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


THE REDUCED BASIS TECHNIQUE AS A COARSE SOLVER FOR PARAREAL IN TIME SIMULATIONS

Liping He 1

1 Department of Mathematics, Shanghai Jiaotong University, Shanghai 200030, China

Received 2007-12-8 Accepted 2009-8-25
Available online 2010-5-1
doi:10.4208/jcm.1003-m2980

Abstract

In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions \emph{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.

Key words: 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 classifications: 52B10, 65D18, 68U05, 68U07


Email: lphe@sjtu.edu.cn
 

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