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

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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.

Author-supplied keywords

  • degrees techniques
  • element approximations
  • finite element and spectral
  • multi-meshes and multi-
  • offline-
  • online procedure
  • parareal in time algorithm
  • reduced basis technique
  • semi-implicit runge-kutta scheme

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