High order lower bounds of eigenvalues for the biharmonic eigenvalue problem by nonconforming finite element methods

In this paper, a new second order nonconforming finite element method (NFEM) on the rectangular grid is proposed for the biharmonic problem. It is proved that this new element produces asymptotic lower bounds of eigenvalues of the biharmonic eigenvalue problem. This is the first high order NFEM which admits such a lower bound property for fourth order problems. In addition, asymptotic lower bounds and upper bounds of eigenvalues are combined to obtain higher accuracy approximations for eigenvalues. Finally, the new element is implemented for three examples whose reference eigenvalues of more than thirty digits were recently computed in [8]. The numerical results show that the new element is very efficient.