Adaptive multilevel methods in space and time for parabolic problems – the periodic case.

*(English)*Zbl 0941.65101Authors’ summary: The aim of this paper is to display numerical results that show the interest of some multilevel methods for problems of parabolic type. These schemes are based on multilevel spatial splittings and the use of different time steps for the various spatial components.

The spatial discretization we investigate is of spectral Fourier type, so the approximate solution naturally splits into the sum of a low frequency component and a high frequence one. The time discretization is of implicit/explicit Euler type of each spatial component.

Based on a posteriori estimates, we introduce adaptive one-level and multilevel algorithms.

Two problems are considered: the heat equation and a nonlinear problem.

Numerical experiments are conducted for both problems using the one-level and the multilevel algorithms. The multilevel method is up to 70% faster than the one-level method.

The spatial discretization we investigate is of spectral Fourier type, so the approximate solution naturally splits into the sum of a low frequency component and a high frequence one. The time discretization is of implicit/explicit Euler type of each spatial component.

Based on a posteriori estimates, we introduce adaptive one-level and multilevel algorithms.

Two problems are considered: the heat equation and a nonlinear problem.

Numerical experiments are conducted for both problems using the one-level and the multilevel algorithms. The multilevel method is up to 70% faster than the one-level method.

Reviewer: S.F.McCormick (Boulder)

##### MSC:

65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

35K15 | Initial value problems for second-order parabolic equations |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |