The optimal order convergence for the lowest order mixed finite element method of the biharmonic eigenvalue problem

In this paper, we present the theoretical analysis of the optimal order convergence for the piecewise linear and continuous finite element method based on the Ciarlet–Raviart mixed formulation of the biharmonic eigenvalue problem associated to the clamped boundary condition. As far as we know, only the convergence of the equal order linear Ciarlet–Raviart finite element method for the eigenvalue problem has been established on convex domains. The aim to this work is to derive the convergence under the minimum regularity requirement and prove an improved convergence rate for the approximate eigenvalues. We introduce the corresponding solution spaces naturally attached to the continuous and discrete problems and prove the spectral approximation and error estimate of the discrete scheme. Some numerical examples are shown for the validation of the theoretical proof.