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Goal-oriented error estimation and adaptivity in MsFEM computations

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Abstract

We introduce a goal-oriented strategy for multiscale computations performed using the Multiscale Finite Element Method (MsFEM). In a previous work, we have shown how to use, in the MsFEM framework, the concept of Constitutive Relation Error (CRE) to obtain a guaranteed and fully computable a posteriori error estimate in the energy norm (as well as error indicators on various error sources). Here, the CRE concept is coupled with the solution of an adjoint problem to control the error and drive an adaptive procedure with respect to a given output of interest. Furthermore, a local and non-intrusive enrichment technique is proposed to enhance the accuracy of error bounds. The overall strategy, which is fully automatic and robust, enables to reach an appropriate trade-off between certified reliability and computational cost in the MsFEM context. The performances of the proposed method are investigated on several illustrative numerical test cases. In particular, the error estimation is observed to be very accurate, yielding a very efficient adaptive procedure.

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Acknowledgements

We thank C. Le Bris for stimulating discussions on the topics addressed in this article. LC thanks Inria for enabling his two-year leave (2014–2016) in the MATHERIALS project team. The work of FL is partially supported by ONR under grant N00014-15-1-2777 and by EOARD under grant FA9550-17-1-0294.

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Authors and Affiliations

  1. Université Paris-Saclay, ENS Paris-Saclay, CNRS, LMT, 4 avenue des Sciences, 91190, Gif-sur-Yvette, France

    Ludovic Chamoin

  2. Institut Universitaire de France (IUF), 1 rue Descartes, 75231, Paris Cedex 5, France

    Ludovic Chamoin

  3. Laboratoire Navier (ENPC, Paris-Est University), 6-8 avenue Blaise Pascal, 77455, Marne-La-Vallée Cedex 2, France

    Frédéric Legoll

  4. Inria Paris, MATHERIALS project-team, 2 rue Simone Iff, CS 42112, 75589, Paris Cedex 12, France

    Frédéric Legoll

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  1. Ludovic Chamoin
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Chamoin, L., Legoll, F. Goal-oriented error estimation and adaptivity in MsFEM computations. Comput Mech 67, 1201–1228 (2021). https://doi.org/10.1007/s00466-021-01990-x

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