Numerical analysis of the quasistatic thermoviscoelastic thermistor problem

Vol. 40, N 2, 2006, pp. 353–366 www.edpsciences.org/m2an

DOI: 10.1051/m2an:2006016

NUMERICAL ANALYSIS OF THE QUASISTATIC THERMOVISCOELASTIC THERMISTOR PROBLEM

Jos´ e R. Fern´ andez

Abstract. In this work, the quasistatic thermoviscoelastic thermistor problem is considered. The thermistor model describes the combination of the effects due to the heat, electrical current conduction and Joule’s heat generation. The variational formulation leads to a coupled system of nonlinear varia- tional equations for which the existence of a weak solution is recalled. Then, a fully discrete algorithm is introduced based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity assumptions, the linear convergence of the scheme is deduced. Finally, some numerical simulations are performed in order to show the behaviour of the algorithm.

Mathematics Subject Classification. 65N15, 65N30, 74D10, 74S05, 74S20.

Received: June 13, 2005.

Introduction

During the last twenty years, the thermistor problem has been studied by many authors (see, e.g., [1–4, 8, 10, 14, 15, 21, 22, 25]), dealing with different issues such as the existence and uniqueness of weak solutions or the degenerate cases, the so-called capacity solutions. Recently, in [5, 16–19, 26] the numerical solution of the thermistor problem has been considered, under different mechanical conditions, using finite differences or the finite element method.

The thermistor model describes the combination of the effects due to the heat, electrical current conduction and Joule’s heat generation. The electrical conductivity is assumed increasing (see [20] for details concerning the physical setting). The novelty of this model is that thermoviscoelastic effects are taken into account. The existence of a unique weak solution for the dynamical problem was established in [20], and its proof is based on the regularization, time-retarding and a fixed point argument. This work continues [20] and it is parallel to [12], where a finite element algorithm is presented and numerical simulations are shown.

Here, our aim is to provide the numerical analysis for the quasistatic thermoviscoelastic thermistor problem.

The paper is structured as follows. In Section 2, the mechanical problem and its variational formulation are

Keywords and phrases. Thermoviscoelastic thermistor, error estimates, finite elements, numerical simulations.

∗This work was partially supported by MCYT-Spain (Project BFM2003-05357). It is also part of the project “New Materials, Adaptive systems and their Nonlinearities; Modelling, Control and Numerical Simulation” carried out in the framework of the European Community program “Improving the Human Research Potential and the Socio-Economic Knowledge Base” (Contract n^◦HPRN-CT-2002-00284).

1Departamento de Matem´atica Aplicada, Facultade de Matem´aticas, Campus Sur s/n, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. jramon@usc.es

c EDP Sciences, SMAI 2006

Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2006016

presented following [12, 20]. An existence and uniqueness result is recalled. Then, a fully discrete scheme is introduced in Section 3, using the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. Error estimates are established on the approximative solutions and, as a consequence, the linear convergence of the algorithm is derived under suitable regularity assumptions.

Finally, in Section 4 some two-dimensional examples are presented in order to show the performance of the method.

1. Mechanical and variational problems

In this section, we follow [12, 20] and we refer the reader there for details. LetS^d denote the space of second order symmetric tensors defined onR^d, and let “:” and| · | represent the inner product and the norm onS^d, respectively.

Let Ω⊂R^d (d= 2,3) be a bounded domain with a smooth boundary Γ =∂Ω, representing the isothermal reference configuration of the thermoviscoelastic body, the thermistor. We assume that Γ is divided into two parts Γ_D and Γ_N such that Γ_D∩Γ_N =∅, where ΓD has positive measure. We denote byν = (ν₁, . . . , ν_d) the outward unit normal vector to Γ.

The body is clamped on Γ_D and moreover, the temperature and electrical potential are prescribed there.

On Γ_N the body is free, electrically insulated and exchanging heat with the environment. Dirichlet conditions are chosen on Γ_D for the three fields, for the sake of simplicity. We let θ denote the temperature field, φthe electrical potential and u= (u₁, . . . , u_d) the displacements field. LetT >0 and set Ω_T = Ω×(0, T).

The constitutive and motion equations are written in Ω_T as follows (see [11]):

ρc_p∂θ

∂t − ∂

∂x_j

k_ij(θ)∂θ

∂x_i

=σ_el(θ)Tr(|∇φ|²)−m_ijΘ_ref ∂²u_i

∂t∂x_j, (1)

− ∂

∂x_j

σ_el(θ)∂φ

∂x_j

= 0, (2)

− ∂

∂x_j(σ_ij) =f_i, (3)

σ_ij =a_ijkl∂u_k

∂x_l +b_ijkl∂²u_k

∂t∂x_l −m_ijθ. (4)

Here and below,i, j, k, l= 1, . . . , dand summation over repeated indices is implied. Tr :R→Ris a truncation operator defined by

Tr(r) =

r if |r| ≤L, L if |r|> L,

whereL >0 is a given constant assumed large enough. Finally, we note that the inertia term was neglected in the motion equation (3), and then the process is quasistatic.

Equation (1) is the energy equation expressed in terms of the temperature. Here,θ is measured with respect to a reference absolute temperature Θ_ref, given in degrees Kelvin. We assume that Θ_ref is also the ambient temperature. The density ρ(> 0) and the heat capacity c_p(> 0) are assumed to be constant; K = K(θ) = {kij(θ)}andM ={mij} are the heat conduction and thermal expansion tensors, respectively, and for the sake of generality it is assumed that the thermal and mechanical properties of the material are anisotropic. The electrical conductivity σ_el =σ_el(θ) is assumed to depend strongly on the temperature. However, we assume that the electrical properties of the material are isotropic.

Equation (2) represents the electrical charge conservation, and (4) is the material constitutive relation. Here, A={aijkl}is the elasticity tensor;B={bijkl}is the tensor of viscosity coefficients;f = (f₁, . . . , f_d) represents the density of body forces. Moreover, if we denote byεthe linearized strain tensor (ε=ε(u) =¹₂(∇u+(∇u)^T)),

we may write the constitutive equation (4) as

σ=Aε(u) +Bε ∂u

∂t

−M θ.

Although different possibilities for σ_el(θ) have been considered in many publications (including the so-called

“capacity solutions” in [8, 22–24]), here it is assumed that the electrical conductivity is nondegenerate and bounded. Thus,

0< σ_∗≤σ_el(θ)≤σ^∗, (5)

forσ_∗sufficiently small, and some constantσ^∗.

To complete the classical formulation of the problem we have to specify the initial and boundary conditions.

For the sake of the simplicity, homogeneous conditions are assumed on Γ_Dfor the displacement and temperature fields (but not for the electrical potential since it would imply thatφ= 0), that is,

θ= 0, φ=φ_b, u=0 on Γ_D×[0, T]. (6)

On Γ_N the body exchanges heat with the environment, it is electrically insulated and stress-free, thus,

−k_ij ∂θ

∂x_iν_j=α(θ−Θ_ref) on Γ_N ×(0, T), (7)

∂φ

∂x_iν_i= 0 on Γ_N×(0, T), (8)

σ_ijν_j= 0 on Γ_N×(0, T), (9)

whereαis the convective heat transfer coefficient.

The initial conditions are,

θ=θ⁰, u=u⁰ in Ω, t= 0, (10)

whereθ⁰,u⁰ denote the initial temperature and the initial displacements, respectively.

The classical formulation of the quasistatic thermoviscoelastic thermistor problem is: find {θ, φ,u} such that (1)–(4), (6)–(10) hold.

We turn now to obtain a weak formulation of the problem. Let us define the following variational spaces:

V =

η∈H¹(Ω) ;η= 0 on Γ_D

, (11)

E=

v∈[H¹(Ω)]^d;v=0on Γ_D

. (12)

The following assumptions are done:

(i) The densityρand the heat capacityc_pare positive constants.

(ii) The electrical conductivityσ_el satisfies (5) and it is Lipschitz continuous.

(iii) The thermal conductivity tensor{kij} is bounded and Lipschitz continuous, satisfying k_ijξ_iξ_j ≥δ|ξ|²,

for all vectorsξ= (ξ_i)^d_i=1∈R^d and some positiveδ.

(iv) The elasticity and viscosity tensors{aijkl}and{bijkl}are bounded and Lipschitz continuous, satisfying a_ijklτ_ijτ_kl ≥α₁|τ|², b_ijklτ_ijτ_kl≥α₂|τ|²,

for all tensorsτ = (τ_ij)^d_i,j=1∈S^d and some positiveα₁, α₂.

(v) The initial temperature and the initial displacements satisfyθ⁰∈V and u⁰∈E, respectively.

(vi) The density of body forces has the regularityf ∈C([0, T]; [L²(Ω)]^d).

(vii) The functionφ_b∈C([0, T];W^1,4(Ω)).

The variational form of (1)–(4), (6)–(10) is: find θ: [0, T]−→V, φ: [0, T]−→H¹(Ω) andu,v : [0, T]−→E such that, fora.e. t∈[0, T] ,

φ(t)−φ_b(t)∈V,

Ωσ_el(θ(t))∇φ(t)· ∇rdx= 0, ∀r∈V, (13)

ρ c_p∂θ

∂t(t), z

VV

+

Ω

k_ij(θ(t))∂θ

∂x_i(t)∂z

∂x_jdx+

ΓN

α(θ(t)−Θ_ref)zdΓ

=

Ωσ_el(θ(t))Tr(|∇φ(t)|²))zdx−

ΩΘ_refM :ε(v(t))zdx, ∀z∈V, (14)

Ω[Bε(v(t)) +Aε(u(t))] : ε(w) dx=

Ωf(t)·wdx+

ΩM θ(t) : ε(w) dx, ∀w∈E, (15) u(t) =u⁰+

_t

0 v(s)ds. (16)

An existence result for a dynamic version of problem (13)–(16) was given in [20]. It is based on time-retarded approximate problems, a priori estimates and compactness arguments. That proof can be modified in order to obtain the existence of a solution to problem (13)–(16), which we summarize in the following.

Theorem 1.1. Let the assumptions (i)–(vii) hold. Then, there exists a solution{u, θ, φ} to problem (13)–(16) with the following regularity:

u∈H¹(0, T;E), φ∈L²(0, T;V), θ∈L²(0, T;V)∩H¹(0, T;V). (17) Remark 1.2. The regularity (17) does not allow to obtain the error estimates provided in the next section.

However, since in this case the problem is quasistatic and the quadratic term in the heat equation was replaced by a Lipschitz function, it seems reasonable to assume that this regularity can be improved using, for instance, Banach fixed point arguments. This problem remains an open question and it will be addressed in the future.

2. A fully discrete scheme: error estimates

In this section, the above variational problem is discretized in time using the backward Euler scheme and in space by a finite element method. The functions u, φ andθ are approximated by continuous piecewise affine functions on a defined mesh. Thus, we obtain a linear discrete problem for the displacement equation coupled with a nonlinear discrete problem for the temperature and the electrical potential which is solved by an iterative algorithm.

For the time discretization, we denote by N the number of time steps and by ∆t=T /N the time step. For a continuous functionF, letFⁿ=F(t_n) witht_n =n∆t, forn= 0, . . . , N.

Associated with a family of regular meshes Th of the domain Ω, we consider the finite element spacesE_h andV_h given by

E_h={vh∈[C(Ω)]^d;v_h|K∈[P₁(K)]^d, ∀K∈ Th, v_h|Γ

D =0}, (18)

V_h={rh∈C(Ω) ;r_h|K∈P₁(K), ∀K∈ Th, r_h|Γ

D = 0}, (19)

whereP₁(K) denotes the space of polynomials of global degree less or equal 1 defined in an elementK.

In order to simplify the writing and the calculations presented below, let us assume that φ_b = 0. It is straightforward to extend the error estimates to the nonhomogeneous case by using the orthogonal projection operator overV_h.

Then, using the backward Euler scheme to discretize the time derivatives, the following fully discretized problem is introduced: Finduh={uⁿ_h}^N_n=0, vh={vⁿ_h}^N_n=0⊂E_h andθ_h={θ_hⁿ}^N_n=0, φ_h ={φⁿ_h}^N_n=0⊂V_h such that u⁰_h=u^0,h,θ⁰_h=θ^0,hand forn= 1, . . . , N,

Ω[Bε(vⁿ_h) + ∆tAε(vⁿ_h)] : ε(wh) dx=

Ωfⁿ·whdx +

Ω

[M θⁿ⁻¹_h −Aε(uⁿ⁻¹_h )] : ε(wh) dx, ∀wh∈E_h, (20)

uⁿ_h=uⁿ⁻¹_h + ∆tvⁿ_h, (21)

Ωσ_el(θⁿ_h)∇φⁿ_h· ∇rhdx= 0, ∀rh∈V_h, (22)

1

∆t

Ωρ c_pθ_hⁿz_hdx+

Ωk_ij(θⁿ_h)∂θⁿ_h

∂x_i

∂z_h

∂x_jdx+

ΓN

α θ_hⁿz_hdΓ

= 1

∆t

Ω

ρ c_pθⁿ⁻¹_h z_hdx+

Ω

σ_el(θⁿ_h)Tr(|∇φⁿ_h|²)z_hdx+

ΓN

αΘ_refz_hdΓ

−

ΩΘ_refM : ε(vⁿ_h)z_hdx, ∀zh∈V_h, (23)

whereu^0,h∈E_handθ^0,h∈V_h are appropriate approximations of the initial conditionsu⁰andθ⁰, respectively.

Remark 2.1. Notice that the discrete problem (20) can be seen as a linear system which is solved using Cholesky’s method and then, displacements are updated from (21). Besides, discretized problem (22)–(23) is a nonlinear system due to the temperature dependence on the physical parameters, as the electrical and thermal conductivities, and the Joule’s source term in (23). It is solved using a penalty-duality algorithm, coupled with fixed point iterations, introduced in [6].

The aim of this paper is to obtain an error estimates for the energy norms θⁿ_h−θⁿV, φⁿ_h−φⁿV and uⁿ_h−uⁿE. Then, let us assume the following regularity conditions on the continuous solution,

φ∈C([0, T];H¹(Ω)∩W^1,∞(Ω)), u∈C¹([0, T]; [H¹(Ω)]^d),

θ∈C([0, T];H¹(Ω)∩W^1,∞(Ω))∩C¹([0, T];L²(Ω)). (24) Remark 2.2. We note that there is a gap between these assumptions, required for obtaining the error estimates presented below, and the regularity of the solution provided in Theorem 1.1. This gap is an open problem that needs to be addressed.

First, we write (13) at time t=t_n. Takingr=r_h∈V_h⊂V and substracting it to (22) we find that

Ω[σ_el(θⁿ)∇φⁿ−σ_el(θⁿ_h)∇φⁿ_h]· ∇r_hdx= 0, ∀r_h∈V_h, and therefore,

Ω[σ_el(θⁿ)∇φⁿ−σ_el(θ_hⁿ)∇φⁿ_h]· ∇(φⁿ−φⁿ_h) dx=

Ω[σ_el(θⁿ)∇φⁿ−σ_el(θ_hⁿ)∇φⁿ_h]· ∇(φⁿ−r_h) dx, ∀rh∈V_h.

Now, keeping in mind that

Ω[σ_el(θⁿ)∇φⁿ−σ_el(θⁿ_h)∇φⁿ_h]· ∇rdx=

Ω[(σ_el(θⁿ)−σ_el(θ_hⁿ))∇φⁿ+σ_el(θⁿ_h)(∇φⁿ− ∇φⁿ_h)]· ∇rdx, for allr∈V, and using the regularityφⁿ ∈W^1,∞(Ω), property (ii) and the Cauchy’s inequality

ab≤εa²+ 1

4εb², a, b, ε >0, (25)

after easy algebraic manipulations we have, for allrⁿ_h∈V_h, cφⁿ−φⁿ_h²_V ≤ −

Ω(σ_el(θⁿ)−σ_el(θⁿ_h))∇φⁿ· ∇(φⁿ−φⁿ_h) dx +

Ω(σ_el(θⁿ)−σ_el(θⁿ_h))∇φⁿ· ∇(φⁿ−rⁿ_h) dx +

Ωσ_el(θ_hⁿ)(∇φⁿ− ∇φⁿ_h)· ∇(φⁿ−rⁿ_h) dx

≤c(φⁿ−r_hⁿ²_V +θⁿ−θ_hⁿ²_L2(Ω)+εφⁿ−φⁿ_h²_V), (26) where εis assumed to be small enough andc >0, here and everywhere below, is a positive constant that may vary from line to line and it is independent of the discretization parametershand ∆t.

Secondly, we rewrite the variational equation (15) at timet=t_nfor allw=wh∈E_h⊂E and we substract it to (20) to obtain

Ω[Bε(vⁿ) +Aε(uⁿ)−Bε(vⁿ_h)−Aε(uⁿ_h)] : ε(wh) dx−

Ω[M θⁿ−M θⁿ⁻¹_h ] : ε(wh) dx= 0, where (21) was employed. Thus,

Ω

[Bε(vⁿ) +Aε(uⁿ)−Bε(vⁿ_h)−Aε(uⁿ_h)] : ε(vⁿ−vⁿ_h) dx−

Ω

[M θⁿ−M θ_hⁿ⁻¹] : ε(vⁿ−vⁿ_h) dx

=

Ω

[Bε(vⁿ)+Aε(uⁿ)−Bε(vⁿ_h)−Aε(uⁿ_h)] : ε(vⁿ−wh) dx−

Ω

[M θⁿ−M θⁿ⁻¹_h ] : ε(vⁿ−wh) dx, ∀wh∈E_h. Since

uⁿ−uⁿ_h²_E≤c

u⁰−u^0,h2_E+ ∆t

n j=1

v^j−v^j_h²_E+I_n²

, whereI_n is the integration error given by

I_n = _tn

0 v(s) ds−∆t

n j=1

v^j E

,

using now assumption (iv) and (25), after some algebra it follows that vⁿ−vⁿ_h²_E≤c

vⁿ−wⁿ_h²_E+ ∆t

n j=1

v^j−v^j_h²_E+θⁿ⁻¹−θⁿ⁻¹_h ²_L2(Ω)

+θⁿ−θⁿ⁻¹²_L2(Ω)+u⁰−u^0,h2_E+I_n²

, ∀wⁿ_h∈E_h. (27)

Finally, we need to estimate the numerical errors on the temperature field. Then, we consider (14) at time t=t_n for allz=z_h∈V_h⊂V. Substracting it to (23), it leads to the following variational equation,

Ωρ c_p ∂θ

∂t(t_n)−θⁿ_h−θⁿ⁻¹_h

∆t

z_hdx+

Ω

k_ij(θⁿ)∂θ

∂x_i(tⁿ)−k_ij(θⁿ_h)∂θⁿ_h

∂x_i ∂z_h

∂x_jdx +

ΩΘ_refM :ε(vⁿ−vⁿ_h)z_hdx−

Ω[σ_el(θⁿ)Tr(|∇φⁿ|²)−σ_el(θⁿ_h)Tr(|∇φⁿ_h|²)]z_hdx= 0, ∀zh∈V_h, which implies

Ωρ c_p ∂θ

∂t(t_n)−θⁿ_h−θⁿ⁻¹_h

∆t

(θⁿ−θ_hⁿ) dx+

ΩΘ_refM :ε(vⁿ−vⁿ_h)(θⁿ−θⁿ_h) dx +

Ω

[k_ij(θⁿ)∂θ

∂x_i(tⁿ)−k_ij(θⁿ_h)∂θⁿ_h

∂x_i

∂(θⁿ−θ_hⁿ)

∂x_j dx

−

Ω

σ_el(θⁿ)Tr(|∇φⁿ|²)−σ_el(θ_hⁿ)Tr(|∇φⁿ_h|²)

(θⁿ−θ_hⁿ) dx

=

Ω

ρ c_p ∂θ

∂t(t_n)−θⁿ_h−θⁿ⁻¹_h

∆t

(θⁿ−z_h) dx+

Ω

Θ_refM :ε(vⁿ−vⁿ_h)(θⁿ−z_h) dx +

Ω

k_ij(θⁿ)∂θ

∂x_i(tⁿ)−k_ij(θⁿ_h)∂θⁿ_h

∂x_i

∂(θⁿ−z_h)

∂x_j dx

−

Ω

σ_el(θⁿ)Tr(|∇φⁿ|²)−σ_el(θ_hⁿ)Tr(|∇φⁿ_h|²)

(θⁿ−z_h) dx, ∀zh∈V_h.

In order to simplify the writing, let us define δθⁿ = (θⁿ−θⁿ⁻¹)/∆t and δθ_hⁿ = (θ_hⁿ−θⁿ⁻¹_h )/∆t. Using the properties (i), (ii) and (iii), the Cauchy’s inequality (25) and the regularityθⁿ∈W^1,∞(Ω), and keeping in mind that

k_ij(θⁿ)∂θ

∂x_i(tⁿ)−k_ij(θⁿ_h)∂θ_hⁿ

∂x_i = (k_ij(θⁿ)−k_ij(θ_hⁿ))∂θ

∂x_i(t_n) +k_ij(θⁿ_h) ∂θ

∂x_i(t_n)−∂θ_hⁿ

∂x_i

, σ_el(θⁿ)Tr(|∇φⁿ|²)−σ_el(θⁿ_h)Tr(|∇φⁿ_h|²) = (σ_el(θⁿ)−σ_el(θ_hⁿ))Tr(|∇φⁿ|²)

+σ_el(θⁿ_h)(Tr(|∇φⁿ|²)−Tr(|∇φⁿ_h|²)), after easy algebraic manipulations we find

Ω(δθⁿ−δθ_hⁿ)(θⁿ−θ_hⁿ) dx+cθⁿ−θ_hⁿ²_V ≤c ∂θ

∂t(t_n)−δθ^{n 2}

L²(Ω)

+εθⁿ−θⁿ_h²_V +vⁿ−vⁿ_h²_E+θⁿ−θⁿ_h²_L2(Ω)+φⁿ−φⁿ_h²_V +θⁿ−z_hⁿ²_V +

Ω

(δθⁿ−δθⁿ_h)(θⁿ−z_hⁿ) dx

, ∀zⁿ_h ∈V_h,

whereε >0 is assumed to be sufficiently small.

Since

Ω(δθⁿ−δθⁿ_h)(θⁿ−θⁿ_h) dx≥ 1 2∆t

θⁿ−θ_hⁿ²_L2(Ω)− θⁿ⁻¹−θⁿ⁻¹_h ²_L2(Ω)

,

proceeding by induction we obtain θⁿ−θ_hⁿ²_L2(Ω)+ ∆t

n j=1

θ^j−θ^j_h²_V ≤c∆t

n j=1

∂θ

∂t(t_j)−δθ^{j 2}

L²(Ω)

+v^j−v^j_h²_E+θ^j−θ^j_h²_L2(Ω)+φ^j−φ^j_h²_V +θ^j−z_h^j2_V

+cθ⁰−θ^0,h2_L2(Ω)+c

n j=1

Ω(θ^j−θ^j−1−(θ^j_h−θ^j−1_h ))(θ^j−z_h^j) dx, {z^j_h}ⁿ_j=1⊂V_h. Using similar ideas to those applied in [7] for estimating the numerical errors in the damage field, we have

n j=1

Ω(θ^j−θ^j−1−(θ_h^j−θ_h^j−1))(θ^j−z_h^j) dx=

Ω(θⁿ−θⁿ_h)(θⁿ−z_hⁿ) dx+

Ω(θ⁰−θ^0,h)(θ¹−z_h¹) dx +

n−1 j=1

Ω(θ^j−θ_h^j)(θ^j−z^j_h−(θ^j+1−z^j+1_h )) dx, and then,

θⁿ−θ_hⁿ²_L2(Ω)+ ∆t

n j=1

θ^j−θ_h^j2_V ≤c∆t

n j=1

∂θ

∂t(t_j)−δθ^{j 2}

L²(Ω)

+v^j−v^j_h²_E+θ^j−θ_h^j2_L2(Ω)+φ^j−φ^j_h²_V +θ^j−z^j_h²_V

+cθ⁰−θ^0,h2_L2(Ω)+cθⁿ−zⁿ_h²_L2(Ω)+cθ¹−z_h¹²_L2(Ω)

+c

n−1 j=1

Ω(θ^j−θ^j_h)(θ^j−z_h^j−(θ^j+1−z_h^j+1)) dx, ∀{z_h^j}ⁿ_j=1⊂V_h. (28) Let us denote by

e_n=θⁿ−θ_hⁿ²_L2(Ω)+ ∆t

n j=1

θ^j−θ_h^j2_V +vⁿ−vⁿ_h²_E+φⁿ−φⁿ_h²_V,

forn= 1, . . . , N, and

e₀=θ⁰−θ^0,h2_L2(Ω)+u⁰−u^0,h2_V the numerical errors, and define, forn= 1, . . . , N,

g_n = ∆t

n j=1

∂θ

∂t(t_j)−δθ^{j 2}

L²(Ω)

+θ^j−z_h^j2_V

+θⁿ−z_hⁿ²_L2(Ω)

+θ¹−z_h¹²_L2(Ω)+φⁿ−r_hⁿ²_V +vⁿ−wⁿ_h²_E+θⁿ−θⁿ⁻¹²_L2(Ω)

+e₀+I_n²+

n j=1

Ω(θ^j−θ^j_h)(θ^j−z_h^j−(θ^j+1−z_h^j+1)) dx, (29) and

g₀=e₀. (30)

Combining now (26), (27) and (28), it leads to the following error estimates, e₀≤g₀, e_n≤c(g_n+ ∆t

n j=1

e_j), forn= 1, . . . , N.

Finally, using a discrete version of Gronwall’s lemma (see [13] for details), the following error estimates result is obtained.

Theorem 2.3. Let the assumptions (i)–(vii) and the regularity conditions (24) hold. Let {u, θ, φ} denote the solution to problem (13)–(16) and{uh, θ_h, φ_h} the solution to discrete problem (20)–(23). Then, the following error estimates are obtained for all {z_hⁿ}^N_n=1⊂V_h,{rⁿ_h}^N_n=1⊂V_h and{wⁿ_h}^N_n=1⊂E_h,

0≤n≤Nmax

θⁿ−θⁿ_h²_L2(Ω)+vⁿ−vⁿ_h²_E+φⁿ−φⁿ_h²_V

+ ∆t

N j=1

θ^j−θ^j_h²_V ≤c max

0≤n≤Ng_n, (31) whereg_n is given by (29)–(30).

We note that (31) is the basis for the analysis of the convergence rate. Therefore, if we assume, for instance, that

θ∈H²(0, T;L²(Ω))∩C([0, T];H²(Ω))∩H¹(0, T;H¹(Ω)), u∈C¹([0, T]; [H²(Ω)]^d)∩H²(0, T; [H¹(Ω)]^d),

φ∈C([0, T];H²(Ω)),

(32)

we have the following corollary which states the linear convergence of the numerical algorithm.

Corollary 2.4. Let the assumptions of Theorem 2.3 hold. Assume that the discrete initial conditions θ^0,h andu^0,h are given by

θ^0,h=π^h(θ⁰), u^0,h= Π^h(u⁰), (33)

where π^h : C(Ω) → V_h is the standard finite element interpolation operator (see [9]), and Π^h = (π_i^h)^d_i=1 : [C(Ω)]^d → E_h. Under the additional regularity conditions (32), the linear convergence of the fully discrete scheme is obtained; that is, there existsc >0, independent of hand∆t, such that

0≤n≤Nmax

θⁿ−θⁿ_hL²(Ω)+uⁿ−uⁿ_hE+φⁿ−φⁿ_hV

≤c(h+ ∆t). (34)

Proof. Sinceθ∈H²(0, T;L²(Ω)) it is easy to check that

N j=1

∆t∂θ

∂t(t_j)−δθ^j2_L2(Ω)≤c(∆t)²θ²_H2(0,T;L²(Ω)),

and, using the approximation properties of the finite element spacesV_handE_hand the regularity (32), it follows that (see [9])

0≤n≤Nmax inf

z_hⁿ∈Vh

θⁿ−z_hⁿ²_V ≤ch²θ²_C([0,T];H2(Ω)),

0≤n≤Nmax inf

r_hⁿ∈Vh

φⁿ−rⁿ_h²_V ≤ch²φ²_C([0,T];H2(Ω)),

0≤n≤Nmax winfⁿ_h∈Eh

vⁿ−wⁿ_h²_E≤ch²u²_C1([0,T];[H²(Ω)]^d).

Keeping in mind thatu∈H²(0, T; [H¹(Ω)]^d) andθ∈H²(0, T;L²(Ω)), we have I_n²≤c(∆t)²u²_H2(0,T;[H¹(Ω)]^d),

0≤n≤Nmax θⁿ−θⁿ⁻¹²_L2(Ω)≤c(∆t)²θ²_H2(0,T;L²(Ω)).

Sinceθ^0,handu^0,hare defined by (33), well-known results (see [9] for details) lead to the following estimates:

θ⁰−θ^0,h2_L2(Ω)≤ch²θ⁰²_H1(Ω)≤ch²θ²_C([0,T];H2(Ω)), u⁰−u^0,h2_V ≤ch²u⁰²_[H2(Ω)]^d≤ch²u²_C1([0,T];[H²(Ω)]^d).

Finally, proceeding as in [7] we obtain that

N−1 j=1

Ω(θ^j−θ_h^j)(θ^j−z^j_h−(θ^j+1−z^j+1_h )) dx≤ch²θ²_H1(0,T;H¹(Ω)),

which concludes the proof.

3. Numerical simulations

In order to verify the performance of the numerical algorithm described in the previous section and to gain insight into the behaviour of the solutions, we performed several numerical experiments. In this section we present two of the results obtained in these numerical simulations.

In all the problems presented below, the body is assumed to be under the plane stress hypothesis with elasticity tensorA,

(Aτ)_αβ= Eκ

1−κ²(τ₁₁+τ₂₂)δ_αβ+ E

1 +κτ_αβ, α, β= 1,2, ∀τ = (τ_αβ)∈S²,

where δ_αβ represents the Kronecker symbol, and E and κ are Young’s modulus and Poisson’s ratio of the material that occupies Ω, respectively.

The viscosity tensorB has a similar form,

(Bτ)_αβ=η₁(τ₁₁+τ₂₂)δ_αβ+η₂τ_αβ, α, β= 1,2, ∀τ = (τ_αβ)∈S²,

where η₁ andη₂ are viscosity coefficients. Finally, the truncation valueL= 1000 is employed and the thermal expansion tensorM is constant and homogeneous, namelyM =m I={m δ_αβ}, withma positive constant.

3.1. First example: convergence of the algorithm in an academical test

As a first example we consider an academical test. We denote by Ω the unit square domain (0,1)×(0,1) which is the cross-section of a three-dimensional viscoelastic body. On the part Γ_D={0,1} ×[0,1] the body is clamped and so the displacement field vanishes there. The temperature field is also assumed to be fixed (and equal to Θ_ref) on Γ_D, and the electrical potential is defined by φ(x, y, t) = 0 if x = 0 and φ(x, y, t) = 1 if x= 1 for allt∈[0,1] (T = 1 s). Finally, the part Γ_N is assumed traction-free, there is no heat exchange (value α= 10⁹is used for its simulation) and no body forces act in the thermistor.

In order to see the convergence behaviour of the scheme, a sequence of numerical solutions is computed based on uniform partitions of the time interval [0,1], and uniform triangulations of the domain [0,1]×[0,1], where