Jiayu Han

Mixed Methods for the Helmholtz Transmission Eigenvalues

The Ciarlet--Raviart mixed finite element method is popular for the biharmonic equations. In this paper, we apply this method to the Helmholtz transmission eigenvalue problem which is quadratic and...

The Lower/Upper Bound Property of the Crouzeix---Raviart Element Eigenvalues on Adaptive Meshes

In this paper we first discover and prove that on adaptive meshes the eigenvalues by the Crouzeix–Raviart element approximate the exact ones from below when the corresponding eigenfunctions are singular. In addition, we use conforming finite elements to do the interpolation postprocessing to get the upper bound of the eigenvalues. Using the upper and lower bounds of eigenvalues we design the control condition of adaptive algorithm, and some numerical experiments are carried out under the package of Chen to validate our theoretical results.

The Shifted-Inverse Iteration Based on the Multigrid Discretizations for Eigenvalue Problems

The shifted-inverse iteration based on the multigrid discretizations developed in recent years is an efficient computation method for eigenvalue problems. In this paper, for general self-adjoint ei...

An Adaptive Finite Element Method for the Transmission Eigenvalue Problem

The classical weak formulation of the Helmholtz transmission eigenvalue problem can be linearized into an equivalent nonsymmetric eigenvalue problem. Based on this nonsymmetric eigenvalue problem, we first discuss the a posteriori error estimates and adaptive algorithm of conforming finite elements for the Helmholtz transmission eigenvalue problem. We give the a posteriori error indicators for primal eigenfunction, dual eigenfunction and eigenvalue. Theoretical analysis shows that the indicators for both primal eigenfunction and dual eigenfunction are reliable and efficient and that the indicator for eigenvalue is reliable. Numerical experiments confirm our theoretical analysis.

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.

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

Non-conforming finite element methods for transmission eigenvalue problem

H^2