Implicit Extrapolation Methods for
                        Variable Coefficient Problems

                                   M. Jung
                           Fakultat fur Mathematik,
                   Technische Universitat Chemnitz-Zwickau
                          D-09107 Chemnitz, Germany
              e-mail:  Dr.Michael.Jung@mathematik.tu-chemnitz.de

                                   U. Ruede
                           Institut fur Informatik,
                        Technische Universitat Munchen
                          D-80290 Munchen 2, Germany
                   e-mail:  ruede@informatik.tu-muenchen.de

                                   Abstract

Implicit extrap olation methods for the solution of partial differential
equations (see [1]) are based on applying the extrapolation principle
indirectly.  Multigrid tau-extrapolation is a special case of this idea.  In
the context of multilevel finite element methods, an algorithm of this type
can be used to raise the approximation order, even when the meshes are non
uniform or locally refined.  For the case of piecewise constant coefficients
this algorithm has been introduced and analyzed in [1].  Here these results
are generalized to the variable coefficient case and thus become applicable
for nonlinear problems.  The analysis is based on studying the local
quadrature formulas for each finite element.  Implicit extrapolation multigrid
is an iteration converging to the solution of a higher order finite element
system.  This is obtained without explicitly constructing higher order
stiffness matrices but by applying extrapolation in a natural form within the
algorithm.  This is easy to implement because it requires only a small change
of a basic low order multigrid method.

                                  References

[1] M. Jung and U. Rude , Implicit extrapolation methods for multilevel finite
element computations, in Preliminary Proceedings of the Colorado Conference on
Iterative Methods, Breckenridge, Colorado, April 4-10, 1994, T. Manteuffel,
ed., 1994.  Accepted for publication in SISC.