Nitsche’s serendipity virtual element method for the eigenvalue problem

Abstract

In this paper, we study the Nitsche’s extended serendipity virtual element method for the eigenvalue problem in two and three dimensions. We start from the introduction of two- and three-dimensional serendipity virtual element spaces, in which the serendipity technique helps us drop all internal-to-face and internal-to-element degrees of freedom with the suitable projection operators fitting into virtual element spaces. Meanwhile, we give the Nitsche’s extended serendipity virtual element scheme of the eigenvalue problem. At the next stage, we prove the spectral approximation and the optimal error estimates of the proposed numerical method. By using the standard interpolation and polynomial approximation properties, we prove the $H^1$-norm error bound of the associated source problem. To consider the $L^2$-norm error bound, the Ritz-Volterra projection based on the formulation of Nitsche’s virtual element bilinear form is defined. Then we rigorously analyze its approximation properties. After that, we build the $L^2$ error estimate of the associated source problem. In the main theorems, we prove the error estimates of eigenfunctions and eigenvalues obtained by the Nitsche’s extended serendipity virtual element method. At the final stage, we extend the Nitsche’s idea to arbitrary curved domains by modifying the virtual element scheme with Taylor expansion terms. Numerical experiments confirm the theoretical results, using the Nitsche’s serendipity virtual element method to solve the Laplacian and Shrödinger eigenvalue problems on plane and curved domains.