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The reduced basis technique as a coarse solver for parareal in time simulations. (English) Zbl 1240.65291

Summary: We extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by J. L. Lions, Y. Maday and G. Turinici [C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 7, 661–668 (2001; Zbl 0984.65085)] 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.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65Y05 Parallel numerical computation
35K55 Nonlinear parabolic equations
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65Y20 Complexity and performance of numerical algorithms

Citations:

Zbl 0984.65085
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