@Article{JCM-28-676, author = {}, 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} }