A C1 stabilization-free Virtual Element Method for the biharmonic equation: Source and spectral problems

In this work, we introduce a novel C1 stabilization-free virtual element discretization for elliptic fourth-order problems, defined on triangular meshes. We propose a discrete approximation of the energy bilinear form that does not require an arbitrary non-polynomial stabilization term, based on the definition of a new polynomial projection, and we address the biharmonic eigenproblem. A representative set of numerical experiments shows the local stability and robustness of the method. Notably, the proposed approach does not produce spurious eigenvalues, which highlights its reliability in spectral approximations.