A Multilevel Elliptic Solver with Applications 
                   to Incompressible Flow and Astrophysics

                               Louis H. Howell
                    Lawrence Livermore National Laboratory
                             Livermore, CA 94550

I will discuss a multilevel algorithm for solving elliptic equations on
three-dimensional adaptive meshes, where refined cells are grouped in to
regular patches to avoid difficulties associated with unstructured grids.  The
emphasis will be on software design, numerical efficiency, and applications.
A flexible object hierarchy written in C++ helps organize the geometry-
dependent operations required along the faces, edges and corners of each
coarse-fine interface.

Two time-dependent applications will be presented, both of which involve
collaborative efforts with other researchers.  For incompressible flow, the
algorithm projects each velocity update on to the space of divergence-free
fields.  The coefficients of the elliptic operator may have strong
discontinuities in some variable-density flow problems.  Fine levels are
advanced with smaller time steps than coarse levels, so projections are
required both on individual levels and to synchronize solutions between
adjacent levels.

For the astrophysics application, the multilevel scheme is coupled to a
Godunov method for gas dynamics to solve problems involving self-gravitating
gas clouds.  It solves Poisson's equation for a gravitational potential, which
then appears as a source term in the advection scheme.  This code is currently
run with all time steps equal, so the Poisson solution must span all levels of
refinement.