NA Seminar, Lea Versbach (Lund), "Introduction to multigrid methods and its application as space-time preconditioner: Ideas and challenges"
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Multigrid methods were developed from well-known iterative methods as e.g. Jacobi or Gauss-Seidel. Since these relaxation methods have different smoothing properties for high- and low-frequency error components, it is of advantage to consider the same problem on different grids.
Multigrid methods were designed in order to solve linear systems coming from discretized PDEs, but can also be used as a preconditioner. We will use multigrid preconditioners for a space-time solver. This adds the problem of having different demands on the multigrid method in space and time. Since space-time discretizations become anisotropic, smoothing does not work equally in the spatial and the temporal directions. Moreover, grid transfer operators in time have to be chosen carefully in order to not have effects as information traveling backwards in time. Hackbusch suggested in  to use a semi-coarseing technique when solving parabolic PDEs. This idea was developed further by other authors in  and  where they construct conditions on the anisotropy parameter by a Fourier mode analysis and suggest a multigrid method switching between full and semi-coarsening depending on these conditions. We will discuss these ideas as well as the challenges they imply when constructing multigrid space-time preconditioners.
 Wolfgang Hackbusch, Parabolic multi-grid methods, Computing Methods in Applied Sciences and Engineering IV (R. Glowinski and Jacques-Louis Lions, eds.), Elsevier Science Publisher B.V., Noth-Holland, 1984, pp. 189–197.
 Graham Horton and Stefan Vandewalle, A space-time multigrid method for parabolic PDEs, SIAM Journal on Scientific Computing, 16 (4), 1993, pp. 848-864.
 Martin J Gander and Martin Neumüller, Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems, SIAM Journal on Scientific Computing, 38 (4), 2016, A2173-A2208.