TY - JOUR T1 - The Reduced Basis Technique as a Coarse Solver for Parareal in Time Simulations 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.