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For elliptic PDEs (Poisson-type), the multigrid method is very sufficient, but how about time-dependent problems (i.e parabolic or hyperbolic PDEs)? Is it efficient to solve such problems using a multigrid method?
Somewhere I read that for very small time steps a classical iterative method (i.e GS) may be a better option than multigrid and for larger time steps multigrid is efficient. Is this statement correct?
This statement seems a bit reductive for what is a rather large and involved problem. But multigrid, although it was developed for and is ideal for elliptic problems, is still just about the best we have for hyperbolic problems and sees widespread use (when people can manage to implement it). The comparison between Gauss-Seidel seems odd as they tend to serve different functions. GS is mainly used as a linear smoother, not a non-linear solver, and multigrid is often used as a non-linear solver (although it can be used as a linear solver or smoother for large scale codes that use Krylov solvers).
