Volume 19, Issue 2
An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems

Ye Li

Commun. Comput. Phys., 19 (2016), pp. 442-472.

Published online: 2018-04

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  • Abstract

In this paper, we propose a 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≪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 $\mathcal{O}$(1/$(NlnN)^\frac{2}{3}$).

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@Article{CiCP-19-442, author = {}, title = {An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems}, journal = {Communications in Computational Physics}, year = {2018}, volume = {19}, number = {2}, pages = {442--472}, abstract = {

In this paper, we propose a 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≪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 $\mathcal{O}$(1/$(NlnN)^\frac{2}{3}$).

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.021114.140715a}, url = {http://global-sci.org/intro/article_detail/cicp/11096.html} }
TY - JOUR T1 - An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems JO - Communications in Computational Physics VL - 2 SP - 442 EP - 472 PY - 2018 DA - 2018/04 SN - 19 DO - http://dor.org/10.4208/cicp.021114.140715a UR - https://global-sci.org/intro/article_detail/cicp/11096.html KW - AB -

In this paper, we propose a 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≪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 $\mathcal{O}$(1/$(NlnN)^\frac{2}{3}$).

Ye Li. (2020). An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems. Communications in Computational Physics. 19 (2). 442-472. doi:10.4208/cicp.021114.140715a
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