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Localized orthogonal decomposition method for the wave equation with a continuum of scales

Authors: Assyr Abdulle and Patrick Henning
Journal: Math. Comp. 86 (2017), 549-587
MSC (2010): Primary 35L05, 65M60, 65N30, 74Q10, 74Q15
Published electronically: May 3, 2016
MathSciNet review: 3584540
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Abstract: This paper is devoted to numerical approximations for the wave equation with a multiscale character. Our approach is formulated in the framework of the Localized Orthogonal Decomposition (LOD) interpreted as a numerical homogenization with an $L^2$-projection. We derive explicit convergence rates of the method in the $L^{\infty }(L^2)$-, $W^{1,\infty }(L^2)$- and $L^{\infty }(H^1)$-norms without any assumptions on higher order space regularity or scale-separation. The order of the convergence rates depends on further graded assumptions on the initial data. We also prove the convergence of the method in the framework of G-convergence without any structural assumptions on the initial data, i.e. without assuming that it is well-prepared. This rigorously justifies the method. Finally, the performance of the method is demonstrated in numerical experiments.

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Additional Information

Assyr Abdulle
Affiliation: Section de Mathématiques, École polytechnique fédérale de Lausanne, 1015 Lausanne, Switzerland

Patrick Henning
Affiliation: Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms- Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany
Address at time of publication: Division of Numerical Analysis, KTH Royal Institute of Technology, Lindstedtsvägen 25, S-10044 Stockholm, Sweden

Keywords: Finite element, wave equation, numerical homogenization, multiscale method, LOD
Received by editor(s): June 24, 2014
Received by editor(s) in revised form: May 12, 2015, September 17, 2015, and September 21, 2015
Published electronically: May 3, 2016
Additional Notes: This work was partially supported by the Swiss National Foundation, Grant No. $200021\_134716/1$, and by the Deutsche Forschungsgemeinschaft, DFG-Grant No. OH 98/6-1.
Article copyright: © Copyright 2016 American Mathematical Society