A priori and a posteriori error estimates for a virtual element spectral analysis for the elasticity equations
We present a priori and a posteriori error analysis of a Virtual Element Method (VEM) to approximate the vibration frequencies and modes of an elastic solid. We analyze a variational formulation relying only on the solid displacement and propose an -conforming discretization by means of VEM. Under standard assumptions on the computational domain, we show that the resulting scheme provides a correct approximation of the spectrum and prove an optimal order error estimate for the eigenfunctions and a double order for the eigenvalues. Since, the VEM has the advantage of using general polygonal meshes, which allows implementing efficiently mesh refinement strategies, we also introduce a residual-type a posteriori error estimator and prove its reliability and efficiency. We use the corresponding error estimator to drive an adaptive scheme. Finally, we report the results of a couple of numerical tests that allow us to assess the performance of this approach.
We analyze in this paper a Virtual Element Method for an eigenvalue problem arising in linear elasticity. The Virtual Element Method (VEM), recently introduced in , is a generalization of the Finite Element Method, which is characterized by the capability of dealing with very general polygonal/polyhedral meshes. In recent years, the interest in numerical methods that can make use of general polygonal/polyhedral meshes for the numerical solution of partial differential equations has undergone a significant growth; this because of the high flexibility that this kind of meshes allow in the treatment of complex geometries. Among the large number of papers on this subject, we cite as a minimal sample .
Although VEM is very recent, it has been applied to a large number of problems; for instance, to Stokes, Brinkman, Cahn-Hilliard, plates bending, advection-diffusion, Helmholtz, parabolic, and hyperbolic problems have been introduced in . Regarding VEM for linear and non-linear elasticity we mention , for spectral problems , whereas a posteriori error analysis for VEM have been developed in .
The numerical approximation of eigenvalue problems for partial differential equations is object of great interest from both, the practical and theoretical points of view, since they appear in many applications. We refer to  and the references therein for the state of the art in this subject area. In particular, this paper focus on the approximation by VEM of the vibration frequencies and modes of an elastic solid. One motivation for considering this problem is that it constitutes a stepping stone towards the more challenging goal of devising virtual element spectral approximations for coupled systems involving fluid-structure interaction, which arises in many engineering problems (see  for a thorough discussion on this topic). Among the existing techniques to solve this problem, various finite element methods have been proposed and analyzed in different frameworks for instance in the following references .
On the other hand, in numerical computations it is important to use adaptive mesh refinement strategies based on a posteriori error indicators. For instance, they guarantee achieving errors below a tolerance with a reasonable computer cost in presence of singular solutions. Several approaches have been considered to construct error estimators based on the residual equations (see  and the references therein). Due to the large flexibility of the meshes to which the VEM is applied, mesh adaptivity becomes an appealing feature since mesh refinement strategies can be implemented very efficiently. However, the design and analysis of a posteriori error bounds for the VEM is a challenging task. References  are the only a posteriori error analyses for VEM currently available in the literature. In , a posteriori error bounds for the -conforming VEM for the two-dimensional Poisson problem are proposed. In  a residual-based a posteriori error estimator for the VEM discretization of the Poisson problem with discontinuous diffusivity coefficient has been introduced and analyzed. Moreover, in , a posteriori error bounds are introduced for the -conforming VEM for the discretization of second order linear elliptic reaction-convection-diffusion problems with non-constant coefficients in two and three dimensions. Finally, in  a posteriori error analysis of a virtual element method for the Steklov eigenvalue problem has been developed.
The aim of this paper is to introduce and analyze an -VEM that applies to general polygonal meshes, made by possibly non-convex elements, for the two-dimensional eigenvalue problem for the linear elasticity equations. We begin with a variational formulation of the spectral problem relying only on the solid displacement. Then, we propose a discretization by means of VEM, which is based on  in order to construct a proper -projection operator, which is used to approximate the bilinear form on the right hand side of the spectral problem. Then, we use the so-called Babuška–Osborn abstract spectral approximation theory (see ) to deal with the continuous and discrete solutions operators which appear as the solution of the continuous and discrete source problems and whose spectra are related with the solutions of the spectral problem. Under rather mild assumptions on the polygonal meshes, we establish that the resulting VEM scheme provides a correct approximation of the spectrum and prove optimal-order error estimates for the eigenfunctions and a double order for the eigenvalues. The second goal of this paper is to introduce and analyze an a posteriori error estimator of residual type for the virtual element approximation of the eigenvalue problem. Since normal fluxes of the VEM solution are not computable, they will be replaced in the estimators by a proper projection. We prove that the error estimator is equivalent to the error and use the corresponding indicator to drive an adaptive scheme. In addition, in this work we address the issue of comparing the proposed a posteriori error estimator with the standard residual estimator for a finite element method.
The outline of this article is as follows: We introduce in Section 2 the variational formulation of the spectral problem, define a solution operator and establish its spectral characterization. In Section 3, we introduce the virtual element discrete formulation, describe the spectrum of a discrete solution operator and establish some auxiliary results. In Section 4, we prove that the numerical scheme provides a correct spectral approximation and establish optimal order error estimates for the eigenvalues and eigenfunctions using the standard theory for compact operators. In Section 5, we establish an error estimate for the eigenfunctions in the -norm, which will be useful in the a posteriori error analysis. In Section 6, we define the a posteriori error estimator and proved its reliability and efficiency. Finally, in Section 7, we report a set of numerical tests that allow us to assess the convergence properties of the method, to confirm that it is not polluted with spurious modes and to check that the experimental rates of convergence agree with the theoretical ones. Moreover, we have also made a comparison between the proposed estimator and the standard residual error estimator for a finite element method,
Throughout the article, is a generic Lipschitz bounded domain of with boundary , we will use standard notations for Sobolev spaces, norms and seminorms. Finally, we employ to denote a generic null vector and to denote generic constants independent of the discretization parameters , which may take different values at different occurrences.
2The spectral problem
We assume that an isotropic and linearly elastic solid occupies a bounded and connected Lipschitz domain . We assume that the boundary of the solid admits a disjoint partition , the structure being fixed on and free of stress on . We denote by the outward unit normal vector to the boundary . Let us consider the eigenvalue problem for the linear elasticity equation in with mixed boundary conditions, written in the variational form:
where is the solid displacement and is the corresponding vibration frequency; is the density of the material, which we assume a strictly positive constant. The constitutive equation relating the Cauchy stress tensor and the displacement field is given by
with being the standard strain tensor and the elasticity operator, which we assume given by Hooke’s law, i.e.,
where and are the Lamé coefficients, which we assume constant.
We introduce the following bounded bilinear forms:
Then, the eigenvalue problem above can be rewritten as follows:
It is easy to check (as a consequence of the Korn inequality) that for all . Then, the bilinear form is -elliptic.
Next, we define the corresponding solution operator:
where is the unique solution of the following source problem:
Thus, the linear operator is well defined and bounded. Notice that solves Problem ? if and only if is an eigenpair of , i.e, if and only if
Moreover, it is easy to check that is self-adjoint with respect to the inner product in .
The following is an additional regularity result for the solution of problem and consequently, for the eigenfunctions of .
The proof follows from the regularity result for the classical elasticity problem (cf. ).
Hence, because of the compact inclusion , is a compact operator. Therefore, we have the following spectral characterization result.
3Virtual elements discretization
We begin this section, by recalling the mesh construction and the shape regularity assumptions to introduce the discrete virtual element space. Then, we will introduce a virtual element discretization of Problem ? and provide a spectral characterization of the resulting discrete eigenvalue problem. Let be a sequence of decompositions of into polygons . Let denote the diameter of the element and . In what follows, we denote by the number of vertices of , and by a generic edge of .
For the analysis, we will make the following assumptions as in : there exists a positive real number such that, for every and every ,
the ratio between the shortest edge and the diameter of is larger than ;
is star-shaped with respect to every point of a ball of radius .
Moreover, for any subset and nonnegative integer , we indicate by the space of polynomials of degree up to defined on .
To continue the construction of the discrete scheme, we need some preliminary definitions. First, we split the bilinear forms and , introduced in the previous section as follows:
Now, we consider a simple polygon and, for , we define
We then consider the following finite dimensional space:
The following set of linear operators are well defined for all :
: The (vector) values of at the vertices.
, for : The edge moments for on each edge of .
, for : The internal moments for on each element .
Now we define the projector for each as the solution of
where for all ,
We note that the second equation in is needed for the problem to be well-posed.
Now, we introduce our local virtual space:
where the space denote the polynomials in that are orthogonal to . We observe that, since , the operator is well defined on and computable only on the basis of the output values of the operators in , and . We note that it can be proved, see  that the set of linear operators , and constitutes a set of degrees of freedom for the local virtual space . Moreover, it is easy to check that . This will guarantee the good approximation properties for the space.
Additionally, we have that the standard -projector operator can be computed from the set of degrees freedom. In fact, for all , the function is defined by:
We can now present the global virtual space: for every decomposition of into simple polygons .
In agreement with the local choice of the degrees of freedom, in we choose the following degrees of freedom:
: the (vector) values of at the vertices of .
, for : The edge moments on each edge .
, for : The internal moments on each element .
On the other hand, let and be symmetric positive definite bilinear forms chosen as to satisfy
for some positive constants , , and depending only on the constant that appears in assumptions and . Then, we introduce on each element the local (and computable) bilinear forms
Now, we define in a natural way
The construction of and guarantees the usual consistency and stability properties of VEM, as noted in the proposition below. Since the proof is simple and follows standard arguments in the Virtual Element literature, it is omitted (see ).
Now, we are in a position to write the virtual element discretization of Problem ?.
We observe that by virtue of , the bilinear form is bounded. Moreover, as is shown in the following lemma, it is also uniformly elliptic.
Thanks to , it is easy to check that the above inequality holds with .
The next step is to introduce the discrete version of operator :
where is the solution of the corresponding discrete source problem:
We deduce from Lemma ?, – and the Lax-Milgram Theorem, that the linear operator is well defined and bounded uniformly with respect to .
Once more, as in the continuous case, solves Problem ? if and only if is an eigenpair of , i.e, if and only if
Moreover, it is easy to check that is self-adjoint with respect to and .
As a consequence, we have the following spectral characterization of the discrete solution operator.
4Spectral approximation and error estimates
To prove that provides a correct spectral approximation of , we will resort to the classical theory for compact operators (see ). With this aim, we recall the following approximation result which is derived by interpolation between Sobolev spaces (see for instance  from the analogous result for integer values of . In its turn, the result for integer values is stated in  and follows from the classical Scott-Dupont theory (see ):
The classical theory for compact operators, is based on the convergence in norm of to as . However, the operator is not well defined for any , since the definition of bilinear form in needs the degrees of freedom and in particular the pointwise values of . To circumvent this drawback, we introduce the projector with range , which is defined by the relation
In our case, the bilinear form correspond to the inner product. Thus, . Moreover,
For the analysis we introduce the following broken seminorm:
which is well defined for every such that for all polygon .
Now, we define . Notice that and the eigenfunctions of and coincide. Furthermore, we have the following result.
Let , for we have that
Now, if we define , thanks to Lemma ?, the definition of (cf ) and those of and , we have
where we have used the consistency property to derive the last equality. We now bound each term , with a constant .
The term can be bounded as follows: Let such that Lemma ? holds true, then by , we have
where we have used the definitions of and , the consistency and stability properties and , respectively, together with Cauchy-Schwarz inequality, Lemma ? and .
To bound , we first use the stability property , Cauchy-Schwarz inequality again and adding and subtracting to obtain
Therefore, by combining the above bounds, we obtain
Hence, the proof follows from the above estimate and .
The next step is to find appropriate term that can be used in the above lemma. Thus, we have the following result.
The proof is identical to that of Theorem 11 from  (in the 2D case), but using the following estimate
Now, we are in a position to conclude that converges in norm to as goes to zero.
The result follows from Lemmas ?– ? and Lemma ?.
As a direct consequence of Corollary ?, standard results about spectral approximation (see , for instance) show that isolated parts of are approximated by isolated parts of and therefore by . More precisely, let be an isolated eigenvalue of with multiplicity and let be its associated eigenspace. Then, there exist eigenvalues of (repeated according to their respective multiplicities) which converge to . Let be the direct sum of their corresponding associated eigenspaces.
We recall the definition of the gap between two closed subspaces and of :
The following error estimates for the approximation of eigenvalues and eigenfunctions hold true.
As a consequence of Corollary ?, converges in norm to as goes to zero. Then, the proof follows as a direct consequence of Theorems 7.1 and 7.3 from .
The theorem above yields error estimates depending on . The next step is to show an optimal-order estimate for this term.
The proof is identical to that of Corollary ?, but using now the additional regularity from Lemma ?(ii).
The error estimate for the eigenvalue of leads to an analogous estimate for the approximation of the eigenvalue of Problem ? by means of the discrete eigenvalues , , of Problem ?. However, the order of convergence in Theorem ? is not optimal for and, hence, not optimal for either. Our next goal is to improve this order.
Let be an eigenfunction corresponding to one of the eigenvalues with . According to Theorem ?, there exists eigenpair of Problem ? such that
From the symmetry of the bilinear forms and the facts that for all (cf. Problem ?) and for all (cf. Problem ?), we have
thus, we obtain the following identity:
The next step is to bound each term on the right hand side above. The first and the second ones are easily bounded using the Cauchy-Schwarz inequality and :
For the third term, let such that . From the definition of (cf ), adding and subtracting and using the consistency property (cf ) we obtain
Then, from the last inequality, Lemma ? and ,we obtain
For the fourth term, repeating similar arguments to the previous case, but using the consistency property (cf ) we have
Then, from the last inequality, Lemma ? and , we have
On the other hand, from the Korn’s inequality and Lemma ?, together with the fact that as goes to zero, we have that
Therefore, the theorem follows from – and the fact that .
5Error estimates for the eigenfunctions in the -norm
Our next goal is to derive an error estimate for the eigenfunctions in the -norm. The main result of this section is the following bound.
The proof of the above result will follow by combining Lemmas ?, ? and ? shown in the sequel.
Let the unique solution of the following problem:
Therefore, , so that according to Lemma ?(i), there exists such that and
Let such that the estimate of Lemma ? holds true. Then, by simple manipulations, we have that
For the second term on the right hand side above, we have the following equality
where we have used , added and subtracted and .
To bound the term , we consider such that and estimate of Lemma ? holds true. Then, using the consistency property (cf ) twice and the stability property (cf ), we obtain
where for the last inequality, we have used Lemmas ? and ?. Then, from , we obtain
For the term , we use the fact that , , , the consistency property twice and the stability property (cf ), to obtain