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

Result code in 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>

Alternative codes found

RIV/00216208:11320/19:10397880

Result on the web

<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>

Result language

angličtina

Original language name

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

Original language description

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.

Czech name

—

Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

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

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2019

Confidentiality

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

Name of the periodical

SIAM Journal on Numerical Analysis

ISSN

0036-1429

e-ISSN

—

Volume of the periodical

57

Issue of the periodical within the volume

3

Country of publishing house

US - UNITED STATES

Number of pages

26

Pages from-to

1369-1394

UT code for WoS article

000473085400017

EID of the result in the Scopus database

2-s2.0-85069917331