Mixed discontinuous Galerkin approximation of the elasticity eigenproblem

Abstract

We introduce a discontinuous Galerkin method for the mixed formulation of the elasticity eigenproblem with reduced symmetry. The analysis of the resulting discrete eigenproblem does not fit in the standard spectral approximation framework since the underlying source operator is not compact and the scheme is nonconforming. We show that the proposed scheme provides a correct approximation of the spectrum and prove asymptotic error estimates for the eigenvalues and the eigenfunctions. Finally, we provide several numerical tests to illustrate the performance of the method and confirm the theoretical results.

A posteriori analysis for a mixed FEM discretization of the linear elasticity spectral problem

“…We mention that the dependency of the constants in the regularity exponents and boundedness on λ is not completely evident. This has been observed in [20,23] when the numerical tests are performed. This motivates us to consider the following assumption along our paper: Assumption 2.1.…”

“…Since the elasticity system depends on the Lamé constants, often denoted by µ and λ, it is well known that when the Poisson ratio is close to 1/2, numerical locking arises since λ → ∞. This motivates the study of the so-called limit eigenvalue problem (see [20,23] for instance). Hence, it is possible to consider two types of estimators: one for the limit eigenproblem and the other for the standard eigenproblem.…”

Mixed methods for the velocity-pressure-pseudostress formulation of the Stokes eigenvalue problem

“…In particular, mixed formulations for eigenvalue problems has been well developed in the past years and the theory to study these problems can be found in [4,22], just for mention the more classic references. On the other hand, concrete applications for mixed formulations in spectral problems can be found in different contexts as, for instance, [1,10,13,12,17,19], where several tools have been implemented as DG methods, VEM methods, FEM, and a posteriori analysis.…”

Displacement-pseudostress formulation for the linear elasticity spectral problem

“…With the aim of approximate the solutions of the linear elasticity equations, several numerical methods have been designed, firstly for the load problem in the past years. We refer to [4,5,7,9,13,18,22,23], and the reference therein, just to mention some results on these subjects.…”

“…For example, in [22] the authors introduce a mixed formulation depending only on the Cauchy stress tensor, where the symmetry is weakly imposed and the displacement can be recovered with a post-process. Analysis with a discontinuous Galerkin method (DG) for this formulation has been also proposed in [18] for the elasticity spectral problem, where the advantages of considering more general meshes are presented. Nevertheless, the main disadvantage of this method lies in the correct choice of the stabilization parameter, since, depending on the configuration of the problem, namely the geometry, boundary conditions or physical quantities, it can generate spurious eigenvalues in the computed spectrum.…”