A priori and a posteriori error estimates for a virtual element spectral analysis for the elasticity equations

摘要

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(?)-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.