| Title:
|
Domain decomposition methods coupled with parareal for the transient heat equation in 1 and 2 spatial dimensions (English) |
| Author:
|
Foltyn, Ladislav |
| Author:
|
Lukáš, Dalibor |
| Author:
|
Peterek, Ivo |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
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65 |
| Issue:
|
2 |
| Year:
|
2020 |
| Pages:
|
173-190 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We present a parallel solution algorithm for the transient heat equation in one and two spatial dimensions. The problem is discretized in space by the lowest-order conforming finite element method. Further, a one-step time integration scheme is used for the numerical solution of the arising system of ordinary differential equations. For the latter, the parareal method decomposing the time interval into subintervals is employed. It leads to parallel solution of smaller time-dependent problems. At each time slice a pseudo-stationary elliptic heat equation is solved by means of a domain decomposition method (DDM). In the $2d$, case we employ a nonoverlapping Schur complement method, while in the $1d$ case an overlapping Schwarz DDM is employed. We document computational efficiency, as well as theoretical convergence rates of FEM semi-discretization schemes on numerical examples. (English) |
| Keyword:
|
domain decomposition method |
| Keyword:
|
parareal method |
| Keyword:
|
finite element method |
| Keyword:
|
heat equation |
| MSC:
|
65F08 |
| MSC:
|
65N30 |
| MSC:
|
65N55 |
| idZBL:
|
07217104 |
| idMR:
|
MR4083463 |
| DOI:
|
10.21136/AM.2020.0219-19 |
| . |
| Date available:
|
2020-05-20T15:45:38Z |
| Last updated:
|
2022-05-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148108 |
| . |
| Reference:
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[1] Bramble, J. H., Pasciak, J. E., Schatz, A. H.: The construction of preconditioners for elliptic problems by substructuring. I.Math. Comput. 47 (1986), 103-134. Zbl 0615.65112, MR 0842125, 10.2307/2008084 |
| Reference:
|
[2] Dai, X., Maday, Y.: Stable parareal in time method for first- and second-order hyperbolic systems.SIAM J. Sci. Comput. 35 (2013), A52--A78. Zbl 1264.65136, MR 3033060, 10.1137/110861002 |
| Reference:
|
[3] Falgout, R. D., Friedhoff, S., Kolev, T. V., MacLachlan, S. P., Schroder, J. B.: Parallel time integration with multigrid.SIAM J. Sci. Comput. 36 (2014), C635--C661. Zbl 1310.65115, MR 3499068, 10.1137/130944230 |
| Reference:
|
[4] Farhat, C., Chandesris, M.: Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid-structure applications.Int. J. Numer. Methods Eng. 58 (2003), 1397-1434. Zbl 1032.74701, MR 2012613, 10.1002/nme.860 |
| Reference:
|
[5] Farhat, C., Roux, F.-X.: A method of finite element tearing and interconnecting and its parallel solution algorithm.Int. J. Numer. Methods Eng. 32 (1991), 1205-1227. Zbl 0758.65075, MR 3618550, 10.1002/nme.1620320604 |
| Reference:
|
[6] Gander, M. J.: 50 years of time parallel time integration.Multiple Shooting and Time Domain Decomposition Methods T. Carraro et al. Contribibutions Mathematical and Computational Sciences 9, Springer, Cham (2015), 69-113. Zbl 1337.65127, MR 3676210, 10.1007/978-3-319-23321-5_3 |
| Reference:
|
[7] Gander, M. J., Halpern, L., Nataf, F.: Optimal Schwarz waveform relaxation for the one dimensional wave equation.SIAM J. Numer. Anal. 41 (2003), 1643-1681. Zbl 1085.65077, MR 2035001, 10.1137/S003614290139559X |
| Reference:
|
[8] Gander, M. J., Jiang, Y.-L., Li, R.-J.: Parareal Schwarz waveform relaxation methods.Domain Decomposition Methods in Science and Engineering XX R. Bank et al. Lectures Notes in Computational Science and Engineering 91, Springer, Berlin (2013), 451-458. Zbl 1416.65007, MR 3243021, 10.1007/978-3-642-35275-1_53 |
| Reference:
|
[9] Gander, M. J., Kwok, F., Mandal, B. C.: Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for parabolic problems.ETNA, Electron. Trans. Numer. Anal. 45 (2016), 424-456. Zbl 1355.65128, MR 3582894 |
| Reference:
|
[10] Gander, M. J., Neumüller, M.: Analysis of a new space-time parallel multigrid algorithm for parabolic problems.SIAM J. Sci. Comput. 38 (2016), A2173--A2208. Zbl 1342.65225, MR 3521549, 10.1137/15M1046605 |
| Reference:
|
[11] Gander, M. J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method.SIAM J. Sci. Comput. 29 (2007), 556-578. Zbl 1141.65064, MR 2306258, 10.1137/05064607X |
| Reference:
|
[12] Gander, M. J., Vandewalle, S.: On the superlinear and linear convergence of the parareal algorithm.Domain Decomposition Methods in Science and Engineering XVI O. B. Widlund et al. Lectures Notes in Computational Science and Engineering 55, Springer, Berlin (2007), 291-298. Zbl 1104.74004, MR 2334115, 10.1007/978-3-540-34469-8_34 |
| Reference:
|
[13] Lions, J.-L., Maday, Y., Turinici, G.: Résolution d'EDP par un schéma en temps ``pararéel''.C. R. Acad. Sci., Paris, Sér. I, Math. 332 (2001), 661-668 French. Zbl 0984.65085, MR 1842465, 10.1016/S0764-4442(00)01793-6 |
| Reference:
|
[14] Lukáš, D., Bouchala, J., Vodstrčil, P., Malý, L.: 2-dimensional primal domain decomposition theory in detail.Appl. Math., Praha 60 (2015), 265-283. Zbl 1363.65215, MR 3419962, 10.1007/s10492-015-0095-5 |
| Reference:
|
[15] Maday, Y.: The `parareal in time' algorithm.Substructuring Techniques and Domain Decomposition Methods F. Magoulès Computational Science, Engineering and Technology Series 24, Saxe-Coburg Publications, Stirling (2010), 19-44. 10.4203/csets.24.2 |
| Reference:
|
[16] Maday, Y., Turinici, G.: The parareal in time iterative solver: a further direction to parallel implementation.Domain Decomposition Methods in Science and Engineering T. J. Barth et al. Lectures Notes in Computational Science and Engineering 40, Springer, Berlin (2005), 441-448. Zbl 1067.65102, MR 2235771, 10.1007/3-540-26825-1_45 |
| Reference:
|
[17] Mandel, J., Brezina, M.: Balancing domain decomposition for problems with large jumps in coefficients.Math. Comput. 65 (1996), 1387-1401. Zbl 0853.65129, MR 1351204, 10.1090/S0025-5718-96-00757-0 |
| Reference:
|
[18] Mercerat, D., Guillot, L., Vilotte, J.-P.: Application of the parareal algorithm for acoustic wave propagation.AIP Conf. Proc. 1168 (2009), 1521-1524. 10.1063/1.3241388 |
| Reference:
|
[19] Neumüller, M.: Space-Time Methods: Fast Solvers and Applications.Monographic Series, Graz University of Technology, Graz (2013). |
| Reference:
|
[20] Schöps, S., Niyonzima, I., Clemens, M.: Parallel-in-time simulation of eddy current problems using parareal.IEEE Trans. Magn. 54 (2018), Article No. 7200604, 1-4. 10.1109/tmag.2017.2763090 |
| Reference:
|
[21] Smith, B. F., Bjørstad, P. E., Gropp, W. D.: Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations.Cambridge University Press, Cambridge (1996). Zbl 0857.65126, MR 1410757 |
| Reference:
|
[22] Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems.Springer Series in Computational Mathematics 25, Springer, Berlin (2006). Zbl 1105.65102, MR 2249024, 10.1007/3-540-33122-0 |
| Reference:
|
[23] Toselli, A., Widlund, O.: Domain Decomposition Methods---Algorithms and Theory.Springer Series in Computational Mathematics 34, Springer, Berlin (2005). Zbl 1069.65138, MR 2104179, 10.1007/b137868 |
| Reference:
|
[24] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/A: Linear Monotone Operators.Springer, New York (1990). Zbl 0684.47028, MR 1033497, 10.1007/978-1-4612-0985-0 |
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