A Multi-level Mixed Element Method for the Eigenvalue Problem of   Biharmonic Equation
  In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits amiable nested discretization. Then, we construct multi-level finite element schemes by implementing the algorithm as in [33] to the nested discretizations on series of nested grids. The multi-level mixed scheme for biharmonic eigenvalue problem possesses optimal convergence rate and optimal computational cost. Both theoretical analysis and numerical verifications are presented. 
Shuo Zhang, Yingxia Xi, Xia Ji
math.NA