Convergence of a finite volume scheme to the eigenvalues of a Schrödinger operator
dc.bibliographicCitation.volume | 1260 | |
dc.contributor.author | Koprucki, Thomas | |
dc.contributor.author | Eymard, Robert | |
dc.contributor.author | Fuhrmann, Jürgen | |
dc.date.accessioned | 2016-03-24T17:38:19Z | |
dc.date.available | 2019-06-28T08:02:46Z | |
dc.date.issued | 2007 | |
dc.description.abstract | We consider the approximation of a Schrödinger eigenvalue problem arising from the modeling of semiconductor nanostructures by a finite volume method in a bounded domain $OmegasubsetR^d$. In order to prove its convergence, a framework for finite dimensional approximations to inner products in the Sobolev space $H^1_0(Omega)$ is introduced which allows to apply well known results from spectral approximation theory. This approach is used to obtain convergence results for a classical finite volume scheme for isotropic problems based on two point fluxes, and for a finite volume scheme for anisotropic problems based on the consistent reconstruction of nodal fluxes. In both cases, for two- and three-dimensional domains we are able to prove first order convergence of the eigenvalues if the corresponding eigenfunctions belong to $H^2(Omega)$. The construction of admissible meshes for finite volume schemes using the Delaunay-Voronoï method is discussed. As numerical examples, a number of one-, two- and three-dimensional problems relevant to the modeling of semiconductor nanostructures is presented. In order to obtain analytical eigenvalues for these problems, a matching approach is used. To these eigenvalues, and to recently published highly accurate eigenvalues for the Laplacian in the L-shape domain, the results of the implemented numerical method are compared. In general, for piecewise $H^2$ regular eigenfunctions, second order convergence is observed experimentally. | eng |
dc.description.version | publishedVersion | eng |
dc.format | application/pdf | |
dc.identifier.issn | 0946-8633 | |
dc.identifier.uri | https://doi.org/10.34657/2165 | |
dc.identifier.uri | https://oa.tib.eu/renate/handle/123456789/1915 | |
dc.language.iso | eng | eng |
dc.publisher | Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik | eng |
dc.relation.ispartofseries | Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik, Volume 1260, ISSN 0946-8633 | eng |
dc.rights.license | This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties. | eng |
dc.rights.license | Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden. | ger |
dc.subject | Schrödinger operator | eng |
dc.subject | eigenvalues | eng |
dc.subject | finite volume schemes | eng |
dc.subject.ddc | 510 | eng |
dc.title | Convergence of a finite volume scheme to the eigenvalues of a Schrödinger operator | eng |
dc.type | report | eng |
dc.type | Text | eng |
dcterms.bibliographicCitation.journalTitle | Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik | eng |
tib.accessRights | openAccess | eng |
wgl.contributor | WIAS | eng |
wgl.subject | Mathematik | eng |
wgl.type | Report / Forschungsbericht / Arbeitspapier | eng |
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