H1-conforming virtual element method for the Laplacian eigenvalue problem in mixed form

Abstract

In this paper, we propose the Stokes-type virtual element method for the Laplacian eigenvalue problem in mixed form, in which the eigenvalue problem is reformulated in terms of the velocity and the pressure of the Darcy flow. The virtual element method is H1-conforming and divergence-free. By applying these properties, the inf-sup condition and the spectral approximation theory of compact operator, we derive the optimal a priori error estimates. Then we focus on the a posteriori error estimates. Based on the superconvergence analysis, the Helmholtz decomposition and the properties of bubble functions, we prove the upper and lower bounds of the residual-type a posteriori error estimator with respect to the approximation error. Finally some numerical results are given to validate the theoretical analysis and present the good performance of the adaptive algorithm.