Numerical approximation of the spectrum of self-adjoint operators in operator preconditioning

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F22%3A10453564" target="_blank" >RIV/00216208:11320/22:10453564 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=eQQhtaSMPX" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=eQQhtaSMPX</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s11075-022-01263-5" target="_blank" >10.1007/s11075-022-01263-5</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Numerical approximation of the spectrum of self-adjoint operators in operator preconditioning

Popis výsledku v původním jazyce

We consider operator preconditioning B-1A, which is employed in the numerical solution of boundary value problems. Here, the self-adjoint operators A,B:H10(Ω)RIGHTWARDS ARROWH-1(Ω) are the standard integral/functional representations of the partial differential operators -NABLADOT OPERATOR   (k(x)NABLAu) and -NABLADOT OPERATOR   (g(x)NABLAu), respectively, and the scalar coefficient functions k(x) and g(x) are assumed to be continuous throughout the closure of the solution domain. The function g(x) is also assumed to be uniformly positive. When the discretized problem, with the preconditioned operator B-1nAn, is solved with Krylov subspace methods, the convergence behavior depends on the distribution of the eigenvalues. Therefore, it is crucial to understand how the eigenvalues of B-1nAn are related to the spectrum of B-1A. Following the path started in the two recent papers published in SIAM J. Numer. Anal. [57 (2019), pp. 1369-1394 and 58 (2020), pp. 2193-2211], the first part of this paper addresses the open question concerning the distribution of the eigenvalues of B-1nAnformulated at the end of the second paper. The approximation of the spectrum studied in the present paper differs from the eigenvalue problem studied in the classical PDE literature which addresses individual eigenvalues of compact (solution) operators.In the second part of this paper, we generalize some of our results to general bounded and self-adjoint operators A,B:VRIGHTWARDS ARROWV#, where V# denotes the dual of V. More specifically, provided that B is coercive and that the standard Galerkin discretization approximation properties hold, we prove that the whole spectrum of B-1A:VRIGHTWARDS ARROWV is approximated to an arbitrary accuracy by the eigenvalues of its finite dimensional discretization B-1nAn.

Název v anglickém jazyce

Numerical approximation of the spectrum of self-adjoint operators in operator preconditioning

Popis výsledku anglicky

We consider operator preconditioning B-1A, which is employed in the numerical solution of boundary value problems. Here, the self-adjoint operators A,B:H10(Ω)RIGHTWARDS ARROWH-1(Ω) are the standard integral/functional representations of the partial differential operators -NABLADOT OPERATOR   (k(x)NABLAu) and -NABLADOT OPERATOR   (g(x)NABLAu), respectively, and the scalar coefficient functions k(x) and g(x) are assumed to be continuous throughout the closure of the solution domain. The function g(x) is also assumed to be uniformly positive. When the discretized problem, with the preconditioned operator B-1nAn, is solved with Krylov subspace methods, the convergence behavior depends on the distribution of the eigenvalues. Therefore, it is crucial to understand how the eigenvalues of B-1nAn are related to the spectrum of B-1A. Following the path started in the two recent papers published in SIAM J. Numer. Anal. [57 (2019), pp. 1369-1394 and 58 (2020), pp. 2193-2211], the first part of this paper addresses the open question concerning the distribution of the eigenvalues of B-1nAnformulated at the end of the second paper. The approximation of the spectrum studied in the present paper differs from the eigenvalue problem studied in the classical PDE literature which addresses individual eigenvalues of compact (solution) operators.In the second part of this paper, we generalize some of our results to general bounded and self-adjoint operators A,B:VRIGHTWARDS ARROWV#, where V# denotes the dual of V. More specifically, provided that B is coercive and that the standard Galerkin discretization approximation properties hold, we prove that the whole spectrum of B-1A:VRIGHTWARDS ARROWV is approximated to an arbitrary accuracy by the eigenvalues of its finite dimensional discretization B-1nAn.

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

2022

Kód důvěrnosti údajů

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

Název periodika

Numerical Algorithms

ISSN

1017-1398

e-ISSN

—

Svazek periodika

91

Číslo periodika v rámci svazku

June

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

25

Strana od-do

301-325

Kód UT WoS článku

000804491400001

EID výsledku v databázi Scopus

2-s2.0-85131295247