In this article, we propose and analyze an alternate proof of a priori error estimates for semidiscrete Galerkin approximations to a general second order linear parabolic initial and boundary value problem with rough initial data. Our analysis is based on energy arguments without using parabolic duality. Further, it follows the spirit of the proof technique used for deriving optimal error estimates for finite element approximations to parabolic problems with smooth initial data and hence, it unifies both theories, that is, one for smooth initial data and other for nonsmooth data. Moreover, the proposed technique is also extended to a semidiscrete mixed method for linear parabolic problems. In both cases, optimal L2-error estimates are derived, when the initial data is in L2. A superconvergence phenomenon is also observed, which is then used to prove L∞-estimates for linear parabolic problems defined on two-dimensional spatial domain again with rough initial data. Copyright © Taylor & Francis Group, LLC.
|Original language||English (US)|
|Number of pages||37|
|Journal||Numerical Functional Analysis and Optimization|
|State||Published - Sep 2011|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: D. G. would like to thank CSIR, Government of India, for the financial support. A. K. P. acknowledges the support provided by the DST (Department of Science and Technology), Government of India project No 08DST012. This publication is also based on work supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). The authors are grateful to the referees for their valuable suggestions and comments, which help to improve the present manuscript.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.