The auxiliary space method and optimal multigrid preconditioning
                      techniques for unstructured grids

                                  Jinchao Xu

                                  Abstract:

An abstract framework of auxiliary space method is proposed and, as an
application, an optimal multigrid technique is developed for general
unstructured grids.  The auxiliary space method is a (nonnested) two level
preconditioning technique based on a simple relaxation scheme (smoother) and
an auxiliary space (that may be roughly understood as a nonnested coarser
space).  An optimal multigrid preconditioner is then obtained for a
discretized partial differential operator defined on an unstructured grid by
using an auxiliary space defined on a more structured grid in which a further
nested multigrid method can be naturally applied.  This new technique make it
possible to apply multigrid methods to general unstructured grids without too
much more programming effort than traditional solution methods.  Some simple
examples are also given to illustrate the abstract theory.

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Jinchao Xu
Department of Mathematics              814-865-1110 (office)
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Penn State University                  814-867-8888 (home)
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