A Two-Scale Discretization Scheme for Mixed Variational Formulation of Eigenvalue Problems

Abstract

This paper discusses highly efficient discretization schemes for mixed variational formulation of eigenvalue problems. A new finite element two-scale discretization scheme is proposed by combining the mixed finite element method with the shifted-inverse power method for solving matrix eigenvalue problems. With this scheme, the solution of an eigenvalue problem on a fine grid $K^h$ is reduced to the solution of an eigenvalue problem on a much coarser grid $K^H$ and the solution of a linear algebraic system on the fine grid $K^h$. Theoretical analysis shows that the scheme has high efficiency. For instance, when using the Mini element to solve Stokes eigenvalue problem, the resulting solution can maintain an asymptotically optimal accuracy by taking $H=O(\sqrt[4]{h})$, and when using the $P_{k+1}$-$P_k$ element to solve eigenvalue problems of electric field, the calculation results can maintain an asymptotically optimal accuracy by taking $H=O(\sqrt[3]{h})$. Finally, numerical experiments are presented to support the theoretical analysis.