A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10492946" target="_blank" >RIV/00216208:11320/24:10492946 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=9CfcF7qTL6" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=9CfcF7qTL6</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/23M1600992" target="_blank" >10.1137/23M1600992</a>
Alternative languages
Result language
angličtina
Original language name
A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators
Original language description
We analyze the spectrum of the operator Delta - 1 [V center dot ( K V u )] subject to homogeneous Dirichlet or Neumann boundary conditions, where Delta denotes the Laplacian and K = K ( x, y ) is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral decomposition K = Q Lambda Q T , where Q = Q ( x, y ) is an orthogonal matrix and Lambda = Lambda ( x, y ) is a diagonal matrix. More precisely, provided that K is continuous, the spectrum equals the convex hull of the ranges of the diagonal function entries of Lambda . The domain involved is assumed to be bounded and Lipschitz. In addition to studying operators defined on infinite -dimensional Sobolev spaces, we also report on recent results concerning their discretized finite -dimensional counterparts. More specifically, even though Delta - 1 [V center dot ( K V u )] is not compact, it turns out that every point in the spectrum of this operator can, to an arbitrary accuracy, be approximated by eigenvalues of the associated generalized algebraic eigenvalue problems arising from discretizations. Our theoretical investigations are illuminated by numerical experiments. The results presented in this paper extend previous analyses which have addressed elliptic differential operators with scalar coefficient functions. Our investigation is motivated by both preconditioning issues (efficient numerical computations) and the need to further develop the spectral theory of second order PDEs (core analysis).
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10102 - Applied mathematics
Result continuities
Project
<a href="/en/project/GC17-04150J" target="_blank" >GC17-04150J: Reliable two-scale Fourier/finite element-based simulations: Error-control, model reduction, and stochastics</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
SIAM Review
ISSN
0036-1445
e-ISSN
1095-7200
Volume of the periodical
66
Issue of the periodical within the volume
1
Country of publishing house
US - UNITED STATES
Number of pages
22
Pages from-to
125-146
UT code for WoS article
001222180700004
EID of the result in the Scopus database
2-s2.0-85187715352