Isoparametric mixed finite element approximation of eigenvalues and eigenvectors of 4th order eigenvalue problems with variable coefficients

Mod´elisation Math´ematique et Analyse Num´erique M2AN, Vol. 36, N^o1, 2002, pp. 1–32 DOI: 10.1051/m2an:2002001

ISOPARAMETRIC MIXED FINITE ELEMENT APPROXIMATION OF EIGENVALUES AND EIGENVECTORS OF 4TH ORDER EIGENVALUE

PROBLEMS WITH VARIABLE COEFFICIENTS

Pulin Kumar Bhattacharyya

and Neela Nataraj

Abstract. Estimates for the combined effect of boundary approximation and numerical integration on the approximation of (simple) eigenvalues and eigenvectors of 4th order eigenvalue problems with variable/constant coefficients in convex domains with curved boundary by an isoparametric mixed finite element method, which, in the particular case of bending problems of aniso-/ortho-/isotropic plates with variable/constant thickness, gives a simultaneous approximation to bending moment tensor field Ψ = (ψij)1≤i,j≤2 and displacement field ‘u’, have been developed.

Mathematics Subject Classification. 35J40, 65N30, 35P99, 74H45.

Received: May 5, 2001.

1. Introduction

In all papers [14, 22, 23, 29] on mixed finite element analysis of 4th order eigenvalue problems, it has been assumed thatneitherany numerical integration is essentialnorany approximation of the boundary is necessary (since the boundary of theconvexdomain is apolygonalone in all the cases, the convexity of the domain being a requirement for the regularity of the solution [18, 21, 24]). But in many situations,we are to consider convex domains with curved boundary Γ. Then an approximation of the curved boundary and possibly numerical evaluation of integrals will be essential, but convergence analysis becomes much more complex and complicated.

Even forclassical, standardfinite element analysis of second order self-adjoint eigenvalue problems in domains with curved boundary we find the situation as stated in ( [40], p. 254): “· · · In contrast to finite element analysis of boundary value problems, in the finite element analysis of eigenvalue problems, there doesnotexist any abstract error estimate consisting of the sum ofthreeterms (error of interpolation, error of approximation of the boundary and error of numerical integration) · · ·”. Hence, in such a situation error analysis for each specific problem can be attempted at and the proofs involved in finding the estimates will be quite complex and too technical in nature due to these additional complicacies introduced by the boundary approximation and obligatory use (for example, in the isoparametric case) of numerical integration. In fact, we find only two papers [25, 40], in which this combined effect of boundary approximation and numerical integration on

Keywords and phrases.Mixed FEM, eigenvalue problem, isoparametric boundary approximation, 4th-order equations, anisotropic plates, convergence analysis, numerical results.

1 Visiting Professor, School of Computer and Systems Sciences, Jawaharlal Nehru University, New Delhi: 110067, India.

e-mail:pulinkum@hotmail.com

2 Lecturer, Department of Mathematics, Indian Institute of Technology, New Delhi, 110016, India.

e-mail:neela@maths.iitd.ernet.in

c EDP Sciences, SMAI 2002

second order self-adjoint eigenvalue approximations using classical isoparametric finite element methods has been estimated, [5] and [6] being the papers which deal with the effect ofonlynumerical integration on eigenvalue approximations. But the situation is still worse in the case of isoparametric mixed finite element analysis of eigenvalue problems, for which error estimates are to be developed again for a specific mixed method formulation (since abstract results for the isoparametric case do not exist even for source problems) and the proofs for the estimates will be much more complex and much more technical in nature. Indeedto our knowledge, [8] is probably thefirstpublication on the estimates for the combined effect of boundary approximation and numerical integration on the mixed finite element approximation of (simple) eigenvalues and eigenvectors of 4th order self-adjoint eigenvalue problems with variable/constant coefficients, many proofs in which, as stated earlier, have remained quite technical in spite of the best efforts of the authors to avoid these technical aspects in some proofs. The present paper, the results of which were announced in [8] (see also [31]), relies heavily on [10] for the corresponding source problem ([9] contains error estimates due to polygonal approximation of the curved boundary along with numerical integration for the same source problem) and also on the results of [4] on the mixed method scheme (see also [33, 36]) for polygonal domains. For other interesting references on eigenvalue approximations, we refer to [2, 15]. Finally, the present paper also contains interesting results of numerical experiments on some problems of practical importance and research interest.

2. The continuous mixed variational eigenvalue problem

Consider the eigenvalue problem: Findλ∈Rfor which∃non-null u such that (P^E) : Λu = λ u in Ω, u|^Γ = (∂u

∂n)|^Γ = 0, (2.1)

where (Λu)(x)≡ X2 i=1

X2 j=1

X2

k=1

X2

l=1

∂²

∂xk∂xl

(aijkl

∂²u

∂xi∂xj

)(x)≡(aijklu,ij),kl(x) ∀x= (x1, x2)∈Ω. (2.2) (In (2.2) and also in the sequel, Einstein’s summation convention with respect to twice repeated indices i, j, k, l= 1,2has been followed unless stated otherwise).

(A1): Ω is a bounded, open, convex domain with Lipschitz continuous boundary Γ which is piecewise ofC^k- class, k≥3 [1, 21, 35, 41];

coefficientsaijkl=aijkl(x) ∀x= (x1, x2)∈Ω¯e ⊂R²,

(A2): Ω being a bounded, open set with boundarye eΓ, which is piecewise of C^k-class, k ≥ 3, such that Ω = Ω¯ ∪Γ⊂Ωe

satisfy the following assumptions: ∀i, j, k, l= 1,2,

(A3): aijkl∈W^2,∞(eΩ),→C¹(Ω);e¯ aijkl≥0; aijkl(x) =aklij(x) =alkij(x) =alkji(x) ∀x∈Ω;e¯ (A4): ∃α >0 such that∀ξ= (ξ11, ξ12, ξ21, ξ22)∈R⁴ withξ21=ξ12, aijkl(x)ξijξkl≥αkξk²_R⁴ ∀x∈Ω.e¯ Then, the correspondingGalerkin Variational Eigenvalue Problem(P^E_G) is defined by:

Find λ∈Rfor which∃non-nullu∈H₀²(Ω) [1, 28] such that

(P^E_G) : a(u, v) =λhu, vi0,Ω ∀v ∈H₀²(Ω) (2.3)

wherea(u, v) =hΛu, vi^0,Ω= Z

Ω

aijklu,ijv,kldΩ =a(v, u) ∀u, v∈H₀²(Ω);

∃α0>0 such thata(v, v)≥α0kvk²2,Ω ∀v∈H₀²(Ω) [3, 20] hu, vi^0,Ω=R

Ω

uvdx.

Sincea(·,·) is continuous andH₀²(Ω)-elliptic, the associatedGalerkin Variational (Source) Problem(PG) defined by: For givenf ∈L²(Ω), findu∈H₀²(Ω) such that

(PG) : a(u, v) =hf, vi^0,Ω ∀v∈H0²(Ω), (2.4) has a unique solution by Lax-Milgram lemma, and we have:

Theorem 2.1 ( [8]). (P^E_G) has a countable non-decreasing system of strictly positive eigenvalues with possibly finite multiplicities and accumulation point at ∞: i.e. 0 <bλ1 ≤λb2 ≤ · · · ≤ bλm ≤ · · · ↑ ∞, and ∃ a system of eigenpairs (bλm,bvm)^∞_m=1 such that the eigensystem (bvm)^∞_m=1 is a Hilbert basis in (H₀²(Ω);hh·,·iia(·,·)) with hhbvm,bvniia(·,·)=a(bvm,bvn) =δmn. Moreover,(

qbλmbvm)^∞_m=1 is a Hilbert basis in L²(Ω).

Now, defining Hilbert spaceHof symmetric tensor-valued functions in Ω by:

H={Φ : Φ = (φij)1≤i,j≤2withφij =φji∈L²(Ω)}withkΦk²H=kΦk²0,Ω= X2

i,j=1

Z

Ω

|φij(x)|²dx

and new coefficients Aijkl =Aijkl(x) ∀x∈ Ω in terms of coefficientse¯ aijkl, the algorithm for which is given in [4], satisfying the following properties: ∀i, j, k, l= 1,2

• ∀x∈Ω, A¯e ijkl(x) =Aklij(x) =Alkij(x) =Alkji(x); (2.5)

•∃α >0 such thatAijkl(x)ξijξkl ≥αkξk²_R⁴ ∀x∈Ω,¯e ∀ξ= (ξij)i,j=1,2∈R⁴withξ12=ξ21; (2.6)

• ∀x∈Ω,e¯ ∀ξ= (ξij)i,j=1,2∈R⁴withξ21=ξ12, ∀ζ= (ζij)i,j=1,2∈R⁴withζ21=ζ12,

Aijkl(x)aijmn(x)ξmnζkl =ξijζij; Aijkl(x)aijmn(x)ξmn=ξkl, and (2.7) (A5): Aijkl∈W^2,∞(eΩ),→C¹(Ω),¯e

we construct an Auxiliary Continuous Mixed Variational Eigenvalue Problem(Q^E_AUX) as follows:

Find λ∈Rfor which∃ non-null (Ψ, u)∈H×H₀²(Ω) (i.e. Ψ6=0, u6= 0) such that

(Q^E_AUX) : A0(Ψ,Φ) +b0(Φ, u) = 0 ∀Φ∈H; −b0(Ψ, v) =λhu, vi^0,Ω ∀v∈H₀²(Ω). (2.8) The associatedSource Problem (QAUX) inContinuous Mixed Variational Formulationis defined by:

For givenf ∈L²(Ω), find (Ψ, u)∈H×H₀²(Ω) such that:

(QAUX) : A0(Ψ,Φ) +b0(Φ, u) = 0 ∀Φ∈H; −b0(Ψ, v) =hf, vi^0,Ω ∀v∈H₀²(Ω), (2.9) whereA0(·,·) andb0(·,·) are continuous bilinear forms defined by:

A0(Ψ,Φ) = Z

Ω

Aijklψijφkldx=A0(Φ,Ψ) with|A0(Ψ,Φ)| ≤M¯0kΨk^HkΦk^Hfor some ¯M0>0,

A0(Φ,Φ)≥αkΦk²H ∀Φ∈Hand for someα >0; (2.10) b0(Φ, v) =−Z

Ω

φijv,ijd Ω ∀Φ∈H∀v∈H₀²(Ω) with|b0(Φ, v)| ≤m¯0kΦk^Hkvk^2,Ωfor some ¯m0>0,

sup

Φ∈H−{0}

|b0(Φ, v)|

kΦkH ≥β0kvk^2,Ω∀v∈H₀²(Ω) for some β0>0. (2.11) As a consequence of (2.10) and (2.11), (QAUX) has a unique solution (Ψ, u)∈H×H0²(Ω) [2, 11, 12], and we defineT~0:f ∈L²(Ω)7→T~0f= (S0f, T0f) = (Ψ, u)∈H×H₀²(Ω) such that

A0(S0f,Φ) +b0(Φ, T0f) = 0 ∀Φ∈H; −b0(S0f, v) =hf, vi^0,Ω ∀v∈H₀²(Ω), (2.12) whereS0∈ L(L²(Ω);H), T0∈ L(L²(Ω);H₀²(Ω)) with

kS0fk^0,Ω+kT0fk^2,Ω≤Ckfk^0,Ω ∀f ∈L²(Ω), S0f = Ψ, T0f =u; S0(·) = ((aijklT0(·),kl)1≤i,j≤2. (2.13) Then,,→ ·T0=T0∈ L(L²(Ω);L²(Ω))≡ L(L²(Ω)) with,→:H₀²(Ω)−→L²(Ω) is acompact, positive, symmetric, linear operator and the eigenvalue problem of T0 ∈ L(L²(Ω)): T0u = µ u is “equivalent” to the eigenvalue problem (Q^E_AUX) withµ= 1/λ >0. Hence, we have:

Theorem 2.2. (Q^E_AUX)has a countable system of strictly positive, non-decreasing system of eigenvalues with possibly finite multiplicities and accumulation point at∞:

0< λ1≤λ2≤ · · · ≤λm≤ · · · ↑ ∞withµm= 1/λm ∀m∈N, (2.14) and∃ eigenpairs(λm; (Ψm, um))^∞m=1 of(Q^E_AUX) :∀m∈N,

A0(Ψm,Φ) +b0(Φ, um) = 0 ∀Φ∈H; −b0(Ψm, v) =λmhum, vi^0,Ω ∀v∈H₀²(Ω), (2.15) (um)^∞_m=1 being a Hilbert basis in L²(Ω) with Ψm = S0(λmum) = (aijklum,kl)i,j=1,2 ∀m ∈ N (see (2.13)).

Moreover, Ψm

√λm

∞ m=1

is an orthonormal system in (H,[·,·]A0(·,·)) with Ψm

√λm

, Ψn

√λn

A₀(·,·)

=A0

Ψm

√λm

, Ψn

√λn

=δmn ∀m, n∈N.

As a consequence of (2.10) and (2.11),∀fixedv∈H₀²(Ω), there existsa uniqueσ∈Hsuch thatA0(σ,Φ) + b0(Φ, v) = 0 ∀Φ∈Hby virtue of Lax-Milgram lemma and this correspondence definesI :v∈H₀²(Ω)7→ Iv= σ∈Hand we set

E=I(H₀²(Ω)) ={σ:σ∈Hfor which∃v∈H₀²(Ω) such thatIv=σ} ⊂H. (2.16) Proposition 2.1. (i) (E; [·,·]A0(·,·)) equipped with inner product[σ, ω]A0(·,·)=A0(σ, ω) ∀σ, ω∈ E is a Hilbert space and∀ eigenpair(λm; (Ψm, um))of (Q^E_AUX), Ψm=Ium=S0(λmum), m∈N.

(ii) I: (H₀²(Ω),hh·,·iia(·,·))−→(E; [·,·]A0(·,·))is a linear, continuous bijection with

hhv, wiia(·,·)= [Iv,Iw]A0(·,·)= [σ, ω]A0(·,·)=A0(σ, ω), (σ, v), (ω, w)being the linked pairsin E ×H₀²(Ω).

SetM=E ×H₀²(Ω) = product space of linked pairs (σ, v) withσ=Iv ∀v∈H₀²(Ω). (2.17)

Rayleigh quotient characterization of eigenpairs.

To (P^E_G) we can associate Rayleigh coefficientR(v) = a(v, v)

hv, vi^0,Ω ∀v∈H²(Ω)− {0}·

But a(v, v) = hhv, vii^a(·,·) = A0(σ, σ) with σ =Iv ∈ E. So ∀v ∈ H₀²(Ω) with σ=Iv∈ E, R(v) = A0(σ, σ) hv, vi^0,Ω· i.e. R(v) is expressed through a linked pair (σ, v) = (Iv, v) ∈ E ×H0²(Ω). Hence, it suggests to define a new Rayleigh quotient <(·,·) on M ≡ E ×H₀²(Ω) by (see also [14]): <(σ, v) = A0(σ, σ)

hv, vi^0,Ω ∀ linked pair (σ, v) = (Iv, v)∈ Msuch that<(σ, v)≡R(v) = a(v, v)

hv, vi^0,Ω ∀v∈H₀²(Ω)− {0}, for which we can apply various extrema [2, 14, 35, 37].

Define a p-dimensional subspace M^p (resp. Up) of M (resp. H0²(Ω)) by: M^p = Span{(Ψm, um)^p_m=1}; Up = Span{(um)^p_m=1}, (λm; (Ψm, um)) ∈ R⁺×(E ×H₀²(Ω)), 1 ≤ m ≤ p, being the first ‘p’ eigenpairs of (Q^E_AUX) with 0< λ1≤λ2≤ · · · ≤λp.

Theorem 2.3 (Min-Max Principle, [2, 37]).

(i) Eigensolutions of (Q^E_AUX)are the stationary points of <(·,·)onM, the corresponding eigenvalues being the values of <(·,·)at these stationary points;

(ii) ∀p∈N, λp= min

Sp⊂M

dim(Sp)=p

(σ,v)max∈Sp

<(σ, v) = max

(σ,v)∈Mp

<(σ, v) =<(Ψp, up). (2.18)

We will need another Rayleigh quotientQ(v) = hT0v, vi^0,Ω

hv, vi^0,Ω ∀v∈H₀²(Ω),whereT0∈ L(L²(Ω)) is compact, positive and symmetric. Hence,

Theorem 2.4 (Max-Min Principle, [2, 37]). ∀p∈N, µp= max

S∗p⊂L2(Ω)

dimS^∗_p=p vmin∈S∗

p

Q(v) = min

v∈Up

U_p=Span{(u_m)^p_m=1}

Q(v) =Q(up), (2.19)

where(µm, um)^p_m=1are the first ‘p’ eigenpairs ofT0corresponding to the first ‘p’ eigenvaluesµ1≥µ2≥ · · · ≥µp

of T0,umbeing the m-th eigenvector of T0 (µm= 1/λm, 1≤m≤p).

But (Q^E_AUX) is not suitable for finite element approximation, since C¹-elements are to be used for con- struction of finite element subspaces of H₀²(Ω). Hence, we construct a new Continuous Mixed Varia- tional Eigenvalue Problem (Q^E), which will be eminently suitable for finite element approximation using C⁰-elements as follows:

Find λ∈Rfor which∃non-null (Ψ, u)∈V×W such that

(Q^E) : A(Ψ,Φ) +b(Φ, u) = 0 ∀Φ∈V, −b(Ψ, v) =λhu, vi0,Ω ∀v∈W, (2.20) whereV={Φ : Φ = (φij)i,j=1,2∈H, φij ∈H¹(Ω)∀i, j= 1,2}withkΦk²V=kΦk²1,Ω=

X2 i=1

X2 j=1

kφijk²1,Ω

W ≡H₀¹(Ω) ={v:v∈H¹(Ω), v|^Γ = 0}withkvk^W =kvk^1,Ω; A(·,·) andb(·,·) are continuous bilinear forms defined by [4]:

A(Ψ,Φ) =A0(Ψ,Φ) ∀Ψ,Φ∈V⊂Hsuch that

|A(Ψ,Φ)| ≤MkΨkVkΦkVfor someM >0; A(Φ,Φ)≥αkΦk²H ∀Φ∈V, for someα >0 [4], (2.21)

b(Φ, v) = Z

Ω

φij,jv,idΩ ∀Φ∈V ∀v∈W withb(Φ, v) =b0(Φ, v)∀Φ∈V∀v∈H₀²(Ω) (2.22)

such that |b(Φ, v)| ≤ mkΦk^Vkvk^W, ∀Φ ∈ V,∀v ∈ W and some m > 0; and ∃β > 0 such that supΦ∈V−{0}|b(Φ,v)|

kΦkV ≥βkvk^W ∀v∈W ( [4]).

If (λ; (Ψ, u))∈R×(V×W) be an eigenpair of (Q^E), thenλ∈R⁺.

Then, the corresponding continuousMixed Variational Source Problem (Q) [4] is defined by:

For givenf ∈L²(Ω), find (Ψ, u)∈V×W such that

(Q) : A(Ψ,Φ) +b(Φ, u) = 0 ∀Φ∈V, −b(Ψ, v) =hf, vi0,Ω ∀v∈W. (2.23) SinceA(·,·) isnot V-elliptic, (Q) is not well-poseda priori. But we have:

Theorem 2.5 ( [4]). Let (A1–A5) hold. If u∈ H³(Ω)∩H₀²(Ω) be the solution of the Galerkin Variational Source Problem (PG)with ψij =aijklu,kl ∈H¹(Ω) ∀i, j = 1,2and Ψ = (ψij)1≤i,j≤2, then (Ψ, u)∈V×W is the unique solution of (Q). Conversely, let(Ψ, u)∈V×W be the solution of(Q). Then, u∈H₀²(Ω)and is the unique solution of(PG)and ψij =aijklu,kl ∀i, j= 1,2; u,kl=Aijklψij ∀k, l= 1,2; Ψ = (ψij)1≤i,j≤2.

Hence, under the assumption that the solutionu∈H0²(Ω) of Galerkin Variational Source Problem (PG) in (2.4) has the additional regularity [18, 21, 24]:

(A6) : u∈H³(Ω)∩H₀²(Ω) withkuk^3,Ω≤Ckfk^0,Ωfor someC >0, (2.24) the correspondence f ∈ L²(Ω) 7→ (Ψ, u) ∈ V ×W with u ∈ H³(Ω) ∩H₀²(Ω) defines an operator Tf~ = (Sf, T f) = (Ψ, u)∈V×W with

A(Sf,Φ) +b(Φ, T f) = 0 ∀Φ∈V, −b(Sf, v) =hf, vi^0,Ω ∀v∈W, (2.25) T :f ∈L²(Ω)7→T f =u∈H³(Ω)∩H₀²(Ω), S: f ∈L²(Ω) 7→Sf = Ψ = (aijkl(T f),kl)i,j=1,2 ∈V,being the solution component operators withS(·) = (aijkl(T(·)),kl)1≤i,j≤2 and kT fk^1,Ω≤Ckfk^0,Ω;kSfk^1,Ω≤Ckfk^0,Ω; kSfk^0,Ω+kT fk^1,Ω≤Ckfk^0,Ω.

Theorem 2.6 ( [8, 31]). Under (A6), the source problems (Q)and(QAUX)are “equivalent” in the sense that these have the same solution (Ψ, u)∈V×W withu∈H³(Ω)∩H₀²(Ω),

ψij =aijklu,kl∈H¹(Ω)∀i, j= 1,2, Ψ = (ψij)1≤i,j≤2∈V⊂H.

Hence under (A6),∀f ∈L²(Ω), Sf =S0f = Ψ∈V⊂H, T f =T0f =u∈H³(Ω)∩H₀²(Ω)⊂W ⊂L²(Ω) and all the results associated with T0 ∈ L(L²(Ω)) and S0 ∈ L(L²(Ω)) will hold for T ∈ L(L²(Ω)) and S ∈ L(L²(Ω);V). Hence, we have the important result:

Theorem 2.7 ( [8]). Under (A6), mixed variational eigenvalue problems(Q^E)and(Q^E_AUX)are equivalent in the sense that both of these eigenvalue problems have the same strictly positive eigenvalues (λm)^∞_m=1 and the same eigenpairs(λm; (Ψm, um))∈R⁺×(V×W)withum∈H³(Ω)∩H₀²(Ω), (um)^∞_m=1 being a Hilbert basis in L²(Ω) and(Ψm/√

λm)^∞_m=1 being an orthonormal system in (H, A(·,·)).

Define a linked pair (σp, χp) = Xp

m=1

cm(Ψm, um)∈ M^p withσp= Xp

m=1

cmΨm∈V, χp=

Xp

m=1

cmum∈H³(Ω)∩H₀²(Ω) cm∈R ∀m= 1,2,· · ·, p,where (Ψm, um)∈V×W with

um∈H³(Ω)∩H₀²(Ω) is an eigenelement of (Q^E) corresponding to the eigenvalueλm, 0< λ1≤λ2≤ · · ·λp.In

particular, for cm= 0 ∀m6=p, cp= 1, σ_p= Ψp, χp=up.Then, ∀(σ_p, χp)∈ M^p with χp∈H³(Ω)∩H₀²(Ω) andσ_p∈V,∃a unique linked pair (σ^∗_p, χ^∗_p)∈ M^p with

χ^∗_p= Xp

m=1

λmcmumandσ^∗_p=Iχ^∗_p= Xp

m=1

λmcmΨm (2.26)

such thatTχ~ ^∗_p = (σ_p, χp) ,T~ being the linear operator defined in (2.25).

Examples [8]: I. Biharmonic Eigenvalue Problem is obtained from (2.2)aijkl defined by: aiiii = 1; a1212 = a2121 = a2112 = a1221 = 1/2; aijkl = 0 otherwise, which satisfy (A3–A4) [3], in Ω. Then, wee have Λ ≡ ∆∆, for which (A6) holds [4]. (Q^E) corresponds to H-H-M (Hellan-Hermann-Miyoshi) mixed method scheme for biharmonic eigenvalue problem [4, 13, 30]. ∀m∈N (λm; (Ψm, um))∈R⁺×(V×W) with um ∈ H³(Ω)∩H0²(Ω) and Ψm = (ψmij)i,j=1,2 is an eigenpair of biharmonic eigenvalue problem in H-H-M mixed method formulation [2, 13, 14, 22]:

Z

Ω

ψmijφijdΩ + Z

Ω

φij,jum,idx= 0∀Φ∈V, −Z

Ω

ψmij,jv,idΩ =λmhum, vi^0,Ω∀v∈W. (2.27)

Remark 2.1. The associated biharmonicsourceproblem corresponds to Stokes problem [34] of fluid mechanics in stream function-vorticity formulation and also to the bending problem of isotropic elastic plates with flexural rigidityD= 1, ν= 0 (see (2.32)).

II. Eigenvalue problems associated with the vibration of elastic plates with variable/

constant thickness. (i) In Anisotropic case [4, 27],

aiiii=Dii, a1212=a1221=a2121=a2112=D66, a1112=a1211=a2111=a1121=D16,

a1222=a2122=a2212=a2221=D26, a2211=a1122=D12, (2.28) Dij = Dij(x1, x2) ∀(x1, x2) ∈ Ω being rigidities [8, 27] for which (A2–A4) hold, and the anisotropic plate¯e bending operator Λ is given by:

Λu≡(D11u,11+ 2D16u,12+D12u,22),11+ 2(D16u,11+ 2D66u,12+D26u,22),12

+ (D12u,11+ 2D26u,12+D22u,22),22. (2.29) Then, coefficientsAijklare defined in terms ofDij’s [4,9,10] and the corresponding bilinear formA(·,·) of (Q^E) is given by: Ψ,Φ∈V,

A(Ψ,Φ) = Z

Ω

4

|A(x)|

{(D22D66−D₂₆² )ψ11+ (D16D26−D12D66)ψ22+ (D12D26−D16D22)ψ12}φ11

+{(D16D26−D12D66)ψ11+ (D11D66−D²16)ψ22+ (D16D12−D11D26)ψ12}φ22

+{(D12D26−D16D22)ψ11+ (D16D12−D11D26)ψ22+ (D11D22−D₁₂² )ψ12}φ12

dx; (2.30) where|A(x)|= 4(D11D22D66−D11D₂₆² −D66D²₁₂−D22D²₁₆+D12D16D26)(x).

(ii) TheOrthotropiccase [3, 27, 38] can be retrieved from the anisotropic case (i) by putting in (2.28)–(2.30), aiiii=Di; a1122=a2211=D12=ν1D2=ν2D1;

a1212=a2121=a2112=a1221=Dτ, aijkl= 0 otherwise, (2.31)

with Di = Di(x1, x2) and Dτ = Dτ(x1, x2) ∀(x1, x2) ∈ Ω, He¯ = D1ν2+ 2Dτ, νi, i = 1,2 being Poisson’s coefficients respectively,Di’s and Dτ being rigidities, assumptions (A3–A4) hold [3] and

(iii) the Isotropiccase is obtained from the Orthotropic case by putting ν1 =ν2 =ν and D1 = D2 =D in all formulae (2.31). Then, the orthotropic (resp. isotropic) plate (bending) operator Λ and the corresponding bilinear formA(·,·) of (Q^E) are given by:

Orthotropic Case: Λu≡(D1u,11+ν2D1u,22),11+ 4(Dτu,12),12+ (ν1D2u,11+D2u,22),22

A(Ψ,Φ) = Z

Ω

1

D1(1−ν1ν2)(ψ11−ν1ψ22)φ11+ 1

D2(1−ν1ν2)(−ν2ψ11+ψ22)φ22+ 1 Dτ

ψ12φ12

dx ∀Ψ,Φ∈V.

Isotropic Case: Λu≡(D(u,11+νu,22)),11+ 2(D(1−ν)u,12),12+ (D(νu,11+u,22)),22.

Then, forD=constant, Λu≡D∆∆u, (A6) will hold [18,21,24]. (2.32)

A(Ψ,Φ) = Z

Ω

1

D(1−ν²)(ψ11−νψ22)φ11+ 1

D(1−ν²)(−νψ11+ψ22)φ22+ 2

D(1−ν)ψ12φ12

dx ∀Ψ,Φ ∈ V.

In aniso-/ortho-/isotropic cases (i–iii), ∀ eigenpair (λm; (Ψm, um)) of (Q^E), um is the deflection mode of the vibrating plate, Ψm = (ψmij)1≤i,j≤2 is the corresponding bending moment tensor, ψmii being the bending moment in thexi direction andψm12=ψm21, being the twisting moment,i.e.

Anisotropic Case: ψmij=aijklu,kl withaijkl’s defined by (2.28),i, j= 1,2;

Orthotropic Case: ψm11=D1(um,11+ν2um,22); ψm22=D2(ν1um,11+um,22), ψm12= 2Dτum,12; Isotropic Case: ψm11=D(um,11+νum,22), ψm22=D(νum,11+um,22), ψm12=ψm21=D(1−ν)um,12. Remark 2.2. In the orthotropic plates with constant thickness, D1 = constant, D2 = constant, H =D1ν2+ 2Dτ = constant and Λu≡D1u,1111+ 2Hu,1122+D2u,2222.

Then, for H =p

D1D2, Λu≡D1u,1111+ 2p

D1D2u,1122+D2u,2222 (2.33) can be reduced to the form (2.32) by introducing a new variableξ2=x2(D1/D2)^1/4, ξ1=x1 ( [38], pp. 366–

367),i.e. Λu^∗≡D1∆∆u^∗ withu^∗=u^∗(ξ1, ξ2), ∆ = _∂ξ^∂22 1+_∂ξ^∂22

2. Hence, (A6) will also hold for (2.33) [18,21,24].

3. Isoparametric mixed finite element eigenvalue problem (Q

)

Isoparametric Triangulation τ_h^ISO: Let{Pi}^Ni=1^c beNc corner points of Γ at whichC^m-smoothness (m≥3) does not hold and {Pj}^NN^hc be the set of possible additional points suitably chosen on Γ such that νh = {Pi}^Ni=1^c ∪ {Pj}^Nj=N^h c+1 ⊂Γ denote the set of boundary vertices of the isoparametric triangulation of ¯Ω under consideration.

I. τ_h^pol: Let τ_h^pol = ˜τ_h^b∪τ_h⁰ be the admissible, regular, quasi-uniform triangulation of the closed polygonal domain ¯Ω^pol_h = Ω^pol_h ∪Γ^pol_h with vertices{Pi}^Ni=1^h into closed triangles ˜T with vertices (a_i,T˜)³_i=1and thestraight sides (∂T˜i)³_i=1, ∂T˜i = [a_i,T˜, a_i+1,T˜] (modulo 3) such that Γ^pol_h = ∪T^˜∈τ˜_h^b ∂T˜1, ∂T˜1 = [a_1,T˜, a_2,T˜] being the boundary side∀T˜∈τ˜_h^b,

˜

τh^b = {T˜: ˜T ∈τ_h^polis a boundary triangle with single boundary side∂T˜1}; (3.1)

τ_h⁰={T˜: ˜T ∈τ_h^polis an interior triangle withat mostone of its vertices lying on Γ}; (3.2) II. τ_h^exact: Keeping allverticesofτ_h^pol and interior triangles ˜T ∈τ_h⁰ in (3.2)undisturbedand replacingeach boundary triangle T˜ ∈ τ˜_h^b by a curved boundary triangle ¯T which is obtained by replacing the straight

boundary side∂T˜1 of ˜T ∈˜τ_h^b by a part∂T¯1of the boundary Γ between the boundary vertices of ˜T ∈τ˜_h^b. Let ¯τ_h^b denote all such curved boundary triangles ¯T. Then,τ_h^exact= ¯τ_h^b∪τ_h⁰ with ¯Ω =∪T^¯∈τ_h^exactT¯and Γ =∪T^¯∈τ˜_h^b∂T¯1. III. τ_h^ISO: Again, we keep all vertices ofτ_h^pol and interior triangles ˜T ∈τ_h⁰ undisturbed. ∀ boundary triangle T˜∈τ˜_h^b in (3.1), define mid-side pointsa_4,T˜ = (a_1,T˜+a_2,T˜)/2, a_5,T˜ = (a_2,T˜+a_3,T˜)/2, a_6,T˜ = (a_3,T˜+a_1,T˜)/2.

Toa_4,T˜∈Γ^pol_h , we associate the pointa^∗_4,T ∈Γ as the point of intersection of the perpendicular bisector of∂T˜1

at a_4,T˜ with Γ. Let ˆT be the reference triangle with vertices ˆa1= (1,0), ˆa2= (0,1), aˆ3 = (0,0) and mid-side nodes ˆa4 = (1/2,1/2), ˆa5= (0,1/2), ˆa6= (1/2,0), sides ∂Tˆi = [ˆai, ˆai+1] (modulo 3) 1≤i≤3. Then, with the help of canonical basis functions ( ˆφi)⁶_i=1 ofP2( ˆT) (i.e. ∀φˆi∈P2( ˆT), φˆi(ˆaj) =δij, 1≤i, j≤2), define the invertibleisoparametric mapping by: ∀xˆ∈T,ˆ

FT(ˆx) = X3 i=1

a_i,T˜φˆi(ˆx) + X6 i=5

a_i,T˜φˆi(ˆx) +a^∗_4,Tφˆ4(ˆx) =x∈T =FT( ˆT), (3.3) such that FT(ˆai) = a_i,T˜ ∈ T˜ ∈ τ˜_h^b, 1 ≤ i 6= 4 ≤ 6, FT(ˆa4) = a^∗_4,T ∈ Γ, FT(∂Tˆi) = ∂Ti, 1 ≤ i ≤ 3.

Then, ∀T˜∈τ˜_h^b in (3.1), we get a curved boundary triangleT =FT( ˆT) with the single curved boundary side

∂T1=FT(∂Tˆ1). Let τ_h^b be all such curved boundary triangles. Then,

τ_h^ISO=τ_h^b∪τ_h⁰with τ_h⁰def ined by(3.2), Ω¯h=∪T∈τ_h^ISOT, Γh=∪T∈τ_h^b∂T1 (3.4) is theIsoparametric Triangulation of ¯Ω, Γh being the approximation of the boundary Γ. For other meth- ods of approximation of boundary Γ, we refer to [7, 41]. Such aτ_h^ISO is regular in the sense of [16]. Ωh isnot convex, Ωh6⊂Ω, Ω6⊂Ωh in general. But by construction, the distance of Γ from Γh tends to 0 ash−→0 and from (A1), ∃Ω with boundarye eΓ, which is piecewise ofC^k-class,k≥3, such that ¯Ω⊂Ω. Hence,e

(A7): ∃h0>0 such that ∀h∈]0, h0[, Ω¯h⊂Ω.e

Then, from (A1) and (A7)∀h∈]0, h0[, Ω¯h⊂Ω,e Ω¯⊂Ω with ( ¯e Ωh∪Ω)¯ ⊂Ω and definee

h=∪T∈τ_h^b,T¯∈τ¯_h^b(T^int−( ¯T^int∩T^int)) = Ωh−(Ω∩Ωh) with meas(h) =O(h³) [10,17]; (3.5) ωh=∪T^¯∈τ¯_h^b, T∈τ_h^b( ¯T^int−( ¯T^int∩T^int)) = Ω−(Ω∩Ωh) with meas(ωh) =O(h³) [10,17], (3.6) whereT^int=int(T), T¯^int=int( ¯T), τ_h^b⊂τ_h^ISO, τ¯_h^b⊂τ_h^exact ∀h∈]0, h0[ withh0>0 .

∀h boundary Γh of Ωh is piecewise of C^∞-class, νh being the set of boundary vertices of τ_h^ISO at which C^∞ smoothness doesnothold. For the properties of the invertibleFT (resp. F_T⁻¹) and its JacobianJ(FT)∈P1( ˆT) with important estimates, we refer to [10, 16, 17].

We will need extensions toR² of functions defined in Ωh (resp. Ω).

Theorem 3.1 ( [32, 39]). Let Dbe a bounded, two-dimensionaldomain with Lipschitz continuous boundary

∂D, which is piecewise of C^k-class,k≥1. Then,

(a)∃ acontinuous, linearextension operator E:H^k(D)−→H^k(R²), i.e. ∃C >0such that

kEukk,R²=ku˜kk,R² ≤Ckuk^k,D ∀fixedk≥1 withEu↓D= ˜u↓D=u∈H^k(D). (3.7) (b) The operatorE is also a linear and bounded extension operator fromH^(k−i)(D)intoH^(k−i)(R²), 1≤i≤k, i.e. ∃C >0such thatkEuk^k−i,R² =ku˜k^k−i,R² ≤Ckuk^k−i,D, 1≤i≤k,withEu↓D= ˜u↓D=u∈H^k−i(D), 1≤ i≤k, and in particular,

ku˜k0,R² ≤Ckuk0,D ∀u∈L²(D) with ˜u∈L²(R²). (3.8)