Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter July 7, 2018

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

Angelos Mantzaflaris ORCID logo, Felix Scholz ORCID logo and Ioannis Toulopoulos

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.

Funding source: Austrian Science Fund

Award Identifier / Grant number: NFN S117

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

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. 10.1017/S096249291400004XSearch in Google 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. 10.1002/9780470749081Search in Google Scholar

[3] C. de Boor, A Practical Guide to Splines, revised ed., Applied Math. Sci. 27, Springer, New York, 2001. Search in 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. 10.1002/nla.1942Search in Google 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. 10.1137/120864210Search in Google Scholar

[6] M. Heroux, An overview of trilinos, Technical Report SAND2003-2927, Sandia National Laboratories, 2003. Search in 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. 10.1553/etna_vol49s126Search in Google 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. 10.1016/j.cma.2004.10.008Search in Google 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. 10.1016/j.cma.2012.08.007Search in Google Scholar

[10] O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Appl. Math. Sci. 49, Springer, New York, 1985. 10.1007/978-1-4757-4317-3Search in 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. 10.1090/mmono/023Search in 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. 10.1007/978-3-319-23315-4_1Search in 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. 10.1016/j.cma.2016.03.042Search in Google 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. 10.1007/978-3-319-22804-4_24Search in 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. 10.1016/j.cma.2016.11.013Search in Google Scholar

[16] A. Mantzaflaris and F. Scholz, G+smo (Geometry plus Simulation modules) v0.8.1, (2017), http://gs.jku.at/gismo. Search in Google Scholar

[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. 10.1016/j.cma.2018.03.026Search in Google Scholar

Received: 2017-10-10
Revised: 2018-01-16
Accepted: 2018-05-02
Published Online: 2018-07-07
Published in Print: 2019-01-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston