In this work, a space-time multigrid method which uses standard coarsening in both temporal and spatial domains and combines the use of different smoothers is proposed for the solution of the heat equation in one and two space dimensions. In particular, an adaptive smoothing strategy, based on the degree of anisotropy of the discrete operator on each grid-level, is the basis of the proposed multigrid algorithm. Local Fourier analysis is used for the selection of the crucial parameter defining such an adaptive smoothing approach. Central differences are used to discretize the spatial derivatives and both implicit Euler and Crank–Nicolson schemes are considered for approximating the time derivative. For the solution of the second-order scheme, we apply a double discretization approach within the space-time multigrid method. The good performance of the method is illustrated through several numerical experiments.

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Keywords Double discretization, Local Fourier analysis, Parabolic partial differential equations, Space-time multigrid
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Journal Applied Mathematics and Computation
Grant This work was funded by the European Commission 7th Framework Programme; grant id h2020/705402 - Efficient numerical methods for deformable porous media. Application to carbon dioxide storage. (poro sos)
Franco, S.R, Gaspar, F.J, Villela Pinto, M.A, & Rodrigo, C. (2018). Multigrid method based on a space-time approach with standard coarsening for parabolic problems. Applied Mathematics and Computation, 317, 1339–1351. doi:10.1016/j.amc.2017.08.043