Methods and Algorithms for Scientific Computing

Efficient Direct Space-Time Finite Element Solvers for Parabolic Initial-Boundary Value Problems in Anisotropic Sobolev Spaces

Abstract

We consider a space-time variational formulation of parabolic initial-boundary value problems in anisotropic Sobolev spaces in combination with a Hilbert-type transformation. This variational setting is the starting point for the space-time Galerkin finite element discretization that leads to a large global linear system of algebraic equations. We propose and investigate new efficient direct solvers for this system. In particular, we use a tensor-product approach with piecewise polynomial, globally continuous ansatz and test functions. The developed solvers are based on the Bartels--Stewart method and on the fast diagonalization method, which result in solving a sequence of spatial subproblems. The solver based on the fast diagonalization method allows us to solve these spatial subproblems in parallel, leading to a full parallelization in time. We analyze the complexity of the proposed algorithms and give numerical examples for a two-dimensional spatial domain, where sparse direct solvers for the spatial subproblems are used.

Keywords

  1. parabolic evolution problems
  2. anisotropic Sobolev spaces
  3. space-time methods
  4. direct solvers
  5. Bartels--Stewart
  6. fast diagonalization

MSC codes

  1. 65F05
  2. 65M60

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References

1.
P. R. Amestoy, A. Buttari, J.-Y. L'Excellent, and T. Mary, Performance and scalability of the block low-rank multifrontal factorization on multicore architectures, ACM Trans. Math. Softw., 45 (2019), pp. 1--26.
2.
P. R. Amestoy, I. S. Duff, J.-Y. L'Excellent, and J. Koster, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 15--41.
3.
R. Andreev, Stability of sparse space-time finite element discretizations of linear parabolic evolution equations, IMA J. Numer. Anal., 33 (2013), pp. 242--260.
4.
R. E. Bank, P. S. Vassilevski, and L. T. Zikatanov, Arbitrary dimension convection-diffusion schemes for space-time discretizations, J. Comput. Appl. Math., 310 (2017), pp. 19--31.
5.
R. H. Bartels and G. W. Stewart, Algorithm 432: Solution of the matrix equation \(AX + XB = C\), Commun. ACM, 15 (1972), pp. 820--826.
6.
M. Costabel, Boundary integral operators for the heat equation, Integral Equations Operator Theory, 13 (1990), pp. 498--552.
7.
T. A. Davis, S. Rajamanickam, and W. M. Sid-Lakhdar, A survey of direct methods for sparse linear systems, Acta Numer., 25 (2016), pp. 383--566.
8.
D. Devaud, Petrov--Galerkin space-time $hp$-approximation of parabolic equations in $H^{1/2}$, IMA J. Numer. Anal., 40 (2020), pp. 2717--2745.
9.
S. Dohr, O. Steinbach, and K. Niino, Space-time boundary element methods for the heat equation, in Space-Time Methods: Application to Partial Differential Equations, U. Langer and O. Steinbach, eds., Radon Ser. Comput. Appl. Math. 25, de Gruyter, Berlin, 2019, pp. 1--60.
10.
I. S. Duff, A. M. Erisman, and J. K. Reid, Direct Methods for Sparse Matrices, 2nd ed., Oxford University Press, Oxford, 2017.
11.
T. Führer and M. Karkulik, Space-time least-squares finite elements for parabolic equations, Comput. Math. Appl., 92 (2021), pp. 27--36.
12.
M. Fontes, Parabolic Equations with Low Regularity, Ph.D. thesis, Lunds Universitet, 1996.
13.
M. Fontes, Initial-boundary value problems for parabolic equations, Ann. Acad. Sci. Fenn. Math., 34 (2009), pp. 583--605.
14.
T. Führer, N. Heuer, and J. S. Gupta, A time-stepping DPG scheme for the heat equation, Comput. Methods Appl. Math., 17 (2017), pp. 237--252.
15.
M. J. Gander, $50$ years of time parallel time integration, in Multiple Shooting and Time Domain Decomposition, T. Carraro, M. Geiger, S. Körkel, and R. Rannacher, eds., Springer-Verlag, Berlin, 2015, pp. 69--114.
16.
J. D. Gardiner, A. J. Laub, J. J. Amato, and C. B. Moler, Solution of the Sylvester matrix equation \(AXB^T+CXD^T=E\), ACM Trans. Math. Softw., 18 (1992), pp. 223--231.
17.
N. I. M. Gould, J. A. Scott, and Y. Hu, A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations, ACM Trans. Math. Softw., 33 (2007), Article 10.
18.
Y. Hardy and W.-H. Steeb, Matrix Calculus, Kronecker Product and Tensor Product: A Practical Approach to Linear Algebra, Multilinear Algebra and Tensor Calculus with Software Implementations, 3rd ed., World Scientific, Singapore, 2019.
19.
C. Hofer, U. Langer, M. Neumüller, and R. Schneckenleitner, Parallel and robust preconditioning for space-time isogeometric analysis of parabolic evolution problems, SIAM J. Sci. Comput., 41 (2019), pp. A1793--A1821.
20.
T. Hughes, L. Franca, and G. Hulbert, A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advection-diffusive equations, Comput. Methods Appl. Mech. Engrg., 73 (1989), pp. 173--189.
21.
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, Vol. 23, American Mathematical Society, Providence, RI, 1968.
22.
U. Langer and M. Neumüller, Direct and iterative solvers, in Computational Acoustics, M. Kaltenbacher, ed., Springer-Verlag, Cham, Switzerland, 2018, pp. 205--251.
23.
S. Larsson and M. Molteni, Numerical solution of parabolic problems based on a weak space-time formulation, Comput. Methods Appl. Math., 17 (2017), pp. 65--84.
24.
S. Larsson and C. Schwab, Compressive Space-Time Galerkin Discretizations of Parabolic Partial Differential Equations, preprint, arXiv:1501.04514, 2015.
25.
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968.
26.
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 2, Travaux et Recherches Mathématiques, No. 18, Dunod, Paris, 1968.
27.
J. W. H. Liu, The multifrontal method for sparse matrix solution: Theory and practice, SIAM Rev., 34 (1992), pp. 82--109.
28.
G. Loli, M. Montardini, G. Sangalli, and M. Tani, Space-time Galerkin isogeometric method and efficient solver for parabolic problems, Comput. Math. Appl., 80 (2020), pp. 2586--2603.
29.
R. Lynch, J. R. Rice, and D. H. Thomas, Direct solution of partial difference equations by tensor product methods, Numer. Math., 6 (1964), pp. 185--199.
30.
P.-G. Martinsson, Fast Direct Solvers for Elliptic PDEs, Vol. 96, SIAM, Philadelphia, 2020.
31.
C. Mollet, Stability of Petrov-Galerkin discretizations: Application to the space-time weak formulation for parabolic evolution problems, Comput. Methods Appl. Math., 14 (2014), pp. 231--255.
32.
M. Neumüller, Space-Time Methods: Fast Solvers and Applications, Monographic Series TU Graz: Computation in Engineering and Science 20, Verlag der Technischen Universität Graz, Graz, Austria, 2013.
33.
M. Neumüller and I. Smears, Time-parallel iterative solvers for parabolic evolution equations, SIAM J. Sci. Comput., 41 (2019), pp. C28--C51.
34.
D. Pardo, M. Paszynski, N. Collier, J. Alvarez, L. Dalcin, and V. M. Calo, A survey on direct solvers for Galerkin methods, SeMA J., 57 (2012), pp. 107--134.
35.
C. Schwab and R. Stevenson, Space-time adaptive wavelet methods for parabolic evolution problems, Math. Comp., 78 (2009), pp. 1293--1318.
36.
C. Schwab and R. Stevenson, Fractional space-time variational formulations of (Navier-) Stokes equations, SIAM J. Math. Anal., 49 (2017), pp. 2442--2467.
37.
V. Simoncini, Computational methods for linear matrix equations, SIAM Rev., 58 (2016), pp. 377--441.
38.
O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements, Springer-Verlag, Berlin 2008.
39.
O. Steinbach, Space-time finite element methods for parabolic problems, Comput. Methods Appl. Math., 15 (2015), pp. 551--566.
40.
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: Application to Partial Differential Equations, Radon Ser. Comput. Appl. Math. 25, U. Langer and O. Steinbach, eds., de Gruyter, Berlin, 2019, pp. 207--248.
41.
O. Steinbach and M. Zank, Coercive space-time finite element methods for initial boundary value problems, Electron. Trans. Numer. Anal., 52 (2020), pp. 154--194.
42.
O. Steinbach and M. Zank, A note on the efficient evaluation of a modified Hilbert transformation, J. Numer. Math., 29 (2021), pp. 47--61.
43.
R. Stevenson and J. Westerdiep, Stability of Galerkin discretizations of a mixed space--time variational formulation of parabolic evolution equations, IMA J. Numer. Anal., 41 (2021), pp. 28--47.
44.
M. Tani, A preconditioning strategy for linear systems arising from nonsymmetric schemes in isogeometric analysis, Comput. Math. Appl., 74 (2017), pp. 1690--1702.
45.
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd ed., Springer-Verlag, Berlin, 2006.
46.
K. Urban and A. T. Patera, An improved error bound for reduced basis approximation of linear parabolic problems, Math. Comput., 83 (2014), pp. 1599--1615.
47.
M. Zank, An exact realization of a modified Hilbert transformation for space-time methods for parabolic evolution equations, Comput. Methods Appl. Math., 21 (2021), pp. 479--496.
48.
M. Zank, Inf-Sup Stable Space-Time Methods for Time-Dependent Partial Differential Equations, Monographic Series TU Graz: Computation in Engineering and Science 36, Verlag der Technischen Universität Graz, Graz, Austria, 2020.

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Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A2714 - A2736
ISSN (online): 1095-7197

History

Submitted: 5 August 2020
Accepted: 14 April 2021
Published online: 4 August 2021

Keywords

  1. parabolic evolution problems
  2. anisotropic Sobolev spaces
  3. space-time methods
  4. direct solvers
  5. Bartels--Stewart
  6. fast diagonalization

MSC codes

  1. 65F05
  2. 65M60

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