Issue |
ESAIM: M2AN
Volume 57, Number 4, July-August 2023
|
|
---|---|---|
Page(s) | 2227 - 2255 | |
DOI | https://doi.org/10.1051/m2an/2023050 | |
Published online | 03 July 2023 |
Least squares solvers for ill-posed PDEs that are conditionally stable
1
Mathematics Department, University of South Carolina, Columbia, SC 29208, USA
2
Korteweg-de Vries (KdV) Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
* Corresponding author: rob.p.stevenson@gmail.com
Received:
15
July
2022
Accepted:
30
May
2023
This paper is concerned with the design and analysis of least squares solvers for ill-posed PDEs that are conditionally stable. The norms and the regularization term used in the least squares functional are determined by the ingredients of the conditional stability assumption. We are then able to establish a general error bound that, in view of the conditional stability assumption, is qualitatively the best possible, without assuming consistent data. The price for these advantages is to handle dual norms which reduces to verifying suitable inf-sup stability. This, in turn, is done by constructing appropriate Fortin projectors for all sample scenarios. The theoretical findings are illustrated by numerical experiments.
Mathematics Subject Classification: 35B30 / 35B35 / 35B45 / 35R25 / 65F08 / 65J20 / 65M12 / 65N12
Key words: Tikhonov regularization / least squares methods / conditional stability / dual norms / inf-sup stability / Fortin projectors / mixed formulations / a posteriori bounds for residuals
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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