Dynamic frictional contact of a viscoelastic beam

Vol. 40, N^o 2, 2006, pp. 295–310 www.edpsciences.org/m2an

DOI: 10.1051/m2an:2006019

DYNAMIC FRICTIONAL CONTACT OF A VISCOELASTIC BEAM

Marco Campo

, Jos´ e R. Fern´ andez

, Georgios E. Stavroulakis

and Juan M. Via˜ no

Abstract. In this paper, we study the dynamic frictional contact of a viscoelastic beam with a de- formable obstacle. The beam is assumed to be situated horizontally and to move, in both horizontal and tangential directions, by the effect of applied forces. The left end of the beam is clamped and the right one is free. Its horizontal displacement is constrained because of the presence of a deformable obstacle, the so-called foundation, which is modelled by a normal compliance contact condition. The effect of the friction is included in the vertical motion of the free end, by using Tresca’s law or Coulomb’s law.

In both cases, the variational formulation leads to a nonlinear variational equation for the horizontal displacement coupled with a nonlinear variational inequality for the vertical displacement. We recall an existence and uniqueness result. Then, by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives, a numerical scheme is proposed. Error estimates on the approximative solutions are derived. Numerical results demonstrate the application of the proposed algorithm.

Mathematics Subject Classification. 65N15, 65N30, 74D05, 74M10, 74M15, 74S05, 74S20.

Received: September 28, 2005.

Introduction

Contact problems involving viscoplastic or viscoelastic materials appear in many industrial problems and everyday life (see, e.g., the monographs[18, 19, 24] and references therein). Recently, one-dimensional contact problems for beams and rods have been studied[1, 3, 12, 17, 20, 21], including the adhesion[15], the wear[11, 22]

or the damage[4]. Moreover, in[2] the adhesive contact problem of a membrane was studied.

In this work, we describe a model for the dynamic frictional contact of a viscoelastic beam with a deformable obstacle, the so-called foundation. The model was introduced in[1], where the existence of a unique weak solution was proved and its regularity was studied. That work was mainly focused on the mathematical aspects provided by the elastic case, that is, when the damping coefficient associated to the viscous effect is neglected.

Keywords and phrases. Dynamic unilateral contact, friction, viscoelastic beam, error estimates, 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 No 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. macampo@usc.es;jramon@usc.es;maviano@usc.es

2 Department of Production Engineering and Management, Technical University of Crete, 73100 Chania, Greece.

gestavr@dpem.tuc.gr

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:2006019

f

L g

f

Figure 1. A viscoelastic beam in frictional contact.

Here, our aim is to study a numerical scheme for the viscoelastic case, based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Moreover, in order to show the accuracy of the proposed algorithm, some numerical simulations are presented.

1. The model and variational formulation

In this section, we follow[1] and we describe the model for the dynamic evolution of a viscoelastic beam in frictional contact with a deformable obstacle. The beam is assumed to be rigidly clamped at its left end, and its right end is free to come into frictional contact with an obstacle situated at a distanceg to the right (see Fig. 1).

Let [0, L], L > 0, be the reference configuration of the beam and let us denote by [0, T], T >0, the time interval of interest. The horizontal and vertical displacements of the beam are represented byu(x, t) and ˜u(x, t), (x, t)∈(0, L)×(0, T), respectively.

The material is assumed viscoelastic and satisfying the Kelvin-Voigt viscoelastic laws[10],

σ_H(x, t) =a₁u_x(x, t) +c₁u_xt(x, t), (x, t)∈(0, L)×(0, T), (1) σ_V(x, t) =−(a₂u˜_xxx(x, t) +c₂u˜_xxxt(x, t)), (x, t)∈(0, L)×(0, T), (2) where, as usual, a subscript denotes a partial derivative with respect to the corresponding variable. Here, a₁, a₂ > 0 are elastic constants, and c₁, c₂ >0 represent the viscosities. We notice that these equations are decoupled because of the assumptions on the symmetry of the cross-section of the beam and the isotropy of its material.

Since we assume that the inertia effect is not negligible, the equations of motion have the following form, ρu_tt(x, t)−(σ_H)_x(x, t) =f_H(x, t), (x, t)∈(0, L)×(0, T), (3) ρ˜u_tt(x, t)−(σ_V)_x(x, t) =f_V(x, t), (x, t)∈(0, L)×(0, T), (4) wheref_H andf_V denote the horizontal and vertical applied forces, respectively. For simplicity, the densityρis assumed to be equal to 1 and

f_H∈W^1,2(0, T;L²(0, L)), f_V ∈ C([0, T];L²(0, L)). (5) Letu₀,v₀, ˜u₀, ˜v₀ be the initial conditions, that is,

u(x,0) =u₀(x), u_t(x,0) =v₀(x), x∈(0, L), (6)

˜

u(x,0) = ˜u₀(x), u˜_t(x,0) = ˜v₀(x), x∈(0, L). (7)

To complete the model, we need to prescribe the boundary conditions. First, since the left end of the beam was supposed rigidly attached, we have

u(0, t) = 0, t∈[0, T], (8)

˜

u(0, t) = ˜u_x(0, t) = 0, t∈[0, T]. (9)

Secondly, the beam is assumed to be located at a distance g from the deformable obstacle. According to[16], the following normal compliance contact condition is considered,

−σ_H(L, t) =c_H(u(L, t)−g)₊, t∈[0, T], (10) where (u(L, t)−g)₊ = max{0, u(L, t)−g} and c_H >0 is a deformability coefficient (i.e. a hard spring or a penalty factor which is used to enforce the nonpenetration unilateral constraint).

Finally, concerning the vertical displacements, we assume that the sum of the moments on the free end is zero,i.e.

a₂u˜_xx(L, t) +c₂u˜_xxt(L, t) = 0, t∈[0, T], (11) and also a friction condition of the form, for a.e. t∈[0, T],

|σV(L, t)| ≤h(t),

if|σ_V(L, t)|=h(t)⇒ ∃λ≥0 ; ˜u_t(L, t) =−λσ_V(L, t), if|σ_V(L, t)|< h(t)⇒u˜_t(L, t) = 0.

⎫⎬

⎭ (12) In (12), functionh(t) represents a friction bound. We will consider two different cases:

(i) h(t) =c_V = constant: it corresponds to the classical Tresca’s conditions.

(ii) h(t) =c_V(u(L, t)−g)₊, forc_V = constant>0: it leads to a particular case of Coulomb’s conditions.

Remark 1.1. We notice that the case (i) results from the case (ii), when the penetrationu(L, t)−gis assumed to be constant. Therefore, only the case (ii) will be considered in the rest of the paper.

The dynamic frictional contact problem of a viscoelastic beam with a deformable obstacle is written as follows.

Find a horizontal displacementu: [0, L]×[0, T]→Rand a vertical displacement u˜: [0, L]×[0, T]→Rwhich satisfy (1)–(12).

In order to derive a weak formulation of the above problem, we define the following variational spaces:

H =L²(0, L),

E={w∈H¹(0, L) ;w(0) = 0}, V ={z∈H²(0, L) ; z(0) =z_x(0) = 0}.

Moreover, we denote by (·,·) the classical inner product defined inL²(0, L) and, for a Hilbert spaceX, let| · |X

represent its norm. If X denotes the dual space of X, then let ·,·XX be the duality pairing between X andX.

Letj_H(u,·) :E→Randj_V(u,·) :V →Rbe the normal compliance and friction forms defined by j_H(u, w) =c_H(u(L, t)−g)₊w(L, t), ∀w∈E,

j_V(u, z) =c_V(u(L, t)−g)₊|z(L, t)|, ∀z∈V.

Integrating by parts the equations of motion (3)–(4) and using the previously given boundary conditions, the variational formulation is then written as follows.

Variational problem VP. Find the horizontal displacement u: [0, T]→ E and the vertical displacement

˜

u: [0, T]→V such that u(0) =u₀,u_t(0) =v₀,u(0) = ˜˜ u₀,u˜_t(0) = ˜v₀ and for a.e. t∈(0, T),

u_tt(t), wEE+a₁(u_x(t), w_x) +c₁(u_xt(t), w_x) +j_H(u(t), w) = (f_H(t), w), ∀w∈E, (13)

u˜_tt(t),(z−u˜_t(t))VV + (a₂u˜_xx(t) +c₂u˜_xxt(t),(z−u˜_t(t))_xx)

+j_V(u(t), z)−j_V(u(t),u˜_t(t))≥(f_V(t),(z−u˜_t(t))), ∀z∈V. (14) The existence of a unique solution to problem VP was proved in[1]. We recall the main result in the following.

Theorem 1.2. Let the assumption (5) hold and assume that u₀ ∈ E, u˜₀ ∈ V and v₀,v˜₀ ∈H. Then, there exists a unique solution {u,˜u}to problem VP withu∈W^1,2(0, T;E),u_tt∈L²(0, T;E),u˜∈W^1,2(0, T;V)and

˜

u_tt∈L²(0, T;V).

2. Numerical approximation

In this section we describe a fully discrete scheme for the variational problem VP. We discretize it in two steps, both the spatial and time variables. We introduce a uniform partition of the time interval with the step size k=T /N and the nodest_n =nk, n= 1, . . . , N. To discretize the spatial variable, we consider a uniform partition of [0, L], denoted by {I_i}^M_i=1, in such a way that [0, L] = ∪^M_i=1I_i. We let h denote the size of the partition. LetV^h andE^h be the following finite element spaces approximatingV andE,

E^h={w^h∈E; w^h_|

Ii∈P₁(I_i), 1≤i≤M}, V^h={z^h∈V ; z_|^h

Ii ∈P₃(I_i), 1≤i≤M},

where P_q(I_i) denotes the polynomial space of degree less or equal to q restricted to I_i. We note that, since z^h∈V ⊂H²(0, L),z^h∈ C¹([0, L]) and thenV^h is composed ofC¹ piecewise cubic functions, whileE^his made of continuous piecewise affine functions.

We use the notation z_n =z(t_n) for a continuous function z(t), and for a sequence {z_n}^N_n=0 we denote by δz_n = (z_n−z_n−1)/kthe divided differences. Moreover, in this section no summation is assumed over a repeated index andc denotes a generic constant which does not depend onk, horn.

For convenience, we will consider our variational problem in terms of the velocity fieldsv(t) = ˙u(t),˜v(t) = ˙˜u(t) given by

u(t) = _t

0 v(s)ds+u₀, u(t) =˜ _t

0 v(s)ds˜ + ˜u₀. (15)

Then, using an Euler scheme to discretize the time derivatives, we introduce the following fully discrete approximation of problem VP.

Fully discrete problem VP^hk. Findv^hk ={v_n^hk}^N_n=0⊂E^handv˜^hk ={˜v_n^hk}^N_n=0⊂V^h, such thatu^hk₀ =u^h₀, v₀^hk=v^h₀,u˜^hk₀ = ˜u^h₀,˜v₀^hk= ˜v^h₀ and, forn= 1, . . . , N,

(δv_n^hk, w^h) +a₁((u^hk_n )_x, w_x^h) +c₁((v_n^hk)_x, w^h_x) +j_H(u^hk_n−1, w^h) = ((f_H)_n, w^h), ∀w^h∈E^h, (16)

(δ˜v_n^hk, z^h−˜v^hk_n ) +a₂((˜u^hk_n )_xx,(z^h−v˜_n^hk)_xx) +c₂((˜v_n^hk)_xx,(z^h−˜v^hk_n )_xx)

+j_V(u^hk_n , z^h)−j_V(u^hk_n ,v˜^hk_n )≥((f_V)_n, z^h−˜v^hk_n ), ∀z^h∈V^h, (17) whereu^h₀,v^h₀,u˜^h₀ andv˜₀^hare appropriate approximations of the initial conditions u₀,v₀,u˜₀ andv˜₀, respectively.

Moreover, u^hk ={u^hk_n }^N_n=0 andu˜^hk ={u˜^hk_n }^N_n=0 denote the displacement fields defined by u^hk_n =u^hk_n−1+kv^hk_n , u˜^hk_n = ˜u^hk_n−1+k˜v_n^hk, n= 1, . . . , N.

By induction, using classical results on variational inequalities (see[13]), we obtain that problem VP^hk admits a unique solution.

Our interest lies in estimating the errors u_n−u^hk_n and ˜u_n −u˜^hk_n . To that end, we make the following assumptions on the regularity of the solution,

u∈ C¹([0, T];E), u¨∈ C([0, T];H), (18)

˜

u∈ C¹([0, T];V), u¨˜∈ C([0, T];H). (19) The numerical analysis corresponding to the horizontal problem was done in [4], where the damage of the material and the thermal effects were also taken into account. Using the notation

I_j = _tj

0 v(s)ds− j l=1

kv_l

E, the following error estimate was obtained.

Theorem 2.1. Let the assumptions of Theorem 1.2 hold. Under the regularity condition (18), the following error estimates are obtained forn= 0, . . . , N,

|v_n−v_n^hk|²_H+ n j=1

k|v_j−v_j^hk|²_E ≤c n j=1

k

|v˙_j−δv_j|²_H+I_j²+ j−1

l=1

k|v_l−v^hk_l |²_E+|u_j−u_j−1|²_E+|v_j−w^h_j|²_E

+c |u₁−u^h₁|²_H+|u₀−u^h₀|²_E+|v₁−w₁^h|²_H+|v_n−w^h_n|²_H +ck⁻¹

n−1

j=1

|v_j−w^h_j − v_j+1−w^h_j+1

|²_H,

∀{w^h_j}ⁿ_j=0⊂V^h. (20) Next, we letz= ˜v_n^hk in (14) at timet=t_n to obtain, after a simple rearrangement,

( ˙˜v_n,v˜_n−˜v^hk_n ) +c₂((˜v_n)_xx,(˜v_n−v˜_n^hk)_xx) +a₂((˜u_n)_xx,(˜v_n−˜v^hk_n )_xx)≤

j_V(u_n,v˜_n^hk)−j_V(u_n,˜v_n) + ((f_V)_n,v˜_n−v˜_n^hk), and the discrete variational inequality (17) is rewritten as follows,

−(δ˜v^hk_n ,v˜_n−˜v_n^hk)−c₂((˜v^hk_n )_xx,(˜v_n−v˜_n^hk)_xx)−a₂((˜u^hk_n )_xx,(˜v_n−˜v_n^hk)_xx)≤

j_V(u^hk_n , z^h_n)−j_V(u^hk_n ,v˜_n^hk)−((f_V)_n, z_n^h−˜v^hk_n )−(δ˜v_n^hk,˜v_n−z_n^h)

−c₂((˜v_n^hk)_xx,(˜v_n−z^h_n)_xx)−a₂((˜u^hk_n )_xx,(˜v_n−z^h_n)_xx), for allz^h_n∈V^h. Adding these two inequalities, we obtain, for allz_n^h∈V^h,

( ˙˜v_n−δ˜v_n^hk,˜v_n−v˜_n^hk) +c₂((˜v_n−˜v_n^hk)_xx,(˜v_n−v˜_n^hk)_xx) +a₂((˜u_n−u˜^hk_n )_xx,(˜v_n−˜v_n^hk)_xx)≤

j_V(u_n,v˜^hk_n )−j_V(u_n,v˜_n) +j_V(u^hk_n , z^h_n)−j_V(u^hk_n ,˜v_n^hk)−(δ˜v_n^hk,˜v_n−z^h_n) + ((f_V)_n,˜v_n−z_n^h)

−c₂((˜v_n^hk)_xx,(˜v_n−z^h_n)_xx)−a₂((˜u^hk_n )_xx,(˜v_n−z^h_n)_xx).

Now, let us define

R(u_n,u˜_n, z_n^h) = ( ˙˜v_n, z_n^h−˜v_n) +c₂((˜v_n)_xx,(z_n^h−v˜_n)_xx)

+a₂((˜u_n)_xx,(z^h_n−v˜_n)_xx) +j_V(u_n, z^h_n)−j_V(u_n,˜v_n)−((f_V)_n, z_n^h−˜v_n).

Writing ˙˜v_n =δ˜v_n−(δ˜v_n−v˙˜_n), after easy manipulations we get

(δ˜v_n−δ˜v_n^hk,v˜_n−˜v^hk_n ) +c₂((˜v_n−v˜^hk_n )_xx,(˜v_n−˜v^hk_n )_xx)

≤j_V(u_n,˜v_n^hk)−j_V(u_n, z^h_n) +j_V(u^hk_n , z_n^h)−j_V(u^hk_n ,v˜^hk_n )

+c₂((˜v_n−˜v^hk_n )_xx,(˜v_n−z_n^h)_xx) +a₂((˜u_n−u˜^hk_n )_xx,(˜v_n^hk−z_n^h)_xx) +( ˙˜v_n−δ˜v_n,˜v_n^hk−z_n^h) +R(u_n,u˜_n, z_n^h) + (δ˜v_n−δ˜v_n^hk,˜v_n−z_n^h).

From the definition ofj_V, it follows that

j_V(u_n,˜v^hk_n )−j_V(u_n, z^h_n) +j_V(u^hk_n , z_n^h)−j_V(u^hk_n ,v˜_n^hk)≤c|un−u^hk_n |E(|˜v_n−˜v_n^hk|V +|˜v_n−z^h_n|V).

Since

(δ˜v_n−δ˜v_n^hk,˜v_n−˜v^hk_n )≥ 1 2k

|˜v_n−˜v^hk_n |²_H− |˜v_n−1−v˜^hk_n−1|²_H ,

using the Cauchy’s inequalityab≤a²+₄¹b² fora, b, >0, the following estimate is obtained,

|˜v_n−v˜_n^hk|²_H+ck|˜v_n−v˜^hk_n |²_V ≤ck

|u_n−u^hk_n |²_E+|v˙˜_n−δ˜v_n|²_H+|˜v_n−z_n^h|²_V

+|˜u_n−u˜^hk_n |²_V + (δ˜v_n−δ˜v_n^hk,˜v_n−z^h_n) +|R(u_n,u˜_n, z_n^h)|

+|˜v_n−1−˜v^hk_n−1|²_H. Then, an induction argument leads to

|˜v_n−v˜_n^hk|²_H+k n j=1

|˜v_j−˜v^hk_j |²_V ≤ck n j=1

|u_j−u^hk_j |²_E+|v˙˜_j−δ˜v_j|²_H+|˜v_j−z^h_j|²_V

+|˜u_j−u˜^hk_j |²_V + (δ˜v_j−δ˜v^hk_j ,v˜_j−z_j^h) +|R(u_j,u˜_j, z_j^h)|

+c|˜v₀−˜v₀^h|²_H.

Keeping in mind that n

j=1

k(δ˜v_j−δ˜v^hk_j ,v˜_j−z_j^h) = (˜v_n−˜v_n^hk,˜v_n−z_n^h) + (˜v^h₀ −v˜₀,v˜₁−z₁^h) +

n−1

j=1

(˜v_j−˜v_j^hk,˜v_j−z_j^h−(˜v_j+1−z_j+1^h )),

and

|˜u_j−˜u^hk_j |²_V ≤c(|˜u₀−u˜^h₀|²_V + ˜I_j²+ j

l=1

k|˜v_l−˜v^hk_l |²_V), (21)

|u_j−u^hk_j |²_E≤c(|u₀−u^h₀|²_E+I_j²+ j l=1

k|v_l−v_l^hk|²_E), (22)

where ˜I_j=_tj

0 ˜v(s)ds−_j

l=1k˜v_l

V is the integration error, the following estimate is obtained:

|v˜_n−v˜_n^hk|²_H+k n j=1

|˜v_j−˜v^hk_j |²_V ≤ck n j=1

_l

k=1

k|v_l−v_l^hk|²_E+I_j²+|v˙˜_j−δ˜v_j|²_H

+|˜v_j−z_j^h|²_V + ˜I_j²+ j

l=1

k|˜v_l−˜v_l^hk|²_V +|R u_j,u˜_j, z_j^h

|

+c|u˜₀−u˜^h₀|²_V +c|v˜₀−˜v^h₀|²_H

+c|˜v₁−z₁^h|²_H+c|˜v_n−z_n^h|²_H+c|u0−u^h₀|²_E+ck⁻¹

n−1

j=1

|˜v_j−z_j^h− v˜_j+1−z_j+1^h

|²_H. (23)

Adding now (20) and (23), we get

|v_n−v^hk_n |²_H+k n j=1

|v_j−v_j^hk|²_E+|v˜_n−˜v^hk_n |²_H+k n j=1

|˜v_j−v˜^hk_j |²_V

≤c ⁿ

j=1

k

|v˙˜_j−δ˜v_j|²_H+|v˙_j−δv_j|²_H+|u_j−u_j−1|²_E+|v_j−w^h_j|²_E

+|˜v_j−z_j^h|²_V + ˜I_j²+I_j²+ j

l=1

k|˜v_l−v˜_l^hk|²_V + j l=1

k|v_l−v^hk_l |²_E +|R(uj,u˜_j, z_j^h)|

+|v0−v₀^h|²_H+|˜v₀−˜v₀^h|²_H+|u0−u^h₀|²_E+|˜u₀−u˜^h₀|²_V +|v₁−w^h₁|²_H+|˜v₁−z₁^h|²_H+|v_n−w^h_n|²_H+|˜v_n−z_n^h|²_H

+k⁻¹

n−1

j=1

|v_j−w_j^h−(v_j+1−w_j+1^h )|²_H+k⁻¹

n−1

j=1

|˜v_j−z_j^h−(˜v_j+1−z^h_j+1)|²_H

. (24)

Let us define the following quantities forn= 1, . . . , N,

e_n=|v_n−v_n^hk|²_H+k n j=1

|v_j−v^hk_j |²_E+|˜v_n−v˜_n^hk|²_H+k n j=1

|˜v_j−˜v_j^hk|²_V,

g_n= n j=1

k

|v˙˜_j−δ˜v_j|²_H+|v˙_j−δv_j|²_H+|u_j−u_j−1|²_E+|v_j−w^h_j|²_E

+|˜v_j−z^h_j|²_V + ˜I_j²+I_j²+|R u_j,˜u_j, z_j^h

|

+|v₀−v^h₀|²_H+|˜v₀−v˜₀^h|²_H +|u₀−u^h₀|²_E+|u˜₀−u˜^h₀|²_V +|v₁−w^h₁|²_H+|˜v₁−z₁^h|²_H

+|vn−w^h_n|²_H+|˜v_n−z^h_n|²_H+k⁻¹

n−1

j=1

|vj−w_j^h− v_j+1−w_j+1^h

|²_H

+k⁻¹

n−1

j=1

|˜v_j−z_j^h− ˜v_j+1−z_j+1^h

|²_H,

and

e₀=g₀=|v₀−v₀^h|²_H+|˜v₀−˜v^h₀|²_H. Therefore, estimate (24) implies that

e₀≤g₀ and e_n ≤cg_n+c n j=1

k_je_j, n= 1, . . . , N,

wherek_j =k,j= 1, . . . , N. Then, we use the following discrete version of the Gronwall’s inequality (see[14]).

Lemma 2.2. Assume that{g_n}^N_n=0,{e_n}^N_n=0and{k_n}^N_n=1are three sequences of nonnegative numbers satisfying

e₀≤g₀, e_n≤cg_n+c

n j=1

k_je_j, n= 1, . . . , N,

wherec is a positive constant. Then, there exists a positive constant d >0such that

0≤n≤Nmax e_n≤d max

0≤n≤Ng_n. Applying Lemma 2.2, we obtain the following result.

Theorem 2.3. Let the assumptions of Theorem 1.2 hold true. Let us assume the additional regularity conditions (18)–(19). Then, the following error estimates are obtained for all {w^h_j}^N_j=0⊂E^h,{z_j^h}^N_j=0⊂V^h,

0≤n≤Nmax

|v_n−v^hk_n |²_H+|˜v_n−v˜_n^hk|²_H +k

N j=1

|v_j−v^hk_j |²_E+|˜v_j−v˜_j^hk|²_V

≤c ^N

j=1

k

|v˙_j−δv_j|²_H+|v˙˜_j−δ˜v_j|²_H+|uj−u_j−1|²_E+|vj−w^h_j|²_E

+|˜v_j−z_j^h|²_V + ˜I_j²+I_j²+|R(u_j,u˜_j, z_j^h)|

+|v₀−v^h₀|²_H+|˜v₀−v˜₀^h|²_H +k⁻¹

N−1 j=1

|v_j−w^h_j −(v_j+1−w^h_j+1)|²_H+k⁻¹

N−1 j=1

|˜v_j−z_j^h−(˜v_j+1−z^h_j+1)|²_H +|u₀−u^h₀|²_E+|˜u₀−˜u^h₀|²_V + max

0≤n≤N|v_n−w_n^h|²_H+ max

0≤n≤N|˜v_n−z_n^h|²_H

. (25)

The inequality (25) is the basis for deriving the convergence rate. Let Π^hand ˜Π^hdenote the standard Lagrange and Hermite interpolation operators associated to the finite element spaces E^h or V^h (see[9]), respectively.

Assume that the initial conditionsu₀, v₀,u˜₀ and ˜v₀ are such that

u₀∈H²(0, L), v₀∈H¹(0, L), u˜₀∈H³(0, L), ˜v₀∈H¹(0, L), and define the discrete initial conditions by

u^h₀ = Π^hu₀, v^h₀ = Π^hv₀, u˜^h₀ = ˜Π^h˜u₀, v˜₀^h= Π^h˜v₀. Then, we have the first error estimate.