The Reduced Basis Technique as a Coarse Solver for Parareal in Time Simulations

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 difierential equation. The flne solver is based on the flnite 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. O†ine-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 flnite element method and a multi-degrees flnite element method are also presented. Some numerical results are reported.