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

In this paper, we analyze the virtual element method for symmetric second-order elliptic eigenvalue problems with variable coefficients in two and three dimensions, which can reduce the degrees of freedom compared with the standard virtual element method. We establish the interpolation theory and carry out stability analysis for serendipity nodal virtual element spaces, and further derive novel stabilization terms for virtual element schemes. Subsequently, we prove spectral approximation properties and derive optimal a priori error estimates. In addition, we design a fully computable residual-based a posteriori error estimator for adaptive serendipity virtual element methods, and prove its upper and lower bounds with respect to approximation errors. Finally, numerical examples are presented to verify the theoretical results and make comparisons between standard virtual element methods and serendipity virtual element methods.