Serendipity Virtual Element Method for the Second Order Elliptic Eigenvalue Problem in Two and Three Dimensions

Abstract

In this paper, we analyze the virtual element method for the symmetric second order elliptic eigenvalue problem with variable coefficients in two and three dimensions, which reduces the number of degrees of freedom of the standard virtual element method. We attempt to prove the interpolation theory and stability analysis for the serendipity nodal virtual element space, which provides new stabilization terms for the virtual element schemes. Then we prove the spectral approximation and the optimal a priori error estimates. Moreover, we construct a fully computable residual-type a posteriori error estimator applied to the adaptive serendipity virtual element method and prove its upper and lower bounds with respect to the approximation error. Finally, we show numerical examples to verify the theoretical results and show the comparison between standard and serendipity virtual element methods.