Finite Element Multigrid Solution of the Initial Value Problem
                           in Numerical Relativity


                               Chia-Ying Huang
                                 Faisal Saied
                        Department of Computer Science
                  University of Illinois at Urbana-Champaign

                                  Ed Seidel
               National Center for Supercomputing Applications
                          and Department of Physics
                  University of Illinois at Urbana-Champaign


Einstein's equations of General Relativity are a complex system of partial
differential equations for which analytical solutions are known only for a few
idealized situations.  There is considerable interest in computing numerical
solutions (spacetimes) to Einstein's equations for modeling black holes and
other problems.  In this paper we consider the Initial Value Problem of
Numerical Relativity.  This problem has traditionally been solved by finite
differences.  We show that this two dimensional non-self-adjoint elliptic
problem can be solved efficiently with the PLTMG package, which uses a
hierarchical basis multigrid method as a preconditioner in conjunction with
adaptive mesh refinement.  We compare the effectiveness of the solution
process in two different coordinate systems and quantify the gains from the
adaptive refinement procedure.