A modal synthesis method for the elastoacoustic vibration problem

Mod´elisation Math´ematique et Analyse Num´erique M2AN, Vol. 36, N 1, 2002, pp. 121–142 DOI: 10.1051/m2an:2002005

A MODAL SYNTHESIS METHOD FOR THE ELASTOACOUSTIC VIBRATION PROBLEM

Alfredo Berm´ udez

, Luis Hervella-Nieto

and Rodolfo Rodr´ ıguez

Abstract. A modal synthesis method to solve the elastoacoustic vibration problem is analyzed. A two-dimensional coupled fluid-solid system is considered; the solid is described by displacement vari- ables, whereas displacement potential is used for the fluid. A particular modal synthesis leading to a symmetric eigenvalue problem is introduced. Finite element discretizations with Lagrangian elements are considered for solving the uncoupled problems. Convergence for eigenvalues and eigenfunctions is proved, error estimates are given, and numerical experiments exhibiting the good performance of the method are reported.

Mathematics Subject Classification. 65N15, 65N25, 74F10.

Received: November 23, 2001.

1. Introduction

The need of computing fluid-solid interactions arises in many important engineering problems. A general overview on the subject can be found in [15], where numerical methods and further references are also given. This paper deals with one of these interactions: the elastoacoustic vibration problem. It concerns the determination of harmonic vibrations of a linear elastic structure interacting with an acoustic (i.e., inviscid, barotropic) fluid.

We will approximate the solutions of this problem using a modal synthesis method.

Let us suppose that we want to approximate the solution of a problem defined on a given domain. The modal synthesis method consists of dividing this domain in several subdomains and calculating the lowest frequency eigenfunctions of the spectral problems associated with the restrictions of the original problem to each subdomain. In some modal synthesis methods, a finite number of functions related to the interfaces between each pair of neighboring subdomains must be calculated too. Then, the solution of the original problem is approximated as a linear combination of all these functions.

Keywords and phrases. Fluid-structure interaction, elastoacoustic, modal synthesis.

1 Departamento de Matem´atica Aplicada, Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Spain.

e-mail:mabermud@usc.es. Partially supported by research project PGIDT00PXI20701PR. Xunta de Galicia (Spain).

2Departamento de Matem´aticas, Universidade da Coru˜na, 15071 A Coru˜na, Spain. Partially supported by FONDAP in Applied Mathematics, Chile.

3 Departamento de Ingenier´ıa Matem´atica, Universidad de Concepci´on, Casilla 160–C, Concepci´on, Chile. Partially supported by FONDECYT Grant 1.990.346 and FONDAP in Applied Mathematics, Chile.

c EDP Sciences, SMAI 2002

The main advantage of this technique is that, instead of solving a complex problem, we first solve several simpler problems and, then, a finite dimensional problem on the whole domain, usually with low dimension and good numerical properties.

These methods have been introduced in the context of dynamical analysis of structures by Hurty in [12]

and improved by Craig and Bampton in [7]. Further, a modal synthesis method without interface associated functions has been proposed by Goldman in [8].

For the application of these methods, we must have into account that the solutions of the problems on each subdomain are not exactly known in most cases. Then, they must be approximated somehow too (for instance, using a finite element technique).

On the other hand, although the modal synthesis methods are very much used in practical computations, they do not appear frequently in the mathematical bibliography. A good introduction to their analysis can be found in [3], where some modal synthesis methods are studied for a 1D-problem. In this reference the functions of the uncoupled problems are supposed to be exactly known. In [4], the analysis is extended to the n-dimensional Laplace problem having into account a finite element discretization.

Other advantage of the component mode synthesis methods is that they allow for a good treatment of problems involving two media with different physical features. This is one reason of their importance in fluid- structure interaction problems, where they are very frequently used (see, for instance [5, 16, 18]).

In [15], a modal synthesis method is introduced to solve the elastoacoustic problem, using the non-symmetric potential/displacement formulation. The solutions are approximated by a linear combination of the lowest- frequency eigenfunctions of the fluid in a rigid cavity, the lowest-frequency eigenfunctions of the solid in vacuo, and the static responses of the fluid to the solid eigenfunctions (i.e., the solutions of the static Neumann problem in the fluid with prescribed normal displacements on the boundary induced by the solid eigenfunctions). Then, the test functions are chosen in a different space not including the static responses. The resulting coupled problem is symmetric, low-dimensional, and with good numerical properties.

In this paper we present a mathematical analysis of this method combined with finite element discretizations based on piecewise linear continuous functions in both the solid and the fluid. The resulting uncoupled problems are classical and easy to solve numerically. We restrict our presentation to two-dimensional domains, for technical reasons, but the techniques can be used in 3D situations.

The outline of the paper is as follows: we introduce the potential/displacement formulation for the elastoa- coustic problem and characterize its spectrum in Section 2. In Section 3, we introduce the spectral uncoupled problems in fluid and solid and their discretization in the corresponding finite element spaces. In Section 4 we introduce the static lifting in the fluid and its discretization. In Section 5 we define the approximate coupled problem with modal synthesis. In Section 6, we prove several intermediate theoretical results that we use for the analysis of this method in the abstract framework of [13]. In Section 7 we prove the convergence for eigen- functions and eigenvalues and obtain error estimates. Finally, in Section 8, we report a numerical test that illustrates the good performance of the method.

2. Statement of the problem

We consider the problem of determining the small-amplitude coupled motions of an inviscid barotropic fluid contained into a linear elastic structure.

Throughout this paper we use the standard notation for Sobolev spaces. We use boldface to represent linear spaces of vector fields.

Let ΩFand ΩSbe the domains occupied by fluid and solid, respectively, as in Figure 1. We suppose both are polygonal domains. Let us denote by ΓI the interface between solid and fluid and by~ν its unit normal vector pointing outwards ΩF. We assume that the exterior boundary of the solid is the union of two parts, ΓD and ΓN, and that the structure is fixed on ΓD and free of stress on ΓN. Finally let~η be the unit outward normal vector along ΓN.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

-~η

-~ν ΩF

ΩS

ΓD

ΓN

ΓI

Figure 1. Fluid and solid domains.

The governing equations for free harmonic small amplitude motions of the coupled system are (see, for instance, [15]),





















∇~p−ω²ρF~u^F=~0 in ΩF, p+ρFc²div~u^F= 0 in ΩF, div [σ(~u)] +ω²ρS~u=~0 in Ω,

~u^F·~ν =~u·~ν on ΓI, σ(~u)~ν =−p~ν on ΓI, σ(~u)~η =~0 on ΓN,

~

u=~0 on ΓD,

(2.1)

whereω is the frequency of the harmonic motion,pis the fluid pressure andcits acoustic velocity;ρF,~u^Fand ρS,~uare the densities and displacements in the fluid and the solid, respectively;σis the stress tensor which is related to~uby Hooke’s law:

σij(~u) =λS

X2 k=1

kk(~u)δij+ 2µSij(~u), i, j= 1,2.

In the previous equation,λS andµS denote the Lam´e coefficients of the solid andij(~u) the components of the infinitesimal strain tensor given by

ij(~u) =1 2

∂ui

∂xj

+∂uj

∂xi

, i, j= 1,2.

According to [15], if we assume~u^F=∇~ϕ, with R

ΩFϕdx= 0, we have p=ρFω²ϕ−ρFc²

|ΩF| Z

Γ_I

~u^F·~νdΓ, simply by using the first two equations in (2.1).

Then we can eliminatepand~u^F in (2.1) to obtain the following potential/displacement formulation,





























−ρF∆ϕ+ ρF

|ΩF| Z

ΓI

~u·~νdΓ =ω²ρF

c²ϕ in ΩF,

−div [σ(~u)] =ω²ρS~u in ΩS,

~

u·~ν = ∂ϕ

∂ν on ΓI,

σ(~u)~ν =

−ρFω²ϕ+ρFc²

|ΩF| Z

ΓI

~ u·~νdΓ

~ ν on ΓI,

~

u=~0 on ΓD,

σ(~u)~η =~0 on ΓN.

(2.2)

We emphasize the coupling condition

~

u·~ν =∂ϕ

∂ν on ΓI, (2.3)

since it plays an important role in the definition of the modal synthesis spaces.

If we denoteλ=ω², by multiplying the first two equations in (2.2) by adequate test functions and integrating by parts, it is straightforward to see that if λand (ϕ, ~u)6=

0,~0

is a solution of (2.2) then it is also a solution of the following variational spectral problem.

VP: Find a real number λ∈Rand 0,~0

6

= (ϕ, ~u)∈V, such that

a((ϕ, ~u),(ψ , ~v)) =λb((ϕ, ~u),(ψ , ~v)) ∀(ψ , ~v)∈V, where

a((ϕ, ~u),(ψ , ~v)) :=

Z

Ω_F

ρF∇~ϕ·∇~ψdx+ Z

Ω_S

σ(~u) :(~v) dx− Z

Γ_I

ρFψ~u·~νdΓ +ρFc²

|ΩF| Z

Γ_I

~ u·~νdΓ

Z

Γ_I

~v·~νdΓ, b((ϕ, ~u),(ψ , ~v)) :=

Z

ΩF

ρF

c²ϕψdx+ Z

ΩS

ρS~u·~vdx+ Z

ΓI

ρFϕ~v·~νdΓ,

and

V:= ˚H¹(ΩF)×H¹_ΓD(ΩS), with ˚H¹(ΩF) being the set of functions ψ in H¹(ΩF) with R

Ω_Fψdx= 0 andH¹_ΓD(ΩS) the set of functions in H¹(ΩS) with null trace on ΓD. We will use the following norms

kψk²F= Z

ΩF

ρF|∇~ψ|²dx forψ∈˚H¹(ΩF), k~vk²S=

Z

ΩS

σ(~v) :(~v) dx for~v∈H¹_ΓD(ΩS),

which are equivalent to the standard ones (see [17]). We will use the product norm inV k(ψ , ~v)k²V=kψk²F+k~vk²S.

We notice that, sinceaandbare not symmetric, the eigenvalues ofVPcould be, in principle, complex numbers.

However, we have the following:

Theorem 2.1. The set of eigenvalues ofVPconsists of a sequence of positive real numbers converging to+∞. All of them have finite multiplicity and their ascent is one.

Furthermore, there exist constants r > ¹₂ ands >0, depending only on the domains ΩF andΩS and on the physical parameters, such that any eigenfunction ofVP,(ϕ, ~u), satisfies

(ϕ, ~u)∈H^1+r(ΩF)×H^1+s(ΩS).

Proof. It is easy to prove (see [11]) that the potential/displacement formulation of the elastoacoustic problem is the adjoint of the pressure/displacement formulation of the same problem. Then the eigenvalues and their ascents coincide for both problems. Thus the results in [19, Sect. 8], apply to prove the characterization of the spectrum.

We can prove the additional regularity by reasoning as in [2, Lem. 6.2].

3. Uncoupled spectral problems

We will use two different finite dimensional spaces to approximateV. In order to define these spaces, we need the lowest frequency eigenfunctions of two uncoupled spectral problems in the fluid and in the solid, respectively.

In this section we introduce these problems and a finite element discretization to approximate their solutions.

We consider the following spectral problem associated with the Laplacian operator in the fluid domain with homogeneous Neumann boundary conditions.

VP^F: Findλ^F∈Rand 06=ϕ∈˚H¹(ΩF), such that Z

Ω_F

ρF∇~ϕ·∇~ψdx=λ^F Z

Ω_F

ρF

c²ϕψdx ∀ψ∈˚H¹(ΩF).

The eigenvalues ofVP^Fform an increasing sequence of real numbers going to infinity. We denote λ^F_i, ϕi

i≥1

the solutions ofVP^F, where the eigenvalues are repeated according to their multiplicities.

We consider the spectral problem associated with the linear elasticity operator in the solid domain with homogeneous Dirichlet conditions on ΓD and homogeneous Neumann conditions on ΓN, namely,

VP^S: Findλ^S∈Rand~06=~u∈H¹_ΓD(ΩS), such that Z

Ω_S

σ(~u) :(~v) dx=λ^S Z

Ω_S

ρS~u·~vdx ∀~v∈H¹Γ_D(ΩS). We denote the solutions of this problem by λ^S_m, ~um

m≥1, where the eigenvalues are repeated according to their multiplicities.

We have the followinga prioriestimate for the solutions of these problems:

Lemma 3.1. Let r andsbe the constants in Theorem 2.1. Then we have

• ϕi∈H^1+r(ΩF),∀i≥1andkϕik^1+r,ΩF ≤Cλ^F_ikϕik^0,ΩF,

• ~um∈H^1+s(ΩS),∀m≥1andk~umk^1+s,ΩS ≤Cλ^S_mk~umk^0,ΩS, where constants are independent of iandm, respectively.

Proof. It is a direct consequence of the usuala priori estimates for the Poisson’s problem and for the linear elasticity problem in a polygon (see [9]) and of the continuity of their solutions with respect to the right-hand side.

We choose the lowest frequency eigenfunctions in each uncoupled problem and define the space V^N,1=

n

ϕi,~0oNF

i=1∪ {(0, ~um)}^Nm=1^S

, whereN = (NF, NS).

Notice that any pair (ϕ, ~u)∈V^N,1 satisfies ^∂ϕ_∂ν|^ΓI= 0, but, in general,~u·ν 6= 0 on ΓI. Thus, the functions in V^N,1 do not satisfy the coupling condition (2.3).

We need to approximate the functions in V^N,1. For so doing we introduce finite element discretizations of problemsVP^F andVP^S.

Let

Th^F and

Th^S be two families of regular triangulations of ΩF and ΩS, respectively. We assume, for simplicity, that for eachhthe triangulationsTh^Fand Th^S are compatible on the contact interface ΓI.

Let

Lh(ΩF) :=

ψh∈H¹(ΩF) : ψh|T ∈ P¹(T), ∀T ∈ Th^F , Lh(ΩS) :=

vh∈H¹(ΩS) : vh|T ∈ P¹(T), ∀T ∈ Th^S · We use the following discrete spaces for fluid and solid,

V^F_h :=

ψh∈Lh(ΩF) : Z

Ω_F

ψhdx= 0

, V^S_h :=

n

~

vh∈Lh(ΩS)²: ~vh|Γ_D=~0 o·

Then the approximate uncoupled spectral problem in the fluid is VP^F_h: Findλ^F_h ∈R, 06=ϕh∈V^F_h, such that

Z

Ω_F

ρF∇~ϕh·∇~ψhdx=λ^Fh

Z

Ω_F

ρF

c²ϕhψhdx ∀ψh∈V^Fh.

Let us remark that it is not necessary to impose the zero-mean condition. More precisely,VP^F_h is equivalent to solve:

Find λ^F_h ∈R,λ^F_h6= 0, ϕh∈Lh(ΩF),ϕh6= 0, such that Z

ΩF

ρF∇~ϕh·∇~ψhdx=λ^F_h Z

ΩF

ρF

c²ϕhψhdx ∀ψh∈Lh(ΩF). (3.1) Indeed, if λ^F_h, ϕh

is solution of (3.1) thenϕhmust be orthogonal to the constant functions, since these functions constitute the eigenspace associated to the eigenvalue 0.

Let ˜N_F^hbe the number of degrees of freedom of V^F_h. We denote the discrete eigenpairs ofVP^F_hby λ^F_ih, ϕih

N^˜_F^h i=1. We define the generalized mass of each eigenfunction as

µ^F_ih= Z

Ω_F

ρF∇~ϕih²dx.

Then the following orthogonality properties are verified, Z

Ω_F

ρF∇~ϕih·∇~ϕjhdx=δijµ^F_ih, (3.2) Z

ΩF

ρF

c²ϕihϕjhdx=δij

µ^F_ih

λ^F_ih· (3.3)

Now we define the approximate uncoupled spectral problem in the solid:

VP^S_h: Find λ^S_h∈R,~06=~uh∈V^S_h, such that Z

ΩS

σ(~uh) :(~vh) dx=λ^S_h Z

ΩS

ρS~uh·~vhdx ∀~vh∈V_h^S.

Let ˜N_S^h be the number of degrees of freedom ofV^S_h. We denote the solutions of this problem λ^S_mh, ~umh

N^˜_S^h m=1. We define the generalized mass for the solid eigenfunctions

µ^S_mh= Z

ΩS

ρS|~umh|² dx.

Then

Z

Ω_S

σ(~umh) :(~unh) dx=δmnλ^S_mhµ^S_mh, (3.4) Z

ΩS

ρS~umh·~unhdx=δmnµ^S_mh. (3.5) We have the following estimates for the distance between the solutions of the continuous and the discrete uncoupled spectral problems.

Lemma 3.2. Let r andsbe the constants in Theorem 2.1. Then there exist constantsC andh0 such that the eigenvectors ϕi,i≥1and~um,m≥1can be chosen so that, for h≤h0,

(i) kϕi−ϕihkF≤Ch^rkϕik1+r,Ω_F, (ii) k~um−~umhkS≤Ch^sk~umk1+s,ΩS.

Proof. It is a direct consequence of Theorem 9.1 in [1].

Finally, we consider theNFlowest frequency eigenmodes of the fluid and theNSlowest frequency eigenmodes of the solid, withNF≤N˜_F^h andNS≤N˜_S^h, and define the finite dimensional space

V^N,1_h =n

ϕih,~0oN_F

i=1∪ {(0, ~umh)}^Nm=1^S

·

4. Static liftings

We have already remarked that the functions inV^N,1 (analogously, the functions inV^N,1_h ) do not satisfy the coupling condition (2.3). Then,V^N,1_h is not a good space to approximate the solutions ofVP. To complete this space we define the static lifting operator. Let us consider the problem

SL:Given a function~u∈H¹_ΓD(ΩS), findϕ^~u·~ν∈H¹(ΩF) as the only function in ˚H¹(ΩF) such that Z

Ω_F

∇~ϕ^~u·~ν·∇~ψdx= Z

Γ_I

ψ~u·~νdΓ ∀ψ∈˚H¹(ΩF). We notice that ^∂ϕ_∂ν^~u·~ν =~u·~ν on ΓI. Functionϕ^~u·~ν will be called static lifting of~u·~ν.

We will use the static liftings of the solid eigenfunctions. In order to simplify the notation we write ϕ^m≡ ϕ^~um·~ν.

Let

V^N,2= n

ϕi,~0oNF

i=1∪ {(ϕ^m, ~um)}^Nm=1^S

· Clearly, any (ϕ, ~u)∈V^N,2 satisfies condition (2.3).

We now introduce the discrete static lifting operator associated with the discrete solid eigenfunctions.

SLh: For 1≤m≤N˜_S^h, letϕ^m_h ∈V^F_h be the solution of Z

ΩF

∇~ϕ^m_h ·∇~ψhdx= Z

ΓI

ψh~umh·~νdΓ ∀ψh∈V_h^F, where V^F_h is the finite element space introduced in Section 3.

In the following lemma we prove ana priori estimate for the solutions of SLand an error estimate for the distance betweenϕ^mandϕ^m_h:

Lemma 4.1. Let r ands be the constants in Theorem 2.1 andt = min{r, s}. There exists C, not depending on m, such that

(i) ϕ^m∈H^1+r(ΩF),∀m≥1, and kϕ^mk1+r,ΩF ≤Ck~umk1,ΩS, (ii) kϕ^m−ϕ^m_hkF≤Ch^tk~umk1+s,ΩS.

Proof. (i) is a consequence of the standarda priori estimate for the Laplace’s equation and of the fact that

~

um·~ν∈H¹²(Γj), for any edge Γj of ΓI.

To prove (ii) we apply Strang Lemma (see, for instance, [6]):

kϕ^m−ϕ^mhkF≤C



 inf

ψ_h∈V^F_hkϕ^m−ψhkF+ sup

ψh∈V^F_h

R

Γ_Iψh~um·~νdΓ−R

Γ_Iψh~umh·~νdΓ kψhkF



·

Using Lemma 3.1 and classical approximation results we get inf

ψ_h∈V^F_hkϕ^m−ψhkF≤Ch^rkϕ^mk1+r,ΩF ≤Ch^rk~umk1,ΩS. On the other hand,

sup

ψ_h∈V^F_h

R

ΓIψh~um·~νdΓ−R

ΓIψh~umh·~νdΓ kψhkF

≤ sup

ψ_h∈V_h^F

kψhk0,ΓIk~um·~ν−~umh·~νk0,ΓI

kψhkF

≤Ck~um−~umhkS≤Ch^sk~umk1+s,ΩS. and (ii) follows sincet= min{r, s}.

We define the finite dimensional space V^N,2_h =

n

ϕih,~0oNF

i=1∪ {(ϕ^m_h, ~umh)}^Nm=1^S

·

5. Modal synthesis

Taking into account the finite dimensional spacesV_h^N,1andV^N,2_h , as defined in the previous sections, we can introduce the approximate coupled problem by modal synthesis.

VP^N_h: Findλ^N_h ∈Rand 0,~0

6

= (ϕh, ~uh)∈V^N,2_h , such that

a((ϕh, ~uh),(ψh, ~vh)) =λ^N_hb((ϕh, ~uh),(ψh, ~vh)) ∀(ψh, ~vh)∈V^N,1_h .

In the remaining of this section we deduce the matrix formulation of this spectral coupled problem (see [11] for more details).

Since (ϕh, ~uh) belongs toV^N,2_h , we have

(ϕh, ~uh) =

NF

X

i=1

αih

ϕih,~0

+

NS

X

m=1

βmh(ϕ^m_h, ~umh) =

NF

X

i=1

αihϕih+

NS

X

m=1

βmhϕ^m_h,

NS

X

m=1

βmh~umh

! ,

for some real coefficientsαihandβmhwhich are the unknowns of our problem. If we introduce this decomposition in VP^N_h and develop the bilinear forms therein, we obtain

N_F

X

i=1

αih

Z

Ω_F

ρF∇~ϕih·∇~ψhdx+

N_S

X

m=1

βmh

Z

Ω_F

ρF∇~ϕ^m_h ·∇~ψhdx+

N_S

X

m=1

βmh

Z

Ω_S

σ(~umh) :(~vh) dx

−

N_S

X

m=1

βmh

Z

ΓI

ρFψh~umh·~νdΓ +

N_S

X

m=1

βmh

ρFc²

|ΩF| Z

ΓI

~

umh·~νdΓ Z

ΓI

~vh·~νdΓ

=λ^N_h

"_NF X

i=1

αih

Z

Ω_F

ρF

c²ϕihψhdx+

NS

X

m=1

βmh

Z

Ω_F

ρF

c²ϕ^m_hψhdx+

NS

X

m=1

βmh

Z

Ω_S

ρS~umh·~vhdx

+

N_F

X

i=1

αih

Z

ΓI

ρFϕih~vh·~νdΓ +

N_S

X

m=1

βmh

Z

ΓI

ρFϕ^m_h~vh·~νdΓ

#

∀(ψh, ~vh)∈V^N,1_h .

Now we take (ψh, ~vh) = ϕjh,~0

, 1 ≤ j ≤ NF, as test function. Taking into account (3.2), (3.3) and the definitions ofϕjh andϕ^m_h, we obtain

αjhµ^F_jh=λ^N_h αjh

µ^F_jh λ^F_jh +

N_S

X

m=1

βmh

1 λ^F_jh

Z

ΩF

ρF∇~ϕ^m_h ·∇~ϕjhdx

! ,

what implies

αjhλ^F_jhµ^F_jh=λ^N_h αjhµ^F_jh+

NS

X

m=1

βmh

Z

Ω_F

ρF∇~ϕ^m_h ·∇~ϕjhdx

!

. (5.1)

Analogously, if we take (ψ , ~v) = (0, ~unh), 1 ≤n≤NS, as test function, use the equalities (3.4) and (3.5) and the definition ofϕ^m_h, we get

βnhλ^Snhµ^Snh+

NS

X

m=1

βmh

ρFc²

|ΩF| Z

Γ_I

~

umh·~νdΓ Z

Γ_I

~

unh·~νdΓ

=λ^N_h

"

βnhµ^S_nh+

N_F

X

i=1

αih

Z

ΩF

ρF∇~ϕih·∇~ϕⁿ_hdx+

N_S

X

m=1

βmh

Z

ΩF

ρF∇~ϕ^m_h ·∇~ϕⁿ_hdx

#

. (5.2) From (5.1) and (5.2) we obtain the following matrix formulation of problemVP^N_h,

K11 0 0 K22

! α β

!

=λ^Nh

M11M12

M₁₂^t M22

! α β

!

, (5.3)

where

• α= (α1h, ..., αN_Fh) andβ = (β1h, ..., βN_Sh),

• (K11)_ij =δijλ^F_ihµ^F_ih, 1≤i, j≤NF,

• (K22)_mn=δmnλ^S_mhµ^S_mh+ρFc²

|ΩF| Z

ΓI

~umh·~νdΓ Z

ΓI

~

unh·~νdΓ, 1≤m, n≤NS,

• (M11)_ij =δijµ^F_ih, 1≤i, j≤NF,

• (M12)_in= Z

Ω_F

ρF∇~ϕih·∇~ϕⁿhdx, 1≤i≤NF, 1≤n≤NS,

• (M22)_mn=δmnµ^Smh+ Z

Ω_F

ρF∇~ϕ^mh ·∇~ϕⁿhdx, 1≤m, n≤NS.

We notice that both block matrices in (5.3) are symmetric. Typically, NS and NF are small numbers in applications. Hence, this is a low dimension eigenproblem.

We prove in the following lemma that the matrix on the left-hand side is positive definite. Then, the numerical solution of (5.3) is very simple.

Lemma 5.1. The matrix

K11 0 0 K22

! ,

as defined in (5.3), is positive definite.

Proof. For anyα∈R^NF andβ∈R^NS, (α, β)6= (0,0), (α β)

K11 0 0 K22

α β

=

NF

X

i=1

α²ihλ^Fihµ^Fih+

NS

X

m=1

βmh² λ^Smhµ^Smh

+

N_S

X

m,n=1

ρFc²

|ΩF|βmhβnh

Z

ΓI

~

umh·~νdΓ Z

ΓI

~

unh·~νdΓ.

The sum of the first two terms is clearly strictly positive, whereas the third one satisfies

NS

X

m,n=1

ρFc²

|ΩF|βmhβnh

Z

Γ_I

~

umh·~νdΓ Z

Γ_I

~

unh·~νdΓ = ρFc²

|ΩF| Z

Γ_I NS

X

m=1

βmh~umh·~νdΓ

!²

≥0. 2

6. Preliminary theoretical results

We will use the results in [13] to estimate the error arising from the approximation of problem VPby the finite dimensional problemVP^N_h. We prove the hypotheses of Theorems 6.3, 6.4, 6.6, and 6.7 in [13]. Then we apply the results from this reference to our problem in Theorems 6.9 and 6.10 below.

To simplify the proofs, we assume that the uncoupled continuous and discrete eigenmodes are now normalized in such a way that

Z

ΩF

ρF

c²ϕ²_idx= 1, Z

ΩF

ρF

c²ϕ²_ihdx= 1, Z

Ω_S

ρS|~um|²dx= 1, Z

Ω_S

ρS|~umh|² dx= 1.

Firstly we prove thatV^N,2_h approximates correctlyV, whenNF, NS→ ∞andh→0.

Lemma 6.1. The linear combinations of the functions n ϕi,~0o

i≥1∪ {(ϕ^m, ~um)}m≥1 are dense inV.

Proof. Let (ϕ, ~u) be an arbitrary element ofV.

Since

√1 λ^S_m~um

m≥1

is a Hilbert basis ofH¹_ΓD(ΩS) with respect to the normk·kS, we have

~u= X∞ m=1

1 λ^S_m

Z

ΩS

σ(~u) :(~um) dx

~ um=

X∞ m=1

Z

ΩS

ρS~u·~umdx

~

um, in H¹(ΩS). We denoteβm=R

ΩSρS~u·~umdx.

Letϕ^~u·~ν be the static lifting associated to ~u·~ν, as defined in SL. From the linearity and continuity of the static lifting operator,ϕ^~u·~ν =P_∞

m=1βmϕ^min H¹(ΩF).

Letαi be the Fourier coefficients of ϕ−ϕ^~u·~ν in the Hilbert basis

√1 λ^F_iϕi

i≥1

:

αi= 1 λ^F_i

Z

ΩF

ρF∇~ ϕ−ϕ^~u·~ν

·∇~ϕidx= Z

ΩF

ρF

c² ϕ−ϕ^~u·~ν ϕidx.

Then we have

(ϕ, ~u) =

ϕ−ϕ^~u·~ν,~0

+ ϕ^~u·~ν, ~u

= X∞ i=1

αi

ϕi,~0

+ X∞ m=1

βm(ϕ^m, ~um) in V.

Lemma 6.2. For any {αi}i≥1 and {βm}m≥1 sequences of real numbers, the numerical series X∞ i=1

α²_iλ^F_i and X∞

m=1

β_m²λ^S_m converge if and only if X∞ i=1

αi

ϕi,~0

and

X∞ m=1

βm(ϕ^m, ~um)converge in V.

Furthermore, there exist two strictly positive constants C1 andC2, such that

C1

"_∞ X

i=1

α²_iλ^F_i + X∞ m=1

β_m²λ^S_m

#

≤

X∞ i=1

αi

ϕi,~0

+ X∞ m=1

βm(ϕ^m, ~um)

2

V

≤C2

"_∞ X

i=1

α²_iλ^F_i + X∞ m=1

β_m²λ^S_m

# .

Proof. For anyψ∈˚H¹(ΩF) and~v∈H¹_ΓD(ΩS) we have kψk²F≤2ψ+ϕ^~v·~ν2

F+ 2ϕ^~v·~ν2

F≤2ψ+ϕ^~v·~ν2

F+Ck~vk²S. Then

kψk²F+k~vk²S≤C˜ψ+ϕ^~v·~ν2

F+k~vk²S

= ˜C ψ+ϕ^~v·~ν, ~v²

V. On the other hand, we have

ψ+ϕ^~v·~ν, ~v²

V= ψ,~0

+ ϕ^~v·~ν, ~v²_V≤2

ψ,~0²_V+ 2 ϕ^~v·~ν, ~v²

V≤C˜˜

kψk²F+k~vk²S

.

From the definition of k · k^F and k · k^S and the normalization of the uncoupled eigenfunctions ϕi and ~um, kϕik^F = p

λ^F_i and k~umk^S = p

λ^Sm. Then the lemma follows by taking C1 = _C¹_˜, C2 =C,˜˜ ψ = PI

i=1αiϕi,

~ v=PM

m=1βm~um, and then letting I, M → ∞.

As a direct consequence of Lemmas 3.2, 4.1, 6.1, and 6.2 we have the following:

Theorem 6.3. Let (ϕ, ~u)∈V, then inf

(ϕ_h,~u_h)∈V^N,2_h k(ϕ, ~u)−(ϕh, ~uh)kV→0, asNF,NS→ ∞and h→0.

In the following theorem we prove that the bilinear formasatisfies two inf-sup conditions.

Theorem 6.4. We have

{(ϕ,~u)∈V:infk(ϕ,~u)kV=1} sup

{(ψ,~v)∈V:k(ψ,~v)kV=1}|a((ϕ, ~u),(ψ , ~v))|=α >0, (6.1) sup

{(ϕ,~u)∈V: (ϕ,~u)6=0}|a((ϕ, ~u),(ψ , ~v))|>0 ∀(ψ , ~v)∈V, (ψ , ~v)6= 0,~0

. (6.2)

Proof. Let (ϕ, ~u)∈V. Let ˜ϕbe the only solution in ˚H¹(ΩF) of the variational problem Z

Ω_F

∇~ϕ˜·∇~ψdx= Z

Ω_F

∇~ϕ·∇~ψdx− Z

Γ_I

ψ~u·~νdΓ ∀ψ∈˚H¹(ΩF).

Then Z

ΩF

∇~ϕ·∇~ψdx= Z

ΩF

∇~ϕ˜·∇~ψdx+ Z

ΓI

ψ~u·~νdΓ ∀ψ∈˚H¹(ΩF), (6.3) and hence there exists a constantC >0 such that

k(ϕ, ~u)kV≤Ck( ˜ϕ, ~u)kV. Thus we have

a

(ϕ, ~u) k(ϕ, ~u)kV

, ( ˜ϕ, ~u) k( ˜ϕ, ~u)kV

= 1

k( ˜ϕ, ~u)kVk(ϕ, ~u)kV

Z

Ω_F

ρF∇~ϕ˜² dx+ Z

Ω_S

σ(~u) :(~u) dx +ρFc²

|ΩF| Z

Γ_I

~ u·~νdΓ

2!

≥ 1

k(ϕ, ~u)kV

k( ˜ϕ, ~u)kV≥ 1 C >0,