A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators

Popis výsledku v původním jazyce

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).

Název v anglickém jazyce

A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators

Popis výsledku anglicky

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).

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

<a href="/cs/project/GC17-04150J" target="_blank" >GC17-04150J: Robustní dvojúrovňové simulace založené na Fourierově metodě a metodě konečných prvků: Odhady chyb, redukované modely a stochastika</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

SIAM Review

ISSN

0036-1445

e-ISSN

1095-7200

Svazek periodika

66

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

22

Strana od-do

125-146

Kód UT WoS článku

001222180700004

EID výsledku v databázi Scopus

2-s2.0-85187715352