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Volume 28, Issue 5
The Reduced Basis Technique as a Coarse Solver for Parareal in Time Simulations

Liping He

DOI: 10.4208/jcm.1003-m2980

J. Comp. Math., 28 (2010), pp. 676-692.

Published online: 2010-10

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  • Abstract

In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions et al. [12] and solve a nonlinear evolutionary parabolic partial differential equation. The fine solver is based on the finite element method or spectral element method in space and a semi-implicit Runge-Kutta scheme in time. The coarse solver is based on a semi-implicit scheme in time and the reduced basis approximation in space. Offline-online procedures are developed, and it is proved that the computational complexity of the on-line stage depends only on the dimension of the reduced basis space (typically small). Parareal in time algorithms based on a multi-grids finite element method and a multi-degrees finite element method are also presented. Some numerical results are reported.

  • Keywords

Finite element and spectral element approximations, Multi-meshes and multi-degrees techniques, Reduced basis technique, Semi-implicit Runge-Kutta scheme, Offline-online procedure, Parareal in time algorithm.

  • AMS Subject Headings

52B10, 65D18, 68U05, 68U07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-28-676, author = {Liping He}, title = {The Reduced Basis Technique as a Coarse Solver for Parareal in Time Simulations}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {5}, pages = {676--692}, abstract = {

In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions et al. [12] and solve a nonlinear evolutionary parabolic partial differential equation. The fine solver is based on the finite element method or spectral element method in space and a semi-implicit Runge-Kutta scheme in time. The coarse solver is based on a semi-implicit scheme in time and the reduced basis approximation in space. Offline-online procedures are developed, and it is proved that the computational complexity of the on-line stage depends only on the dimension of the reduced basis space (typically small). Parareal in time algorithms based on a multi-grids finite element method and a multi-degrees finite element method are also presented. Some numerical results are reported.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1003-m2980}, url = {http://global-sci.org/intro/article_detail/jcm/8543.html} }
TY - JOUR T1 - The Reduced Basis Technique as a Coarse Solver for Parareal in Time Simulations AU - Liping He JO - Journal of Computational Mathematics VL - 5 SP - 676 EP - 692 PY - 2010 DA - 2010/10 SN - 28 DO - http://doi.org/10.4208/jcm.1003-m2980 UR - https://global-sci.org/intro/article_detail/jcm/8543.html KW - Finite element and spectral element approximations, Multi-meshes and multi-degrees techniques, Reduced basis technique, Semi-implicit Runge-Kutta scheme, Offline-online procedure, Parareal in time algorithm. AB -

In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions et al. [12] and solve a nonlinear evolutionary parabolic partial differential equation. The fine solver is based on the finite element method or spectral element method in space and a semi-implicit Runge-Kutta scheme in time. The coarse solver is based on a semi-implicit scheme in time and the reduced basis approximation in space. Offline-online procedures are developed, and it is proved that the computational complexity of the on-line stage depends only on the dimension of the reduced basis space (typically small). Parareal in time algorithms based on a multi-grids finite element method and a multi-degrees finite element method are also presented. Some numerical results are reported.

Liping He. (2010). The Reduced Basis Technique as a Coarse Solver for Parareal in Time Simulations. Journal of Computational Mathematics. 28 (5). 676-692. doi:10.4208/jcm.1003-m2980
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