Laplacian Preconditioning of Elliptic PDEs: Localization of the Eigenvalues of the Discretized Operator

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F19%3A00505754" target="_blank" >RIV/67985807:_____/19:00505754 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/00216208:11320/19:10397880

Výsledek na webu

<a href="http://dx.doi.org/10.1137/18M1212458" target="_blank" >http://dx.doi.org/10.1137/18M1212458</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1137/18M1212458" target="_blank" >10.1137/18M1212458</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Laplacian Preconditioning of Elliptic PDEs: Localization of the Eigenvalues of the Discretized Operator

Popis výsledku v původním jazyce

In [IMA J. Numer. Anal., 29 (2009), pp. 24--42], Nielsen, Tveito, and Hackbusch study the operator generated by using the inverse of the Laplacian as the preconditioner for second order elliptic PDEs $- abla cdot (k(x) abla u) = f$. They prove that the range of $k(x)$ is contained in the spectrum of the preconditioned operator, provided that $k(x)$ is continuous. Their rigorous analysis only addresses mappings defined on infinite dimensional spaces, but the numerical experiments in the paper suggest that a similar property holds in the discrete case. Motivated by this investigation, we analyze the eigenvalues of the matrix ${L}^{-1}{A}$, where ${L}$ and ${{A}}$ are the stiffness matrices associated with the Laplace operator and second order elliptic operators with a scalar coefficient function, respectively. Using only technical assumptions on $k(x)$, we prove the existence of a one-to-one pairing between the eigenvalues of ${L}^{-1}{A}$ and the intervals determined by the images under $k(x)$ of the supports of the finite element nodal basis functions. As a consequence, we can show that the nodal values of $k(x)$ yield accurate approximations of the eigenvalues of ${L}^{-1}{A}$. Our theoretical results, including their relevance for understanding how the convergence of the conjugate gradient method may depend on the whole spectrum of the preconditioned matrix, are illuminated by several numerical experiments.

Název v anglickém jazyce

Laplacian Preconditioning of Elliptic PDEs: Localization of the Eigenvalues of the Discretized Operator

Popis výsledku anglicky

In [IMA J. Numer. Anal., 29 (2009), pp. 24--42], Nielsen, Tveito, and Hackbusch study the operator generated by using the inverse of the Laplacian as the preconditioner for second order elliptic PDEs $- abla cdot (k(x) abla u) = f$. They prove that the range of $k(x)$ is contained in the spectrum of the preconditioned operator, provided that $k(x)$ is continuous. Their rigorous analysis only addresses mappings defined on infinite dimensional spaces, but the numerical experiments in the paper suggest that a similar property holds in the discrete case. Motivated by this investigation, we analyze the eigenvalues of the matrix ${L}^{-1}{A}$, where ${L}$ and ${{A}}$ are the stiffness matrices associated with the Laplace operator and second order elliptic operators with a scalar coefficient function, respectively. Using only technical assumptions on $k(x)$, we prove the existence of a one-to-one pairing between the eigenvalues of ${L}^{-1}{A}$ and the intervals determined by the images under $k(x)$ of the supports of the finite element nodal basis functions. As a consequence, we can show that the nodal values of $k(x)$ yield accurate approximations of the eigenvalues of ${L}^{-1}{A}$. Our theoretical results, including their relevance for understanding how the convergence of the conjugate gradient method may depend on the whole spectrum of the preconditioned matrix, are illuminated by several numerical experiments.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure 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

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2019

Kód důvěrnosti údajů

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

Název periodika

SIAM Journal on Numerical Analysis

ISSN

0036-1429

e-ISSN

—

Svazek periodika

57

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

26

Strana od-do

1369-1394

Kód UT WoS článku

000473085400017

EID výsledku v databázi Scopus

2-s2.0-85069917331