# Variational and Numerical Analysis of a Static Thermo-Electro-Elastic Problem with Friction

Authors: Othman Baiz; Hicham Benaissa; Driss El Moutawakil; Rachid Fakhar
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

We consider a mathematical model which describes a static frictional contact between a piezoelectric body and a thermally conductive obstacle. The constitutive law is supposed to be thermo-electro-elastic and the contact is modeled with normal compliance and a version of Coulomb’s friction law. We derive a variational formulation of the problem and we prove the existence and uniqueness of its solution. The proof is based on some results of elliptic variational inequalities and fixed point arguments. Furthermore, a finite element approximation and a priori error estimates are obtained.


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## Body

## 1. Introduction

Contact problems involving thermopiezoelectricity arise when there is an interaction between the mechanical, electrical, and thermal properties of the considered material. This kind of problems has a considerable interest in various fields of modern industries. Indeed, the thermal effects such as thermal deformations and pyroelectric effects are important to many smart ceramic materials. Thus, for some materials, it may be impossible to predict the electromechanical behavior without taking into account the thermal effects; see for example, [1–4]. There is a considerable interest in contact problems involving piezoelectric materials when thermal effects are considered. These so-called thermopiezoelectric materials can operate effectively as distributed sensors and actuators for controlling structural response. In sensor applications, mechanically or thermally induced disturbances can be determined from measurement of the induced electric potential difference (direct piezoelectric effect), whereas in actuator applications deformation or stress can be controlled through the introduction of an appropriate electric potential difference (converse piezoelectric effect). Among the numerous applications of these materials, we cite accelerometers, microphones, ultrasonic transducers, and so on. A good example is the use of this class of materials as sensors and actuators in microelectromechanical systems, for instance, the piezoelectric accelerometer which triggers an airbag in ten thousandths of a second during an accident.Some general models and their analysis have been established for elastic-piezoelectric materials [5–8], for thermoelastic materials [9–11], and for electro/thermoviscoelastic bodies [12, 13]. The mathematical treatment of static contact process for thermopiezoelectric materials is recent. The reason lies in the considerable difficulties of the nonlinear evolutionary inequalities modeling the static contact problems present in the variational analysis. Existence and uniqueness results in the study of static contact problems can be found, for instance, in [14–16].The present paper is a continuation of this kind of models. It deals with a new and nonstandard mathematical model, which is in a form of a system coupling a nonlinear variational inequality for the displacement field and two nonlinear variational equations for the electric potential and the temperature field. The boundary conditions on the contact surface used in this paper is described with a general normal compliance law and the associated Coulomb’s friction law, taking into account the electrical and thermal conductivity of the foundation. This work serves two purposes. The first purpose is to provide the variational analysis of the mechanical problems and to show the existence of a unique solution to each model. The second is to study a discrete scheme based on finite element method for numerical solutions and to establish the unique solvability of the scheme and derive an optimal order error estimate.The rest of the paper is structured as follows. In Section2, we present the model of our frictional thermopiezoelectric contact problem and we derive its variational formulation, given as a coupled system for the displacement field, the electric potential, and the temperature fields. In Section 3, we provide the assumptions on the data and we state the main existence and uniqueness theorem together with its proof. In Section 4, we present the main error estimate results for the finite element approximation of the problem.


## 2. The Mathematical Model

We consider a thermopiezoelectric body that occupies a bounded domainΩ in Rd (d=2,3) with a Lipschitz continuous boundary Γ. In the sequel, we decompose Γ into three open and disjoint parts Γ1, Γ2, and Γ3 such that Γ=Γ¯1∪Γ¯2∪Γ¯3 and meas(Γ1)>0 on the one hand and we consider a partition of Γ¯1∪Γ¯2 into two sets Γ¯a and Γ¯b with disjoint, relatively open sets Γa and Γb such that meas(Γa)>0 on the other hand.We assume that the body is fixed onΓ1 where the displacement field vanishes and it is subject to a volume forces of density f0, a volume electric charges of density ϕ0, and a heat source term per unit volume ϑ0 in Ω. We assume also that a surface tractions of density f2 and a strength of heat source ϑ2 act on Γ2 and a surface electric charges of density ϕ2 acts on Γb. Finally, we suppose that the electrical potential vanishes in Γa and the temperature is maintained constant at the part Γ1 of the boundary, set to be θb. Over the contact surface Γ3, the body comes in a frictional contact with a thermally conductive foundation.Here and below, the indicesi, j, k, and l run between 1 and d, the summation convention over repeated indices is adopted, and the index that follows a comma indicates a partial derivative with respect to the corresponding component of the spatial variable; for example, ui,j=∂ui/∂xj. In the sequel, let Sd be the space of second order symmetric tensors on Rd. The canonical inner products and norms on Rd and Sd are given by(1)u·v=ui·vi,v=v·v1/2∀u=ui,v=vi∈Rd,σ·τ=σij·τij,τ=τ·τ1/2∀σ=σij,τ=τij∈Sd.To present the mathematical model which describes the physical setting above, we denote byu, σ, φ, D, θ, and q the displacement field, the stress tensor, the electric potential field, the electric displacement field, the temperature field, and the heat flux vector, respectively. These are functions which depend on the spatial variable x. Nevertheless, in what follows we do not indicate explicitly the dependence of these quantities on x; that is, we write σ instead of σ(x). Also, ε(u) denotes the linearized strain tensor and E(φ)=-∇φ is the electric field.We assume that the process is static, then the equations of stress equilibrium, the equation of the quasistationary electric field, and the heat conduction equation are(2)Divσ+f0=0inΩ,(3)divD=ϕ0inΩ,(4)divq=ϑ0inΩ,where Divσ=σij,j and divD=Dj,j are the divergence operator for tensor and vector field. The material is assumed to be thermopiezoelectric and satisfies the following constitutive laws [1, 4]:(5)σ=Fεu-E∗Eφ-θMinΩ,D=Eεu+βEφ-θPinΩ,where F is the elasticity tensor, E is the third-order piezoelectric tensor, β represents the electric permittivity tensor, M is the thermal expansion, and P denotes the pyroelectric tensor. Moreover, E∗ is the transpose of E and it satisfies(6)Eσ·v=σ·E∗v,∀σ∈Sd,v∈Rd.For the heat flux, we adopt the following Fourier-type law:(7)q=-K∇θinΩ,where K denotes the thermal conductivity tensor. We use classical decomposition in the normal and the tangential components of the displacement u and of the stress σ on Γ; that is,(8)uν=u·ν,uτ=u-uνν,σν=σν·ν,στ=σν-σνν,where ν is the outward unit normal vector on Γ and the physical setting to complete our model with the following boundary conditions:(9)u=0onΓ1,(10)σν=f2onΓ2,(11)φ=0onΓa,(12)D·ν=ϕ2onΓb,(13)θ=θbonΓ1,(14)q·ν=ϑ2onΓ2.We model the frictional contact on Γ3 with the following reduced normal compliance condition:(15)σν=-cnuν-g+mnonΓ3,(16)στ≤cTuν-g+mTστ<cTuν-g+mT⟹uτ=0στ=cTuν-g+mT⟹στ=-λuτforsomeλ≥0onΓ3,where cn, mn, cT, and mT are material interface parameters and (uν-g)+=max⁡uν-g,0 represents the penetration approach; for more detail see [17–19]. Here (15) is the normal compliance power law and (16) is a variant of Coulomb’s friction law. Furthermore, the thermoelectric contact is described with the following regularized conditions (see [13, 14]):(17)D·ν=0onΓ3(18)q·ν=kcuν-gϕLθ-θFonΓ3,where kc:r→kc(r) is the thermal conductance function, supposed to be zero for r<0 and positive otherwise, nondecreasing, and Lipschitz continuous, and θF is the foundation temperature. The truncate function ϕL is defined by(19)ϕLs=sifs≤L,ssLifs>L,where L is a large positive constant. At last, we note that condition (17) describes the fact that the foundation is supposed to be a perfect electrical insulator. Under all these conditions, the classical formulation of our problem is as follows.Problem (
P
). Find a displacement field u:Ω→Rd, an electric potential φ:Ω→R, and a temperature field θ:Ω→R such that (2)–(7) and (9)–(18) hold.Note that once the triplet(u,φ,θ) which solve (P) is known, then the stress tensor σ, the electric displacement field D, and the heat flux q can be obtained from (5) and (7). In order to derive the variational formulation of the problem P, we need the following Hilbert spaces:(20)H=L2Ω,Rd,H=L2Ω,Sd,H1=H1Ω,RdH1=σ∈H;Divσ∈H,W=D∈H;divD∈L2Ω,endowed with the following inner products:(21)u,vH=∫Ωuividx,σ,τH=∫Ωσijτjidx,u,vH1=u,vH+εu,εvH,σ,τH1=σ,τH+Divσ,DivτH,D,EW=D,EH+divD,divEL2Ω.Next, we note that, by the Sobolev trace theorem, we can define the trace γv of a function v∈H1 on Γ such that γv=v|Γ if v∈H1∩C(Ω¯,Rd). For simplicity, for an element v∈H1 we still denote by v its trace γv on Γ. Let HΓ=H1/2(Γ,Rd) and L2(Γ)d=L2(Γ,Rd), then the trace operator γ:H1→HΓ⊂L2(Γ)d is a linear continuous operator; that is, there exists a positive constant c, depending only on Ω, such that(22)vL2Γd≤cvH1,∀v∈H1.Keeping in mind condition (9), we introduce the closed subspace V of H1 given by(23)V=v∈H1;v=0onΓ1.Since meas(Γ1)>0, Korn’s inequality holds; there exists a positive constant cK which depends only on Ω and Γ1 such that (see, e.g., [7, 20])(24)εvH≥cKvH1,∀v∈V.We define over the space V, the following inner product, and its associated norm:(25)u,vV=εu,εvH,vV=εvH.It comes from (24) that the norms ·V and ·H1 are equivalent on V and therefore V,vV is a real Hilbert space. Finally, note that from (22) and (24), we deduce that there exists a positive constant c0 depending on Ω, Γ1, and Γ3 such that(26)vL2Γ3d≤c0vV,∀v∈V.According to the boundary conditions (11) and (13), the electric potential field and the temperature field are, respectively, to be found in the closed subspace W and Q of H1(Ω) given by(27)W=ψ∈H1Ω;ψ=0onΓa,Q=θ∈H1Ω;θ=0onΓ1.Since meas(Γa)>0, the Friedrichs-Poincaré inequality holds and thus there exists a positive constant cF depending only on Ω and Γa such that(28)∇ψH≥cFψH1Ω∀ψ∈W.We introduce on W the inner product (φ,ψ)W=(∇φ,∇ψ)H and its associated norm ψW. It follows from (28) that the norms ·H1(Ω) and ·W are equivalent on W and therefore, the space (W,·W) is a real Hilbert. Since meas(Γ1)>0, we can prove in an analogous way that the norm ·Q associated with the inner product (θ,ξ)Q=(∇θ,∇ξ)H is equivalent to the usual norm ·H1(Ω) on Q and thus, (Q,·Q) is a real Hilbert space. Using the Sobolev trace theorem, we get that there exists a positive constant c1 depending only on Ω, Γa, and Γ3 such that(29)ψL2Γ3≤c1ψW∀ψ∈Wand a positive constant c2 which depends only on Ω, Γ1, and Γ3 such that(30)ξL2Γ3≤c2ξQ∀ξ∈Q.In the study of the mechanical problem(P), the following assumptions will be needed:(h1)
The elasticity operatorF=(fijkl):Ω×Sd→Sd, the electric permittivity tensor β=(βij):Ω×Rd→Rd, and the thermal conductivity tensor K=(kij):Ω×Rd→Rd satisfy the usual properties of symmetry, boundedness, and ellipticity(31)fijkl=fjikl=fklij∈L∞Ω,βij=βji∈L∞Ω,kij=kji∈L∞Ωand there exist positive constantsmF, mβ, and mK such that(32)fijklξijξkl≥mFξ2∀ξ=ξij∈Sd,βijζiζj≥mβζ2,kijζiζj≥mKζ2∀ζ=ζi∈Rd.(h2)
The piezoelectric tensorE=(eijk):Ω×Sd→Rd, the thermal expansion tensor M=(mij):Ω×Rd→Rd, and the pyroelectric tensor P=(pi):Ω→Rd satisfy(33)eijk=eikj∈L∞Ω,mij=mji∈L∞Ω,pi∈L∞Ω.(h3)
The thermal conductancekc:Γ3×R→R+ satisfies the following conditions:(34)kcx,u≤Mk∀u∈R,x∈Γ3whereMkis  a  positive  constant,x⟼kcx,uis  mesurable  onΓ3∀u∈Rand  is  zero∀u≤0.(h4)
The functionu↦kc(x,u) is a Lipschitz function on R for all x∈Γ3; that is,(35)kcx,u1-kcx,u2≤Lku1-u2∀u1,u2∈RwithLk>0is  a  constant.(h5)
The forces, the traction, the charges densities, and the strength of the heat source satisfy(36)f0∈L2Ωd,f2∈L2Γ2d,ϕ0∈L2Ω,ϕ2∈L2Γb,ϑ0∈L2Ω,ϑ2∈L2Γ2.(h6)
The foundation temperature satisfies(37)θF∈L2Γ3.(h7)
The friction bound function and the coefficient of friction satisfy(38)cn,cT∈L∞Γ3,cn,cT≥0.(h8)
The material interface parameters satisfy(39)1≤mn,mT<∞ifd=2,1≤mn,mT≤2ifd=3.Using the Riesz representation theorem, we can define f∈V, ϕ∈W, and ϑ∈Q as follows:(40)f,vV=∫Ωf0·vdx+∫Γ2f2·vda,∀v∈V,ϕ,ξW=∫Ωϕ0ξdx-∫Γbϕ2ξda,∀ξ∈W,ϑ,ηQ=∫Ωϑ0ηdx+∫Γ2ϑ2ηda,∀η∈Q.We consider the functionals jn,jT:V×V→R and χ:V×Q×Q→R defined by(41)jnu,v=∫Γ3cnuν-g+mnvνda,∀u,v∈V,(42)jTu,v=∫Γ3cTuν-g+mTvτda,∀u,v∈V,(43)χu,θ,η=∫Γ3kcuν-gϕLθ-θFηda,∀u∈V,∀θ,η∈Q.It follows from (h3) and (h5)–(h8) that the integrals above are well defined. We note that if σ∈H1 and Ψ∈W are sufficiently regular, the following Greens formulas hold:(44)∫Γσν·vda=σ,εvH+Divσ,vH,∀v∈H1,∫ΓΨ·νψda=Ψ,∇ψH+divΨ,vL2Ω,∀v∈H1Ω.Using the previous Greens formulas, it is straightforward to see that if (u,φ,θ) are sufficiently regular functions which satisfy (2)–(7) and (9)–(18), then(45)σ,εv-εuH+jnu,v-u+jTu,v-jTu,u≥f,v-uV,D,∇ξH+ϕ,ξW=0,q,∇ηH=χu,θ,η-ϑ,ηQ,for all v∈V, ξ∈W, and η∈Q. We use (5), (7), and the notation E(φ)=-∇φ to obtain the following variational formulation of our problem.Problem
(
P
V
). Find a displacement field u∈V an electric potential φ∈W and a temperature field θ∈Q such that(46)Fεu,εv-εuH+E∗∇φ,εv-εuH-Mθ,εv-εuHjnu,v-u+jTu,v-jTu,u≥f,v-uV,∀v∈V,(47)β∇φ,∇ξH-Eεu,∇ξH-Pθ,∇ξH=ϕ,ξW,∀ξ∈W,(48)K∇θ,∇ηH+χu,θ,η=ϑ,ηQ,∀η∈Q.


## 3. Existence and Uniqueness of Weak Solution

The main existence and uniqueness result in the study of the problem(PV) are as follows.Theorem 1.
Assume (h1)–(h3) and (h5)–(h8) hold. Then one has the following:(1)
The problem(PV) has at least one solution.(2)
Under (h4), there exists a constant L∗>0 such that if(49)mT+P+M+LkL+Mk<L∗,then the problem(PV) has a unique solution.
Here the norms of the tensorsP=(pi) and M=(mij) are given by(50)P=max1⩽i⩽d⁡pi,M=max1⩽i,j⩽d⁡mij.The proof will be carried out in several steps and it is based on arguments of variational inequalities and fixed point techniques. We assume in what follows that (h1)–(h3) and (h5)–(h8) hold and for every z=(z1,z2) where z1, z2 ≥ 0∈L2(Γ3), we define(51)χ1z,η=∫Γ3z1ηda,∀η∈Q,(52)χ2z,v=∫Γ3z2vτda,∀v∈V.For all z supposed to be known, we consider the following auxiliary problem.Problem
(
P
V
z
). Find the elements uz∈V, φz∈W, and θz∈Q such that(53)Fεuz,εv-εuzH+E∗∇φz,εv-εuzH-Mθz,εv-εuzHjnuz,v-uz+χ2z,v-χ2z,uz≥f,v-uzV,∀v∈V,(54)β∇φz,∇ξH-Eεuz,∇ξH-Pθz,∇ξH=ϕ,ξW,∀ξ∈W,(55)K∇θz,∇ηH+χ1z,η=ϑ,ηQ,∀η∈Q.In the study of this problem, the two following results will be needed.Lemma 2.
Letα be a positive real number. For all x,y∈R, one has(56)x+α-y+αx-y≥0and consequently for all g∈R, one has(57)x-g+α-y-g+αx-y≥0,∀x,y∈R.Proof.
This lemma can be obtained by examining the two casesxy≥0 and xy≤0.Theorem 3.
LetΩ be open bounded set of Rd with a Lipschitz boundary Γ and 1<p<∞. The trace operator γ:W1,p(Ω)→Lr(Γ) satisfies the following results:(1)
Ifp<d, the map γ is compact for any 1≤r<dp-p/(d-p). Then, there exists Nr>0 such that(58)vLrΓ≤NrvW1,pΩ∀v∈W1,pΩ.(2)
Ifp≥d, the map γ is compact for any r≥1 and then, there exists Nr>0 such that(59)vLrΓ≤NrvW1,pΩ∀v∈W1,pΩ.Proof.
For the proof of the trace Theorem3, we can refer to [21, 22].Lemma 4.
Under the assumptions (h1)–(h3) and (h5)–(h8), the problem (PVz) has a unique solution (uz,φz,θz)∈V×W×Q which depends Lipschitz continuously on z.Proof.
We use Riesz’s representation theorem to define the elementϑz of Q(60)ϑz,ηQ=ϑ,ηQ-χ1z,η∀η∈Qand the operator T:Q→Q such that(61)Tθz,ηQ=K∇θz,∇ηL2Ωd,∀η∈Q.Thus equation (55) will be(62)Findθz∈Qsuch  thatTθz,ηQ=ϑz,ηQ,∀η∈Q.Using the assumptions of K (see (h1)), we can deduce that T is a linear symmetric and positive definite operator. Hence, T is linear continuous and invertible operator on Q and let C denote its inverse. Thus, by the Lax-Milgram theorem, we get that problem (62) has a unique solution(63)θz=Cϑz∈Q.It follows from the properties of the operators T, M, and P that MC and PC are linear continuous operators. Moreover, we apply Riesz’s representation theorem to define the element ϕz∈W and the operators B:W→W and C:V→W as follows:(64)ϕz,ξW=ϕ,ξW+PCϑz,∇ξH,∀ξ∈W,(65)Bφ,ξW=β∇φ,∇ξH,∀φ,ξ∈W×W,(66)Cv,ξW=Eεv,∇ξH,∀v,ξ∈V×W.Let C∗ be the adjoint operator of C and then, it comes from (66) that(67)C∗ξ,vW=E∗∇ξ,εvH∀v∈Vξ∈W.Next, we replace (64), (65), and (66) in (54) to obtain(68)Bφz=ϕz+Cuz.Keeping in mind the properties of β (see (h1)), we get that B is a linear symmetric and positive definite operator. Hence, the operator B is invertible and let B-1 be its inverse. We have(69)φz=B-1Cuz+B-1ϕz∈W.It comes from (63), (69) that inequality (53) is equivalent to find uz∈V such that(70)Fεuz,εv-εuzH+C∗B-1Cuz,v-uzV+jnuz,v-uz+χ2z,v-χ2z,uz≥f,v-uzV+MCϑz,εv-εuH+C∗B-1ϕz,v-uzV∀v∈V.The variational problem above is equivalent to the following minimization problem:(71)Finduz∈Vsuch  thatJzuz=infv∈V⁡Jzv,where the functional Jz:V→R is defined as follows:(72)Jzv=12Fεv,εvH+12C∗B-1Cv,vV+∫Γ3cnmn+1vν-g+mn+1da+χ2z,v-f,vV-MCϑz,εvH-C∗B-1ϕz,vV,∀v∈V.We consider the function j~n(v)=∫Γ3cn/mn+1vν-g+mn+1da whose derivative is(73)Dj~nv,w=∫Γ3cnvν-g+mnwnda,∀w∈V.Applying Lemma 2, we get for all v and w of V that(74)Dj~nv-Dj~nw,v-w=∫Γ3cnvν-g+mn-wν-g+mnvν-wνda≥0.Hence the function j~n is convex. In plus, it follows from the strict convexity of Jz-j~n that the functional Jz=(Jz-j~n)+j~n is strictly convex on V. Moreover, since Jz-j~n is coercive and Jz(v)≥Jz(v)-j~n(v), we deduce that Jz is coercive. Consequently, the minimization problem has a unique solution uz∈V. Therefore, keeping in mind (63) and (69), we conclude that the variational problem (PVz) has a unique solution (uz,φz,θz) of V×W×Q.
For the second part of Lemma4, let z=(z1,z2) and z′=(z1′,z2′) be two given elements of the reflexive space L2(Γ3)2 such that z2≥0 and z2′≥0. We consider (uz,φz,θz), (uz′,φz′,θz′), the unique solution of the problems (PVz), (PVz′), respectively. Then, the variational inequality (53) leads to(75)Fεuz,εv-εuzH+E∗∇φz,εv-εuzH-Mθz,εv-εuzH+jnuz,v-uz+χ2z,v-χ2z,uz≥f,v-uzV,∀v∈V,Fεuz′,εv-εuz′H+E∗∇φz′,εv-εuz′H-Mθz′,εv-εuz′H+jnuz′,v-uz′+χ2z′,v-χ2z′,uz′≥f,v-uz′V,∀v∈V.We take v=uz′ in the first inequality and v=uz in the second to get(76)Fεuz,εuz′-εuzH+E∗∇φz,εuz′-εuzH-Mθz,εuz′-εuzH+jnuz,uz′-uz+χ2z,uz′-χ2z,uz≥f,uz′-uzV,Fεuz′,εuz-εuz′H+E∗∇φz′,εuz-εuz′H-Mθz′,εuz-εuz′H+jnuz′,uz-uz′+χ2z′,uz-χ2z′,uz′≥f,uz-uz′Vand by adding the two induced inequalities, we obtain(77)Fεuz-Fεuz′,εuz-εuz′H+E∗∇φz-E∗∇φz′,εuz-εuz′H≤χ2z,uz-χ2z,uz′+χ2z′,uz′-χ2z′,uz+jnuz′,uz-uz′-jnuz,uz-uz′+Mθz-Mθz′,εuz-εuz′H.Moreover, the definition of jn and Lemma 2 imply that(78)jnuz′,uz-uz′-jnuz,uz-uz′=-∫Γ3cnuz′ν-g+mn-uzν-g+mnuz′ν-uzνda≤0.Then, we have(79)Fεuz-Fεuz′,εuz-εuz′H+E∗∇φz-E∗∇φz′,εuz-εuz′H≤χ2z,uz-χ2z,uz′+χ2z′,uz′-χ2z′,uz+Mθz-Mθz′,εuz-εuz′H.It follows from (26) and the definition of χ2 that(80)χ2z,uz-χ2z,uz′+χ2z′,uz′-χ2z′,uz=∫Γ3z1-z1′uzτ-uzτ′da≤c0z1-z1′L2Γ3uz-uz′V.In addition, the variational equation (54) leads to(81)β∇φz,∇ξH-Eεuz,∇ξH-Pθz,∇ξH=ϕ,ξW,(82)β∇φz′,∇ξH-Eεuz′,∇ξH-Pθz′,∇ξH=ϕ,ξW.After taking ξ=φz-φz′ in (81) and ξ=φz′-φz in (81), we add the induced equations(83)β∇φz-∇φz′,∇φz-∇φz′H-Eεuz-Eεuz′,∇φz-∇φz′H=Pθz-Pθz′,∇φz-∇φz′H.Using (79), (80), (83), (30), and (6), the strong monotonicity of F, and the ellipticity of β and after some algebra, we find that there exists a constant cb>0 such that(84)uz-uz′V+φz-φz′W≤cbz1-z1′L2Γ3+θz-θz′Q.From (60), (51), and (30), we obtain(85)ϑz′-ϑz,ηQ=χ1z′,η-χ1z,η(86)≤c2z1′-z1L2Γ3ηQ,and from (61), (62), (85), and the ellipticity of K, we find that there exists ck>0 such that(87)θz-θz′Q≤ckz2-z2′L2Γ3.Finally, we combine (84) and (87) to find that there exists a constant c3>0 such that(88)uz-uz′V+φz-φz′W+θz-θz′Q≤c3z1-z1′L2Γ3+z2-z2′L2Γ3.Hence the second part of Lemma 4 is proved.Remark 5.
The second part of Lemma4 implies that the function z↦(uz,φz,θz) where the triplet (uz,φz,θz) is the solution of (PVz) is a continuous function from L2(Γ3)2 to V×W×Q.Lemma 6.
If the triplet(uz,φz,θz)∈V×W×Q is a solution of problem (PVz), then there exists a positive constant c~1 such that(89)uzV≤c~1fV+ϕW+1mKϑQ+c2z1L2Γ3.Proof.
Takingη=θz in the variational equation (55), we get(90)K∇θz,∇θzH=ϑ,θzQ-χ1z,θz.Using (43), (30), and the ellipticity of the operator K, we find(91)θQ≤1mKϑQ+c2z1L2Γ3.Moreover, if we take v=0∈V in (53) and ξ=φz∈W in (54), we have(92)Fεuz,εuzH+E∗∇φz,εuzH-Mθz,εuzH+jnuz,uz+χ2z,uz≤f,uzV,β∇φz,∇φzH-Eεuz,∇φzH-Pθz,∇φzH=ϕ,φzW.Keeping in mind (6), the ellipticity of F and β, the positivity of jn(uz,uz) and χ2(z,uz), and the properties of M and P, we deduce that there exists a constant c~1>0 such that(93)uzV+φzW≤c~1fV+ϕW+θzQ.We combine the two inequalities (91) and (93) to obtain(94)uzV≤c~1fV+ϕW+1mKϑQ+c2z1L2Γ3,which finishes the proof.Remark 7.
Using the same argument as in the proof of in Lemma6, we have that if the triplet (u,φ,θ)∈V×W×Q is a solution of problem (PV), then(95)uV≤c~1fV+ϕW+1mKϑQ+c2MkLmeasΓ31/2=G,where c~1 is the same constant as in (94).In this step, we consider the operatorΛ:L2(Γ3)2→L2(Γ3)2 defined by(96)Λz=cnuzν-g+mn,kcuzν-gϕLθz-θF.Now, we shall prove that the operator Λ has a unique fixed point z∗∈L2(Γ3)2. For that, we need to introduce the two closed convex subsets of L2(Γ3)(97)K1=z1≥0∈L2Γ3,z1L2Γ3≤k1,K2=z2∈L2Γ3,z2L2Γ3≤k2,where k1 and k2 will be defined below.Lemma 8.
For a specified values ofk1 and k2, the operator Λ has at least one fixed point.Proof.
Letz=(z1,z2)∈K1×K2. We have z1L2(Γ3)≤k1 and z2L2(Γ3)≤k2 and then(98)zL2Γ32≤k1+k2.On another hand, it follows from the definition of Λ that(99)ΛzL2Γ32≤cnuzν-g+mnL2Γ3+kcuzν-gϕLθz-θFL2Γ3≤cnL∞Γ3uzν-gL2mnΓ3mn+kcuzν-gϕLθz-θFL2Γ3≤cnL∞Γ3uzνL2mnΓ3+gL2mnΓ3mn+kcuzν-gϕLθz-θFL2Γ3.Using (h3), (h8), and Theorem 3, we deduce that there exists a constant c~>0 such that(100)ΛzL2Γ32≤c~cnL∞Γ3uzV+gL2mnΓ3mn+MkLmeasΓ31/2.Since z1L2(Γ3)≤k1, it becomes from Lemma 6 that there exists a constant c~1>0 such that(101)uzV≤c~1fV+ϕW+1mKϑQ+c2k1=G~.If we choose k1=MkLmeas(Γ3)1/2 and k2=c~cnL∞Γ3(G~+gL2mn(Γ3))mn, we get(102)ΛzL2Γ32≤k1+k2.Hence Λ is an operator of K1×K2 into itself. Since K1×K2 is a nonempty, convex, and closed subset of the reflexive space L2(Γ3)×L2(Γ3), then K1×K2 is weakly compact. Using the continuity of the functions ϕL and kc and Remark 5, we deduce that Λ is a continuous operator. Hence, by Schauder’s fixed point theorem the operator Λ has a fixed point.Proof of Theorem1.
Existence. Let z∗ be the fixed point of the operator Λ obtained in Lemma 8. We denote by (u∗,φ∗,θ∗) the solution of the problem (PVz) for z=z∗. It follows from the definition of Λ and (PVz) that (u∗,φ∗,θ∗) is a solution of (PV) which concludes the proof of the existence part.
Uniqueness. Here, we will show that there exists a positive constant L∗ such that (PV) has a unique solution if mT+P+M+LkL+Mk<L∗. We consider (u1,φ1,θ1) and (u2,φ2,θ2) two solutions of (PV). From (46), we have(103)Fεu1,εv-εu1H+E∗∇φ1,εv-εu1H-Mθ1,εv-εu1H+jnu1,v-u1+jTu1,v-jTu1,u1≥f,v-u1V,∀v∈V,Fεu2,εv-εu2H+E∗∇φ2,εv-εu2H-Mθ2,εv-εu2H+jnu2,v-u2+jTu2,v-jTu2,u2≥f,v-u2V,∀v∈V.After taking v=u2 in the first inequality and v=u1 in the second and adding the resulting inequalities, it follows from the positivity of jn(u1-u2,u1-u2) that(104)Fεu1-Fεu2,εu1-εu2H+E∗∇φ1-E∗∇φ2,εu1-εu2H-Mθ1-Mθ2,εu1-εu2H≤G1=jTu1,u2-jTu1,u1+jTu2,u1-jTu2,u2.Plus the variational equation (47) leads to(105)β∇φ1,∇ξH-Eεu1,∇ξH-Pθ1,∇ξH=ϕ,ξW,∀ξ∈W,β∇φ2,∇ξH-Eεu2,∇ξH-Pθ2,∇ξH=ϕ,ξW,∀ξ∈W.We take ξ=φ1-φ2 in the first equation and ξ=φ2-φ1 in the second to obtain(106)β∇φ1-β∇φ2∇φ1-∇φ2H-Eεu1-Eεu2,∇φ1-∇φ2H-Pθ1-Pθ2,∇φ1-∇φ2H=0.Hence, the addition of (104) and (106) implies that(107)Fεu1-Fεu2,εu1-εu2H+β∇φ1-β∇φ2,∇φ1-∇φ2H-Pθ1-Pθ2,∇φ1-∇φ2H-Mθ1-Mθ2,εu1-εu2H≤G1.Moreover, we use the variational equation (48) to deduce(108)K∇θ1,∇ηH+χu1,θ1,η=ϑ,ηQ,∀η∈Q,K∇θ2,∇ηH+χu2,θ2,η=ϑ,ηQ,∀η∈Q.We substitute η by θ1-θ2 and we subtract the two induced equations to obtain(109)K∇θ1-K∇θ2,∇θ1-∇θ2H+χu1,θ1,θ1-θ2-χu2,θ2,θ1-θ2=0.Let us consider Δ1=χ(u1,θ1,θ1-θ2)-χ(u2,θ2,θ1-θ2). Then, we have(110)Δ1=∫Γ3kcu1ν-gϕLθ1-θF-kcu2ν-gϕLθ2-θFθ1-θ2da≤Δ2L2Γ3θ1-θ2L2Γ3,where(111)Δ2=kcu1ν-gϕLθ1-θF-kcu2ν-gϕLθ2-θF=kcu1ν-gϕLθ1-θF-kcu2ν-gϕLθ1-θF+kcu2ν-gϕLθ1-θF-kcu2ν-gϕLθ2-θF.Taking in mind (26), (30), and the assumptions (h3) and (h4), we get(112)Δ2L2Γ3≤c0LkLu1-u2V+c2Mkθ1-θ2Q.Hence, it comes from (109), (110), and (112) and the ellipticity of K that(113)mKθ1-θ2Q2≤LkLc0c2u1-u2Vθ1-θ2Q+Mkc22θ1-θ2Q2.Recalling ab⩽(1/4)a2+b2, forall  a,b∈R, we deduce(114)mKθ1-θ2Q2≤54LkLc0c2+Mkc22u1-u2V2+θ1-θ2Q2.Furthermore, the definition of jT implies that(115)G1=jTu1,u2-jTu1,u1+jTu2,u1-jTu2,u2=∫Γ3cTu1ν-g+mT-u2ν-g+mTu1τ-u2τda≤cTL∞Γ3u1ν-g+mT-u2ν-g+mTL4/3Γ3u1-u2L4Γ3d.Using the following mathematical inequalities (see [23])(116)am-bm≤ma-bam-1-bm-1,∀a,b≥0,∀m>1,a+-b+≤a-b,∀a,b∈R.Then, we obtain(117)u1ν-g+mT-u2ν-g+mTL4/3Γ3=∫Γ3u1ν-g+mT-u2ν-g+mT4/3da3/4≤mTu1ν-u2νL4Γ3u1ν-g+mT-1+u2ν-g+mT-1L2Γ3≤mTu1-u2L4Γ3du1ν-g+mT-1L2Γ3+u2ν-g+mT-1L2Γ3.Keeping in mind Theorem 3 and Remark 7, we obtain(118)G1≤mTRu1-u2V2,where R=2N42cTL∞(Γ3)[N2(mT-1)mT-1GmT-1+gmT-1L2(Γ3)].
Using (107), (114), and (118), after some algebra it follows that there exists C~>0 such that(119)u1-u2V2+φ1-φ2W2+θ1-θ2Q2≤C~mT+P+M+LkL+Mku1-u2V2+φ1-φ2W2+θ1-θ2Q2,where(120)C~=max⁡1,R,c0c2,c22min⁡mF,mβ,mK.Choose L∗=1/C~. Then, if mT+P+M+LkL+Mk<L∗ holds, we will conclude that u1=u2, φ1=φ2, and θ1=θ2 which leads to the uniqueness part of Theorem 1.


## 4. Discrete Approximation

This section deals with the discrete approximation of the problem(PV). We assume that the conditions (h1)–(h8) hold, then the problem (PV) has a unique solution (u,φ,θ)∈V×W × Q. Let Th=(Tr)r∈Gh be a family of regular triangulations of the polygonal domain Ω such that(121)⋃rT¯r=Ω¯,Tr∩Tr′=∅,∀r≠r′∈Gh.Here and below h>0 is a discretization parameter. We define the following finite dimensional subspaces Vh, Wh, and Qh which approximate, respectively, the spaces V, W, and Q by(122)Vh=vh∈CΩ¯d;∀r∈Gh,vTrh∈P1Trd,vh=0onΓ1,Wh=ξh∈CΩ¯;∀r∈Gh,ξTrh∈P1Tr,ξh=0onΓa,Qh=ηh∈CΩ¯;∀r∈Gh,ηTrh∈P1Tr,ηh=0onΓ1,where P1(Tr) denotes the space of polynomials of a degree lower or equal to one on Tr. Then, the discrete approximation of the problem (PV) is as follows.Problem (
P
V
h
). Find the displacement uh∈Vh, the electric potential φh∈Wh, and the temperature θh∈Qh such that(123)Fεuh,εvh-εuhH+E∗∇φh,εvh-εuhH-Mθ,εvh-εuhH+jnuh,vh-uh+jTuh,vh-jTuh,uh≥f,vh-uhV,∀vh∈Vh,(124)β∇φh,∇ξhH-Eεuh,∇ξhH-Pθh,∇ξhH=ϕ,ξhW,∀ξh∈Wh,(125)K∇θh,∇ηhH+χuh,θh,ηh=ϑ,ηhQ,∀ηh∈Qh.Under the assumptions of Theorem 1 and with the same arguments, we can prove that the discrete problem (PVh) has a unique solution (uh,φh,θh) in Vh×Wh×Qh. Now, we proceed to derive some error estimates for the discrete solution. In the sequel, c denotes positive constants which are independent of the discretization parameter h.Theorem 9.
Assume the conditions of Theorem1 hold. One has the following error estimate:(126)u-uhV2+φ-φhW2+θ-θhQ2≤cu-vhV2+u-vhV+φ-ξhW2+θ-ηhQ2.Proof.
Takingξ=ξh∈Wh⊂W in the second equation of the problem (PV) and subtracting the obtained equation to the second equation of the problem (PVh), we obtain (127)β∇φ-β∇φh,∇ξhH-Eεu-Eεuh,∇ξhH-Pθ-Pθh,∇ξhH=0.We substitute ξh by ξh-φh=(ξh-φ)+(φ-φh) to find after simplifications(128)Eεu-Eεuh,∇φ-∇φhH=β∇φ-β∇φh,∇φ-∇φhH+β∇φ-β∇φh,∇ξh-∇φH-Pθ-Pθh,∇φ-∇φhH-Pθ-Pθh,∇ξh-∇φH-Eεu-Eεuh,∇ξh-∇φH.Take v=uh∈Vh⊂V in the first inequality of the problem (PV) to deduce(129)Fεu,εuh-εuH+E∗∇φ,εuh-εuH-Mθ,εuh-εuH+jnu,uh-u+jTu,uh-jTu,u≥f,uh-uV.By using vh-uh=(u-uh)-(u-vh), it follows from the first inequality of (PVh) that(130)Fεuh,εu-εuhH+E∗∇φh,εu-εuhH-Mθh,εu-εuhH+jnuh,u-uh-jnuh,u-vh+jTuh,vh-jTuh,uh≥Fεuh,εu-εvhH-Mθh,εu-εvhH+E∗∇φh,εu-εvhH+f,vh-uhV,∀vh∈Vh.From Lemma 2, we know that jn(uh,u-uh)+jn(u,uh-u)≥0. Then, keeping in mind t (128) and (6), the addition of (129) and (130) gives(131)Fεu-Fεuh,εu-εuhH+β∇φ-β∇φh,∇φ-∇φhH≤Pθ-Pθh,∇φ-∇φhH+Mθ-Mθh,εu-εuhH+I1+I2,where(132)I1=Fεuh,εvh-εuH+Eεu-Eεuh,∇ξh-∇φH-β∇φ-β∇φh,∇ξh-∇φH+Pθ-Pθh,∇ξh-∇φH-jnuh,u-vh+Mθh,εu-εvhH-E∗∇φh,εu-εvhH-f,vh-uV.I2=jTuh,vh-jTuh,uh+jTu,uh-jTu,u.Next, we rewrite (48) for η=ηh∈Qh⊂Q and we subtract it from (125) to obtain(133)K∇θh-K∇θ,∇ηhH+∫Γ3kuνh-gϕLθh-θF-kuν-gϕLθ-θFηhda=0.We replace ηh by ηh-θh=(ηh-θ)+(θ-θh). Hence, we have(134)K∇θh-K∇θ,∇θ-∇θhH=K∇θh-K∇θ,∇θ-∇ηhH+∫Γ3kuνh-gϕLθh-θF-kuν-gϕLθ-θFθ-ηhda-∫Γ3kuνh-gϕLθh-θF-kuν-gϕLθ-θFθ-θhda.By adding the two results (134) and (131), we deduce the following inequality:(135)Fεu-Fεuh,εu-εuhH+β∇φ-β∇φh,∇φ-∇φhH+K∇θ-K∇θh,∇θ-∇θhH≤I1+I2+I3+I4,where I3 and I4 are given by(136)I3=K∇θ-K∇θh,∇θ-∇ηhH+∫Γ3kuνh-gϕLθh-θF-kuν-gϕLθ-θFηh-θda,I4=Pθ-Pθh,∇φ-∇φhL2Ωd+Mθ-Mθh,εu-εuhH-∫Γ3kuνh-gϕLθh-θF-kuν-gϕLθ-θFθh-θda.Now, we begin to estimate the quantities I1, I2, I3, and I4. We have(137)I1≤Fu-uhVu-vhV+FuVu-vhV+Eu-uhVξh-φW+βφ-φhWξh-φW+Pθ-θhQξh-φW+fVvh-uV+Mθ-θhQu-vhV+MθQu-vhV+E∗φWu-vhV+cnuνh-g+mnL4/3Γ3vh-uL4Γ3d+E∗φ-φhWu-vhV.We recall that there exists a positive constant c independent of h such that(138)cnuνh-g+mnL4/3Γ3vh-uL4Γ3d≤ccnL∞Γ3uνh-gL4mn/3Γ3mnvh-uV≤ccnL∞Γ3N4mn/3mnuhVmn+gL4mn/3Γ3mnvh-uVand similar to Remark 7, we can find that uhV≤G. Thus(139)cnuνh-g+mnL4/3Γ3vh-uL4Γ3d≤ccnL∞Γ3N4mn/3mnGmn+gL4mn/3Γ3mnvh-uV.Finally, remembering (95), we find that there exists a constant α1>0 such that(140)I1≤α1vh-uV+uh-uVvh-uV+uh-uVξh-φW+vh-uVθh-θQ+φ-φhWξh-φW+θh-θQξh-φW+vh-uVφh-φW.Using the same process as in the proof of (118), we can deduce(141)I2=jTuh,vh-jTuh,uh+jTu,uh-jTu,u=jTu,uh-jTuh,uh+jTuh,u-jTu,u+jTuh,vh-jTu,vh+jTu,u-jTuh,u+jTu,vh-jTu,u=∫Γ3cTuν-g+mT-uνh-g+mTuτh-uτda+∫Γ3cTuν-g+mT-uνh-g+mTvτh-uτda+∫Γ3cTuν-g+mTvτh-uτda≤mTRuh-uV2+mTRuh-uVvh-uV+cTL∞Γ3N2mTmTGmT+gL2mTΓ3mTvh-uL2Γ3d≤mTRuh-uV2+α2uh-uVvh-uV+vh-uV,where α2 is a positive constant. Moreover, we have(142)I3≤Kθ-θhQθ-ηhQ+Mkc2θh-θQθ-ηhL2Γ3+LLkc0uh-uVθ-ηhL2Γ3≤α3θ-θhQθ-ηhQ+θh-θQθ-ηhQ+uh-uVθ-ηhQ,where α3 is a positive constant. Finally, we have(143)I4≤Pθ-θhQφ-φhW+Mθ-θhQu-uhV+Mkc22θ-θhQ2+LLkc1c2θ-θhQu-uhV≤Pθ-θhQ2+φ-φhW2+Mθ-θhQ2+u-uhV2+Mkc22θ-θhQ2+LLkc1c2θ-θhQ2+u-uhV2≤α4u-uhQ2+φ-φhW2+θ-θhQ2,where α4 is a positive constant. Keeping in mind the following α-inequality ab≤αa2+(1/α)b2, it follows from (140)–(143) that if mT+P+M+LkL+Mk<L∗, then there exists a positive constant c such that(144)u-uhV2+φ-φhW2+θ-θhQ2≤cu-vhV2+u-vhV+θ-ηhQ2+φ-ξhW2,and that finishes the proof of the Theorem 9.As a result of the previous theorem, the following corollary is about the convergence order error estimates for the fully discrete approximations with the previous subspacesVh, Wh, and Qh. Let Πh be, as usual, the interpolation operator Πh:V×W×Q→Vh×Wh×Qh. Using the standard finite element interpolation error estimates, we have the following approximations:(145)u-ΠhuV≤chuH2Ωd,φ-ΠhφW≤chφH2Ω,θ-ΠhθQ≤chθH2Ω.Corollary 10.
Assume that conditions of Theorem9 hold. Under the regularity conditions(146)u∈H2Ωd,φ∈H2Ω,θ∈H2Ω,there exists positive constant c>0, independent of h, such that(147)u-uhV2+φ-φhW2+θ-θhQ2≤ch2+h.This corollary gives an estimation of the numerical errors of the problem(PV) and its proof is based on the above approximation properties of the finite element spaces Vh, Wh, and Qh.Now, we investigate the particular case wheremT=0 and 1≤mn<2. Indeed, under some conditions on στ and cT, we can prove the following result.Theorem 11.
LetmT=0 and 1≤mn<2. One assumes that, for some s>3/2, one has(148)στ∈Hs-3/2Γ3,cT∈Hs-3/2Γ3.Then, the following error estimate can be obtained:(149)u-uhV2+φ-φhW2+θ-θhQ2≤cu-vhV2+u-vhH2-sΩd+u-vhL2Γ3d2+u-vhL2Γ3d+θ-ηhQ2+θ-ηhL2Γ32+φ-ξhW2.Proof.
SincemT is zero, the functional jT becomes(150)jT0v=∫Γ3cTvτda.Then, the variational inequalities (46) and (123) can be written(151)σ,εv-εuH+jnu,v-u+jT0v-jT0u≥f,v-uV,∀v∈V,(152)σh,εvh-εuhH+jnuh,vh-uh+jT0vh-jT0uh≥f,vh-uhV,∀vh∈Vh.We take v=uh in (151) to obtain(153)-σ,εuh-εuH≤jnu,uh-u+jT0uh-jT0u-f,uh-uV,and from (152), we conclude(154)σh,εuh-εuH≤σh-σ,εvh-εuH+σ,εvh-εuH+jnuh,vh-uh+jT0vh-jT0uh-f,vh-uhV,∀vh∈Vh.Adding (153) and (154), we deduce(155)σh-σ,εuh-εuH≤σh-σ,εvh-εuH+σ,εvh-εuH+jnuh,vh-uh+jnu,uh-u+jT0vh-jT0u-f,vh-uV,∀vh∈Vh.We recall that σ=Fεu-E∗Eφ-Mθ and σh=Fε(uh)-E∗E(φh)-Mθh, then(156)Fεu-Fεuh,εuh-εuH+E∗∇φ-E∗∇φh,εuh-εuH-Mθ-Mθh,εuh-εuH≤σh-σ,εvh-εuH+σ,εvh-εuH+jnuh,vh-uh+jnu,uh-u+jT0vh-jT0u-f,vh-uV,∀vh∈Vh.Remembering (128) and (6), we replace (E∗∇φ-E∗∇φh,ε(uh)-ε(u))H by its value. Hence(157)Fεu-Fεuh,εuh-εuH+β∇φ-β∇φh,∇φ-∇φhH≤σh-σ,εvh-εuH+σ,εvh-εuH+jnuh,vh-uh+jnu,uh-u+jT0vh-jT0u-f,vh-uV-β∇φ-β∇φh,∇ξh-∇φH+Pθ-Pθh,∇φ-∇φhH+Pθ-Pθh,∇ξh-∇φH+Eεu-Eεuh,∇ξh-∇φH+Mθ-Mθh,εuh-εuH.It follows from the sum of (134) and (157) that, for all (vh,ξh,ηh) of Vh×Wh×Qh, we have(158)Fεu-Fεuh,εuh-εuH+β∇φ-β∇φh,∇φ-∇φhH+K∇θ-K∇θh,∇θ-∇θhH≤σh-σ,εvh-εuH+σ,εvh-εuH+jnuh,vh-uh+jnu,uh-u+jT0vh-jT0u-f,vh-uV-β∇φ-β∇φh,∇ξh-∇φH+Pθ-Pθh,∇φ-∇φhH+Pθ-Pθh,∇ξh-∇φH+Eεu-Eεuh,∇ξh-∇φH+Mθ-Mθh,εuh-εuH+K∇θ-K∇θh,∇θ-∇ηhH-∫Γ3kuνh-gϕLθh-θF-kuν-gϕLθ-θFθ-θhda+∫Γ3kuνh-gϕLθh-θF-kuν-gϕLθ-θFθ-ηhda.To simplify the calculations, let us consider the following quantities:(159)I~1=σh-σ,εvh-εuH-β∇φ-β∇φh,∇ξh-∇φH+Pθ-Pθh,∇ξh-∇φH+Eεu-Eεuh,∇ξh-∇φH+K∇θ-K∇θh,∇θ-∇ηhH+∫Γ3kuνh-gϕLθh-θF-kuν-gϕLθ-θFθ-ηhda,(160)I~2=σ,εvh-εuH+jT0vh-jT0u-f,vh-uV,(161)I~3=jnuh,vh-uh+jnu,uh-u=∫Γ3cnuνh-g+mnvνh-uνhda+∫Γ3cnuν-g+mnuνh-uνda,(162)I~4=Pθ-Pθh,∇φ-∇φhH+Mθ-Mθh,εuh-εuH-∫Γ3kuνh-gϕLθh-θF-kuν-gϕLθ-θFθ-θhda.Taking θh-θF=(θh-θ)+(θ-θF) in the integral term of (159), it comes from (26), (30), and the assumptions (h1)–(h4) that(163)I~1≤σh-σvh-uV-βφ-φhWξh-φW+Pθ-θhQξh-φW+Eu-uhVξh-φW+Kθ-θhQθ-ηhQ+Mkc2θ-θhQθ-ηhL2Γ3+LLkc0u-uhVθ-ηhL2Γ3.Using the Green formula and recalling (2), (9), (10), (15), and (16), we obtain(164)I2′=∫Ωσεvh-εuda+jT0vh-jT0u-f,vh-uV=∫Γσνvh-uda-∫ΩDivσvh-udx+jT0vh-jT0u-f,vh-uV=∫Γ2f2vh-uda+∫Γ3σνvνh-uνda+∫Γ3στvτh-uτda+∫Ωf0vh-udx+jT0vh-jT0u-f,vh-uVand since (f,vh-u)V=∫Ω‍f0(vh-u)dx+∫Γ2‍f2(vh-u)da, we have(165)I~2=∫Γ3σνvνh-uνda+∫Γ3στvτh-uτda+jT0vh-jT0u=∫Γ3-cnuν-g+mnvνh-uνda+∫Γ3στvτh-uτda+∫Γ3cTvτh-uτda.Hence(166)I~2≤cnuν-g+mnL2Γ3vνh-uνL2Γ3+στHs-3/2Γ3+cTHs-3/2Γ3vτh-uτH-s+3/2Γ3≤cnuν-g+mnL2Γ3vh-uL2Γ3d+στHs-3/2Γ3+cTHs-3/2Γ3vh-uH-s+2Ωd.Moreover, taking vνh-uνh=(vνh-uν)+(uν-uνh) in the first integral of (161) and using the Lemma 2, we have(167)I~3=∫Γ3cnuνh-g+mnvνh-uνda+∫Γ3cnuνh-g+mn-uν-g+mnuν-uνhda≤∫Γ3cnuνh-g+mnvνh-uνdaand then we conclude(168)I~3≤cnuνh-g+mnL2Γ3vνh-uνL2Γ3≤cnL∞Γ3uνh-g+mnL2Γ3vνh-uνL2Γ3≤cnL∞Γ3uνh-gL2mnΓ3vνh-uνL2Γ3≤cnL∞Γ3uh-uL2mnΓ3d+uL2mnΓ3d+gL2mnΓ3vνh-uνL2Γ3≤cnL∞Γ3N2mnuh-uV+N2mnuV+gL2mnΓ3vh-uL2Γ3d.However, we take θh-θF=(θh-θ)+(θ-θF) in the integral term of (162) and we use conditions (h2)–(h4) and (26), (30) to find(169)I~4≤Pθ-θhQφ-φhW+Mθ-θhQuh-uV+Mkc22θ-θhQ2+LLkc0c2uh-uVθ-θhQ≤C~P+M+Mk+LLkθ-θhQ2+φ-φhW2+uh-uV2,where C~=max⁡(1,c22,c0c2). If we suppose mT+P+M+Mk+LkL<L∗ with(170)L∗=max⁡R,C~min⁡mF,mβ,mK,then(171)C~P+M+Mk+LLk<1.Keeping in mind (h1) and (158), (159), (160), (161), (162), (163), (166), (168), (169), and (171) we apply several times the α-inequality to conclude that there exists a constant c>0 such that, for all (uh,ξh,ηh) of Vh×Wh×Qh, we have(172)u-uhV2+φ-φhW2+θ-θhQ2≤cu-vhV2+u-vhH-s+2Ωd+u-vhL2Γ3d2+u-vhL2Γ3d+θ-ηhQ2+θ-ηhL2Γ32+φ-ξhW2.


## 5. Conclusion

In this work, we presented a model for the static process of frictional contact between a piezoelectric body and an electrically thermally conductive foundation. The constitutive relation of the material is assumed to be thermo-electro-elastic. The contact was modeled with the normal compliance condition and the associated Coulomb’s friction law, including the electrical and thermal conductivity conditions. The existence of the unique weak solution for the problem was established by using arguments from the theory of variational inequalities and a fixed point theorem. A discrete scheme by finite element method was used to approach the problem and an optimal order error estimate was derived. A numerical validation of the convergence result included in this method will be provided in a forthcoming paper.



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*Source: 8346891-2018-01-28.xml*