Space–time least–squares isogeometric method and efficient solver for parabolic problems

Montardini, Monica; Negri, Matteo; Sangalli, Giancarlo; Tani, Mattia

doi:10.1090/mcom/3471

Space–time least–squares isogeometric method and efficient solver for parabolic problems
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by Monica Montardini, Matteo Negri, Giancarlo Sangalli and Mattia Tani HTML | PDF

Math. Comp. 89 (2020), 1193-1227 Request permission

Abstract:

In this paper, we propose a space-time least-squares isogeometric method to solve parabolic evolution problems, well suited for high-degree smooth splines in the space-time domain. We focus on the linear solver and its computational efficiency: thanks to the proposed formulation and to the tensor-product construction of space-time splines, we can design a preconditioner whose application requires the solution of a Sylvester-like equation, which is performed efficiently by the fast diagonalization method. The preconditioner is robust w.r.t. spline degree and mesh size. The computational time required for its application, for a serial execution, is almost proportional to the number of degrees-of-freedom and independent of the polynomial degree. The proposed approach is also well-suited for parallelization.

References

Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient flows in metric spaces and in the space of probability measures, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. MR 2401600
Jean-Pierre Aubin, Applied functional analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1979. Translated from the French by Carole Labrousse; With exercises by Bernard Cornet and Jean-Michel Lasry. MR 549483
L. Beirão da Veiga, A. Buffa, G. Sangalli, and R. Vázquez, Mathematical analysis of variational isogeometric methods, Acta Numer. 23 (2014), 157–287. MR 3202239, DOI 10.1017/S096249291400004X
L. Beirão da Veiga, D. Cho, and G. Sangalli, Anisotropic NURBS approximation in isogeometric analysis, Comput. Methods Appl. Mech. Engrg. 209/212 (2012), 1–11. MR 2877950, DOI 10.1016/j.cma.2011.10.016
Brent C. Bell and Karan S. Surana, A space-time coupled $p$-version least-squares finite element formulation for unsteady fluid dynamics problems, Internat. J. Numer. Methods Engrg. 37 (1994), no. 20, 3545–3569. MR 1300113, DOI 10.1002/nme.1620372008
Brent C. Bell and Karan S. Surana, A space-time coupled $p$-version least squares finite element formulation for unsteady two-dimensional Navier-Stokes equations, Internat. J. Numer. Methods Engrg. 39 (1996), no. 15, 2593–2618. MR 1403687, DOI 10.1002/(SICI)1097-0207(19960815)39:15<2593::AID-NME968>3.0.CO;2-2
Pavel B. Bochev and Max D. Gunzburger, Least-squares finite element methods, Applied Mathematical Sciences, vol. 166, Springer, New York, 2009. MR 2490235, DOI 10.1007/b13382
A. Bressan and E. Sande, Approximation in FEM, DG and IGA: A Theoretical Comparison, arXiv preprint arXiv:1808.04163 (2018).
H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, 5 (1973), North-Holland Mathematics Studies, No. 5. Notas de Matemática (50).
Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382
Nathan Collier, Lisandro Dalcin, David Pardo, and V. M. Calo, The cost of continuity: performance of iterative solvers on isogeometric finite elements, SIAM J. Sci. Comput. 35 (2013), no. 2, A767–A784. MR 3035485, DOI 10.1137/120881038
J. Austin Cottrell, Thomas J. R. Hughes, and Yuri Bazilevs, Isogeometric analysis, John Wiley & Sons, Ltd., Chichester, 2009. Toward integration of CAD and FEA. MR 3618875, DOI 10.1002/9780470749081
J. A. Cottrell, T. J. R. Hughes, and A. Reali, Studies of refinement and continuity in isogeometric structural analysis, Computer Methods in Applied Mechanics and Engineering 196 (2007), no. 41, 4160–4183.
Carl de Boor, A practical guide to splines, Revised edition, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York, 2001. MR 1900298
M. O. Deville, P. F. Fischer, and E. H. Mund, High-order methods for incompressible fluid flow, Cambridge Monographs on Applied and Computational Mathematics, vol. 9, Cambridge University Press, Cambridge, 2002. MR 1929237, DOI 10.1017/CBO9780511546792
C. A. Dorao and H. A. Jakobsen, A parallel time-space least-squares spectral element solver for incompressible flow problems, Appl. Math. Comput. 185 (2007), no. 1, 45–58. MR 2298429, DOI 10.1016/j.amc.2006.07.009
John A. Evans, Yuri Bazilevs, Ivo Babuška, and Thomas J. R. Hughes, $n$-widths, sup-infs, and optimality ratios for the $k$-version of the isogeometric finite element method, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 21-26, 1726–1741. MR 2517942, DOI 10.1016/j.cma.2009.01.021
Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
I. Fried, Finite-element analysis of time-dependent phenomena., AIAA Journal 7 (1969), no. 6, 1170–1173.
Martin J. Gander, 50 years of time parallel time integration, Multiple shooting and time domain decomposition methods, Contrib. Math. Comput. Sci., vol. 9, Springer, Cham, 2015, pp. 69–113. MR 3676210
Pierre Grisvard, Elliptic problems in nonsmooth domains, Classics in Applied Mathematics, vol. 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. Reprint of the 1985 original [ MR0775683]; With a foreword by Susanne C. Brenner. MR 3396210, DOI 10.1137/1.9781611972030.ch1
J. H. Nguyen and Reynen, A space-time least-square finite element scheme for advection-diffusion equations, Computer Methods in Applied Mechanics and Engineering 42 (1984), no. 3, 331–342.
C. Hofer, U. Langer, and M. Neumüller, Robust Preconditioning for Space-Time Isogeometric Analysis of Parabolic Evolution Problems, arXiv preprint arXiv:1802.09277 (2018).
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. MR 2152382, DOI 10.1016/j.cma.2004.10.008
J. C. Bruch Jr and G. Zyvoloski, Transient two-dimensional heat conduction problems solved by the finite element method, International Journal for Numerical Methods in Engineering 8 (1974), no. 3, 481–494.
Tamara G. Kolda and Brett W. Bader, Tensor decompositions and applications, SIAM Rev. 51 (2009), no. 3, 455–500. MR 2535056, DOI 10.1137/07070111X
Arne Morten Kvarving and Einar M. Rønquist, A fast tensor-product solver for incompressible fluid flow in partially deformed three-dimensional domains: parallel implementation, Comput. & Fluids 52 (2011), 22–32. MR 2971993, DOI 10.1016/j.compfluid.2011.08.007
O. A. Ladyženskaja and N. N. Ural′ceva, Équations aux dérivées partielles de type elliptique, Monographies Universitaires de Mathématiques, No. 31, Dunod, Paris, 1968 (French). Traduit par G. Roos. MR 0239273
Ulrich Langer, Stephen E. Moore, and Martin Neumüller, Space-time isogeometric analysis of parabolic evolution problems, Comput. Methods Appl. Mech. Engrg. 306 (2016), 342–363. MR 3502571, DOI 10.1016/j.cma.2016.03.042
U. Langer, M. Neumüller, and I. Toulopoulos, Multipatch space-time isogeometric analysis of parabolic diffusion problems, Large-scale scientific computing, Lecture Notes in Comput. Sci., vol. 10665, Springer, Cham, 2018, pp. 21–32. MR 3763512, DOI 10.1007/978-3-319-73441-5_{2}
Robert E. Lynch, John R. Rice, and Donald H. Thomas, Direct solution of partial difference equations by tensor product methods, Numer. Math. 6 (1964), 185–199. MR 213047, DOI 10.1007/BF01386067
M. Montardini, G. Sangalli, and M. Tani, Robust isogeometric preconditioners for the Stokes system based on the fast diagonalization method, Comput. Methods Appl. Mech. Engrg. 338 (2018), 162–185. MR 3811623, DOI 10.1016/j.cma.2018.04.017
J. T. Oden, A general theory of finite elements. I. Topological considerations, International Journal for Numerical Methods in Engineering 1 (1969), no. 2, 205–221.
J. T. Oden, A general theory of finite elements. II. Applications, International Journal for Numerical Methods in Engineering 1 (1969), no. 3, 247–259.
Giancarlo Sangalli and Mattia Tani, Isogeometric preconditioners based on fast solvers for the Sylvester equation, SIAM J. Sci. Comput. 38 (2016), no. 6, A3644–A3671. MR 3572372, DOI 10.1137/16M1062788
G. Sangalli and M. Tani, Matrix-free weighted quadrature for a computationally efficient isogeometric $k$-method, Comput. Methods Appl. Mech. Engrg. 338 (2018), 117–133. MR 3811621, DOI 10.1016/j.cma.2018.04.029
Filippo Santambrogio, {Euclidean, metric, and Wasserstein} gradient flows: an overview, Bull. Math. Sci. 7 (2017), no. 1, 87–154. MR 3625852, DOI 10.1007/s13373-017-0101-1
Christoph Schwab and Rob Stevenson, Space-time adaptive wavelet methods for parabolic evolution problems, Math. Comp. 78 (2009), no. 267, 1293–1318. MR 2501051, DOI 10.1090/S0025-5718-08-02205-9
Farzin Shakib and Thomas J. R. Hughes, A new finite element formulation for computational fluid dynamics. IX. Fourier analysis of space-time Galerkin/least-squares algorithms, Comput. Methods Appl. Mech. Engrg. 87 (1991), no. 1, 35–58. MR 1103416, DOI 10.1016/0045-7825(91)90145-V
L. Sorber, M. Van Barel, and L. De Lathauwer, Tensorlab v2. 0, Available online, URL: www.tensorlab.net (2014).
Olaf Steinbach, Space-time finite element methods for parabolic problems, Comput. Methods Appl. Math. 15 (2015), no. 4, 551–566. MR 3403450, DOI 10.1515/cmam-2015-0026
Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
Thomas J. R. Hughes, Leopoldo P. Franca, and Gregory M. Hulbert, A new finite element formulation for computational fluid dynamics. VIII. The Galerkin/least-squares method for advective-diffusive equations, Comput. Methods Appl. Mech. Engrg. 73 (1989), no. 2, 173–189. MR 1002621, DOI 10.1016/0045-7825(89)90111-4
K. Takizawa, T. E. Tezduyar, A. Buscher, and S. Asada, Space–time fluid mechanics computation of heart valve models, Computational Mechanics 54 (2014), no. 4, 973–986.
Kenji Takizawa, Tayfun E. Tezduyar, Yuto Otoguro, Takuya Terahara, Takashi Kuraishi, and Hitoshi Hattori, Turbocharger flow computations with the space-time isogeometric analysis (ST-IGA), Comput. & Fluids 142 (2017), 15–20. MR 3577974, DOI 10.1016/j.compfluid.2016.02.021
R. Vázquez, A new design for the implementation of isogeometric analysis in Octave and Matlab: GeoPDEs 3.0, Comput. Math. Appl. 72 (2016), no. 3, 523–554. MR 3521056, DOI 10.1016/j.camwa.2016.05.010
Eugene L. Wachspress, Generalized ADI preconditioning, Comput. Math. Appl. 10 (1984), no. 6, 457–461 (1985). MR 783519, DOI 10.1016/0898-1221(84)90076-2