A priori and a posteriori error estimates for a virtual element spectral analysis for the elasticity equations. (30th October 2018)

• Record Type: Journal Article
• Authors: Mora, David; Rivera, Gonzalo
• Is Part Of: IMA journal of numerical analysis. Volume 40:Number 1(2020)
• Page Start: 322
• Page End: 357
• Publication Date: 2018-10-30
• Subjects: virtual element method -- elasticity equations -- eigenvalue problem -- a priori error estimates -- a posteriori error analysis -- polygonal meshes; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/dry063
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: We present a priori and a posteriori error analyses of a virtual element method (VEM) to approximate the vibration frequencies and modes of an elastic solid. We analyse a variational formulation relying only on the solid displacement and propose an $H^{1}(\Omega )$ -conforming discretization by means of the 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 efficient implementation of 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.