Parallel and Robust Preconditioning for Space-Time Isogeometric Analysis of Parabolic Evolution Problems
Abstract
We propose and investigate new robust preconditioners for space-time Isogeometric Analysis (IgA) of parabolic evolution problems. These preconditioners are based on a time-parallel multigrid method. We consider a decomposition of the space-time cylinder into time-slabs which are coupled via a discontinuous Galerkin technique. The time-slabs provide the structure for the time-parallel multigrid solver. The most important part of the multigrid method is the smoother. We utilize the special structure of the involved operator to decouple its application into several spatial problems by means of generalized eigenvalue or Schur decompositions. Some of these problems have a symmetric saddle point structure, for which we present robust preconditioners. Finally, we present numerical experiments confirming the robustness of our space-time IgA solver.
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References
1.
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed., Pure Appl. Math. (Amst.) 140, Elsevier/Academic Press, Amsterdam, 2003.
2.
R. Andreev and C. Tobler, Multilevel preconditioning and low rank tensor iteration for space-time simultaneous discretizations of parabolic PDEs, Numer. Linear Algebra Appl., 22 (2015), p. 317--337, https://doi.org/10.1002/nla.1951.
3.
L. Beira͂o da Veiga, A. Buffa, G. Sangalli, and R. Vázquez, Mathematical analysis of variational isogeometric methods, Acta Numer., 23 (2014), pp. 157--287, https://doi.org/10.1017/S096249291400004X.
4.
L. Beira͂o da Veiga, L. F. Pavarino, S. Scacchi, O. B. Widlund, and S. Zampini, Adaptive selection of primal constraints for isogeometric BDDC deluxe preconditioners, SIAM J. Sci. Comput., 39 (2017), pp. A281--A302, https://doi.org/10.1137/15M1054675.
5.
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften 223, Springer-Verlag, Berlin, New York, 1976.
6.
A. N. Brooks and T. J. R. Hughes, Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32 (1982), pp. 199--259.
7.
J. A. Cottrell, T. J. R. Hughes, and Y. Bazilevs, Isogeometric Analysis, Toward Integration of CAD and FEA, John Wiley & Sons, Chichester, West Sussex, UK, 2009.
8.
M. Donatelli, C. Garoni, C. Manni, S. Serra-Capizzano, and H. Speleers, Symbol-based multigrid methods for Galerkin B-spline isogeometric analysis, SIAM J. Numer. Anal., 55 (2017), pp. 31--62, https://doi.org/10.1137/140988590.
9.
R. Falgout, S. Friedhoff, T. Kolev, S. MacLachlan, and J. Schroder, Parallel time integration with multigrid, SIAM J. Sci. Comput., 36 (2014), pp. C635--C661, https://doi.org/10.1137/130944230.
10.
M. J. Gander, 50 years of time parallel time integration, in Multiple Shooting and Time Domain Decomposition, Contrib. Math. Comput. Sci., 9, T. Carraro, M. Geiger, S. Körkel, and R. Rannacher, eds., Springer, Cham, 2015, pp. 69--114; available online at www.unige.ch/ gander/Preprints/50YearsTimeParallel.pdf.
11.
M. Gander and M. Neumüller, Analysis of a new space-time parallel multigrid algorithm for parabolic problems, SIAM J. Sci. Comput., 38 (2016), pp. A2173--A2208, https://doi.org/10.1137/15M1046605.
12.
M. Gander and S. Vandewalle, Analysis of the parareal time-parallel time-integration method, SIAM J. Sci. Comput., 29 (2007), pp. 556--578, https://doi.org/10.1137/05064607X.
13.
W. Hackbusch, Parabolic multigrid methods, in Computing Methods in Applied Sciences and Engineering VI, R. Glowinski and J.-L. Lions, eds., North-Holland, Amsterdam, 1984, pp. 189--197.
14.
C. Hofer, Parallelization of continuous and discontinuous Galerkin dual-primal isogeometric tearing and interconnecting methods, Comput. Math. Appl., 74 (2017), pp. 1607--1625, https://doi.org/10.1016/j.camwa.2017.06.051.
15.
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), pp. 126--150.
16.
C. Hofer and S. Takacs, A parallel multigrid solver for multi-patch isogeometric analysis, in Advanced Finite Element Methods with Applications. Selected Papers from the 30th Chemnitz Finite Element Symposium 2017, T. Apel, U. Langer, A. Meyer, and O. Steinbach, eds., Lect. Notes Comput. Sci. Eng. 128, Springer, Cham, 2019, pp. 191--206.
17.
C. Hofreither and S. Takacs, Robust multigrid for isogeometric analysis based on stable splittings of spline spaces, SIAM J. Numer. Anal., 4 (2017), pp. 2004--2024, https://doi.org/10.1137/16M1085425.
18.
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), pp. 4135--4195.
19.
B. Jüttler, U. Langer, A. Mantzaflaris, S. E. Moore, and W. Zulehner, Geometry + simulation modules: Implementing isogeometric analysis, PAMM, 14 (2014), pp. 961--962, https://doi.org/10.1002/pamm.201410461.
20.
O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Nauka, Moscow, 1973 (in Russian); Appl. Math. Sci. 49, Springer-Verlag, New York, 1985 (in English).
21.
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967 (in Russian); Transl. Math. Mongr. 23, AMS, Providence, RI, 1968 (in English).
22.
J. Lang, Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Theory, Algorithm, and Applications, Lect. Notes Comput. Sci. Eng. 16, Springer-Verlag, Berlin, Heidelberg, 2001.
23.
U. Langer, S. Moore, and M. Neumüller, Space-time isogeometric analysis of parabolic evolution equations, Comput. Methods Appl. Mech. Engrg., 306 (2016), pp. 342--363, https://doi.org/10.1016/j.cma.2016.03.042.
24.
U. Langer and I. Toulopoulos, Analysis of multipatch discontinuous Galerkin IgA approximations to elliptic boundary value problems, Comput. Vis. Sci., 17 (2015), pp. 217--233, https://doi.org/10.1007/s00791-016-0262-6.
25.
J.-L. Lions, Y. Maday, and G. Turinici, A parareal in time discretization of PDEs, C.R. Acad. Sci. Paris, Sér. I Math., 332 (2001), pp. 661--668.
26.
A. Mantzaflaris, C. Hofer, et al., G+Smo (Geometry + Simulation Modules) v0.8.1, http://gs.jku.at/gismo, 2015.
27.
E. McDonald, J. Pestana, and A. Wathen, Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations, SIAM J. Sci. Comput., 40 (2018), pp. A1012--A1033, https://doi.org/10.1137/16M1062016.
28.
M. Neumüller, Space-Time Methods: Fast Solvers and Applications, Monographic Series TU Graz: Computation in Engineering and Science 20, TU Graz, 2013, https://doi.org/10.3217/978-3-85125-290-3.
29.
M. Neumüller and I. Smears, Time-parallel iterative solvers for parabolic evolution equations, SIAM J. Sci. Comput., 41 (2019), p. C28--C51, https://doi.org/10.1137/18M1172466.
30.
G. Sangalli and M. Tani, Isogeometric preconditioners based on fast solvers for the Sylvester equation, SIAM J. Sci. Comput., 38 (2016), pp. A3644--A3671, https://doi.org/10.1137/16M1062788.
31.
O. Steinbach, Space-time finite element methods for parabolic problems, Comput. Methods Appl. Math., 15 (2015), pp. 551--566, https://doi.org/10.1515/cmam-2015-0026.
32.
O. Steinbach and H. Yang, Comparison of algebraic multigrid methods for an adaptive space-time finite-element discretization of the heat equation in 3D and 4D, Numer. Linear Algebra Appl., 25 (2018), e2143, https://doi.org/10.1002/nla.2143.
33.
O. Steinbach and H. Yang, Space--time finite element methods for parabolic evolution equations: Discretization, a posteriori error estimation, adaptivity and solution, in Space-Time Methods: Applications to Partial Differential Equations, Radon Series on Computational and Applied Mathematics 25, U. Langer and O. Steinbach, eds., Walter de Gruyter, Berlin, 2019.
34.
M. Stynes, Steady--state convection--diffusion problems, Acta Numer., 14 (2005), pp. 445--508.
35.
S. Takacs, Robust approximation error estimates and multigrid solvers for isogeometric multi-patch discretizations, Math. Models Methods Appl. Sci., 28 (2018), pp. 1899--1928, https://doi.org/10.1142/S021820251850046X.
36.
M. Tani, A preconditioning strategy for linear systems arising from nonsymmetric schemes in isogeometric analysis, Comput. Math. Appl., 74 (2017), pp. 1690--1702, https://doi.org/10.1016/j.camwa.2017.06.013.
37.
V. Thomeé, Galerkin Finite Element Methods for Parabolic Problems, 2nd ed., Springer Ser. Comput. Math. 25, Springer-Verlag, Berlin, 2006.
38.
S. Vandewalle, Parallel Multigrid Wavefrom Relaxation for Parabolic Problems, Teubner Skripten zur Numerik, Teubner, 1993.
39.
T. Warburton and J. Hesthaven, On the constants in $hp$-finite element trace inverse inequalities, Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 2765--2773, https://doi.org/10.1016/S0045-7825(03)00294-9.
40.
M. Wolfmayr, Multiharmonic Finite Element Analysis of Parabolic Time-Periodic Simulation and Optimal Control Problems, Ph.D. thesis, Johannes Kepler University, Institute of Computational Mathematics, Linz, Austria, 2014, http://www.numa.uni-linz.ac.at/Teaching/PhD/Finished/wolfmayr.
41.
W. Zulehner, Nonstandard norms and robust estimates for saddle point problems, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 536--560, https://doi.org/10.1137/100814767.
Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A1793 - A1821
DOI: 10.1137/18M1208794
ISSN (online): 1095-7197
Copyright
© 2019, Society for Industrial and Applied Mathematics.
History
Submitted: 21 August 2018
Accepted: 19 March 2019
Published online: 4 June 2019
Keywords
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Funding Information
Austrian Science Fund https://doi.org/10.13039/501100002428 : W1214, DK4, NFN S117-03
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