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.