Andrey B. Andreev

Superconvergence Postprocessing for Eigenvalues

Abstract The main goal of this paper is to present a new strategy of increasing the convergence rate for the numerical solution of the linear finite element eigenvalue problems. This is done by introducing a postprocessing technique for eigenvalues. The postprocessing technique deals with solving a corresponding linear elliptic problem. We prove that the proposed algorithm has the superconvergence property of the eigenvalues and this improvement is attained at a small computational cost. Thus, good finite element approximations for eigenvalues are obtained on the coarse mesh. The numerical examples presented and discussed here show that the resulting postprocessing method is computationally more efficient than the method to which it is applied.

One-Dimensional patch-recovery finite element method for fourth-order elliptic problems

Interpolated one-dimensional finite elements are constructed and applied to the fourth-order self-adjoint elliptic boundary-value problems. A superconvergence postprocessing approach, based on the patch-recovery method, is presented. It is proved that the rate of convergence depends on the different variational forms related to the variety of the corresponding elliptic operators. Finally, numerical results are presented.

Supercloseness between the elliptic projection and the approximate eigenfunction and its application to a postprocessing of finite element eigenvalue problems

An estimate confirming the supercloseness between the Ritz projection and the corresponding eigenvectors, obtained by finite element method, is hereby proved. This result is true for a large class of self-adjoint 2m–order elliptic operators. An application of this theorem to the superconvergence postprocessing patch-recovery technique for finite element eigenvalue problems is also presented. Finally, the theoretical investigations are supported by numerical experiments.

Optimal Order FEM for a Coupled Eigenvalue Problem on 2D Overlapping Domains

In this paper we present a numerical approach to a nonstandard second-order elliptic eigenvalue problem defined on two overlapping rectangular domains with a nonlocal (integral) boundary condition. Usually, for this class of problems error estimates are suboptimal. By introducing suitable degrees of freedom and a corresponding interpolation operator we derive optimal order finite element approximation. Numerical results illustrate the efficiency of the proposed method.

Superconvergent Finite Element Postprocessing for Eigenvalue Problems with Nonlocal Boundary Conditions

We present a postprocessing technique applied to a class of eigenvalue problems on a convex polygonal domain i¾?in the plane, with nonlocal Dirichlet or Neumann boundary conditions on

On the Postprocessing Technique for Eigenvalue Problems

\Gamma_1 \subset \partial \Omega

Nonconforming Rectangular Morley Finite Elements

. Such kind of problems arise for example from magnetic field computations in electric machines. The postprocessing strategy accelerates the convergence rate for the approximate eigenpair. By introducing suitable finite element space as well as solving a simple additional problem, we obtain good approximations on a coarse mesh. Numerical results illustrate the efficiency of the proposed method.

Properties and estimates of an integral type nonconforming finite element

We present a new strategy of accelerating the convergence rate for the finite element solutions of the large class of linear eigenvalue problems of order 2m. The proposed algorithms have the superconvergence properties of the eigenvalues, as well as of the eigenfunctions. This improvement is obtained at a small computational cost. Solving a more simple additional problem, we get good finite element approximations on the coarse mesh. Different ways for calculating the postprocessed eigenfunctions are considered. The case where the spectral parameter appears linearly in the boundary conditions is discussed. The numerical examples, presented here, confirm the theoretical results and show the efficiency of the postprocessing method.

A Zienkiewicz-type finite element applied to fourth-order problems

We analyze some approximation properties of modified rectangular Morley elements applied to fourth-order problems. Degrees of freedom of integrals type are used which yields superclose property. Further asymptotic error estimates for biharmonic solutions are derived. Some interesting and new numerical results concerning plate vibration problems are also presented.

Acceleration of convergence for eigenpairs approximated by means of non-conforming finite element methods

This paper is intended to provide an investigation to the application of an extended Crouzeix-Raviart (EC-R) nonconforming finite element. Integral degrees of freedom are used, which yields some superclose properties. The considered finite element basis contains an integral type bubble function. The approximate eigenvalues obtained by means of this nonconforming method give asymptotically lower bounds of the exact eigenvalues. It is considerable easier to obtain upper bounds for eigenvalues using variational numerical methods. That is why approximations from below are very valued and useful. Finally, computational aspects are discussed and numerical examples are presented.