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Article: A multiscale finite element method for the Schrodinger equation with multiscale potentials

TitleA multiscale finite element method for the Schrodinger equation with multiscale potentials
Authors
Issue Date2019
PublisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://epubs.siam.org/sam-bin/dbq/toclist/SISC
Citation
SIAM Journal on Scientific Computing, 2019, v. 41 n. 5, p. B1115–B1136 How to Cite?
AbstractIn recent years, an increasing attention has been paid to quantum heterostructures with tailored functionalities, such as heterojunctions and quantum metamaterials, in which quantum dynamics of electrons can be described by the Schrödinger equation with multiscale potentials. The model, however, cannot be solved by asymptotic-based approaches where an additive form of different scales in the potential term is required to construct the prescribed approximate solutions. In this paper, we propose a multiscale finite element method to solve this problem in the semiclassical regime. The localized multiscale basis functions are constructed using sparse compression of the Hamiltonian operator and thus are “blind' to the specific form of the potential term. After an one-shot eigendecomposition, we solve the resulting system of ordinary differential equations explicitly for the time evolution. In our approach, the spatial mesh size is $ H=mathcal{O}(epsilon)$, where $epsilon$ is the semiclassical parameter and the time stepsize $ k$ is independent of $epsilon$. Numerical examples in one dimension with a periodic potential, a multiplicative two-scale potential, and a layered potential, and in two dimension with an additive two-scale potential and a checkboard potential are tested to demonstrate the robustness and efficiency of the proposed method. Moreover, first-order and second-order rates of convergence are observed in $H^1$ and $L^2$ norms, respectively. © 2019, Society for Industrial and Applied Mathematics Read More: https://epubs.siam.org/doi/abs/10.1137/19M1236989
Persistent Identifierhttp://hdl.handle.net/10722/275053
ISSN
2019 Impact Factor: 1.976
2015 SCImago Journal Rankings: 2.166

 

DC FieldValueLanguage
dc.contributor.authorChen, J-
dc.contributor.authorMa, D-
dc.contributor.authorZhang, Z-
dc.date.accessioned2019-09-10T02:34:28Z-
dc.date.available2019-09-10T02:34:28Z-
dc.date.issued2019-
dc.identifier.citationSIAM Journal on Scientific Computing, 2019, v. 41 n. 5, p. B1115–B1136-
dc.identifier.issn1064-8275-
dc.identifier.urihttp://hdl.handle.net/10722/275053-
dc.description.abstractIn recent years, an increasing attention has been paid to quantum heterostructures with tailored functionalities, such as heterojunctions and quantum metamaterials, in which quantum dynamics of electrons can be described by the Schrödinger equation with multiscale potentials. The model, however, cannot be solved by asymptotic-based approaches where an additive form of different scales in the potential term is required to construct the prescribed approximate solutions. In this paper, we propose a multiscale finite element method to solve this problem in the semiclassical regime. The localized multiscale basis functions are constructed using sparse compression of the Hamiltonian operator and thus are “blind' to the specific form of the potential term. After an one-shot eigendecomposition, we solve the resulting system of ordinary differential equations explicitly for the time evolution. In our approach, the spatial mesh size is $ H=mathcal{O}(epsilon)$, where $epsilon$ is the semiclassical parameter and the time stepsize $ k$ is independent of $epsilon$. Numerical examples in one dimension with a periodic potential, a multiplicative two-scale potential, and a layered potential, and in two dimension with an additive two-scale potential and a checkboard potential are tested to demonstrate the robustness and efficiency of the proposed method. Moreover, first-order and second-order rates of convergence are observed in $H^1$ and $L^2$ norms, respectively. © 2019, Society for Industrial and Applied Mathematics Read More: https://epubs.siam.org/doi/abs/10.1137/19M1236989-
dc.languageeng-
dc.publisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://epubs.siam.org/sam-bin/dbq/toclist/SISC-
dc.relation.ispartofSIAM Journal on Scientific Computing-
dc.rightsSIAM Journal on Scientific Computing. Copyright © Society for Industrial and Applied Mathematics.-
dc.rights© [year] Society for Industrial and Applied Mathematics. First Published in [Publication] in [volume and number, or year], published by the Society for Industrial and Applied Mathematics (SIAM).-
dc.titleA multiscale finite element method for the Schrodinger equation with multiscale potentials-
dc.typeArticle-
dc.identifier.emailZhang, Z: zhangzw@hku.hk-
dc.identifier.authorityZhang, Z=rp02087-
dc.description.naturelink_to_subscribed_fulltext-
dc.identifier.doi10.1137/19M1236989-
dc.identifier.hkuros304205-
dc.identifier.volume41-
dc.identifier.issue5-
dc.identifier.spageB1115–B1136-
dc.identifier.epageB1115–B1136-
dc.publisher.placeUnited States-

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