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

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Yidu Yang ◽  
Wei Jiang ◽  
Yu Zhang ◽  
Wenjun Wang ◽  
Hai Bi
Keyword(s):  
Finite Element ◽  
Theoretical Analysis ◽  
Eigenvalue Problem ◽  
Variational Formulation ◽  
Eigenvalue Problems ◽  
Discretization Scheme ◽  
Fine Grid ◽  
Asymptotically Optimal ◽  
Mixed Variational Formulation ◽  
Optimal Accuracy

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 gridKhis reduced to the solution of an eigenvalue problem on a much coarser gridKHand the solution of a linear algebraic system on the fine gridKh. 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 takingH=O(h4), and when using thePk+1-Pkelement to solve eigenvalue problems of electric field, the calculation results can maintain an asymptotically optimal accuracy by takingH=O(h3). Finally, numerical experiments are presented to support the theoretical analysis.

A Two-Grid Discretization Scheme for a Sort of Steklov Eigenvalue Problem

2012 ◽  
Vol 557-559 ◽  
pp. 2087-2091
Author(s):  
Chao Xia ◽  
Yi Du Yang ◽  
Hai Bi
Keyword(s):  
Eigenvalue Problem ◽  
Algebraic System ◽  
Discretization Scheme ◽  
Linear Algebraic System ◽  
Fine Grid ◽  
Asymptotically Optimal ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Optimal Accuracy ◽  
Numer Anal

On the basis of Yang and Bi’s work (SIAM J Numer Anal 49, p.1602-1624), this paper discusses a discretization scheme for a sort of Steklov eigenvalue problem and proves the high effiency of the scheme. With the 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. And the resulting solution can maintain an asymptotically optimal accuracy. Finally, the numerical results are provided to support the theoretical analysis..

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

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Yidu Yang ◽  
Yu Zhang ◽  
Hai Bi
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Electric Field ◽  
Adaptive Algorithm ◽  
Variational Formulation ◽  
High Efficiency ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Rayleigh Quotient ◽  
Rayleigh Quotient Iteration

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.

Finite element approximation of eigenvalue problems

Acta Numerica ◽  
2010 ◽  
Vol 19 ◽  
pp. 1-120 ◽  
Author(s):  
Daniele Boffi
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Differential Forms ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Galerkin Approximations ◽  
Laplace Eigenvalue ◽  
Main Emphasis ◽  
Adjoint Operators

We discuss the finite element approximation of eigenvalue problems associated with compact operators. While the main emphasis is on symmetric problems, some comments are present for non-self-adjoint operators as well. The topics covered include standard Galerkin approximations, non-conforming approximations, and approximation of eigenvalue problems in mixed form. Some applications of the theory are presented and, in particular, the approximation of the Maxwell eigenvalue problem is discussed in detail. The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations. Several examples and numerical computations complete the paper, ranging from very basic exercises to more significant applications of the developed theory.

An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems

2016 ◽  
Vol 19 (2) ◽  
pp. 442-472
Author(s):  
Ye Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problem ◽  
Singularly Perturbed ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Eigenvalue

AbstractIn this paper, we propose an uniformly convergent adaptive finite element method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation (BEC) and quantum chemistry. We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations for the problem are reviewed to confirm the asymptotic behaviors of the solutions in the boundary/interior layer regions. By using the normalized gradient flow, we propose an adaptive finite element with hybrid basis to solve the singularly perturbed nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively to the small parameter ε. Extensive numerical results are reported to show the uniform convergence property of our method. We also apply the AFEM-HB to compute the ground and excited states of BEC with box/harmonic/optical lattice potential in the semiclassical regime (0 <ε≪C 1). In addition, we give a detailed error analysis of our AFEM-HB to a simpler singularly perturbed two point boundary value problem, show that our method has a minimum uniform convergence order

Modified Finite Element Transfer Matrix Method for Eigenvalue Problem of Flexible Structures

2010 ◽  
Vol 78 (2) ◽  
Author(s):  
Bao Rong ◽  
Xiaoting Rui ◽  
Guoping Wang
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Eigenvalue Problems ◽  
Flexible Structures ◽  
System Matrix ◽  
Element Transfer ◽  
High Computational Efficiency

The speedy computation of eigenvalue problems is the key point in structure dynamics. In this paper, by combining transfer matrix method and finite element method, the modified finite element-transfer matrix method and its algorithm for eigenvalue problems are presented. By using this method, the speedy computation of eigenvalue problem of flexible structures can be realized, and the repeated eignvalue problem can be solved simply and conveniently. This method has the low order of system matrix, high computational efficiency, and stability. Formulations of this method, as well as some numerical examples, are given to validate the method.

scholarly journals An efficient finite element method and error analysis for eigenvalue problem of Schrödinger equation with an inverse square potential on spherical domain

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yubing Sui ◽  
Donghao Zhang ◽  
Junying Cao ◽  
Jun Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Original Problem ◽  
Eigenvalue Problems ◽  
Weak Form ◽  
Spherical Coordinate ◽  
One Dimensional ◽  
Spherical Domain ◽  
Element Method

Abstract We provide an effective finite element method to solve the Schrödinger eigenvalue problem with an inverse potential on a spherical domain. To overcome the difficulties caused by the singularities of coefficients, we introduce spherical coordinate transformation and transfer the singularities from the interior of the domain to its boundary. Then by using orthogonal properties of spherical harmonic functions and variable separation technique we transform the original problem into a series of one-dimensional eigenvalue problems. We further introduce some suitable Sobolev spaces and derive the weak form and an efficient discrete scheme. Combining with the spectral theory of Babuška and Osborn for self-adjoint positive definite eigenvalue problems, we obtain error estimates of approximation eigenvalues and eigenvectors. Finally, we provide some numerical examples to show the efficiency and accuracy of the algorithm.

A FINITE ELEMENT APPROXIMATION OF AN IMAGE SEGMENTATION PROBLEM

1999 ◽  
Vol 09 (02) ◽  
pp. 243-259 ◽  
Author(s):  
G. CORTESANI
Keyword(s):  
Finite Element ◽  
Image Segmentation ◽  
Variational Formulation ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Discretization Scheme ◽  
Numerical Discretization ◽  
Segmentation Problem

We study a numerical discretization scheme for the Mumford–Shah functional, which is introduced to give a variational formulation of image segmentation problems, based on the nonlocal approximation proposed by Braides–Dal Maso.

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