Multigrid methods provide highly efficient, numerically optimal
solvers or preconditioners for
many kinds of mathematical problems.
One of the areas for which the multigrid development is most mature
is the solution of partial differential equations.
The various existing multigrid approaches have in common that they use
a hierarchy of coarser grids (levels, scales) to accelerate the (slow) convergence
of standard (one-level) iterative methods, such as Jacobi or Gauss-Seidel.
Whereas in geometric multigrid usually a predefined "natural" hierarchy
of coarser grids is used, in algebraic multigrid all coarser levels are constructed
automatically.
In this tutorial an introduction to the basic multigrid principles is given. In particular,
the interplay between smoothing and coarse-grid correction and the basic
multigrid cycling are explained. Furthermore, local refinement methods and,
especially in this context, the full approximation scheme (FAS) are discussed.
The talk is concluded by an outlook on modern approaches,
including some remarks on AMG and eigenvalue problems.
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