On the finite element method for hyperbolic-parabolic ...

On the finite element method for hyperbolic-parabolic equations with nonlinearity of Kirchhoff-Carrier type with moving boundary M. Mbehou Department of Mathematical Sciences, University of South Africa, Pretoria 0003, South Africa. Department of Mathematics, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon. Email: [email protected]/[email protected]

Abstract This paper is devoted to the analysis of the finite element method for the mixed problem for the Kirchhoff nonlinear model given by the hyperbolic-parabolic equations with moving boundaries. With the use of the coordinate transformation which fixes the boundaries, the semidiscrete formulation is presented and the convergence and error bounds in the energy norm and for the first time derivative in the L2 -norm are established. In particular, the error in the energy norm and for the first time derivative in the L2 -norm is shown to converge with the optimal order O(hr ) with respect to the mesh size h and the polynomial degree r ≥ 1. To obtain the fully discrete solution, the generalized-α method is adapted to the semidiscrete formulation . Finally, numerical simulations that validate the theoretical findings are exhibited. Keywords: Kirchhoff model, moving boundaries, optimal error estimate, Newmark schemes, generalized-α method, Galerkin finite element method. AMS Subject Classification: 65N30

1

Introduction

Moving boundary problems occur in many physical applications involving diffusion, like in heat transfer where a phase transition occurs; in moisture transport, such as swelling grains or polymers; and in deformable porous media problems where solid displacement is governed by diffusion. These problems have been extensively studied by several authors such as Santos [1], Benabidallah [2], Briozzo [3] and Ferreira and Lar’kin [4]. The Kirchhoff’s time-retarded potential boundary integral equation has been used in the numerical calculation of the transient response in an enclosure with rigid or absorbent boundary. The sound pulse reflected from the absorbent boundary, consisted of a porous layer on a rigid backing which is calculated by using the time-dependent ”impulse admittance”. Kirchhoff equations can also be used as string vibration models that take into account tension growth due to string extension as mentioned in [5, 6, 7, 8] and references therein. 1

In this paper, we study the following Kirchhoff-Carrier model for the hyperbolic-parabolic equations  R 2 2   (ρ1 ut )t + ρ2 ut − (1 + M (t, Ωt |Du| dx))D u = f (x, t) in Qt  (1.1) u(α(t), t) = u(β(t), t) = 0 on (0, T )    u(x, 0) = u (x) u (x, 0) = u (x) in Ω = (α(0), β(0)), 0 t 1 0

where Qt is a bounded noncylindrical domain defined by Qt = {(x, t) ∈ R2 : α(t) < x < β(t), Ωt = {x ∈ R : α(t) < x < β(t),

for all

0 < t < T },

0 < t < T }.

ρ1 (.) and ρ2 (.) are two positive functions. α(.) and β(.) are two functions such that ∂k α(t) < β(t) for all t ∈ [0, T ]. D k (k ∈ N) denotes the differential operator ∂z k (z is a generic one spatial dimensional variable). In [9], Bisognin proved the existence of local solution of (1.1) in a bounded or unbounded domain of Rn . The existence of global solutions to problem (1.1) with analytic initial data was firstly investigated by Pokhozhaev [10] and Arosio and Spagnolo [11]. The behavior as t tends to ∞ of solutions to problem (1.1) in a cylinder with some modifications was studied by Nishihara [12]. Benabidallah and Ferreira [2] investigated the global existence, uniqueness and asymptotic behavior of regular solutions to problem (1.1). In addition to that, Cavalcanti et al. [13] extended the results obtained by Benabidallah and Ferreira [2] to the case where the length γ(t) = β(t) − α(t) tends to ∞. The finite element method has been widely studied and applied to nonlinear hyperbolic or parabolic problems in fixed domains as discussed [14, 15, 16, 17, 18, 19]. However, the analysis and implementations of numerical methods for Kirchhoff equations and moving boundary problems are quite few. We can refer to the works of Baines et al. [20] and Boffi and Gastaldi [21]. In [22], Robalo et al. applied a moving mesh method to a nonlocal parabolic equation with moving boundaries while Almeida et al. [23] presented the convergence analysis of the fully discretized in the finite element method in space variables and the Crank-Nicolson method in time variables for the same problems. The study of Liu and Rincon in [8] which involves problem (1.1) with ρ2 = 0 is solved by means of the explicit Lax-Friedrichs difference scheme and the stability of this scheme is studied assuming that the nonlinear term is neglected. Peradze [6] however, applied the Galerkin method in space variables and Crank-Nicolson difference scheme in time variables to the nonlinear Kirchhoff string equation   Z 2 π 2 utt − λ + (Du(x, t)) dx D 2 u(x, t) = 0. π 0 Hence, the goal of this research article is firstly, to use the coordinate transformation to fix the boundaries, then propose and analyze the Galerkin finite element method in the fixed domain problem. In particular, the optimal a priori error bounds in the energy norm and 2

for the first time derivative in the L2 -norm shall be derived. Secondly, by the help of the generalized-α method, the fully discrete scheme is obtained. The main interested shall be focused on the numerical aspects of the model and these results are new for the nonlinear Kirchhoff models with moving boundaries. The plan of this paper is organized as follows. Section 2 is concerned with the weak variational formulation of the problem and the hypotheses on the data. In Section 3, we propose and analyze the semidiscrete solution and its convergence while in Section 4, we present the fully discrete scheme by the mean of the generalized-α method. Finally numerical results are presented in Section 5 to demonstrate the theoretical analysis.

2

Preliminaries and weak formulation

We consider problem (1.1) in which we assume that the real functions ρ1 (·), ρ1 (·), α(·), β(·) and M (·, ·) satisfy the following conditions:    M ∈ C 1 ([0, ∞) × [0, ∞)),   (2.1) 0 < m0 ≤ M (t, λ) ≤ m1 for all (t, λ) ∈ [0, T ] × [0, ∞),    M (t, λ) ≤ 0 for all (t, λ) ∈ [0, T ] × [0, ∞). t

 ρ ∈ W 2,∞ (0, T ), ρ ∈ W 1,∞ (0, T ), 1 2 ρ (t) − 1 |ρ′ (t)| ≥ δ > 0 for all t ∈ [0, ∞), 2

2

(2.2)

0

1

   α, β ∈ C 3 ([0, T ]), γmin ≤ γ = β − α ≤ γmax   α′ (t) < 0, β ′ (t) > 0, for all t ∈ [0, T ],    (α′ (t) + γ ′ (t)y)2 < m0 for all (y, t) ∈ [0, 1] × [0, T ],

(2.3)

2

where m0 , m1 , γmin , γmax and δ0 are given positive numbers. Benabidallah and Ferreira in [2] established the existence, uniqueness and asymptotic behavior of strong solutions of problem (1.1) using the coordinate transformation which fixes the boundaries. That is, problem (1.1) is transformed into an equivalent problem in a fixed rectangular domain Q = (0, 1) × (0, T ), using the change of variables (x, t) ∈ Qt 7→ (y, t) ∈ Q,

y=

x − α(t) . γ(t)

The change of variables u(x, t) = v(y, t) and f (x, t) = g(y, t) with x = α(t) + γ(t)y transforms (1.1) into the following problem:   ˜ (t, 1 kDvk2 ))D 2 v − D(aDv) + bDvt + cDv = g(y, t) in Q  (ρ v ) + ρ2 vt − γ12 (1 + M  γ  1 t t v(0, t) = v(1, t) = 0 on (0, T )    v(y, 0) = v0 (y) vt (y, 0) = v1 (y) in

(0, 1),

(2.4) 3

where v0 (y) = u0 (α(0) + γ(0)y) and v1 (y) = u1 (α(0) + γ(0)y). The coefficients a b, c and ˜ are defined by M  ′    ′ α + γ′y 2 α + γ′y m0 , b(y, t) = −2ρ1 , a(y, t) = 2 − 2γ γ γ   α′′ + γ ′′ y α′ + γ ′ y ˜ (t, λ) = M (t, λ) − m0 . c(y, t) = − ρ1 + (ρ′1 + ρ2 ) , M γ γ 2 We have the following result which can be found in [2]. Theorem 2.1 Let Ω = (0, 1). We assume that conditions (2.1)-(2.3) hold. Given v0 ∈ H02 (Ω),

v1 ∈ H01 (Ω)

and

g ∈ L2 (0, T ; H01 (Ω)),

then there exists T ∗ > 0 such that problem (2.4) admits in [0, T ∗ ] a unique solution v : Q → R satisfying the following conditions: v ∈ L∞ (0, T ∗ , H01 (Ω) ∩ H 2 (Ω)), vt ∈ L2 (0, T ∗ , H01 (Ω)), √ ρ1 vt ∈ L∞ (0, T ∗ , H 1 (Ω)), (ρ1 vt )t ∈ L2 (0, T ∗ , L2 (Ω)), and u satisfies (2.4) in the sense of L2 (0, T ∗ , L2 (Ω)). Our numerical analysis will be based on the equivalent problem (2.4) in the domain Q = (0, 1) × (0, T ). In what follows, without loss of generality, we assume that ρ1 and ρ2 are constants. From Theorem 2.1, we have the following definition of the weak formulation. Definition 2.1 A function v is said to be a weak solution of problem (2.4) if: v ∈ L∞ (0, T ∗ , H01 (Ω) ∩ H 2 (Ω)), vt ∈ L2 (0, T ∗ , H01 (Ω)), vtt ∈ L2 (0, T ∗ , L2 (Ω)), (2.5) Z Z Z Z 1 ˜ (t, 1 kDvk2 )) vt φdy + 2 (1 + M vtt φdy + ρ2 ρ1 aDvDφdy DvDφdy + γ γ Ω Ω Ω Ω Z Z Z gφdy ∀ φ ∈ H01 (Ω), t ∈ (0, T ), (2.6) cDvφdy = + bDvt φdy + Ω

v(y, 0) = v0 (y)

Ω

Ω

vt (y, 0) = v1 (y)

in

Ω.

(2.7)

(2.6) must be understood as an equality in D ′ (Ω). For the numerical analysis, besides the hypotheses above, we will also assume that the following conditions are satisfied |M (t, λ1 ) − M (t, λ2 )| ≤ Clip |λ1 − λ2 | ∀λ1 , λ2 ∈ [0, ∞) d M (t, h(t)) ≤ 0 for all (t, h(t)) ∈ [0, T ] × [0, ∞), with h ∈ C 1 ([0, T ]). dt For all r ≥ 1,

(2.8) (2.9)

kv0 kH r+1 + kvkL∞ (H r+1 (Ω)) + kvt kL2 (H r+1 (Ω)) + kvtt kL2 (H 1 (Ω)) + kvttt kL2 (L2 (Ω)) ≤ C, (2.10) where kwkLp (X) = kwkLp (0,T,X) for 1 < p ≤ ∞. 4

3

Spatial discretization

We define Th as a partition of Ω into finite disjoint intervals Ti , i = 1, ..., nt . We denote h = max{diam(Ti ), i = 1, ..., nt }. Let Vh be the finite dimensional subspace of H01 (Ω), which consists of continuous piecewise polynomials of degree r ≥ 1 on Th . Let Πh be an interpolation operator and Rh : H01 (Ω) → Vh be a Ritz projection operator defined by Z Ω

∀w ∈ H01 (Ω).

D(u − Rh u)Dwdx = 0

(3.1)

Then we have the following Lemma. Lemma 3.1 ([14]) If u ∈ H r+1 (Ω) ∩ H01 (Ω), then ku − Πh uk + hkD(u − Πh u)k ≤ Chr+1 kukH r+1

ku − Rh uk + hkD(u − Rh u)k ≤ Chr+1 kukH r+1 ,

(3.2) (3.3)

where C is a positive constant which does not depend on h and r. To simplify the notations, we will denote (wh )t = w˙h , (wh )tt = w¨h . The semidiscrete problem associated to (2.6) is given by finding vh ∈ Vh , for t ≥ 0, such that Z Z Z Z 1 1 2 ˜ v˙h φh dy + 2 (1 + M (t, kDvh k )) v¨h φh dy + ρ2 ρ1 aDvh Dφh dy Dvh Dφh dy + γ γ Ω Ω Z ZΩ ZΩ gφh dy, ∀ φh ∈ Vh , t ∈ (0, T ) (3.4) cDvh φh dy = + bD v˙h φh dy + Ω

vh (y, 0) = Πh v0

Ω

Ω

v˙ h (y, 0) = Πh v1 .

(3.5)

We have the following Lemma. Lemma 3.2 The semidiscrete problem (3.4) admits a unique solution vh ∈ Vh and kv˙h (t)k + kDvh (t)k ≤ C,

for all

t ∈]0, T ]

(3.6)

where C is a generic constant which does not depend on h. Proof. If we take φh = v˙h in (3.4), one has 1 d 1 ˜ (t, 1 kDvh k2 )) d kDvh k2 (ρ1 kv˙h k2 ) + ρ2 kv˙h k2 + 2 (1 + M (3.7) 2 dt 2γ γ dt Z Z Z Z d 1 2 g v˙h dy. cDvh v˙h dy = bD v˙h v˙h dy + a |Dvh | dy + + 2 Ω dt Ω Ω Ω On the other hand, 1 ˜ 1 d 1 ˆ γ′ ˜ 1 1 1 2 d 2 2 kDv k ) kDv k = ( kDv k )) + M (t, M (t, kDvh k2 ) M (t, h h h 2γ 2 γ dt 2 dt γ γ 2γ 2 γ Z 1 kDvh k2 ′ γ 1 ˜ t (t, 1 kDvh k2 )ds + γ M ˜ (t, 1 kDvh k2 )kDvh k2 , M − 2γ 0 γ 2γ 3 γ 5

(3.8)

ˆ (t, λ) = where M

Rλ 0

˜ (t, s)ds, M

1 d d 1 γ′ 2 2 kDv k = ( kDv k ) + kDvh k2 h h 2γ 2 dt dt γ 2 γ3 Z Z Z d 1d 1 1 a |Dvh |2 dy = a|Dvh |2 dy − at |Dvh |2 dy 2 Ω dt 2 dt Ω 2 Ω Z γ′ bD v˙h v˙h dy = ρ1 kv˙ h k2 . γ Ω

(3.9) (3.10) (3.11)

Taking (3.8)-(3.11) into (3.7), we have   Z 1 1 1 d 1 ˆ 2 2 2 2 a|Dvh | dy + ρ1 kv˙h k + M (t, kDvh k ) + 2 kDvh k + 2 dt γ γ γ Ω Z 1 kDvh k2 γ γ′ ˜ 1 1 2 ˜ t (t, 1 kDvh k2 )ds + M (t, kDvh k ) − M 2 2γ γ 2γ 0 γ ′ ′ γ 1 γ′ γ ˜ 2 2 2 kDv k )kDv k + )kv˙ h k2 = I, M (t, kDv k + (ρ + ρ + 2 1 h h h 2γ 3 γ γ3 γ where I = ≤

Z Z Z 1 2 g v˙h dy cDvh v˙h dy + at |Dvh | dy − 2 Ω Ω Ω ρ kv˙h k2 + CkDvh k2 + Ckgk2 . 2 γ′

min Using the hypotheses (2.1), (2.9) and denoting ρ by ρ = ρ2 + ρ1 γmax , we get   Z 1 d 1 1 1 ˆ 2 2 2 2 a|Dvh | dy ≤ CkDvh k2 + Ckgk2 . ρ1 kv˙h k + M (t, kDvh k ) + 2 kDvh k + 2 dt γ γ γ Ω

Integrating the equation above with respect to t ∈ (0, s), for all s ∈ (0, T ), we obtain, with the aid of Gronwall’s lemma, the inequality kv˙h (s)k2 + kDvh (s)k2 ≤ C, where C = C(kv0 kL∞ (0,T,L2 (Ω)) , kv1 kL∞ (0,T,L2 (Ω)) , kgkL2 (0,T,L2 (Ω)) ). We now state and prove one of the the main result of this work about the convergence of the spatial discrete solution vh of (3.4). Theorem 3.1 If v is the solution of (2.6) and vh the solution of (3.4), then kvt − v˙h k2 + kDv − Dvh k2 ≤ Ch2r ,

for all

t ∈]0, T ],

where C does not depend on h. Proof. Let us write vh − v = (vh − Rh v) + (Rh v − v) = χh + η. By Lemma 3.1, we have kηt k + hkDηk ≤ Chr+1 (kvt kH r+1 + kvkH r+1 ). 6

(3.12)

To estimate kχ˙h k + kDχh k, from (2.6), (3.4) and using (3.1), we note that χh satisfies the following equations Z Z Z 1 1 2 ˜ Dχh Dφh dy χ˙h φh dy + 2 (1 + M (t, kDvh k )) χ¨h φh dy + ρ2 LHS ≡ ρ1 γ γ Ω Ω Ω Z Z Z cDχh φh dy bD χ˙h φh dy + + aDχh Dφh dy + Ω ΩZ Ω Z Z 1 ˜ 1 ˜ (t, 1 kDvh k2 )) = −ρ1 ηtt φh dy − ρ2 ηt φh dy + 2 (M DvDφh dy (t, kDvk2 ) − M γ γ γ Ω Ω Ω Z Z Z cDηφh dy ≡ RHS. bDηt φh dy − − aDηDφh dy − Ω

Ω

Ω

If we take φh = χ˙h , LHS = ρ1

Z

χ¨h χ˙h dy + ρ2

Z

2

χ˙h dy + I1 + I2 + I3 +

Ω

Ω

Z

cDχh χ˙h dy, Ω

where Z 1 ˜ 1 2 I1 = M (t, kDvh k ) Dχh D χ˙h dy γ2 γ Ω   1 ˜ 1 1 γ′ ˜ d 2 2 M (t, kDv k )kDχ k M (t, kDvh k2 )kDχh k2 + = h h 2 3 dt 2γ γ γ γ 1 d 1 ˜ (t, kDvh k2 ) − kDχh k2 M 2γ 2 dt γ Z Z Z d 1 1 aDχh D χ˙h dy = I2 = a|Dχh |2 dy − at |Dχh |2 dy dt 2 2 Ω Ω Ω Z Z ′ γ χ˙h 2 dy (by applying integration by parts). bD χ˙h χ˙h dy = ρ1 I3 = γ Ω Ω That is  Z 1 ˜ 1 2 2 2 LHS = a|Dχh | dy χ˙h dy + 2 M (t, kDvh k )kDχh k + ρ1 γ γ Ω Ω Z γ′ γ′ ˜ d ˜ 1 1 1 + (ρ2 + ρ1 ) χ˙h 2 dy + 3 M (t, kDvh k2 )kDχh k2 − 2 kDχh k2 M (t, kDvh k2 ) γ Ω γ γ 2γ dt γ Z Z 1 cDχh χ˙h dy at |Dχh |2 dy + − 2 Ω Ω 1 d 2 dt



Z

2

d ˜ M (t, γ1 kDvh k2 ) ≥ 0, we can droop it from Note that from hypothesis (2.9), − 2γ1 2 kDχh k2 dt γ′

′

LHS. If we denote by ρ = ρ2 + ρ1 γγmax and m = γ 3min m20 , using the lower bound of M , we min max have the following   Z Z 1 ˜ 1 1 d 2 2 2 2 a|Dχh | dy χ˙h dy + 2 M (t, kDvh k )kDχh k + LHS ≥ ρ1 2 dt γ γ Ω Ω Z Z Z 1 + ρ χ˙h 2 dy + mkDχh k2 − cDχh χ˙h dy. (3.13) at |Dχh |2 dy + 2 Ω Ω Ω RHS = −ρ1

Z

Ω

ηtt χ˙h dy − ρ2

Z

Ω

ηt χ˙h dy + J1 + J2 − 7

Z

Ω

bDηt χ˙h dy −

Z

cDη χ˙h dy Ω

where J1 = =

Z 1 1 ˜ 2 ˜ (t, 1 kDvh k2 )) DvD χ˙h dy kDvk ) − M ( M (t, γ2 γ γ Ω Z Z 1 ˜ 1 1 d 2 2 ˜ Dvt Dχh dy). (M (t, kDvk ) − M (t, kDvh k ))( DvDχh dy − γ2 γ γ dt Ω Ω Z Z Z d J2 = − aDηD χ˙h dy = − aDηDχh dy + (aDη)t Dχh dy dt Ω Ω Ω

If we apply the Lipschitz- continuity of M and Cauchy Schwartz inequalities, we have Z d DvDχh dy J1 ≤ C(kDvkL∞ (L2 (Ω)) , kDvh kL∞ (L2 (Ω)) )kDv − Dvh k dt Ω + C(kDvkL∞ (L2 (Ω)) , kDvh kL∞ (L2 (Ω)) , kDvt kL∞ (L2 (Ω)) )kDv − Dvh kkDχh k Z d J2 ≤ C(kDηk + kDηt k)kDχh k − aDηDχh dy. dt Ω Applying Cauchy-Schwartz and Young’s inequalities to RHS, we have Z Z Z ρ ρ ρ χ˙h 2 dy + Ckηt k2 + χ˙h 2 dy + CkDηt k2 + χ˙h 2 dy RHS ≤ Ckηtt k2 + 8 Ω 8 Ω 8 Ω Z m ρ 2 χ˙h 2 dy + kDχh k2 + CkDv − Dvh k2 (3.14) + CkDηk + 8 Ω 4 Z Z d m d + CkDv − Dvh k DvDχh dy + kDχh k2 − aDηDχh dy. dt Ω 4 dt Ω Also notice that

kDv − Dvh k2 ≤ C{kDηk2 + kDχh k2 }. Combining (3.13), (3.14) and taking the two last terms of (3.13) to the right hand side of (3.14),   Z Z 1 ˜ 1 1 d 2 2 2 2 a|Dχh | dy ≤ CkDχh k2 + C{kηtt k2 + χ˙h dy + 2 M (t, kDvh k )kDχh k + ρ1 2 dt γ γ Ω ΩZ Z d d 2 2 2 kηt k + kDηt k + kDηk } + CkDv − Dvh k DvDχh dy + aDηDχh dy. (3.15) dt dt Ω

Ω

Integrating (3.15) with respect to t ∈ [0, s] for s ∈ (0, T ], we have

p 1 (ρ1 kχ˙h (s)k2 + m2 kDχh (s)k2 + k a(s)Dχh (s)k2 ) ≤ C(kχ˙h (0)k2 + kDχh (0)k2 ) 2 Z T +C (kηtt k2 + kηt k2 + kDηt k2 + kDηk2 )dt + I p p p 0 p +k a(s)Dχh (s)kk a(s)Dη(s)k + k a(0)Dχh (0)kk a(0)Dη(0)k, (3.16)

d R Rs where m2 = 2γm2 0 and I = C 0 (kDv − Dvh k dt Ω DvDχh dy )dt. max Applying the Young’s inequality to the two last terms and bound a(0),

+C

Z

T 0

ρ1 kχ˙h (s)k2 + m2 kDχh (s)k2 ≤ C(kχ˙h (0)k2 + kDχh (0)k2 ) (kηtt k2 + kηt k2 + kDηt k2 + kDηk2 )dt + I + CkDη(s)k2 + CkDη(0)k2 . 8

(3.17)

To conclude the proof, we have to bound I. I

≤ CkDv − Dvh kL∞ (0,T,L2 (Ω)) kDvkL∞ (0,T,L2 (Ω)) kDχh kL∞ (0,T,L2 (Ω)) m2 ≤ C(kDvkL∞ (0,T,L2 (Ω)) )kDηk2L∞ (0,T,L2 (Ω)) + kDχh k2L∞ (0,T,L2 (Ω)) . 2

(3.18)

Taking (3.18) into (3.17) and use the fact that since the inequality holds for any s ∈ (0, T ], it also holds for the maximum over (0, T ], that is kχ˙h k2L∞ (0,T,L2 (Ω)) + kDχh k2L∞ (0,T,L2 (Ω)) ≤ C(kχ˙h (0)k2 + kDχh (0)k2 + kDη(0)k2 ) Z T +C (kηtt k2 + kηt k2 + kDηt k2 + kDηk2 )dt + CkDηk2L∞ (0,T,L2 (Ω)) . (3.19) 0

By Lemma 3.1, kχ˙h (0)k2 + h2 kDχh (0)k2 + h2 kDη(0)k2 ≤ C(kvt kH r+1 (Ω) , kvkH r+1 (Ω) )h2(r+1) . Z

T 0

(kηtt k2 + kηt k2 + h2 kDηt k2 + h2 kDηk2 )dt ≤ Ch2(r+1) ,

where the constant C = C(kvtt kL2 (H r+1 (Ω)) , kvt kL2 (H r+1 (Ω)) , kvkL2 (H r+1 (Ω)) ). kDηk2L∞ (0,T,L2 (Ω)) ≤ C(kvkL∞ (H r+1 (Ω)) )h2r . Hence kχ˙h k2L∞ (0,T,L2 (Ω)) + kDχh k2L∞ (0,T,L2 (Ω)) ≤ Ch2r Therefore adding the estimate of kηt k + kDηk, we obtain the desired result.

4

Numerical scheme

To obtain the fully discrete discretization of (2.6)-(2.7), we adapt to our spatial discretization (3.4)-(3.5) the second-order generalized-α scheme in time presented in [24, 25]. Let Np be the Lagrange basis of Vh , with Np the number of nodes. We can represent the (φk )k=1 solution vh ∈ Vh of the semidiscrete problem (3.4)-(3.5) as vh =

Np X

βk (t)φk .

k=1

Therefore (3.4)-(3.5) is equivalent to the following nonlinear second-order system of ordinary differential equations. ρ1 M V ′′ (t) + (ρ2 M + D b )V ′ (t) + A(V (t))V (t) + (Aa + Dc )V (t) = F (t)

(4.1)

with the initial conditions V (0) = v h0 ;

V ′ (0) = v h1 , 9

(4.2)

with V = (β1 , · · ·, βNp )T ; v h0 and v h1 are respectively the vector components of v0 and v1 in the basis (φi ). Z Z Z ωDφj Dφi dx ωDφj φi dx; (Aω )ij = φj φi dx; (D ω )ij = (M )ij = Ω

Ω

Ω

Np

X 1 ˜ (t, 1 k A(V (t))ij = 2 (1 + M βk (t)Dφk k2 )) γ (t) γ(t) k=1

Z

Dφj Dφi dx; F i = Ω

Z

gφi dx.

Ω

Let {tn | tn = nδ; 0 ≤ n ≤ N } be a uniform partition of [0, T ] with the time step δ = T /N . If we denote by V n , X n and W n the approximation of V (tn ), V ′ (tn ) and V ′′ (tn ) respectively, the ’adapted’ generalized-α scheme is given by: find W n , V n , X n ∈ RNp such that    ρ M W n+1−αm + CX n+1−αf + K(V n )V n+1−αf = F (tn+1−αf )   1 (4.3) V n+1 = V n + δX n + δ2 ((1/2 − µ)W n + µW n+1 )    X = X + δ((1 − λ)W + λW ), n+1

n

n

n+1

with the initial conditions  V = v h , X = v h 0 0 0 1 [ρ M + δλC + δ2 µK(V )]W = F (0) − CX − K(V )V , 1 0 0 0 0 0

(4.4)

where

W n+1−αm = (1 − αm )W n+1 + αm W n

(4.5)

X n+1−αf = (1 − αf )X n+1 + αf X n

(4.6)

V n+1−αf = (1 − αf )V n+1 + αf V n

(4.7)

tn+1−αf = (1 − αf )tn+1 + αf tn ,

(4.8)

and C = (ρ2 M + Db ), K(V n ) = (A(V n ) + Aa + D c ). In the above equations, λ, µ, αm and αf are the algorithm parameters related with the accuracy and the stability of the scheme. From Equations (4.3)-(4.7), W n+1 can be solved in terms of V n , X n and W n , that is W n+1 = D(V n )−1 [F (tn+1−αf ) − αm ρ1 M W n − CX pn+1 − K(V n )V pn+1 ],

(4.9)

where D(V n ) = (1 − αm )ρ1 M + (1 − αf )δλC + (1 − αf )δ2 µK(V n )

X pn+1 = X n + (1 − αf )(1 − λ)δW n

V pn+1 = V n + (1 − αf )δX n + (1 − αf )(1/2 − µ)δ2 W n .

Remark 4.1 The generalized-α method is the second-order accurate and is unconditionally stable provided the following relationships (see [24] for more details). 1 1 1 1 µ ≥ + (αm − αf ), λ = − αm + αf and αm ≤ αf ≤ (4.10) 4 2 2 2 Note that when αm = αf = 0, we have the Newmark family scheme and its analysis can be found for instance in [26]. 10

Implementation Combining equations (4.3)-(4.7) and (4.9), the time-integration algorithm can be implemented. It involves quantities X pn+1 and V pn+1 which are the predictor terms depending on the previous step n.

Algorithm Require: λ, µ, αm , αf Require: Initial conditions V 0 , X 0 W 0 ← (ρ1 M + δλC + δ2 µK(V 0 ))−1 (F (0) − CX 0 − K(V 0 )V 0 ) for n = 0 to N do X pn+1 ← X n + (1 − αf )(1 − λ)δW n V pn+1 ← V n + (1 − αf )δX n + (1 − αf )(1/2 − µ)δ2 W n W n+1 ← D(V n )−1 (F (tn+1−αf ) − αm ρ1 M W n − CX pn+1 − K(V n )V pn+1 ) V n+1 ← V n + δX n + δ2 ((1/2 − µ)W n + µW n+1 ) X n+1 ← X n + δ((1 − λ)W n + λW n+1 ) End for

5

Numerical simulation

In this section, we check through numerical simulations our theoretical analysis. All computations are performed using Matlab. We consider problem (1.1) in Qt = (α(t), β(t)) × (0, 1) with constant parameters ρ1 = ρ2 = 1 and α(t) = −

t , t+1

β(t) =

2t + 1 , t+1

1 M (t, s) = 5(γ(t) + 1 + s) 2

and u0 , u1 and f are chosen such that the analytical solution is given by u(x, t) =

1 sin(πΥ)cos(πt), π2

where Υ =

(t + 1)x + t . 3t + 1

In all our tests, the algorithm parameters are: λ = 21 , µ = 14 and αm = αf = 21 . Figure 1 shows the evolution of the spatial domain versus time (left) and illustrates the movement of the two boundaries α(t) and β(t). This domain has been choosing with respect to the hypothesis (2.3). Our goal in Figures 2 and 3 is to compare the exact solution and the approximate solution. Figure 2 (left) illustrates the evolution in time of the exact solution v(y, t) after applying the transformation τ from the moving domain Qt to the fixed domain Q = (0, 1) × (0, T ) while Figure 2 (right) gives the approximate solution vh (y, t) in the domain Q. This solution was calculated with quadratic finite element approximation, h = 10−3 and δ = 10−2 . The picture in Figure 3 represents the exact solution u(x, t) in the moving domain. 11

1 1.5

0.9

β 0.8 1

0.7

0.5

0.5

x

t

0.6

0.4 0.3

0 α

0.2 0.1 0

−0.5 −0.5

0

0.5 x

1

1.5

0

0.2

0.4

0.6

0.8

1

t

Figure 1: Nodal movement in spatial domain (left) and motion of the boundaries (right).

0.2

0.2

0.1 vh(y,t)

v(y,t)

0.1 0

0

−0.1

−0.1 −0.2 1

−0.2 1 1

1

0.8 0.5

0.8

0.5

0.6

0.6

0.4

0.4

0.2 t

0

0

0

t

y

0.2 0

y

Figure 2: Evolution in time of the exact solution v(y, t) (left) and the approximate solution vh (y, t) (right) in the fixed boundary problem.

0.2

U(t,x)

0.1 0 −0.1 −0.2 1

0.5

t

0

−1

−0.5

0.5

0

1

1.5

2

x

Figure 3: Evolution in time of the exact solution u(x, t) in the moving boundary problem. To analyze the convergence rates, we simulated the same problem with different values of h and r with δ = 10−3 and the error results are represented in Figure 4. The error has been calculated at t = 0.1. Figure 4 (left) shows the error estimate in the energy norm 12

and the error is in the form of O(hr ) which confirms our theoretical analysis. Figure 4 (right) shows the L2 -norm error estimate and the error is in the form of O(hr+1 ). Even though we do not prove the L2 -norm error estimate theoretically, we think that this result is in accordance with what we shall be expecting. Convergence for L2− norm

Convergence for energy norm norm −6.5

−7.5 err P1 err P2 slope 1 slope 2

−7 −7.5

−9 log(|uex−unh|0)

−8 log(|uex−unh|V)

err P1 err P2 slope 2 slope 3

−8 −8.5

−8.5 −9 −9.5

−9.5 −10 −10.5

−10

−11

−10.5

−11.5

−11

−12

−11.5

−12.5

−3.4

−3.2

−3

−2.8 log(h)

−2.6

−2.4

−2.2

−3.4

−3.2

−3

−2.8 log(h)

−2.6

−2.4

−2.2

Figure 4: Convergence rates: Energy norm (left) and L2 -norm (right).

Conclusion We have presented and analyzed the Galerkin finite element method for the mixed problem for the Kirchhoff nonlinear model given by the hyperbolic-parabolic equations with moving boundaries. We have carried out an a priori error analysis of the semidiscrete solution and derived optimal error estimate in the energy norm and for the first time derivative in the L2 -norm using sufficient conditions on the exact solution. We presented some numerical experiments on Matlab’s environment and our numerical results confirm the theoretical analysis. compress]ieeetr

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