Finite element analysis for the Navier-Lam\'e eigenvalue problem

TL;DR

This paper analyzes the eigenvalue problem for the Navier-Lamé system in elasticity, proposing a finite element method with convergence, error estimates, and numerical validation for eigenfrequency approximation.

Contribution

It introduces a finite element approach for the Navier-Lamé eigenvalue problem, including convergence analysis, error estimates, and an a posteriori error estimator.

Findings

Finite element method effectively approximates eigenfrequencies and eigenfunctions.

Convergence and error estimates are established for the proposed method.

Numerical tests confirm the reliability and efficiency of the error estimators.

Abstract

The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so called Navier-Lam\'e system is considered. Such a system introduces the displacement, rotation and pressure of some linear and elastic structure. The analysis of the spectral problem is based in the compact operators theory. A finite element method based in polynomials of degree $k\geq 1$ are considered in order to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimate are presented. An a posteriori error analysis is performed, where the reliability and efficiency of the proposed estimator is proved. We end this contribution reporting a series of numerical tests in order to assess the performance of the proposed numerical method, for the a priori and a posteriori estimates.