Abstract

This work develops and analyzes Morley-type virtual element schemes for the two-dimensional biharmonic eigenvalue problem, providing a unified treatment of several boundary conditions, including the nonstandard Cahn-Hilliard case as well as the classical clamped and simply supported conditions. The analysis is carried out on simply connected polygonal domains, possibly non-convex, under standard mesh regularity assumptions. By means of a suitable enriching operator, we establish the spectral correctness of the proposed nonconforming discretizations. Under minimal regularity assumptions on the exact solution, we derive optimal-order error estimates for the eigenfunctions and quadratic convergence rates for the eigenvalues. Numerical experiments are presented to validate the theoretical findings and to illustrate the performance of the method on a variety of polygonal meshes and boundary conditions. Furthermore, additional numerical tests on perforated domains are included to explore the behavior of the proposed schemes in multiply connected configurations, which are not covered by the present theoretical analysis.