Local And Parallel Finite Element Method For Solving The Biharmonic Eigenvalue Problem Of Plate Vibration

The study for the plate vibration and buckling is one of the important fields of re-search in applied mechanics and has attracted attentions from academic circle.Local and parallel computation of finite element is one of the hot spots in scientific computing today and can be applied to a variety of partial differential equations,and it performs well especially for the problems with local singularity or local low smoothness.This paper devotes to study the local and parallel finite element method for the bihar-monic eigenvalue problem of plate vibration.The local error estimations of finite element solution for the biharmonic equation/eigenvalue problem are first proved in this paper,then,combining the local defect correction and shifted-inverse iteration,a three-grid dis-cretization scheme based on local computation and shifted-inverse iteration is established for solving the biharmonic eigenvalue problem of plate vibration.The error estimation of the three-grid discretization scheme is followed.Based on the three-grid scheme,a local and parallel finite element algorithm and its error estimates are given.Finally,numerical experiments on the L-shaped and slit domain are reported.From theoretical analysis and numerical experiments it can be seen that,when the diameters of three grids satisfy H4=v(w2)=v(h),the approximate solutions obtained by three-grid scheme can achieve the asymptotically optimal accuracy.And comparing with Morley element and C0IPG method,one can use the three-grid scheme in this paper to obtain the same accurate approximate eigenvalues with relatively few degrees of freedom.Theoretical analysis and numerical experiments indicate that the local and parallel finite element method estab-lished in this paper is suitable and efficient for solving the biharmonic eigenvalue problem of plate vibration.