John
C. Strikwerda

Department of Computer Sciences

University of Wisconsin-Madison

**Abstract:**
In solving linear systems of equations arising from discretizations
of elliptic systems of equations, it is common to precondition the
system so that the resulting system is easier to solve by an iterative
method. What this means is that the linear system *Ax=b* is replaced
by the system *PA x = Pb*, where *P*, the preconditioning matrix, is chosen
to mimic the inverse of *A* in some way. For example, the eigenvalues of
*PA* may be clustered much more closely than the eigenvalues of *A* itself.
If *A* results for the discretization of an elliptic differential
operator *E*, then one way to choose *P* is as an approximation to the
inverse of a differential operator *G* so that *G ^{-1} E*
is a bounded
operator. For example, if