Implicit Extrapolation Methods for Multilevel Finite Element Computations
        Theory and Applications

        Michael Jung  and  Ulrich Ruede  

        Fakultaet fuer Mathematik,
        Technische Universitaet Chemnitz-Zwickau,
        D-09009 Chemnitz 
        Germany,
        e-mail: dr.michael.jung@mathematik.tu-chemnitz.de,
                ruede@mathematik.tu-chemnitz.de

        SPC-Preprint 94_11, Juni 1994


        ABSTRACT

Extrapolation methods for the solution of partial differential equations are
commonly based on the existence of error expansions for the approximate
solution.  Implicit extrapolation, in the contrast, is based on applying
extrapolation indirectly, by using it on quantities like the residual.  In the
context of multigrid methods, a special technique of this type is known as
tau-extrapolation.  For finite element systems this algorithm can be shown to
be equivalent to higher order finite elements.  The analysis is local and does
not use global expansions, so that the implicit extrapolation technique may be
used on unstructured meshes and in cases where the solution fails to be
globally smooth.  Furthermore, the natural multilevel structure can be used to
construct efficient multigrid and multilevel preconditioning techniques.  The
effectivity of the method is demonstrated for heat conduction problems and
problems from elasticity theory.

Keywords:        Finite Elements, Extrapolation, Multigrid, Elasticity.

AMS(MOS) Subject Classification:        65F10, 65F50, 65N22, 65N50, 65N55.
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