Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications

DOI:10.1051/cocv/2012036 www.esaim-cocv.org

UNIFORMLY EXPONENTIALLY OR POLYNOMIALLY STABLE APPROXIMATIONS FOR SECOND ORDER EVOLUTION EQUATIONS

AND SOME APPLICATIONS

Farah Abdallah

, Serge Nicaise

, Julie Valein

and Ali Wehbe

Abstract.In this paper, we consider the approximation of second order evolution equations. It is well known that the approximated system by finite element or finite difference is not uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal is to damp the spurious high frequency modes by introducing numerical viscosity terms in the approximation scheme. With these viscosity terms, we show the exponential or polynomial decay of the discrete scheme when the continuous problem has such a decay and when the spectrum of the spatial operator associated with the undamped problem satisfies the generalized gap condition.

By using the Trotter–Kato Theorem, we further show the convergence of the discrete solution to the continuous one. Some illustrative examples are also presented.

Mathematics Subject Classification. 65M60, 35L05, 35L15.

Received December 21, 2011. Revised May 7, 2012.

Published online June 3, 2013.

1. Introduction and main results

Let H be a complex Hilbert space with norm and inner product denoted respectively by. and (., .). Let A : D(A) → H be a densely defined self-adjoint and positive operator with a compact inverse in H. Let V =D(A¹²) be the domain ofA¹².Denote byD(A¹²) the dual space ofD(A¹²) obtained by means of the inner product inH.

Furthermore, let U be a complex Hilbert space (which will be identified to its dual space) with norm and inner product denoted respectively by._U and (., .)_U and letB∈ L(U, H).We consider the closed loop system

¨

ω(t) +Aω(t) +BB^∗ω(t) = 0,˙

ω(0) =ω0,ω(0) =˙ ω1, (1.1)

Keywords and phrases.Stability, wave equation, numerical approximations.

1 Universit´e de Valenciennes et du Hainaut Cambr´esis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, 59313 Valenciennes Cedex 9, France. The Ph.D. studies of F. Abdallah are financed by the association Azm and Saad´e.Farah.Abdallah@meletu.univ-valenciennes.fr

2 Universit´e de Valenciennes et du Hainaut Cambr´esis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, 59313 Valenciennes Cedex 9, France.Serge.Nicaise@univ-valenciennes.fr

3 Institut Elie Cartan Nancy (IECN), Nancy-Universit´e & INRIA (Project-Team CORIDA), 54506 Vandoeuvre-l`es-Nancy Cedex France.Julie.Valein@iecn.u-nancy.fr

4 Universit´e Libanaise, Ecole Doctorale des Sciences et de Technologie, Hadath, Beyrouth, Liban.ali.wehbe@ul.edu.lb

Article published by EDP Sciences c EDP Sciences, SMAI 2013

where t∈[0,∞) represents the time,ω : [0,∞)→H is the state of the system. Most of the linear equations modeling the vibrations of elastic structures with feedback control (corresponding to collocated actuators and sensors) can be written in the form (1.1), whereω represents the displacement field.

We define the energy of system (1.1) at timet by E(t) = 1

2

ω(t)˙ ²+A¹²ω(t)²

. Simple formal calculations give

E(0)−E(t) = _t

0

(BB^∗ω(s),˙ ω(s)) ds,˙ ∀t≥0.

This obviously means that the energy is non-increasing.

In many applications, the system (1.1) is approximated by finite dimensional systems but usually if the continuous system is exponentially or polynomially stable, the discrete ones do no more inherit of this property due to spurious high frequency modes. Several remedies have been proposed and analyzed to overcome this difficulties. Let us quote the Tychonoff regularization [18,19,31,34], a bi-grid algorithm [16,28], a mixed finite element method [6,10,11,17,27], or filtering the high frequencies [22,25,35] (both methods providing good numerical results).

As in [31,34] our goal is to damp the spurious high frequency modes by introducing a numerical viscosity in the approximation schemes. Though our paper is inspired from [31], it differs from that paper on the following points:

(i) Contrary to [31] where the standard gap condition is required, we only assume that the spectrum of the operatorA^1/2 satisfies the generalized gap condition, allowing to treat more general concrete systems;

(ii) we analyze the polynomial decay of the discrete schemes when the continuous problem has such a decay;

(iii) we prove a result about uniform polynomial stability for a family of semigroups of operators;

(iv) by using a general version of the Trotter–Kato Theorem proved in [23], we show that the discrete solution tends to the solution of (1.1) as the discretization parameter goes to zero and if the discrete initial data are well chosen.

Before stating our main results, let us introduce some notations and assumptions.

We denote by._V the norm

ϕ_V =

(A¹²ϕ, A¹²ϕ),∀ϕ∈V.

Remark that

ϕ_V =

(Aϕ, ϕ),∀ϕ∈ D(A).

We now assume that (V_h)_h>0 is a sequence of finite dimensional subspaces ofD(A¹²). The inner product inV_h is the restriction of the inner product of H and it is still denoted by (., .) (sinceV_h can be seen as a subspace ofH). We define the operatorA_h : V_h→V_h by

(A_hϕ_h, ψ_h) = (A¹²ϕ_h, A¹²ψ_h),∀ϕ_h, ψ_h∈V_h. (1.2) Leta(., .) be the sesquilinear form on V_h×V_h defined by

a(ϕ_h, ψ_h) = (A¹²ϕ_h, A¹²ψ_h), ∀(ϕh, ψ_h)∈V_h×V_h. (1.3) We also define the operatorsB_h: U →V_hby

B_hu=j_hBu, ∀u∈U, (1.4)

wherej_his the orthogonal projection ofH into V_h with respect to the inner product inH.

The adjointB_h^∗ ofB_h is then given by the relation

B_h^∗ϕ_h=B^∗ϕ_h, ∀ϕ_h∈V_h.

We also suppose that the family of spaces (V_h)_h approximates the space V = D(A¹²). More precisely, if π_h denotes the orthogonal projection ofV =D(A¹²) ontoV_h, we suppose that there existθ >0, h^∗>0 andC0>0 such that, for allh∈(0, h^∗), we have:

π_hϕ−ϕ_V ≤C0h^θAϕ,∀ϕ∈ D(A), (1.5) π_hϕ−ϕ ≤C0h^2θAϕ,∀ϕ∈ D(A). (1.6) Assumptions (1.5) and (1.6) are, in particular, satisfied in the case of standard finite element approximations of Sobolev spaces.

Denote by{λ_k}_k≥1the set of eigenvalues ofA¹² counted with their multiplicities (i.e.we repeat the eigenvalues according to their multiplicities). We further rewrite the sequence of eigenvalues{λ_k}k≥1 as follows:

λ_k1 < λ_k2 < . . . < λ_ki< . . .

wherek1= 1, k2is the lowest index of the second distinct eigenvalue,k3is the lowest index of the third distinct eigenvalue, etc. For alli∈N^∗, letl_i be the multiplicity of the eigenvalueλ_ki,i.e.

λ_ki−1 < λ_ki =λ_ki+1=. . .=λ_ki+li−1< λ_ki+li=λ_ki+1.

We havek1= 1, k2= 1+l1, k3= 1+l1+l2,etc. Let{ϕ_ki+j}0≤j≤li−1be the orthonormal eigenvectors associated with the eigenvalueλ_ki.

Now, we assume that the following generalized gap condition holds:

∃M ∈N^∗,∃γ0>0,∀k≥1, λ_k+M −λ_k≥M γ0. (1.7) Fix a positive real numberγ₀ ≤γ0 and denote byA_k, k= 1, . . . , M the set of natural numbers k_msatisfying (see for instance [5]) ⎧

⎨

⎩

λ_km−λ_km−1 ≥γ₀

λ_kn−λ_kn−1 < γ₀ form+ 1≤n≤m+k−1, λ_km+k−λ_km+k−1 ≥γ₀.

Then one easily checks that

{km+j+l|km∈A_k, k∈ {1, . . . , M}, j∈ {0, . . . , k−1}, l∈ {0, . . . , lm+j−1}}=N^∗.

Notice that some setsA_k may be empty because, for the generalized gap condition, the choice ofM takes into account multiple eigenvalues. Fork_n∈A_k, we defineB_kn= (B_{kn, ij})1≤i, j≤k the matrix of sizek×kby

B_{kn, ij} =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

n+j−1

q=n+i−1q=n

(λ_kn+i−1−λ_kq)⁻¹if i≤j,(i, j)= (1,1),

1 if (i, j) = (1,1),

0 else.

More explicitly, we have

B_kn=

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎝

1 _λ ¹

kn−λ_kn+1 1

(λkn−λ_kn+1)(λkn−λ_kn+2) · · · _(λkn−λ ¹

kn+1)···(λkn−λ_kn+k−1)

0 _λ ¹

kn+1−λkn

1

(λ_kn+1−λkn)(λ_kn+1−λ_kn+2) · · · (λ_kn+1−λkn)···(¹λ_kn+1−λ_kn+k−1)

0 0 _(λ ¹

kn+2−λ_kn)(λ_kn+2−λ_kn+1) · · · _(λ ¹

kn+2−λ_kn)···(λ_kn+2−λ_kn+k−1)

... ... . .. ...

0 0 0 · · · _{(λkn+k−1−λkn)···(}¹_{λkn+k−1−λkn+k−2)}

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎠ .

Lemma 1.1. The inverse matrix of B_kn is given by

B⁻_kn¹_{, ij} =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

n+i−2 q=n

(λ_kn+j−1−λ_kq)if i≤j, i= 1,

1 if i= 1,

0 else,

that is to say

B_k⁻_n¹=

⎛

⎜⎜

⎜⎜

⎝

1 1 · · · 1

0 (λ_kn+1−λ_kn)· · · (λ_kn+k−1−λ_kn)

0 0 · · · (λ_kn+k−1−λ_kn)(λ_kn+k−1−λ_kn+1)

... ... . .. ...

0 0 · · · (λ_kn+k−1−λ_kn)· · ·(λ_kn+k−1−λ_kn+k−2)

⎞

⎟⎟

⎟⎟

⎠,

and therefore

B_k⁻_n¹→

⎛

⎜⎜

⎝

1 1· · · 1 0 0· · · 0 ... . .. ...

0 · · · 0

⎞

⎟⎟

⎠,when n→+∞.

Proof. The form of B_k⁻_n¹ is obtained by induction on the size k of B_kn. The generalized gap condition (1.7) implies thatλ_kn+j−λ_kn→0 asn→+∞,∀0≤j≤k−1.This leads to the convergence ofB⁻_kn¹.

Now, fork_n∈A_k, we define the matrixΦ_kn with coefficients inU and sizek×L_n, whereL_n= k i=1

l_n+i−1,as follows: for alli= 1, . . . , k, we set

(Φ_kn)_ij =

B^∗ϕ_kn+i−1+j−Ln, i−1−1ifL_{n, i−}1< j≤L_{n, i},

0 else,

where

L_n,0= 0, L_{n, i}= i i=1

l_n+i−1fori≥1. (1.8)

For a vectorc= (c_l)^m_l=1 in U^m,we setc_U,2 its norm inU^mdefined by c²_U,2=

m l=1

c_l²_U.

In this paper, we prove two results. The first result gives a necessary and sufficient condition to have the exponential stability of the family of systems

¨

ω_h(t) +A_hω_h(t) +B_hB^∗_hω˙_h(t) +h^θA_hω˙_h(t) = 0

ω_h(0) =ω0h∈V_h,ω˙_h(0) =ω1h∈V_h, (1.9) in the absence of the standard gap condition assumed in [31]. Here and belowω0h(resp.ω1h) is an approximation ofω0(resp.ω1) inV_h. For that purpose, we need to make the following assumption

∃α0>0,∀k∈ {1, . . . , M},∀k_n∈A_k,∀C∈R^Ln,B_k⁻¹

nΦ_knC

U,2≥α0C2, (1.10) where.₂is the euclidian norm. The first main result is the following

Theorem 1.2. Suppose that the generalized gap condition (1.7)and the assumption(1.10)are verified. Assume that the family of subspaces (V_h) satisfies(1.5)and (1.6). Then the family of systems (1.9)is uniformly expo- nentially stable, in the sense that there exist constantsM, α, h^∗>0 (independent ofh, ω0h, ω1h) such that for allh∈(0, h^∗):

ω˙_h(t)²+a(ω_h(t), ω_h(t))≤Me^−αt(ω1h²+a(ω0h, ω0h)),∀t≥0.

Remark 1.3. If the standard gap condition

∃γ0>0,∀n≥1, λ_kn+1−λ_kn ≥γ0 (1.11) holds, thenA1=N^∗ andB1= 1.In this case, the assumption (1.10) becomes

∃α0>0,∀k_n≥1,∀C∈R^Ln,Φ_knC_U ≥α0C2.

Moreover, if the standard gap condition (1.11) holds and if the eigenvalues are simple, the assumption (1.10) becomes

∃α0>0,∀k≥1,B^∗ϕ_kU ≥α0. (1.12) These assumptions are assumed in [31].

Remark 1.4. Note that Theorem1.2 is the discrete counterpart of the exponential decay of the solution of the continuous problem (1.1) under the assumptions (1.7) and (1.10), which follows Theorem 2.2 of [3] (see also [29]). Note that the assumption (H) from [3] here holds since A is a positive selfadjoint operator with a compact resolvent andB is bounded.

Remark 1.5. The uniform exponential stability of the family of systems (1.9) has been already proved in Theorem 7.1 of [14] without any assumption on the spectrum ofAand the dimension of the space. The proof of this theorem is based on decoupling of low and high frequencies. More precisely, the author combines a uniform observability estimate for filtered initial data corresponding to low frequencies (see [14], Thm. 1.3) together with a result of [15]. Indeed, in [15], after adding the numerical viscosity term, another uniform observability estimate is obtained for the high frequency components. The two established observability inequalities yield the uniform exponential decay of (1.9).

If the condition (1.10) is not satisfied, we may look at a weaker version. Namely if we assume that

∃l∈N^∗,∃α0>0,∀k∈ {1, . . . , M},∀kn ∈A_k, ∀C∈R^Ln,B_k⁻_n¹Φ_knC

U,2≥ α0

λ^l_k

n

C2, (1.13) then we will obtain a polynomial stability for the family of systems

¨

ω_h(t) +

1 +h^θ₋2

I+h^θA_h^2l 2

A_hω_h(t) +

I+h^θA_h^l2 B_hB_h^∗+h^θA¹⁺_h ^2l I+h^θA_h^2l ₋1

˙

ω_h(t) = 0, ω_h(0) =ω0h∈V_h,ω˙_h(0) =

1 +h^θ₋1

I+h^θA_h^l2

ω1h∈V_h.

(1.14)

The structure of the above discrete system has been inspired from the one introduced in [31] for the exponential stability case where the authors have used system (1.9) corresponding to l = 0. In both cases, this choice is motivated by the corresponding observability estimates. The numerical viscosity term (I+h^θA_h^2l)(B_hB^∗_h+ h^θA¹⁺_h ^l2)(I+h^θA_h^l2)⁻¹ω˙_h(t) is added to damp the high frequency modes and as the set of high frequency modes is larger in the polynomial case, the viscosity term is naturally stronger. In the casel≥2 the powers of (I+h^θA_h^l2) have been added to guarantee the boundedness of the resolvent of ˜A_l,h (defined below) near zero. The question of the optimality of these viscosity terms remains open.

The second main result of our paper is the following one.

Theorem 1.6. Suppose that the generalized gap condition (1.7) and the Assumption (1.13) are verified with l∈N^∗even. Assume that the family of subspaces(V_h)satisfies(1.5)and(1.6). Then the family of systems(1.14) is uniformly polynomially stable, in the sense that there exist constantsC, h^∗>0(independent of h, ω0h, ω1h) such that for all h∈(0, h^∗):

I+h^θA_h^2l −1

˙ ω_h(t)

²+a(ω_h(t), ω_h(t))≤ C

t²(ω0h, ω1h)²_D( ˜A^2l_l,h),

I+h^θA_h^2l ₋1

˙ ω_h(t)

²+a(ω_h(t), ω_h(t))≤ C

t^1l (ω0h, ω1h)²_D(A^˜l,h),∀t >0, ∀(ω0h, ω1h)∈V_h×V_h, where forq∈N^∗,._D( ˜A^q_l,h) is the graph norm of the matrix operatorA˜^q_l,h given in(4.1) of Section4 below.

For a technical reason, we assumel to be even (see Lem. 4.4). If (1.13) holds forl odd, then it is also true forl+ 1 and we can apply the previous result withl+ 1.

Remark 1.7. If the standard gap condition (1.11) holds, the Assumption (1.13) becomes

∃l∈N^∗,∃α0>0,∀kn ≥1,∀C∈R^Ln,ΦknC_U ≥ α0

λ^l_k

n

C2.

Moreover, if the standard gap condition (1.11) holds and if the eigenvalues are simple, the Assumption (1.13) becomes

∃l∈N^∗,∃α0>0,∀k≥1,B^∗ϕ_kU ≥ α0

λ^l_k· (1.15)

Remark 1.8. As before, Theorem1.6 is the discrete counterpart of the polynomial decay of the solution of the continuous problem (1.1) under the Assumptions (1.7) and (1.13), that follows from Theorem 2.4 of [3] (see also [29]).

The paper is organized as follows: In Section2, we show that the generalized gap condition and the observ- ability conditions (1.10) and (1.13) remain valid for filtered eigenvalues. Section3 first recalls a result about uniform exponential stability for a family of semigroup of operators, and then extends such a result to the case of uniform polynomial stability. Some technical lemmas are proved in Section4. Sections5and6are devoted to the proof of Theorem1.2and1.6respectively. In Section7, we show that the solutionω_h (resp.

I+h^θA_h^2l)⁻¹ω˙_h tends toω, the solution of (1.1), (resp. ˙ω) inV (resp. inH) ashgoes to zero and if the discrete initial data are well chosen. Finally, we illustrate our results by presenting different examples in Section8.

2. Spectral analysis of the discretized problem

The eigenvalue problem of the discretized problem is the following one: findλ_{k, h}∈]0,+∞[, ϕk, h ∈V_h,such that

a(ϕ_{k, h}, ψ_h) =λ²_{k, h}(ϕ_{k, h}, ψ_h),∀ψ_h∈V_h. (2.1) Let N(h) be the dimension ofV_h. We denote by {λ²_{k, h}}1≤k≤N(h) the set of eigenvalues of (2.1) counted with their multiplicities. Let{ϕk, h}1≤k≤N(h) be the orthonormal eigenvectors associated with the eigenvalueλ²_{k, h}.

In this Section, we show that the generalized gap condition (1.7) and the observability conditions (1.10) and (1.13) still hold for the approximate problem (uniformly inh), provided that we consider only “low fre- quencies”. More precisely, we first have the following result:

Proposition 2.1. Suppose that the generalized gap condition (1.7) and the Assumption (1.10) are verified.

Then, there exist two constants >0 andh^∗ >0, such that, for all 0< h < h^∗ and for all k∈ {1, . . . , N(h)}

satisfying

h^θλ²_k≤, (2.2)

we have

∃M ∈N^∗,∃γ >0, λ_k+M, h−λ_{k, h}≥M γ (2.3) and

∃α >0,∀p∈ {1, . . . , M},∀k_n∈A_p,h,∀C∈R^Ln,B_k⁻¹

nΦ_{kn, h}C

U,2≥αC₂, (2.4) where α is independent of h, and where the matrix Φ_{kn, h} ∈ Mp, Ln(U), with coefficients in U, is defined as follows: for all i= 1, . . . , p, we set

(Φ_{kn, h})_ij =

B^∗_hϕ_kn+i−1+j−Ln, i−1−1, hif L_{n, i−}1< j≤L_{n, i},

0 else,

whereL_{n, i−}1 is defined by (1.8) and

A_p,h={kn∈A_p satisfying(2.2)and s.t.k_n+p−1+l_n+p−1−1≤N(h)}.

For the proof of this proposition, we need a result proved by Babuska and Osborn in [4]. For that purpose, we introduce_h(n, j) such that

_h(n, j) = inf

ϕ∈Mj(λkn) inf

vh∈Vhϕ−v_hV,

where M_j(λ_kn) = {ϕ ∈ M(λ_kn) : a(ϕ, ϕ_{kn, h}) = . . . = a(ϕ, ϕ_kn+j−2, h) = 0} and M(λ_kn) = {ϕ : ϕis an eigenvector ofA¹² corresponding toλ_kn,ϕ = 1}. The restrictions a(ϕ, ϕ_{kn, h}) = . . . = a(ϕ, ϕ_kn+j−2, h) = 0 are not imposed if j = 1. Then, we have the following estimate about the eigenvalue and eigenvector errors for the Galerkin method in terms of the approximability quantities_h(n, j).

Theorem 2.2. There are positive constantsC andh0 such that

λ_kn+j, h−λ_kn+j≤C²_h(n, j), ∀0< h≤h0, j= 0, . . . , l_n−1, k_n+j ≤N(h), n∈N^∗ (2.5) and such that the eigenvectors{ϕkn+j}0≤j≤ln−1 ofA¹² can be chosen so that

ϕ_kn+j, h−ϕ_kn+j_V ≤C_h(n, j), ∀0< h≤h0, j= 0, . . . , l_n−1, k_n+j≤N(h), n∈N^∗. (2.6) This result is proved by Babuska and Osborn in [4], p. 702. because

λ²_{kn+j, h}−λ²_kn+j = (λ_kn+j, h−λ_kn+j)(λ_kn+j, h+λ_kn+j)≥2λ1(λ_kn+j, h−λ_kn+j).

Remark 2.3. Notice that for everyϕ∈M_j(λ_kn) we have _h(n, j)≤ inf

vh∈Vhϕ−v_hV

≤C0h^θAϕ by (1.5)

≤C0h^θλ²_knϕ=C0h^θλ²_kn+j.

(2.7) Proof of Proposition2.1. We begin with the proof of the generalized gap condition for the approximate eigen- valuesλ_{k, h}.First, we use the Min-Max principle (see [32]) to obtain

λ_k ≤λ_{k, h},∀k∈ {1, . . . , N(h)}. (2.8) Second, we use the estimates (2.5) and (2.7) and we have

λ_{k, h}≤λ_k+C(C0h^θλ²_k)²≤λ_k+C(C0)²≤λ_k+CC₀², (2.9) for allk∈ {1, . . . , N(h)} verifying (2.2) and≤1. Therefore, we may write

λ_k+M, h−λ_{k, h}≥λ_k+M −λ_k−CC₀²≥M γ0−CC₀²≥Mγ0

2 =:M γ for allk∈ {1, . . . , N(h)} satisfying (2.2) and for≤ ₂^Mγ_CC⁰2

0·

Now, we prove the estimate (2.4) which is the approximated version of (1.10). Notice that Φkn, h−Φ_knU ≤C max

i=0,...,p−1 ln+i−1

j=0

B^∗ϕ_kn+i+j, h−B^∗ϕ_kn+i+j

U

≤C max

i=0,...,p−1 ln+i−1

j=0

B^∗_L(H, U)ϕ_kn+i+j, h−ϕ_kn+i+j

≤C max

i=0,...,p−1 ln+i−1

j=0

B^∗_L(H, U)ϕ_kn+i+j, h−ϕ_kn+i+j

V

≤C max

i=0,...,p−1 ln+i−1

j=0

_h(n+i, j) by (2.6)

≤C max

i=0,...,p−1 ln+i−1

j=0

h^θλ²_kn+i+jby (2.7).

Thus, by (2.2), we get

Φ_{kn, h}−Φ_knU ≤C. (2.10)

Therefore the triangular inequality leads to B_k⁻_n¹Φ_{kn, h}C

U,2=B⁻_kn¹Φ_knC+B_k⁻_n¹(Φ_{kn, h}−Φ_kn)C

U,2

≥B⁻_kn¹Φ_knC

U,2−B_k⁻_n¹(Φ_{kn, h}−Φ_kn)C

U,2

≥α0C₂−B_k⁻_n¹(Φ_{kn, h}−Φ_kn)C

U,2

by (1.10). But, asB_k⁻_n¹=

⎛

⎜⎜

⎝

1 1· · · 1 0 0· · · 0 ... . .. ...

0 · · · 0

⎞

⎟⎟

⎠+R_kn,withR_kn→0,whenk_n→+∞(see Lem.1.1), we obtain

B_k⁻_n¹(Φ_{kn, h}−Φ_kn)C

U,2≤

⎛

⎜⎜

⎝

1 1· · · 1 0 0· · · 0 ... . .. ...

0 · · · 0

⎞

⎟⎟

⎠(Φ_{kn, h}−Φ_kn)C

U,2

+R_kn(Φ_{kn, h}−Φ_kn)C_U,2

≤CΦ_{kn, h}−Φ_knUC₂+η_nΦ_{kn, h}−Φ_knUC₂

≤C(1 +η_n)C2,

(2.11)

whereη_n=R_kn →0.Thus

B_k⁻_n¹Φ_{kn, h}C

U,2≥(α0−C(1 +η_n))C2≥ α0

2 C2

for≤ _2C(1+max^α0

n (1 +η_n))·

For the polynomial stability, we have the same kind of result, but more filtering is necessary in order to have the discrete counterpart of the observability condition (1.13) (uniformly inh).

Proposition 2.4. Suppose that the generalized gap condition (1.7) and the Assumption (1.13) are verified.

Then, there exist two constants >0 and h^∗ >0, such that, for all0< h < h^∗ and for all k∈ {1, . . . , N(h)}, satisfying

h^θλ²_k ≤

λ^l_k, (2.12)

we have (2.3)and

∃α >0, ∀p∈ {1, . . . , M},∀k_n ∈A⁽_p,h^l),∀C∈R^Ln,B⁻_kn¹Φ_{kn, h}C

U,2≥ α

λ^l_knC₂, (2.13) whereA⁽_p,h^l) ={k_n∈A_p satisfying (2.12)and s.t. k_n+p−1+l_n+p−1−1≤N(h)}.

Proof. The generalized gap condition for the approximate eigenvaluesλ_{k, h} is a consequence of Proposition2.1, becauseλ_k ≥λ1>0.

To prove the estimate (2.13) we notice that Φ_{kn, h}−Φ_knU ≤C max

i=0,...,p−1 ln+i−1

j=0

h^θλ²_kn+i+j ≤C h^θλ²_kn+p−1. Moreover, by the triangular inequality and (1.13), we have

B_k⁻¹

nΦ_{kn, h}C

U,2≥ _λ^αl⁰

kn C2−B_k⁻¹

n(Φ_{kn, h}−Φ_kn)C

U,2. By (2.11) and (2.12), we obtain

B_k⁻_n¹Φ_{kn, h}C

U,2≥

α0

λ^l_kn −^C_λ⁽¹⁺l ^ηn) kn+p−1

C2

≥

α0

λ^l_kn −_λl^C

kn+ρn(1 +η_n)

C2,withρ_n=λ^l_kn+p−1−λ^l_kn→0

≥ _λ^αl kn C₂

for an appropriate choice of >0.

3. Uniform stability results

3.1. Exponential stability result

The Proof of Theorem1.2is based on the following result (see Thm. 7.1.3 in [26]):

Theorem 3.1. Let (T_h)_h>0 be a family of semigroups of contractions on the Hilbert spaces (X_h)_h>0 and let ( ˜A_h)_h>0 be the corresponding infinitesimal generators. The family (T_h)_h>0 is uniformly exponentially stable, that is to say there exist constantsM >0, α >0 (independent ofh∈(0, h^∗)) such that

T_h(t)_L(Xh)≤Me^−αt,∀t≥0, if and only if the two following conditions are satisfied:

(i) For allh∈(0, h^∗), iR is contained in the resolvent setρ( ˜A_h)of A˜_h, (ii) sup

h∈(0,h^∗),ω∈R

(iω−A˜_h)⁻¹

L(Xh)<+∞.

3.2. Polynomial stability result

The proof of Theorem1.6 is based on the results presented in this section by adapting the results from [9]

and from [24] to obtain the (uniform) polynomial stability of the discretized problem (1.14). Throughout this section, let (T_h(t)) t≥0

h∈(0,h^∗)

be a family of uniformly boundedC0 semigroups on the associated Hilbert spaces (X_h)_h∈(0,h∗) (i.e., ∃M > 0, ∀h ∈ (0, h^∗), T_h(t)_L(Xh) ≤ M ) and let ( ˜A_h)_h∈(0,h^∗) be the corresponding infinitesimal generators.

In the following, for shortness, we denote byR(λ,A˜_h) the resolvent (λ−A˜_h)⁻¹; moreover, for any operator mapping X_h intoX_h, we skip the indexL(X_h) in its norm, since in the whole section we work inX_h.

Definition 3.2. Assuming that

iR⊆ρ( ˜A_h), ∀h∈(0, h^∗), (3.1)

and that for allm≥1,there existsc=c(m)>0 such that sup

h∈(0,h∗)

|s|≤m

R(is,A˜_h)_L(Xh)≤c, (3.2)

we define the fractional power ˜A^−α_h forα >0 andh∈(0, h^∗), according to [2] and [13], as A˜^−α_h = 1

2πi

Γλ^−α(λ−A˜_h)⁻¹dλ,

where λ^−α = e^−αlogλ and R⁺ is taken as the cut branch of the complex log function and where the curve Γ =Γ1∪Γ2 is given by

Γ ={−+te^iθ, t∈[0,+∞)} ∪ {−−te^−iθ, t∈(−∞,0]} (3.3) for some >0 small enough independent ofhandθis a fixed angle in

0,π

4

.

Remark 3.3. Throughout this section, whenever ˜A^−α_h is mentioned, the Assumptions (3.1) and (3.2) are directly taken into consideration since otherwise ˜A^−α_h is not well defined.

In fact, under the Assumptions (3.1) and (3.2), for allm >0 there exists=(m)>0 such that

−μ+iβ∈ρ( ˜A_h), ∀h∈(0, h^∗), ∀0≤μ≤,∀|β| ≤m.

Indeed, for all m >0 such that|β| ≤m, we have

(−μ+iβ−A˜_h)⁻¹= (iβ−A˜_h)⁻¹[I_h−μ(iβ−A˜_h)⁻¹]⁻¹ and

μ(iβ−A˜_h)⁻¹ ≤μc.

Hence, if|β| ≤mandμ≤≤ ₂¹_c, then (−μ+iβ−A˜_h) is invertible and we have

(−μ+iβ−A˜_h)⁻¹ ≤2(iβ−A˜_h)⁻¹ ≤2c, ∀h∈(0, h^∗). (3.4) We choose m = (−+te^iθ) = tanθ when (−+te^iθ) = 0, i.e. when t = _cosθ· Therefore, by (3.4), Assumptions (3.1) and (3.2) imply that there exists >0 independent ofhsuch that the curveΓ is included in ρ( ˜A_h) for anyh∈(0, h^∗), and hence ˜A^−α_h is well defined. In fact, ifξ∈Γ such thatξ >0, then, by the Hille Yosida theorem,ξ∈ρ( ˜A_h), while if−≤ξ≤0, then, by (3.4),ξ∈ρ( ˜A_h).

Proposition 3.4. If, in addition to Assumptions (3.1)and(3.2), we have

h∈sup(0,h^∗)

R(is,A˜_h)_L(Xh)=O(|s|^α), |s| → ∞, (3.5)

thenA˜^−α_h is uniformly bounded independently ofh∈(0, h^∗).