An adaptive ﬁnite element scheme to solve ﬂuid-structure vibration problems on non-matching grids

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Introduction
The computation of the vibration modes of an elastic solid interacting with an acoustic fluid (i.e., the elastoacoustic vibration problem) occurs in many engineering applications.A general overview of this and other fluid-structure interaction problems can be found in the monographs by Morand and Ohayon [16], and Ohayon and Soize [17], where numerical methods and further references are also given.
The present paper deals with an adaptive finite element scheme to solve the elastoacoustic vibration problem on meshes for fluid and solid domains which do not necessarily match on the fluid-solid interface.Such non-matching grids turn out particularly useful when adequately refined meshes are used, since different refinement levels are usually needed on each medium.This is specially relevant when the solution has interface singularities.In fact, such singularities use to appear at the interface corners and, obviously, the reentrant corners of the fluid domain are convex for the solid, and conversely.Thus, any adaptive scheme producing correctly refined grids would require differently refined meshes on each medium.It is clear that this can be more efficiently done if no grid matching is imposed on the interface.
We consider a method introduced in [10] to solve the elastoacoustic vibration problem.It consists of discretizing a pure displacement formulation with different elements for each medium: Raviart-Thomas elements for the fluid and standard piecewise linear elements for the solid, the coupling of both on the interface being of non conforming type.
The use of a displacement formulation presents many attractive features (see for instance [22]), but one main drawback: standard finite element methods produce spurious modes with positive vibration frequencies interspersed among the real ones (see [13,15]).Instead, the method we use has been proved in [5] to be free of spurious modes.Optimal order a priori error estimates for this method have been also proved in [5] and [21], when the meshes on each domain match on the interface.Some extensions to other fluid-structure interaction problems can be found in [6,7,9], but in all cases for matching grids.Numerical experiments showing the effectiveness of this method on non-matching grids have just been recently reported in [8].
In the present paper we consider an extension of this method to non-matching grids and prove optimal order a priori error estimates.We also report numerical experiments confirming the theoretical results and giving evidence of the advantage of using differently refined meshes for each medium.Finally, we introduce an adaptive scheme to automatically design correctly refined meshes in order to attain a prescribed accuracy with lowest computational effort.
This scheme is based on an a posteriori error indicator introduced and analyzed in [1].In this reference, an error estimator for Raviart-Thomas eigenvalue problems ( [3,11]) is combined with other one for elastic vibrations ( [2]) on matching grids.This combined error indicator has been proved in [1] to be efficient and reliable; i.e., it provides an upper estimate of the global error and local lower error estimates, up to multiplicative constants independent of the mesh-size and of the particular eigenvalue.Thus, it can be safely used to guide an adaptive procedure.
However, to obtain maximum advantage of such procedure, the adaptive strategy should allow for grids not matching on the interface.We consider a straightforward extension of the error indicator in [1] to this kind of meshes and present numerical results showing that this approach is highly efficient.
The outline of the paper is as follows.In Sect. 2 we recall the pure displacement formulation of the elastoacoustic vibration problem and the theoretical results we need about spectral characterization and further regularity.In Sect. 3 we present our finite element method for non-matching grids and prove optimal order a priori error estimates for the approximate eigenfunctions and a double order for the eigenvalues.In Sect. 4 we discuss some implementation issues and report numerical experiments on non-matching grids.In Sect. 5 we present the a posteriori error indicator and the adaptive scheme; we also report numerical experiments where an optimal order of convergence is attained.

Statement of the problem
We consider the problem of determining the free vibration modes of a linear elastic structure containing an ideal acoustic (barotropic, inviscid, and compressible) fluid.Our model problem consists of a 2D polygonal vessel completely filled with fluid as that in Fig. 1.Throughout this paper we will use the standard notation for Sobolev spaces, norms and seminorms.We also denote H 1

Γ D
(Ω S ) the closed subspace of functions in H 1 (Ω S ) vanishing on Γ D , and We use the following notations for the physical magnitudes; in the fluid: -u: the displacement vector, p: the pressure, -ρ F : the density, c: the acoustic speed, and in the solid: -w: the displacement vector, -ρ S : the density, -λ S and µ S : the Lamé coefficients, -ε(w): the strain tensor defined by σ(w): the stress tensor which is related to the strain tensor by Hooke's law: The classical elastoacoustics model for small amplitude motions yields the following eigenvalue problem for the free vibration modes of the coupled system and their corresponding vibration frequencies ω (see, for instance, [16]): 2 , and p ∈ H 1 (Ω F ), (u, w, p) = (0, 0, 0), such that: The coupling between fluid and structure is taken into account by equations ( 4) and (5).The first one relates the solid normal stresses on the interface with the pressure exerted by the fluid.The second one means that fluid and solid are in contact at the interface.Both equations can be understood in the L 2 sense since p| Γ I and w • n, both belong to L 2 (Γ I ).
We introduce appropriate functional spaces to obtain a variational formulation for this problem.Let 2 , both endowed with the respective product norms.We denote by the closed subspace of X corresponding to the admissible coupled fluid-solid displacements.
Let a and b be the symmetric continuous bilinear forms defined by a ((u, w), (v, z)) := 2 , respectively, using (2) to eliminate the pressure p in terms of the fluid displacement field u, and denoting λ = ω 2 , the following symmetric variational spectral problem is obtained: Problem 2. Find λ ∈ IR and (u, w) ∈ V, (u, w) = (0, 0), such that:

By integrating by parts (1) and (3) multiplied by
It was shown in [5] that the problem above has two types of solutions: 1. λ 0 = 0, with corresponding eigenspace 2. a sequence of finite multiplicity eigenvalues λ n > 0, n ∈ IN, diverging to +∞, with corresponding eigenfunctions The infinite dimensional eigenspace K associated to λ 0 = 0 consists of pure rotational motions which induce neither fluid pressure variations nor structural vibrations.They are solution of the present formulation of our coupled problem, because no irrotational constraint was imposed on the fluid displacements allowed in the model.
The second set of eigenfunctions (i.e., those corresponding to λ n > 0) is a complete orthogonal system of the subspace G of V consisting of all the coupled displacement fields which are conservative in the fluid, namely: Notice that G and K are orthogonal in both, H and X inner products.
Given (u, w) ∈ G, let ϕ ∈ H 1 (Ω F ) be such that u = ∇ϕ; then, the fluid potential ϕ is a solution of the compatible Neumann problem: This problem attains a unique solution ϕ ∈ H 1 (Ω F )/IR and there holds As an immediate consequence of this estimate and Korn's inequality, the bilinear form a is X elliptic on G. Hence, we are able to define the linear bounded operator T : It is simple to show that λ is a positive eigenvalue of Problem 2 if and only if µ = 1 λ is a positive eigenvalue of the operator T , and the corresponding associated eigenfunctions coincide.
The following theorem, which shows that T is a regularizing operator, was proved in [5]: Theorem 1.There exist constants s ∈ ( 1 2 , 1], t ∈ (0, 1], and 2 , and The constant s is either 1, if Ω F is convex, or any s < π θ − , with θ being the largest reentrant corner and any positive number, otherwise.The constant t depends on the reentrant angles of Ω S , on the angles between Γ D and Γ N , and on the Lamé coefficients (see [12]).
The theorem above yields, as a by-product, a similar estimate for the eigenfunctions with ( f , g) replaced by 1  λ (u, w), which, in its turn, yields the equivalence of Problems 1 and 2 (see [5]).Remark 1.The additional regularity of u stated in Theorem 1, actually holds for any (u, w) ∈ G.In fact, let ϕ ∈ H 1 (Ω F )/IR be the solution of ( 9)- (10).Then, because of the standard a priori estimate for this Neumann problem, ϕ ∈ H 1+s (Ω F )/IR and with s ∈ ( 1 2 , 1] as described above (see [12]).

Finite element discretization on non-matching grids
In spite of the fact that the rotational eigenmodes corresponding to λ 0 = 0 are not physically relevant, any suitable numerical approximation of Problem 2 should take care of them.Otherwise spurious modes might appear.This is what happens, for instance, when continuous piecewise linear finite elements are used for both, fluid and solid displacements (see [13,15]).
A finite element method free of spurious modes was introduced in [10] and further analyzed in [5] and [21], where optimal a priori error estimates were obtained.It consists of using meshes on both domains matching on the interface and lowest-order Raviart-Thomas elements for the fluid, coupled in a non conforming way with piecewise linear elements for the solid.
However, when adequately refined meshes are used on each medium, the use of matching triangulations is a severe constraint.In what follows we extend the results in the above mentioned papers to the case of non-matching grids.
Let T F h and T S h be two families of regular triangulations of Ω F and Ω S , respectively, such that the end points of Γ D coincide with vertices of T S h .We denote by h the maximum mesh-size of both triangulations.Since we do not assume that T F h and T S h match on the fluid-solid interface, then both meshes induce different partitions of Γ I .We denote by T I h the one induced by T F h , namely: For each component of the solid displacements we use the standard piecewise linear finite element space and, for the fluid, the Raviart-Thomas space ( [20]) The degrees of freedom in R h (Ω F ) are the (constant) values of the normal component of u h along each edge of the triangulation.The discrete analogue of X is The conforming finite element spaces V ∩ X h are too restrictive for this problem, since any pair of displacements in V ∩ X h has constant normal components along each whole edge of the polygonal interface Γ I .Instead we use as discrete space Therefore, we obtain the following discrete analogue of Problem 2: Let us remark that, for and the method turns out to be non conforming.
Proceeding as in [5], it is simple to prove that Problem 3 has also two kind of solutions: 1. λ 0 = 0, with corresponding eigenspace 2. a finite set of positive eigenvalues λ hn , with corresponding eigenfunctions (u hn , w hn ) ∈ G h , where G h is the orthogonal complement of K h in V h .Notice that that K h and G h are also orthogonal in the H inner product and, hence, The null space of the discrete problem, K h , provides a good approximation of that of the continuous one, K.In fact, K and K h both consist of fluid displacements of the form curl ζ, with ζ being constant on each connected component of Non existence of spurious modes and optimal-order spectral convergence of the solutions of Problem 2 to those of Problem 3 have been proved in [5] and [21], in the case of triangulations matching on the fluid-solid interface.In what follows we extend these results to the case of non-matching grids.
In order to introduce a discrete analogue of the operator T , we will use the following lemma regarding a Helmholtz decomposition of functions in with (∇ξ, w h ) ∈ G and div χ = 0.Moreover, there exists a constant C, independent of h and the particular and where r := min{s, t}, with s and t being the constants in Theorem 1. Proof.It is essentially identical to those of decomposition (5.5) in [5] and the corresponding estimates, which are given in the proofs of Theorem 5.4 and Lemma 5.5 of that reference.
As an immediate consequence of the previous lemma, the bilinear form a is X elliptic on G h .Therefore, we are able to define the (uniformly) bounded linear operators As in the continuous case, it is simple to show that λ h is a positive eigenvalue of Problem 3 if and only if µ h = 1 λ h is a positive eigenvalue of the operator T h , and the corresponding associated eigenfunctions coincide.
To prove that the eigenvalues and eigenfunctions of Problem 3 approximate those of Problem 2, we will make use of the standard spectral approximation theory stated in [4].To this goal we first need to show that T h converge in norm to T .Since ( 11) is a non-conforming discretization of (8), we will need the estimates for the corresponding consistency and approximation terms provided in the following two lemmas.From now on, r denotes the constant in Lemma 1 and C a strictly positive constant, not necessarily the same at each occurrence, but always independent of the mesh-size h.
and the lemma follows from estimate (13).
Proof.Let w I ∈ L h,Γ D (Ω S ) 2 be the Lagrange interpolant of w.
The standard error estimate yields the latter because of Theorem 1 (t being the constant in that theorem).Let u I ∈ R h (Ω F ) be the Raviart-Thomas interpolant of u, conveniently modified on the edges ∈ T I h to enforce (u I , w I ) to belong to V h .That is, for all the edges of triangles T ∈ T F h , we define the degrees of freedom of u I by: where n is a unit normal to .Proceeding as in Sect. 5 of [5], it is simple to prove that the latter because of Theorem 1 (s and t being the constants in that theorem).Let P : V h −→ K h ⊂ V h be the projection onto K h in the inner product of X.Let ( u, w) := (u I , w I ) − P(u I , w I ) ∈ G h ; then ( u, w) and P(u I , w I ) are orthogonal in X.On the other hand, since P(u I , w I ) ∈ K h ⊂ K, then (u, w) and P(u I , w I ) are also orthogonal in X.Hence, the latter because of estimates ( 15) and ( 16).Now we are able to conclude the convergence of T h to T : Lemma 4.There exists a positive constant C such that Proof.This is easily obtained from the two previous lemmas and standard techniques for non conforming methods.In fact, let (u, w) = T( f , g) and (u h , w h ) = T h ( f , g).Let ( u, w) ∈ G h as in Lemma 3.There holds Because of the X ellipticity of a on G h , we have the latter because of the continuity of a and Lemma 2. By combining the two previous inequalities we obtain the last inequality because of Lemma 3.
As an immediate consequence of the lemma above, we obtain that T − T h X and T − T h H , both converge to zero as h goes to zero.Thus, we are in order to use the spectral approximation theory to prove optimal order convergence for the eigenfunctions.
Let µ > 0 be a fixed eigenvalue of T with algebraic multiplicity m.Let E be the corresponding associated eigenspace.Since T − T h X → 0, then there exist m eigenvalues of T h , µ (1)  h , . . ., µ (m)  h (repeated accordingly to their respective multiplicities) converging to µ (see [14]).Let E h be the direct sum of the corresponding associated invariant subspaces.
We recall the definition of the gap δ between two closed subspaces, Y and Z, of X: The following theorem implies an optimal order of convergence for the eigenfunctions: Theorem 2. There exists a positive constant C such that Proof.It is an immediate consequence of Theorem 7.1 in [4] and Lemma 4.
In order to prove an optimal-order error estimate for the approximate eigenvalues, we will use the following lemma: Consider the Helmholtz decompositions as in Lemma 1.Then, proceeding as in the proof of Lemma 2 we obtain On the other hand, since div χ = 0, (∇ξ, w h ) ∈ G, and a and b are symmetric, we have Thus, subtracting the first equation above from the second one, we obtain The first term in the right hand side above is easily bounded by using the continuity of a and Lemma 4: Therefore, we only need to prove similar estimates for the two integrals to conclude the lemma.Let us consider the first one (for the second integral the proof runs identically).Since the latter because (∇ ξ, w h ) ∈ G ⊂ V. Now we proceed essentially as in the proof of Lemma 3.1 in [21].Let and P be the orthogonal projection of L 2 (Γ Hence, by using this in the previous equation and standard properties of the L 2 projection, we obtain Hence, because of the error estimate for the L 2 projection and the trace theorem, we have On the other hand, using once more the error estimate for the L 2 projection and the trace theorem, we have the last inequality because of Theorem 1, Lemma 4, and the fact that r ≤ t ≤ min{1, 1 2 + t}.Thus, as a consequence of the last three inequalities, which allows us to conclude the lemma.Now we may prove a double order of convergence for the eigenvalues.Let λ = 1/µ be a positive eigenvalue of Problem 2, and λ (i)  h = 1/µ (i) h , i = 1, . . ., m, be the eigenvalues of Problem 3 converging to λ.The following error estimate holds: Theorem 3.There exists a positive constant C such that Proof.It is an immediate consequence of the previous lemma, Lemma 4, and Remark 7.5 in [4].

Numerical results
In this section we present a numerical test to confirm the theoretical results of the previous section.This test also illustrates the advantage of using non-matching grids.First, we discuss some implementation issues.
In order to solve Problem 3, the kinematic constraint h in the definition of the discrete spaces V h can be conveniently imposed by means of a piecewise constant Lagrange multiplier.In fact, consider the following discrete mixed spectral problem where C h is the space defined by ( 17): Remark 2. Proceeding as in Sect.8 of [6], it can be shown that γ h is as an approximation of the interface pressure p| Γ I .This will be used in the next section to define an a posteriori error indicator on the triangles with edges lying on Γ I .
The following lemma shows the equivalence of Problems 4 and 3

Lemma 6. (λ h , (u h , w h )) ∈ IR × V h is a solution of Problem 3 if and only if there exist (a unique)
Proof.Clearly, any solution of ( 18)-( 19) provides a solution of (11).
To prove the converse, let (λ h , (u h , w h )) be a solution of (11); equation ( 19) is satisfied since (u h , w h ) ∈ V h , whereas equation ( 18) is also true for (v h , z h ) ∈ V h , independently of the particular value of γ h ∈ C h .Let ϕ denote the nodal basis functions of R h (Ω F ) associated with the edge .Then Thus, it is enough to prove that there exists a unique γ h ∈ C h such that (18) holds for To prove this, let {χ : ∈ T I h } denote the canonical basis of C h (i.e., χ | ≡ 1 and χ | Γ I \ ≡ 0).Then, by writing the unknown multiplier in this basis, is clear that it is enough to verify that there are coefficients c such that (18) holds for and these equations are always uniquely solvable.
Problem 4 is the one we have implemented for our experiments.The only difference with the implementation of the same method on matching grids is the computation of the terms Γ I γ h z h • n.In fact, γ h is constant on each edge ⊂ Γ I of triangles T ∈ T F h , whereas z h • n is linear on each edge ⊂ Γ I of triangles T ∈ T S h .Thus, two different partitions of Γ I have to be taken into account to compute this integral (T I h and the one induced by T S h on Γ I ).We have tested our method on different non-matching grids with a simple structural-acoustic vibration problem taken from [10].We compare the obtained results with those of this reference, where the same method had been used on matching grids.
We have computed the lowest-frequency vibration modes of a closed (two-dimensional) steel vessel completely filled with water.Figure 2 shows the geometrical data of the vessel.Figure 3 shows the initial meshes for both domains.
All the meshes we have used for this test have been obtained by uniformly subdividing these initial meshes.The refinement parameters N F and N S indicate that each triangle in the initial mesh has been uniformly subdivided into N 2 F (resp.N 2 S ) similar triangles (see Fig. 4 for an example).Notice that when N F = N S the triangulations match on the fluid-solid interface.These are the meshes used in [10] (where a static condensation technique has been used to impose the kinematic constraint, instead of the Lagrange multiplier) and we have exactly reproduced the results in this reference with our approach.
We have considered three families of meshes: .. The first one, M1, corresponds to matching grids.
For each mode, we have computed the vibration frequency ω h = √ λ h on these meshes.To estimate the error of ω h , we have used as "exact" eigenfrequency, a very accurate approximation ω ex , obtained by a least squares fitting of the model ω h ≈ ω ex + CM −α to the values ω h computed on highly refined meshes, with M being the number of degrees of freedom (d.o.f.).
We show the results obtained for two different vibration modes: the ones denoted S 1 and F 11 in [10].The first one is a "solid mode" (i.e., a vibration mode attained for the solid in vacuo, that in our coupled problem appears modified by the presence of the fluid).The second one is a "fluid mode" (i.e., a vibration mode attained for the fluid in a rigid cavity, that in our coupled problem appears modified by the elastic response of the structure).Figures 5 and 6 show the deformed structure, the fluid displacement field, and the pressure field for each of these two modes.It can be clearly observed from Figs. 7 and 8 that the order of convergence is the same for the three choices of meshes (i.e., the slopes of the curves in the log-log plot are the same for large values of the number of d.o.f.).However, the nonmatching grids produce significantly better results than the matching meshes.
This behavior can be theoretically anticipated since the solid domain has reentrant corners and varying boundary conditions on Γ D ∪ Γ N .Instead, the fluid domain is convex.Therefore, the proof of Theorem 1 (see [5,21]) shows that at least u ∈ H 1 (Ω F ) 2 and div u ∈ H 2 (Ω F ), whereas w ∈ H 1+t (Ω S ) 2 for t ≈ 0.68 (see [12]).Therefore, Raviart-Thomas elements are able to approximate u with errors O(h) in • div,Ω F , whereas piecewise linear elements can only approximate w with errors O(h t ) in • 1,Ω S .Thus, more refined meshes in  the solid should improve the results, and this is what actually happens as can be seen in Figs.7 and 8.
Anyway, since the solid displacement field is singular at the corners of Ω S , the best strategy would be to combine the method with an a posteriori error indicator to design adequately refined meshes.This topic will be considered in more detail in the following section.

An adaptive scheme for non-matching grids
The a priori estimates proved in Sect. 3 are valid for any regular family of meshes.However, from the point of view of applications, it is highly important to be able to design meshes correctly refined to reduce the approximation errors as much as possible with the lowest computational effort.
The standard approach to attain this goal is to compute an approximation of the eigenpair of interest on an initial coarse mesh, and then to use it to compute estimates of some local measure of the error for each element, to know which of them should be further refined.
We use a weighted H(div, Ω F ) × H 1 (Ω S ) 2 norm on X to measure the error of the computed eigenfunction (u h , w h ): The weight in the H(div, Ω F ) norm is chosen to control the error of the approximate pressure in L 2 (Ω F ) norm: p 0,Ω F = ρ F c 2 div (u − u h ) 0,Ω F .The other weight is chosen to scale accordingly the H 1 (Ω S ) seminorm of the solid displacements.
An error estimator for this method on matching grids was introduced and analyzed in [1].This definition extends directly to non-matching grids.The local error indicators η T for each element T are defined differently on each medium: -for each element T ∈ T F h , we estimate the local error ρ F c 2 u − u h div,T by means of the error indicator η T = η F T defined by , with E T := { edge of T } and where t is a unit vector tangent to and • denotes the jump across the edge .
-for each element T ∈ T S h , we estimate the local error 2µ S w − w h 1,T by means of the error indicator η T = η S T defined by , with E T defined as above and where n is a unit vector normal to the edge .Notice that the term J S is also a residual for ⊂ Γ I , since, according to Remark 2, γ h is an approximation of the interface pressure p| Γ I .
When dealing with non-matching grids some extra work is needed to compute the jumps on the fluid-solid interface.In fact, in the computation of J F for ⊂ Γ I , we have to integrate w h • n on , which is in general piecewise linear on this edge.The same care must be taken to compute J S for ⊂ Γ I , since γ h is only piecewise constant on .
It was proved in [1] that these indicators satisfy the following properties, when meshes matching on the fluid-solid interface are used: -they provide upper estimates of the global error: where h.o.t.denotes higher order terms (i.e., terms which become negligible in comparison with the other ones in the estimate, when the mesh-size becomes smaller); -they provide local lower error estimates, which allow us to know the elements that should be refined: where T is the union of T and the neighboring elements in T F h or T S h sharing an edge with T ; -the effective computation of η S T and η F T is inexpensive in comparison with the overall computation of (u h , w h ) and λ h .The constants C 1 and C 2 in the estimates above are strictly positive and independent of the corresponding eigenvalue and of the mesh-size, provided the used meshes satisfy the usual minimum angle condition.We refer to [1] for further details.
We have combined these error estimators with a standard refinement strategy to design an adaptive scheme.This strategy consists of subdividing by the longest edge those elements T such that with β ∈ (0, 1) being a parameter fixed in advance (see [19] for details).For our tests we have chosen β = 0.7.
As usual, to preserve the conformity of each separate mesh, T F h and T S h , whenever a triangle T with longest edge ⊂ ∂Ω F (resp.⊂ ∂Ω S ) is subdivided, the triangle T sharing the edge is subdivided too (see [19] again for details).Since the meshes T F h and T S h do not need to match on the fluid-solid interface, this has not been done when the longest edge ⊂ Γ I .
As a first test, we have applied this adaptive scheme to compute the lowest-frequency vibration mode S 1 of the problem in the previous section.We have used as initial mesh the uniform matching grids corresponding to N F = N S = 2.After several iteration steps of this adaptive procedure, no refinement of the mesh on the fluid domain was needed.In fact, the largest indicators always occurred on elements in the solid domain.Figure 9 shows the adaptively created mesh T S h at the end of the process.
Figure 10 shows a log-log plot of the error of the computed eigenvalues |λ h − λ ex | in terms of the number M of d.o.f.The "exact" eigenvalue λ ex and the order of convergence α have been again obtained by a least squares fitting of the model λ h ≈ λ ex + CM −α to the computed values λ h .To do this, we have neglected the approximate eigenvalues computed with the coarsest meshes.Figure 10 also shows the fitted line, whose negative slope is α = 1.069.This almost coincide with the optimal value α = 1 (namely, the one corresponding to uniform meshes for a regular solution) showing Fig. 9. Mode S 1 : final mesh on the solid domain that the adaptive procedure converges with an optimal order O(M −1 ).
Figure 11 shows the evolution of the squared global error estimator The slope of the linear fit is in this case α = 0.984, which also shows an optimal order of convergence for the approximation of the eigenfunction.
As stated above, no mesh refinement on the fluid domain has been needed in this test, since the fluid displacements are quite regular because of the geometry of this domain.In order to assess the quality of the estimator for the fluid variables, a second test has been performed on a different geometry.
We have applied our adaptive scheme to other vibration elastoacoustic problem with eigenfunctions having strong singularities in both domains.We have considered the L-shaped domains for fluid and solid shown in Fig. 12.
We have also considered more general boundary conditions on the fluid domain, so that the eigenfunctions attain stronger singularities.Firstly, one part of the boundary, Γ R , is assumed to be perfectly rigid.Secondly, other part, Γ F , is assumed to be a free surface of the fluid.As we show below, the formulation, discretization and error estimators can be readily extended to this case.
Notice that, in spite of the fact that Γ F coincides with part of the solid Neumann boundary Γ N , no fluid contact is assumed there.Thus, the fluid domain has a reentrant corner on the vertex Γ I ∩ Γ F with different boundary conditions on the   The definition of the terms J F extends to include these boundary conditions as follows (see [18] for further details): We have used the same physical parameters as in the previous example and coarse initial uniform matching meshes for fluid and solid domains (70 d.o.f. in T F h and 60 d.o.f. in T S h ).We have applied several steps of our adaptive scheme to compute the lowest-frequency vibration mode of the coupled system, which we denote L 1 .
Figures 13 and 14 show the meshes T F h and T S h , respectively, attained at the end of the process.It can be seen that, in this test, both meshes have been effectively refined, particularly in the neighborhood of the respective reentrant corners.Figure 14 also shows the deformed structure, whereas Fig. 15 shows the fluid pressure field computed with these meshes.Figure 17 shows the evolution of the squared global error estimator.The slope of the linear fit is α = 0.945, showing again an optimal order of convergence.
It can be observed that the estimator lost monotonicity and began to behave somewhat erratically at the end of the process.However, this does not affect significantly the approximation of the eigenvalue depicted in Fig. 16.

Conclusions
We have extended the analysis of a spurious-modes free finite element method for the computation of elastoacoustic vibration modes, to cover the case of meshes non-matching on the fluid-solid interface.The obtained theoretical results coincide with those previously known for matching grids.
We have reported some numerical experiments exhibiting the advantage of avoiding mesh matching.This is particularly useful when adequately refined meshes are used to solve the singularities of the solution on each domain (which in general do not coincide).
We have adapted an a posteriori error indicator for this problem to be used with non-matching grids.We have reported numerical experiments showing the effectiveness of an adaptive scheme based on these indicators, although there are no proofs yet of their efficiency and reliability.This should be a subject of future research.

Figures 7
Figures 7 and 8 show log-log plots of the relative error of the computed vibration frequency, |ω h − ω ex |/ω ex , versus the number M of d.o.f. for all the meshes.It can be clearly observed from Figs. 7 and 8 that the order of convergence is the same for the three choices of meshes (i.e., the slopes of the curves in the log-log plot are the same for large values of the number of d.o.f.).However, the nonmatching grids produce significantly better results than the matching meshes.This behavior can be theoretically anticipated since the solid domain has reentrant corners and varying boundary conditions on Γ D ∪ Γ N .Instead, the fluid domain is convex.Therefore, the proof of Theorem 1 (see[5,21]) shows that at least u ∈ H 1 (Ω F ) 2 and div u ∈ H 2 (Ω F ), whereas w ∈ H 1+t (Ω S ) 2 for t ≈ 0.68 (see[12]).Therefore, Raviart-Thomas elements are able to approximate u with errors O(h) in • div,Ω F , whereas piecewise linear elements can only approximate w with errors O(h t ) in • 1,Ω S .Thus, more refined meshes in

Fig. 12 .
Fig. 12. Geometrical data for the second test

Figure 16
Figure16shows the log-log plot of the error of the computed eigenvalues in terms of the number of d.o.f. and the linear fit obtained as in the previous test.The slope of the linear fit is in this case α = 1.030, showing once more that the adaptive procedure converges with optimal order.Figure17shows the evolution of the squared global error estimator.The slope of the linear fit is α = 0.945, showing again an optimal order of convergence.It can be observed that the estimator lost monotonicity and began to behave somewhat erratically at the end of the process.However, this does not affect significantly the approximation of the eigenvalue depicted in Fig.16.