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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr


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Volume 19, Issue 1

Issues

Low-Rank Space-Time Decoupled Isogeometric Analysis for Parabolic Problems with Varying Coefficients

Angelos MantzaflarisORCID iD: https://orcid.org/0000-0001-7135-1084 / Felix ScholzORCID iD: https://orcid.org/0000-0003-3339-0079 / Ioannis Toulopoulos
  • Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, A-4040 Linz, Austria
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Published Online: 2018-07-07 | DOI: https://doi.org/10.1515/cmam-2018-0024

Abstract

In this paper we present a space-time isogeometric analysis scheme for the discretization of parabolic evolution equations with diffusion coefficients depending on both time and space variables. The problem is considered in a space-time cylinder in d+1, with d=2,3, and is discretized using higher-order and highly-smooth spline spaces. This makes the matrix formation task very challenging from a computational point of view. We overcome this problem by introducing a low-rank decoupling of the operator into space and time components. Numerical experiments demonstrate the efficiency of this approach.

Keywords: Isogeometric Analysis; Low-Rank Approximation; Parabolic Initial-Boundary Value Problems; B-Splines; Isogeometric Matrix Assembly

MSC 2010: 65D07; 65M12; 65M15; 65M60

References

  • [1]

    L. Beirão da Veiga, A. Buffa, G. Sangalli and R. Vázquez, Mathematical analysis of variational isogeometric methods, Acta Numer. 23 (2014), 157–287. CrossrefWeb of ScienceGoogle Scholar

  • [2]

    J. A. Cottrell, T. J. R. Hughes and Y. Bazilevs, Isogeometric Analysis. Toward integration of CAD and FEA, John Wiley & Sons, Chichester, 2009. Google Scholar

  • [3]

    C. de Boor, A Practical Guide to Splines, revised ed., Applied Math. Sci. 27, Springer, New York, 2001. Google Scholar

  • [4]

    S. Dolgov and B. Khoromskij, Simultaneous state-time approximation of the chemical master equation using tensor product formats, Numer. Linear Algebra Appl. 22 (2015), no. 2, 197–219. Web of ScienceCrossrefGoogle Scholar

  • [5]

    S. V. Dolgov, B. N. Khoromskij and I. V. Oseledets, Fast solution of parabolic problems in the tensor train/quantized tensor train format with initial application to the Fokker–Planck equation, SIAM J. Sci. Comput. 34 (2012), no. 6, A3016–A3038. Web of ScienceCrossrefGoogle Scholar

  • [6]

    M. Heroux, An overview of trilinos, Technical Report SAND2003-2927, Sandia National Laboratories, 2003. Google Scholar

  • [7]

    C. Hofer, U. Langer, M. Neumüller and I. Toulopoulos, Time-multipatch discontinuous Galerkin space-time isogeometric analysis of parabolic evolution problems, Electron. Trans. Numer. Anal. 49 (2018), 126–150. CrossrefGoogle Scholar

  • [8]

    T. J. R. Hughes, J. A. Cottrell and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 39–41, 4135–4195. CrossrefGoogle Scholar

  • [9]

    S. K. Kleiss, C. Pechstein, B. Jüttler and S. Tomar, IETI—isogeometric tearing and interconnecting, Comput. Methods Appl. Mech. Engrg. 247/248 (2012), 201–215. CrossrefGoogle Scholar

  • [10]

    O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Appl. Math. Sci. 49, Springer, New York, 1985. Google Scholar

  • [11]

    O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 23, American Mathematical Society, Providence, 1968. Google Scholar

  • [12]

    U. Langer, A. Mantzaflaris, S. E. Moore and I. Toulopoulos, Multipatch discontinuous Galerkin isogeometric analysis, Isogeometric Analysis and Applications—IGAA 2014, Lect. Notes Comput. Sci. Eng. 107, Springer, Cham (2015), 1–32. Google Scholar

  • [13]

    U. Langer, S. E. Moore and M. Neumüller, Space-time isogeometric analysis of parabolic evolution problems, Comput. Methods Appl. Mech. Engrg. 306 (2016), 342–363. CrossrefGoogle Scholar

  • [14]

    A. Mantzaflaris, B. Jüttler, B. N. Khoromskij and U. Langer, Matrix generation in isogeometric analysis by low rank tensor approximation, Curves and Surfaces, Lecture Notes in Comput. Sci. 9213, Springer, Cham (2015), 321–340. Google Scholar

  • [15]

    A. Mantzaflaris, B. Jüttler, B. N. Khoromskij and U. Langer, Low rank tensor methods in Galerkin-based isogeometric analysis, Comput. Methods Appl. Mech. Engrg. 316 (2017), 1062–1085. CrossrefGoogle Scholar

  • [16]

    A. Mantzaflaris and F. Scholz, G+smo (Geometry plus Simulation modules) v0.8.1, (2017), http://gs.jku.at/gismo.

  • [17]

    F. Scholz, A. Mantzaflaris and B. Jüttler, Partial tensor decomposition for decoupling isogeometric Galerkin discretizations, Comput. Methods Appl. Mech. Engrg. 336 (2018), 485–506. CrossrefGoogle Scholar

About the article

Received: 2017-10-10

Revised: 2018-01-16

Accepted: 2018-05-02

Published Online: 2018-07-07

Published in Print: 2019-01-01


Funding Source: Austrian Science Fund

Award identifier / Grant number: NFN S117

This work was supported by the Austrian Science Fund (FWF) under the grant NFN S117.


Citation Information: Computational Methods in Applied Mathematics, Volume 19, Issue 1, Pages 123–136, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0024.

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