A New Family of Mixed Method for the Biharmonic Eigenvalue Problem Based on the First Order Equations of Hellan–Herrmann–Johnson Type

In this paper, we consider the numerical approximation of a biharmonic eigenvalue problem by introducing a new family of the mixed method. This method is based on a formulation where the fourth-order eigenproblem is recast as a system of four first-order equations. The optimal convergence rates with 2 k+ 2 (k≥ 0 is the degree of the polynomials) of eigenvalue approximation are theoretically derived and numerically verified. The optimal or sub-optimal convergences of the other unknowns are theoretically proved. The new numerical schemes based on the deduced problems can be of lower complicacy, and the framework is fit for various fourth-order eigenvalue problems.