Gauss' adaptive relaxation for the multilevel solution of partial differential
equations on sparse grids

C. Pflaum and U. R\"{u}de
Institut f\"{u}r Informatik, Technische Universit\"{a}t
D-80290 M\"{u}n\-chen, Germany
e-mail: pflaum/ ruede@informatik.tu-muenchen.de

Abstract:  In combination with the multilevel principle, relaxation methods
are among the most efficient numerical solution techniques for elliptic
partial differential equations.  Typical methods used today are derivations of
the Gau\ss -Seidel or Gau\ss -Jacobi method.  Recently it has been recognized
that in the context of multilevel algorithms, the original method suggested by
Gau\ss\ has specific advantages.  For this method the iteration is
concentrated on unknowns where fast convergence can be obtained by
intelligently monitoring the residuals.  We will present this algorithm in the
context of a sparse grid multigrid algorithm.  Using sparse grids the
dimension of the discrete approximation space can be reduced additionally.