Felipe Lepe

Felipe Lepe, Salim Meddahi, David Mora et al.

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.

Arbaz Khan, Felipe Lepe, Jesus Vellojin

We analyze a novel locking-free mixed formulation for the elasticity eigenvalue problem in both two and three dimensions, expressed exclusively in terms of the pseudostress tensor. An important feature of this formulation is that it does not require the enforcement of symmetry, either in a weak or strong sense. The displacement of the structure is recovered via a postprocess of the computed pseudostress. We introduce a mixed finite element method based in the tensorial version of the standard families of finite elements to discretize the space $\boldsymbol{\mathcal{H}}(\bdiv)$. We prove convergence and a priori error estimates under the theory of non-compact operators. Additionally, we perform an a posteriori error analysis for the problem, proving reliability and efficiency of the proposed indicator. We validate our theoretical results with numerical tests on different geometrical and physical configurations.

Felipe Lepe, Salim Meddahi, David Mora et al.

In this paper we analyze a finite element method for solving a quadratic eigenvalue problem derived from the acoustic vibration problem for a heterogeneous dissipative fluid. The problem is shown to be equivalent to the spectral problem for a noncompact operator and athorough spectral characterization is given. The numerical discretization of the problem is based on Raviart-Thomas finite elements. The method is proved to be free of spurious modes and to converge with optimal order. Finally, we report numerical tests which allow us to assess the performance of the method.