Hehu Xie

A multi-level correction scheme for eigenvalue problems

AbstractIn this paper, a new type of multi-level correction scheme is proposed forsolving eigenvalue problems by finite element method. With this new scheme,the accuracy of eigenpair approximations can be improved after each correc-tion step which only needs to solve a source problem on finer finite elementspace and an eigenvalue problem on the coarsest finite element space. Thiscorrection scheme can improve the efficiency of solving eigenvalue problemsby finite element method.Keywords. Eigenvalue problem, multi-level correction scheme, finite ele-ment method, multi-space, multi-grid.AMS subject classifications. 65N30, 65B99, 65N25, 65L15 1 Introduction The purpose of this paper is to propose a new type of multi-level correction schemebased on finite element discretization to solve eigenvalue problems. The two-gridmethod for solving eigenvalue problems has been proposed and analyzed by Xu andZhou in [21]. The idea of the two-grid comes from [19, 20] for nonsymmetric orindefinite problems and nonlinear elliptic equations. Since then, there have existedmany numerical methods for solving eigenvalue problems based on the idea of two-grid method ([1, 6, 17]).

Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods

We introduce some ways to compute the lower and upper bounds of the Laplace eigenvalue problem. By using the special nonconforming finite elements, i.e., enriched Crouzeix-Raviart element and extended Q1rot, we get the lower bound of the eigenvalue. Additionally, we use conforming finite elements to do the postprocessing to get the upper bound of the eigenvalue, which only needs to solve the corresponding source problems and a small eigenvalue problem if higher order postprocessing method is implemented. Thus, we can obtain the lower and upper bounds of the eigenvalues simultaneously by solving eigenvalue problem only once. Some numerical results are also presented to demonstrate our theoretical analysis.

A Multigrid Method for Helmholtz Transmission Eigenvalue Problems

In this paper, we analyze the convergence of a finite element method for the computation of transmission eigenvalues and corresponding eigenfunctions. Based on the obtained error estimate results, we propose a multigrid method to solve the Helmholtz transmission eigenvalue problem. This new method needs only linear computational work. Numerical results are provided to validate the efficiency of the proposed method.

Lower bounds of the discretization error for piecewise polynomials

Assume that Vh is a space of piecewise polynomials of a degree less than r ≥ 1 on a family of quasi-uniform triangulation of size h. There exists the well-known upper bound of the approximation error by Vh for a sufficiently smooth function. In this paper, we prove that, roughly speaking, if the function does not belong to Vh, the upper-bound error estimate is also sharp. This result is further extended to various situations including general shape regular grids and many different types of finite element spaces. As an application, the sharpness of finite element approximation of elliptic problems and the corresponding eigenvalue problems is established.

A multigrid method for eigenvalue problem

A multigrid method is proposed to solve the eigenvalue problem by the finite element method based on the combination of the multilevel correction scheme for the eigenvalue problem and the multigrid method for the boundary value problem. In this scheme, solving eigenvalue problem is transformed to a series of solutions of boundary value problems by the multigrid method on multilevel meshes and a series of solutions of eigenvalue problems on the coarsest finite element space. Besides the multigrid scheme, all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems. The total computational work of this scheme can reach the optimal order as same as solving the corresponding boundary value problem. Therefore, this type of iteration scheme improves the overall efficiency of the eigenvalue problem solving. Some numerical experiments are presented to validate the efficiency of the method.

Discontinuous Galerkin Methods for Delay Differential Equations of Pantograph Type

This paper is concerned with the application of the discontinuous Galerkin method to delay differential equations with vanishing delay

High-Order Convergence of Spectral Deferred Correction Methods on General Quadrature Nodes

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Collocation Methods for General Volterra Functional Integral Equations with Vanishing Delays

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Superconvergence of Discontinuous Galerkin Solutions for Delay Differential Equations of Pantograph Type

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Error Analysis of a Mixed Finite Element Method for the Molecular Beam Epitaxy Model

). Our aim is to establish optimal global and local superconvergence results on uniform meshes and compare these with analogous estimates for collocation methods. The theoretical results are illustrated by a broad range of numerical examples.