Summing Logarithmic Expansions for Singularly Perturbed Eigenvalue Problems

1993 ◽  
Vol 53 (3) ◽  
pp. 799-828 ◽  
Author(s):  
Michael J. Ward ◽  
William D. Heshaw ◽  
Joseph B. Keller
Keyword(s):  
Eigenvalue Problems ◽  
Singularly Perturbed ◽  
Logarithmic Expansions

scholarly journals Asymptotic methods for solving boundary value eigenvalue problems

2020 ◽  
Vol 164 ◽  
pp. 09022
Author(s):  
Galina Zhukova
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Numerical Integration ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Asymptotic Methods ◽  
Singularly Perturbed ◽  
Asymptotic Behavior Of Solutions ◽  
Diagram Method

The aim of the study is an approximate construction with a given accuracy of solutions of boundary value problems for eigenvalues under various types of boundary conditions. It is shown that the problem of finding approximate large eigenvalues of boundary value problems is reduced to the analysis and solution of singularly perturbed differential equations with variable coefficients. Methods used: asymptotic diagram method developed to construct the asymptotic behavior of solutions of singularly perturbed differential equations and systems; methods of numerical integration of boundary value problems. The main results obtained are: the asymptotics of the required accuracy are constructed in the analytical form for the eigenvalues and eigenfunctions of the boundary value problems under various boundary conditions; analysis of the computational capabilities of the practical use of the constructed asymptotics in comparison with the results of numerical integration.

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

Singularly Perturbed Eigenvalue Problems

1987 ◽  
Vol 47 (6) ◽  
pp. 1131-1149 ◽  
Author(s):  
Herbert Steinrück
Keyword(s):  
Eigenvalue Problems ◽  
Singularly Perturbed
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