Hai Bi

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

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 is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid . 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 , and when using the - element to solve eigenvalue problems of electric field, the calculation results can maintain an asymptotically optimal accuracy by taking . Finally, numerical experiments are presented to support the theoretical analysis.

Multigrid Discretization and Iterative Algorithm for Mixed Variational Formulation of the Eigenvalue Problem of Electric Field

This paper discusses highly finite element algorithms for the eigenvalue problem of electric field. Combining the mixed finite element method with the Rayleigh quotient iteration method, a new multi-grid discretization scheme and an adaptive algorithm are proposed and applied to the eigenvalue problem of electric field. Theoretical analysis and numerical results show that the computational schemes established in the paper have high efficiency.

A C0IPG method and its error estimates for the Helmholtz transmission eigenvalue problem

Abstract The interior penalty methods using C 0 Lagrange elements ( C 0 IPG ) developed in the last decade for the fourth order problems are an interesting topic in academia at present. In this paper, we apply the methods to the Helmholtz transmission eigenvalue problem which is quadratic and non self-adjoint and has important applications in the inverse scattering theory. We propose a discrete scheme, present a complete error analysis and first prove the C 0 IPG method can capture smooth eigenpairs efficiently. Using the argument in this paper we can prove that the C 0 IPG methods for the fourth order eigenvalue problems in the existing literature can also capture smooth eigenpairs efficiently. Numerical examples confirm our theoretical analysis.

A new multigrid finite element method for the transmission eigenvalue problems

Numerical methods for the transmission eigenvalue problems are hot topics in recent years. Based on the work of Lin and Xie (2015), we build a multigrid method to solve the problems. With our method, we only need to solve a series of primal and dual eigenvalue problems on a coarse mesh and the associated boundary value problems on the finer and finer meshes. Theoretical analysis and numerical results show that our method is simple and easy to implement and is efficient for computing real and complex transmission eigenvalues.

The lower bound property of the Morley element eigenvalues

In this paper, we prove that the Morley element eigenvalues approximate the exact ones from below on regular meshes, including adaptive local refined meshes, for the fourth-order elliptic eigenvalue problems with the clamped boundary condition in any dimension. And we implement the adaptive computation with the Morley element to obtain lower bounds of eigenvalues for the vibration problem of clamped plate under tension and a fourth-order elliptic eigenvalue problem with variable coefficients.

Local and Parallel Finite Element Algorithms for the Transmission Eigenvalue Problem

Based on the work of Xu and Zhou (Math Comp 69:881–909, 2000), we establish local and parallel algorithms for the Helmholtz transmission eigenvalue problem. For the

A two-grid discretization scheme of non-conforming finite elements for transmission eigenvalues

H^2

A note on the residual type a posteriori error estimates for finite element eigenpairs of nonsymmetric elliptic eigenvalue problems

H2-conforming finite element and the spectral element approximations, we prove the local error estimates and the efficiency of local and parallel algorithms. Numerical experiments indicate that our algorithms are easy to implement on the existing packages, and can be used to solve the transmission eigenvalue problem with local low smooth eigenfunctions efficiently.