The two-grid discretization of Ciarlet–Raviart mixed method for biharmonic eigenvalue problems

Abstract

In this paper, for biharmonic eigenvalue problems with clamped boundary condition in Rn which include plate vibration problem and plate buckling problem, we primarily study the two-grid discretization based on the shifted-inverse iteration of Ciarlet–Raviart mixed method. With our scheme, the solution of a biharmonic eigenvalue problem on a fine mesh πh can be reduced to the solution of an eigenvalue problem on a coarser mesh πH and the solution of a linear algebraic system on the fine mesh πh. With a new argument which is not covered by existing work, we prove that the resulting solution still maintains an asymptotically optimal accuracy when H>h≥O(H2). The surprising numerical results show the efficiency of our scheme.