Preconditioning plays an important role in the iterative solution of algebraic problems arising from the numerical discretization of differential operators. It is well known that for many common differential operators the application of standard finite element or finite difference schemes leads to linear systems whose condition number behaves like as , where is a typical edge length in the finite element/difference grid. As the grid becomes very fine therefore the importance of preconditioning grows significantly.
A popular class of preconditioning is based upon a geometric decomposition
of the grid using additive or multiplicative Schwarz methods. This is
because such an approach is naturally amenable to implementation on a
parallel computing platform due to the inherent decomposition of the
problem. This talk will describe work with three different collaborators,
R. Bank, S. Nadeem and S. Nepomnyaschikh, on the development and application
of a new variant of a two level additive Schwarz preconditioner that has
many of the properties of more complex multilevel preconditioners of the
same family. In particular it can be shown that this preconditioner yields a
condition number that is independent of the mesh spacing . Examples of
the application of this approach will be provided for problems arising from
the discretization of a number of three-dimensional PDEs using linear
tetrahedral finite elements.