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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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On the approximations of derivatives of integrated semigroups. II

1Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China.

2Scientific Research Computer Center, Lomonosov Moscow State University, Vorobjevy Gory, Moscow 119991, Russia.

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 19, Issue 4-5, Pages 643–688, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/jiip.2011.053, September 2011

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This paper is devoted to the regularization in the process of approximation of values of unbounded operators in Banach spaces. More precisely, we consider the approximations of unbounded generators of C 0-semigroups and integrated semigroups. We consider M. M. Lavrentiev's method. Some iterative procedures are also investigated. It is shown that in general Banach spaces the approximation of the values of unbounded operators converges in strong sense. The presentation is given in the abstract framework of the discrete approximation scheme, which includes finite element methods, finite difference schemes and projection methods.

Keywords.: Abstract differential equations; Banach spaces; C 0-semigroups; integrated semigroups; Trotter–Kato theorem; discretization methods; discrete semigroups; stability of difference schemes; Lavrentiev's method; regularization procedure; iterative methods; ill-posed problems

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