A Generalized Multiscale Virtual Element Method for the Biharmonic Eigenvalue Problem

We present a Generalized Multiscale Virtual Element Method (GMsVEM) for the numerical approximation of the eigenvalue problem associated with the biharmonic operator with clamped boundary conditions. The proposed approach combines the geometric flexibility of the Virtual Element Method (VEM) on polygonal meshes with a multiscale enrichment strategy based on local spectral problems defined on coarse neighborhoods. The multiscale basis functions are constructed by solving local eigenvalue problems and incorporating the dominant modes into a reduced approximation space through a partition of unity. This strategy significantly reduces the number of degrees of freedom while preserving the accuracy of the spectral approximation. Numerical experiments show that the proposed method accurately approximates the first five  eigenvalues and eigenfunctions of the biharmonic operator, exhibiting clear convergence toward reference solutions computed on fine meshes as the number of multiscale basis functions increases.