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Weakly Consistent Regularisation Methods for Ill-Posed Problems

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Numerical Methods for PDEs

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 15))

Abstract

This Chapter takes its origin in the lecture notes for a 9 h course at the Institut Henri Poincaré in September 2016. The course was divided in three parts. In the first part, which is not included herein, the aim was to first recall some basic aspects of stabilised finite element methods for convection-diffusion problems. We focus entirely on the second and third parts which were dedicated to ill-posed problems and their approximation using stabilised finite element methods. First we introduce the concept of conditional stability. Then we consider the elliptic Cauchy-problem and a data assimilation problem in a unified setting and show how stabilised finite element methods may be used to derive error estimates that are consistent with the stability properties of the problem and the approximation properties of the finite element space. Finally, we extend the result to a data assimilation problem subject to the heat equation.

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Authors and Affiliations

  1. Department of Mathematics, University College London, London, UK

    Erik Burman & Lauri Oksanen

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  1. Erik Burman
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  2. Lauri Oksanen
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Correspondence to Erik Burman .

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Editors and Affiliations

  1. Institut Montpelliérain Alexander Grothendieck, CNRS, Université Montpellier, Montpellier, France

    Daniele Antonio Di Pietro

  2. Universit Paris Est, CERMICS (ENPC), Marne la Vallée, France

    Alexandre Ern

  3. MOX - Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Luca Formaggia

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Burman, E., Oksanen, L. (2018). Weakly Consistent Regularisation Methods for Ill-Posed Problems. In: Di Pietro, D., Ern, A., Formaggia, L. (eds) Numerical Methods for PDEs. SEMA SIMAI Springer Series, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-94676-4_7

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