A priori and a posteriori error estimates for a virtual element spectral analysis for the elasticity equations
This work provides rigorous error analysis for VEM in spectral elasticity, enabling adaptive mesh refinement with general polygonal meshes, which is important for computational mechanics.
The paper develops a priori and a posteriori error estimates for a virtual element method (VEM) applied to eigenvalue problems in elasticity, proving optimal convergence rates and introducing a reliable and efficient residual-type error estimator for adaptive mesh refinement.
We present a priori and a posteriori error analysis of a Virtual Element Method (VEM) to approximate the vibration frequencies and modes of an elastic solid. We analyze a variational formulation relying only on the solid displacement and propose an $H^1(Ω)$-conforming discretization by means of 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 implementing efficiently 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.