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Meshfree Methods for Partial Differential Equations VII pp 185–204Cite as

Multiscale Partition of Unity

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 100)

Abstract

We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite element mesh. The method modifies a given partition of unity such that optimal convergence is achieved independent of oscillation or discontinuities of the diffusion coefficient. The modification is based on an orthogonal decomposition of the solution space while preserving the partition of unity property. This precomputation involves the solution of independent problems on local subdomains of selectable size. We deduce quantitative error estimates for the method that account for the chosen amount of localization. Numerical experiments illustrate the high approximation properties even for ‘cheap’ parameter choices.

Keywords

  • Partition of unity method
  • Multiscale method
  • LOD
  • Upscaling
  • Homogenization

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Author information

Authors and Affiliations

  1. ANMC, Section de Mathématiques, École polytechnique fédérale de Lausanne, Lausanne, Switzerland

    Patrick Henning

  2. Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany

    Philipp Morgenstern & Daniel Peterseim

Authors
  1. Patrick Henning
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  2. Philipp Morgenstern
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  3. Daniel Peterseim
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Corresponding author

Correspondence to Philipp Morgenstern .

Editor information

Editors and Affiliations

  1. Institut für Numerische Simulation, Universität Bonn, Bonn, Germany

    Michael Griebel

  2. Institut für Numerische Simulation, Universität Bonn, Bonn, Germany

    Marc Alexander Schweitzer

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Henning, P., Morgenstern, P., Peterseim, D. (2015). Multiscale Partition of Unity. In: Griebel, M., Schweitzer, M. (eds) Meshfree Methods for Partial Differential Equations VII. Lecture Notes in Computational Science and Engineering, vol 100. Springer, Cham. https://doi.org/10.1007/978-3-319-06898-5_10

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