• Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

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ETNA - Electronic Transactions on Numerical Analysis



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Verlag der Österreichischen Akademie der Wissenschaften
Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2,
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400
https://verlag.oeaw.ac.at, e-mail: bestellung.verlag@oeaw.ac.at
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Conditioning of linear systems arising from penalty methods

    William Layton, Shuxian Xu

ETNA - Electronic Transactions on Numerical Analysis, pp. 394-401, 2023/04/25

doi: 10.1553/etna_vol58s394

doi: 10.1553/etna_vol58s394


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doi:10.1553/etna_vol58s394



doi:10.1553/etna_vol58s394

Abstract

Penalizing incompressibility in the Stokes problem leads, under mildassumptions, to matrices with condition numbers $\kappa =\mathcal{O}(\varepsilon ^{-1}h^{-2})$, with $\varepsilon =$ penalty parameter $\ll1$ and$h= $ meshwidth $<1$. Although $\kappa =\mathcal{O}(\varepsilon^{-1}h^{-2}) $ is large, practical tests seldom report difficulty in solvingthese systems. In the SPD case, using the conjugate gradient method, thisis usually explained by spectral gaps occurring in the penalized coefficientmatrix. Herein we point out a second contributing factor. Since the solutionis approximately incompressible, solution components in the eigenspacesassociated with the penalty terms can be small. As a result, the effective condition number can be much smaller than the standard condition number.

Keywords: penalty method, effective condition number

Published Online: 2023/04/25 09:46:19

Object Identifier: 0xc1aa5572 0x003e3388

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