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On multiscale methods in Petrov–Galerkin formulation

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Abstract

In this work we investigate the advantages of multiscale methods in Petrov–Galerkin (PG) formulation in a general framework. The framework is based on a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space with good approximation properties and a high dimensional remainder space, which only contains negligible fine scale information. The multiscale space can then be used to obtain accurate Galerkin approximations. As a model problem we consider the Poisson equation. We prove that a Petrov–Galerkin formulation does not suffer from a significant loss of accuracy, and still preserve the convergence order of the original multiscale method. We also prove inf-sup stability of a PG continuous and a discontinuous Galerkin finite element multiscale method. Furthermore, we demonstrate that the Petrov–Galerkin method can decrease the computational complexity significantly, allowing for more efficient solution algorithms. As another application of the framework, we show how the Petrov–Galerkin framework can be used to construct a locally mass conservative solver for two-phase flow simulation that employs the Buckley–Leverett equation. To achieve this, we couple a PG discontinuous Galerkin finite element method with an upwind scheme for a hyperbolic conservation law.

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Acknowledgments

We would like to thank the anonymous referees for their valuable comments and their constructive feedback on the original manuscript which helped us to improve this article.

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Author notes

  1. Patrick Henning

    Present address: University of Münster, 48149, Münster, Germany

Authors and Affiliations

  1. Department of Information Technology, Uppsala University, Box 337, 751 05, Uppsala, Sweden

    Daniel Elfverson

  2. Department of Mathematics, University of Wyoming, Laramie, WY, 82071, USA

    Victor Ginting

  3. Section de Mathématiques, École polytechnique fédérale de Lausanne, 1015, Lausanne, Switzerland

    Patrick Henning

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  1. Daniel Elfverson
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  2. Victor Ginting
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Correspondence to Patrick Henning.

Additional information

D. Elfverson and P. Henning were partially supported by the Göran Gustafsson Foundation and the Swedish Research Council.

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Elfverson, D., Ginting, V. & Henning, P. On multiscale methods in Petrov–Galerkin formulation. Numer. Math. 131, 643–682 (2015). https://doi.org/10.1007/s00211-015-0703-z

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  • DOI: https://doi.org/10.1007/s00211-015-0703-z

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