Virtual element method for the Laplacian eigenvalue problem with Neumann boundary conditions

The Laplacian eigenvalue problem is fundamental to the free vibration problem. The finite element method (FEM) is a classical numerical method for solving eigenvalue problems. However, for some complex problems, the flexibility of mesh generation in the FEM is not strong. The virtual element method (VEM), an extension of the FEM to polygonal meshes, offers more flexibility in mesh generation and has significant advantages when dealing with complex regions with high regularity. Additionally, the VEM does not require the explicit expression of basis functions, which simplifies the computational process. In this paper, we use the VEM to solve the Laplacian eigenvalue problem with Neumann boundary conditions. Considering standard assumptions on polygonal meshes, the source problem of the Laplacian eigenvalue problem in variational form by defining the solution operator is obtained. We then construct the virtual element space and the degrees of freedom to define the appropriate projection operator. To approximate the exact solutions, we construct discrete bilinear form and right-hand side. Furthermore, we apply the spectral approximation theory to prove the error estimates of the eigenvalues. To support the convergence analysis, a series of numerical examples is reported. For the Laplacian eigenvalue problem with Neumann boundary conditions on a square domain, experiments demonstrate the convergence behavior of approximate solutions and extrapolated values. The results indicate that the errors of the eigenvalues are converged with the refinement of the grid. Numerical experiments also verify the effectiveness of the theoretical analysis in solving eigenvalue problems under different meshes.