφn−φnhkW2+mKθn−θnhkV2≤Snvnh,wn+Sϱhk+Sℓhk+Sχhk+Shk,where(133)Shk=cwnhk−wn,vnh−wn+aunhk−un,vnh−wnhk+evnh−wnhk,φn−1hk−φn−mθn−1hk−θn,vnh−wnhk+bφn−φnhk,φn−ξh+eun−unhk,ξh−φnhk+pθn−θnhk,ξh−φnhk+dθn−θnhk,θn−ηh+δθn,θn−ηh−δθn,θn−θnhk−δθnhk,θn−ηh+δθnhk,θn−θnhk.Now, we estimate each of the terms in (132).Thanks to
HP
1
−
HP
3, (107), (117), (111),and(134)vnh−wnhk≤vnh−wn+wn−wnhk,we get(135)Shk≤αhkwn−wnhkV2+un−unhkV2φn−φnhkW2+θn−θnhkQ2+φn−φn−1hkW2+θn−θn−1hkQ2+θn−1−θn−1hkQ2+vnh−wnV2+ξh−φnW2+θn−ηhQ2,(136)Sϱhk≤cϱ′un−unhkV2+wn−wnhkV2+vnh−wnV2,(137)Sℓhk≤αℓφn−φnhkW2+un−unhkV2+φn−ξhW2,(138)Sχhk≤αχθn−θnhkQ2+un−unhkV2+θn−ηhQ2.Combining (132) and (135)–(138), we have the following estimate:(139)wn−wnhkV+φn−φnhkW+θn−θnhkQ≤cφn−φn−1hkW+θn−θn−1hkQ+θn−1−θn−1hkQ+Snvnh,wn1/2+vnh−wnV+ξh−φnW+ηh−θnQ.We estimate now the termsφn−φn−1hk, θn−θn−1hk, and θn−1−θn−1hk, for the first term; we have(140)φn−1hk−φn=∑j=1n−1kjδφjhk+φ0h−∫0tnδφsds−φ0=∑j=1n−1kjδφjhk−δφj+φ0h−φ0+∑j=1n−1δφjkj−∫tj−1tjδφsds−∫tn−1tnδφsds,where δφj=1/kjφj−φj−1 and δφjhk=1/kjφjhk−φj−1hk, and(141)∑j=1n−1δφjkj−∫tj−1tjδφsdsW=∑j=1n−1∫tj−1tjδφj−δφsdsW≤∑j=1n−1∫tj−1tjδφj−δφsWds=Ikδφ,(142)∫tn−1tnδφsds≤∫tn−1tnδφsWds≤kδφC0,T;W.Thus,(143)φn−1hk−φnW≤∑j=1n−1kjδφjkh−δφjW+φ0h−φ0W+Ikδφ+kφC0,T;W.Similar to (143), we deduce that(144)θn−1hk−θnQ≤∑j=1n−1kjδθjkh−δθjQ+θ0h−θ0Q+Ikδθ+kθC10,T;Q,(145)θn−1hk−θn−1Q≤∑j=1n−1kjδθjkh−δθjQ+θ0h−θ0Q+Ikδθ.To proceed, we need the following discrete version of the Grönwall’s inequality presented in [12].Lemma 5.
Assuming thatgnn=1N and enn=1N are the two sequences of nonnegative numbers satisfying(146)en≤cgn+c∑j=1n−1kjej.Then,(147)en≤cgn+∑j=1n−1kjgj,n=1,…,N.Therefore,(148)max1≤n≤Nen≤cmax1≤n≤Ngn.Also, the following result.Lemma 6.
Let us define the map by(149)Ikv=∑j=1n−1∫tj−1tjvj−vsXds.Then,Ikv converges to zero as k⟶0, for all v∈C0,T;X.Proof.
Sincev∈C0,T;X, t⟼vt is uniformly continuous on 0,T. Thus, for any ε>0, there exists a k0>0 such that if k<k0, we have(150)vt−vsX<εT∀s,t∈0,T,t−s<k.
Then, we have(151)Ikv≤∑j=1N∫tj−1tjεTds=ε.
Hence,Ikv converges to zero as k⟶0.We now combine the inequalities (139) and (143)–(145) and applying the Lemma 5, we obtain the following estimate:(152)max1≤n≤Nwn−wnhkV+un−unhkV+φn−φnhkW+θn−θnhkQ≤max1≤n≤Ninfvh∈Vhξh∈Whηh∈QhcSnvnh,wn1/2+vnh−wnQ+ξh−φnW+ηh−θnQ+cφ0−φ0hW+θ0−θ0hQ+cIkδφ+Ikδφ+kφC10,T;W+kθC10,T;Q.Proof of Theorem 3.
We takevnh=Πhwn, ξnh=Πhφn, and ηnh=Πhθn in (152).
The convergence result follows from Lemma 6 together with an argument similar to the proof of Theorem 2.


## 3.1. Proof of Theorem 1

The proof of Theorem 1 is based on fixed point argument, Galerkin method, and compactness method, similar to those used in [14, 15] but with a different choice of the operators.We turn now the following existence and uniqueness result.Letℱ∈C0,T;V given by(86)ℱt,v−u˙ℱtV=ev−u˙ℱt,φℱt−mθℱt,v−u˙ℱt.In the first step, we consider the intermediate ProblemPVdf.ProblemPVdf. Find uℱ∈Vad for, a.e., t∈0,T such that(87)cu˙ℱt,v−u˙ℱt+auℱt,v−u˙ℱt+ℱt,v−u˙ℱtV+1εϱuℱt,v−u˙t+jv−ju˙ℱt≥ft,v−u˙ℱt,uℱ0=u0,∀v∈V.We have the following result forPVdf.Lemma 1.
For allv∈Vad and for, a.e., t∈0,T, the Problem PVdf has a unique solution uℱ∈C10,T;V.Proof.
Using Riesz’s representation theorem, we define the operatorA:V⟶V and the element fℱt∈V by(88)fℱt,vV=ft,vV−ℱt,vV,(89)Auℱt,v=auℱt,v+1εϱuℱt,v.
Then, ProblemPVdf can be written in the following form:(90)cu˙ℱt,v−u˙ℱt+Auℱt,v−u˙ℱt+jv−ju˙ℱt≥fℱt,v−u˙ℱt,uℱ0=u0.
For allu,v∈V, there exists a constant c>0 which depends only on Mℑ, Cs1, and ϵ such that(91)Au,v≤cuVvV.
The assumptionHP4 and ℱ∈C0,T;V imply that fℱ∈C0,T;V and by HP1−HP2, the operator c is continuous and coercive.
We use now the result presented in pp. 61–65 in [16], and we conclude Problem PVdf has a unique solution uℱ∈C10,T;V.
Next, we use the displacement fielduℱ obtained in the first step and we consider the following lemma proved in [15].Lemma 2.
(a)
For allη∈Q and, a.e., t∈0,T, the problem
(92)
d
θ
ℱ
t
,
η
+
θ
˙
ℱ
t
,
η
Q
+
χ
u
ℱ
t
,
θ
ℱ
t
,
η
=
q
th
t
,
η
Q
,
∀
η
∈
Q
,
θ
ℱ
0
=
θ
0
,
has a unique solutionθℱ∈L20,T;Q.
(b)
For allξ∈W and for, a.e., t∈0,T, the problem
(93)
b
φ
ℱ
t
,
ξ
−
e
u
ℱ
t
,
ξ
−
p
θ
ℱ
t
,
ξ
+
ℓ
u
ℱ
t
,
φ
ℱ
t
,
ξ
=
q
e
t
,
ξ
W
,
φ
ℱ
0
=
φ
0
,has a unique solution φℱ∈L20,T;W.In the last step, forℱ∈L20,T;V, φℱ and θℱ are the functions obtained in Lemma 2 and we consider the operator ℒ:C0,T;V⟶C0,T;V defined by(94)ℒℱt,vV=ev,φℱt−mθℱt,v,for all v∈V and for, a.e., t∈0,T.For the operatorℒ, we have the following result obtained in [15].Lemma 3.
There exists a uniqueℱ˜∈C0,T;V such that ℒℱ˜=ℱ˜.We now turn to a proof of Theorem 1.Existence. Let ℱ˜∈C0,T;V be the fixed point of the operator ℒ and us denote x˜=u˜ℱ,φ˜ℱ,θ˜ℱ the solution of variational Problem PVℱ, for ℱ˜=ℱ; the definition of ℒ and Problem PVℱ prove that x˜ is a solution of Problem PV.Uniqueness. The uniqueness of the solution follows from the uniqueness of the fixed point of the operator ℒ.


## 3.2. Proof of Theorem 2

To prove Theorem 2, we need the following resultLemma 4.
Assume thatHP1−HP3. Then, we have the estimate as follows:(95)ut−uhtL20,T;V+φt−φhtL20,T;W+θt−θhtL20,T;Q≤c∗infvh∈L20,T;Vhξh∈L20,T;Whηh∈L20,T;Qhu˙t−vhL20,T;V+φt−ξhL20,T;W+θt−ηhL20,T;Q+St,vh,u˙tL20,T1/2+u0−u0hV+θ0−θ0hQ,for a positive constant c∗.Proof.
Takev=u˙ht in (64) and add the inequality to (71), we have(96)cu˙t−u˙ht,u˙t−u˙ht≤cu˙ht,vh−u˙t+aut,u˙ht−u˙t+auht,vh−u˙ht+eu˙ht−u˙t,φt+evh−u˙ht,φht−mθt,u˙ht−u˙t−mθht,vh−u˙ht+1εϱut,u˙ht−u˙t+ϱuht,vh−u˙ht+jvh−ju˙ht+ju˙ht−ju˙t−f,vh−u˙t−f,u˙h−u˙t.
Then,(97)cu˙t−u˙ht,u˙t−u˙ht≤St,vh,u˙t+SΦh+cu˙ht−u˙t,vh−u˙t+auht−ut,vh−u˙ht+evh−u˙ht,φht−φt−mθht−θt,vh−u˙ht,where(98)St,vh,u˙t=cu˙t,vh−u˙t+aut,vh−u˙t+evh−u˙t,φt−mθt,vh−u˙t+jvh−eu˙t−f,vh−u˙t,(99)Sϱh=1εϱut,u˙ht−u˙t+ϱuht,vh−u˙ht−ϱut,vh−u˙t.
We takeξ=ξh in (65) and η=ηh in (66) and by subtracting to (72) and to (73), respectively, we deduce that(100)bφt−φht,φt−φht=Sℓh+bφt−φht,φt−ξh−eut−uht,φt−ξh−eut−uht,φht−φt−pθt−θht,φt−ξh−pθt−θht,φht−φt,(101)dθt−θht,θt−θht=Sχh+dθt−θht,θt−ηh−θ˙t−θ˙ht,ηh−θt−θ˙t−θ˙ht,θt−θht,where(102)Sℓh=−ℓut,φt,ξh−φt−ℓut,φt,φt−φht+ℓuht,φht,ξh−φt+ℓuht,φht,φt−φht, (103)Sχh=χuht,θht,ηh−θt+χuht,θht,θt−θht−χut,θt,ηh−θt−χut,θt,θt−θht.
We now add (87), (100), and (101); we obtain(104)cu˙t−u˙ht,u˙t−u˙ht+bφt−φht,φt−φht+dθt−θht,θt−θht≤St,vh,u˙t+Sϱh+Sℓ+Sχ+cu˙ht−u˙t,vh−u˙t+auht−ut,vh−u˙ht+evh−u˙ht,φht−φt−mθht−θt,vh−u˙ht,bφt−φht,φt−ξh−eut−uht,φt−ξh−eut−uht,φht−φt−pθt−θht,φt−ξh−pθt−θht,φht−φt+dθt−θht,θt−ηh−θ˙t−θ˙ht,ηh−θt−θ˙t−θ˙ht,θt−θht.
FromHP1 and the previous inequality, it follows that(105)mcu˙t−u˙htV2+mβφt−φhtW2+mKθt−θhtQ2≤St,vh,u˙t+Sϱh+Sℓh+Sχh+Sh,where(106)Sh=cu˙ht−u˙t,vh−wt+auht−ut,vh−u˙ht+evh−u˙ht,φht−φt−mθht−θt,vh−u˙htbφt−φht,φt−ξh−eut−uht,φt−ξh−eut−uht,φht−φt−pθt−θht,φt−ξh−pθt−θht,φht−φt+dθt−θht,θt−ηh−θ˙t−θ˙ht,ηh−θt−θ˙t−θ˙ht,θt−θht.
Let us estimate each of the terms in (105), HP1−HP3, and(107)xy≤βx2+14βy2,∀β>0,allowing us to obtain(108)Sh≤αhu˙t−u˙htV2+ut−uhtV2+φt−φhtW2+θt−θhtQ2+θ˙t−θ˙htQ2+u˙ht−vhV2+φht−ξhW2+θt−ηhQ2.
Using the assumptionsHP3, (38), and (58), we find(109)Sℓh≤−ℓut,φt,ξh−φt−ℓut,φt,φt−φht+ℓuht,φht,ξh−φt+ℓuht,φht,φt−φht≤∫ΓCψutν−gϕLφt−φFξ−φtda+∫ΓCψutν−gϕLφt−φFφ−φhtda≤∫ΓCψutν−gϕLφt−φFξ−φtda+∫ΓCψutν−gϕLφt−φFφ−φhtda+∫ΓCψuhtν−gϕLφht−φFξh−φtda+∫ΓCψuhtν−gϕLφht−φFφ−φhtda≤Mψ∫ΓCφt−φht2da+LLψ∫ΓCφt−φhtuνt−uνhtda≤MψCs1φt−φhtW2+LLψCs1Cs2ut−uhtVφt−φhtW+MψCs2φt−φhtWφt−ξhL2ΓC+LLψc0ut−uhtVφt−ξhL2ΓC≤αℓφt−φhtW2+ut−uhtV2+φt−ξhW.
Similarly, we find(110)Sχh≤αχθt−θhtW2+ut−uhtV2+θt−ηhQ2.
By combining the inequality(111)x+−y+≤x−y,with (107), we observe that(112)Sϱh≤cϱut−uhtV2+u˙t−u˙htV2+vh−u˙tV2.
Thus, by (106) and (108)–(110), we have the inequality(113)u˙t−u˙htV2+φt−φhtW2+θt−θhtQ2≤c1∗ut−uhtV2+θ˙t−θ˙htQ2+Rt;vh,u˙t+u˙t−vhV2+φt−ξhW2+θt−ηhQ2.
Then, by (75) and (76), we have(114)ut−uht=∫0tu˙s−u˙hsds+u0−u0h,θt−θht=∫0tθ˙s−θ˙hsds+θ0−θ0h,and so(115)ut−uhtV2≤c2∗∫0tu˙s−u˙hsV2ds+u0−u0hV2,θt−θhtQ2≤c3∗∫0tθ˙s−θ˙hsQ2ds+θ0−θ0hQ2.
Consequently, from the previous inequalities and Grönwall's inequality in (113), we find (95).Proof of Theorem 2.
To estimate the error provided by the approximation of the finite element spaceVh,Wh,andQh,we useΠhu,Πhφ,andΠhθ, the standard finite element interpolation operator of u,φandθ, respectively. We then have the interpolation error estimate [16](116)u˙−Πhu˙V≤chu˙L20,T;V,φ−ΠhφW≤chφL20,T;W,θ−ΠhθQ≤chθL20,T;Q.
We bound now the termS.;vh.u˙..Using the properties of c, a, e, m, f, and (98), there exists positive constant c depending on Mc, Mℑ, Mℰ, Mℳ, fL20,T;V, uL20,T;V, φL20,T;V, and θL20,T;V such that(117)St,vh,u˙t≤cvht−u˙tV+jvht−ju˙t.
Taking (116) in (95), we have(118)u−uhL20,T;V+φ−φhL20,T;W+θ−θhL20,T;Q≤c+u˙−Πhu˙L20,T;Vφ−ΠhφL20,T;W+θ−ΠhθL20,T;Q+u˙−Πhu˙L20,T;V1/2+jΠhu˙−ju˙L20,T;V1/2+u0−u0hV+θ0−θ0hQ.
From (116), the constant c is independent of h, u, φ, and θ. This implies(119)u˙−Πhu˙V⟶0,φ−ΠhφW⟶0θ−ΠhθQash⟶0.
FromHP4 and (53), it follows that,(120)jΠhu˙−ju˙L20,T;V⟶0.
Finally, we find the result (78).


## 3.3. Proof of Theorem 3

Add (80) with vh=vnh and (64) with v=wnhk at t=tn, we find(121)cwnhk,vnh−wnhk+cwn,wnhk−wn+aunhk,vnh−wnhk+aun,wnhk−wn+evnh−wnhk,φn−1hk+ewnhk−wn,φn−mθn−1hk,vnh−wnhk−mθn,wnhk−wn+1εϱunhk,vnh−wnhk+ϱun,wnhk−wn+jvnh−jwnhk+jwnhk−jwn≥fn,vnh−wnhk+fn,wnhk−wn.This equality,(122)cwn−wnhk,wn−wnhk=cwnhk,wnhk−vnh+cwn,wn−vnh−cwnhk,wn−vnh+cwn,vnh−wnkh,allows us to obtain(123)cwn−wnhk,wn−wnhk≤Rvnh,wn+Sϱhk+cwnhk−wn,vnh−wn+aunhk−un,vnh−wnhk+evnh−wnhk,φn−1hk−φn−mθn−1hk−θn,vnh−wnhk,where(124)Snvnh,wn=cwn,vnh−wn+aun,vnh−wn+evnh−wn,φn−mθn,vnh−wn+jvnh−jwn−fn,vnh−wn, (125)Sϱhk=1εϱunhk,vnh−wnhk+ϱun,wnhk−wn−ϱun,vnh−wn.For alln≥1, we subtract (65) from (81) at t=tn; we obtain that(126)bφn−φnhk,ξh−eun−unhk,ξh−pθn−θnhk,ξh+ℓun,φn,ξh−ℓunhk,φnhk,ξh=0.We replaceξh by φn−φnhk and φn−ξh in (126); we find that(127)bφn−φnhk,φn−φnhk≤Sℓhk+bφn−φnhk,φn−ξh+eun−unhk,ξh−φnhk+pθn−θnhk,ξh−φnhk,where(128)Sℓhk=ℓun,φn,φn−ξh−ℓunhk,φnhk,φn−ξh−ℓun,φn,φn−φnhk+ℓunhk,φnhk,φn−φnhk.For alln≥1 and similar to (127) and (128), we have(129)dθn−θnhk,θn−θnhk≤Sχhk+dθn−θnhk,θn−ηh+δθn,θn−ηh−δθn,θn−θnhk−δθnhk,θn−ηh+δθnhk,θn−θnhk,where δθn=1/knθn−θn−1 and δθnhk=1/knθnhk−θn−1hk, and(130)Sχhk=χun,θn,θn−ηh−χunhk,θnhk,θn−ηh+χunhk,θnhk,θn−θnhk−χun,θn,θn−θnhk.Adding (124), (128), and (130), we find(131)cwn−wnhk,wn−wnhk+bφn−φnhk,φn−φnhk+dθn−θnhk,θn−θnhk≤Snvnh,wn+Sϱhk+Sℓhk+Sχhk+cwnhk−wn,vnh−wn+aunhk−un,vnh−wnhk+evnh−wnhk,φn−1hk−φn−mθn−1hk−θn,vnh−wnhk+bφn−φnhk,φn−ξh+dθn−θnhk,θn−ηh+eun−unhk,ξh−φnhk+pθn−θnhk,ξh−φnhk+δθn,θn−ηh−δθn,θn−θnhk−δθnhk,θn−ηh+δθnhk,θn−θnhk.The ellipticity of operatorsc, b, d, and the previous inequality follows that(132)mCwn−wnhkV2+mβφn−φnhkW2+mKθn−θnhkV2≤Snvnh,wn+Sϱhk+Sℓhk+Sχhk+Shk,where(133)Shk=cwnhk−wn,vnh−wn+aunhk−un,vnh−wnhk+evnh−wnhk,φn−1hk−φn−mθn−1hk−θn,vnh−wnhk+bφn−φnhk,φn−ξh+eun−unhk,ξh−φnhk+pθn−θnhk,ξh−φnhk+dθn−θnhk,θn−ηh+δθn,θn−ηh−δθn,θn−θnhk−δθnhk,θn−ηh+δθnhk,θn−θnhk.Now, we estimate each of the terms in (132).Thanks to
HP
1
−
HP
3, (107), (117), (111),and(134)vnh−wnhk≤vnh−wn+wn−wnhk,we get(135)Shk≤αhkwn−wnhkV2+un−unhkV2φn−φnhkW2+θn−θnhkQ2+φn−φn−1hkW2+θn−θn−1hkQ2+θn−1−θn−1hkQ2+vnh−wnV2+ξh−φnW2+θn−ηhQ2,(136)Sϱhk≤cϱ′un−unhkV2+wn−wnhkV2+vnh−wnV2,(137)Sℓhk≤αℓφn−φnhkW2+un−unhkV2+φn−ξhW2,(138)Sχhk≤αχθn−θnhkQ2+un−unhkV2+θn−ηhQ2.Combining (132) and (135)–(138), we have the following estimate:(139)wn−wnhkV+φn−φnhkW+θn−θnhkQ≤cφn−φn−1hkW+θn−θn−1hkQ+θn−1−θn−1hkQ+Snvnh,wn1/2+vnh−wnV+ξh−φnW+ηh−θnQ.We estimate now the termsφn−φn−1hk, θn−θn−1hk, and θn−1−θn−1hk, for the first term; we have(140)φn−1hk−φn=∑j=1n−1kjδφjhk+φ0h−∫0tnδφsds−φ0=∑j=1n−1kjδφjhk−δφj+φ0h−φ0+∑j=1n−1δφjkj−∫tj−1tjδφsds−∫tn−1tnδφsds,where δφj=1/kjφj−φj−1 and δφjhk=1/kjφjhk−φj−1hk, and(141)∑j=1n−1δφjkj−∫tj−1tjδφsdsW=∑j=1n−1∫tj−1tjδφj−δφsdsW≤∑j=1n−1∫tj−1tjδφj−δφsWds=Ikδφ,(142)∫tn−1tnδφsds≤∫tn−1tnδφsWds≤kδφC0,T;W.Thus,(143)φn−1hk−φnW≤∑j=1n−1kjδφjkh−δφjW+φ0h−φ0W+Ikδφ+kφC0,T;W.Similar to (143), we deduce that(144)θn−1hk−θnQ≤∑j=1n−1kjδθjkh−δθjQ+θ0h−θ0Q+Ikδθ+kθC10,T;Q,(145)θn−1hk−θn−1Q≤∑j=1n−1kjδθjkh−δθjQ+θ0h−θ0Q+Ikδθ.To proceed, we need the following discrete version of the Grönwall’s inequality presented in [12].Lemma 5.
Assuming thatgnn=1N and enn=1N are the two sequences of nonnegative numbers satisfying(146)en≤cgn+c∑j=1n−1kjej.Then,(147)en≤cgn+∑j=1n−1kjgj,n=1,…,N.Therefore,(148)max1≤n≤Nen≤cmax1≤n≤Ngn.Also, the following result.Lemma 6.
Let us define the map by(149)Ikv=∑j=1n−1∫tj−1tjvj−vsXds.Then,Ikv converges to zero as k⟶0, for all v∈C0,T;X.Proof.
Sincev∈C0,T;X, t⟼vt is uniformly continuous on 0,T. Thus, for any ε>0, there exists a k0>0 such that if k<k0, we have(150)vt−vsX<εT∀s,t∈0,T,t−s<k.
Then, we have(151)Ikv≤∑j=1N∫tj−1tjεTds=ε.
Hence,Ikv converges to zero as k⟶0.We now combine the inequalities (139) and (143)–(145) and applying the Lemma 5, we obtain the following estimate:(152)max1≤n≤Nwn−wnhkV+un−unhkV+φn−φnhkW+θn−θnhkQ≤max1≤n≤Ninfvh∈Vhξh∈Whηh∈QhcSnvnh,wn1/2+vnh−wnQ+ξh−φnW+ηh−θnQ+cφ0−φ0hW+θ0−θ0hQ+cIkδφ+Ikδφ+kφC10,T;W+kθC10,T;Q.Proof of Theorem 3.
We takevnh=Πhwn, ξnh=Πhφn, and ηnh=Πhθn in (152).
The convergence result follows from Lemma 6 together with an argument similar to the proof of Theorem 2.


## 4. Conclusion

The paper presents a numerical analysis of the contact process between a thermo-electro-viscoelastic body and a conductive foundation. A spatially semidiscrete scheme and a fully discrete scheme to approximate the contact problem were derived. We can also study this problem with more general friction laws. The numerical simulation with the same algorithm is an interesting direction for future research.



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