 
Summary: Adaptive Finite Elements and Colliding Black Holes
Douglas N. Arnold, Arup Mukherjee, and Luc Pouly
Abstract According to the theory of general relativity, the relative acceleration of
masses generates gravitational radiation. Although gravitational radiation has not yet
been detected, it is believed that extremely violent cosmic events, such as the collision
of black holes, should generate gravity waves of su cient amplitude to detect on earth.
The massive Laser Interferometer Gravitationalwave Observatory, or LIGO, is now
being constructed to detect gravity waves. Consequently there is great interest in the
computer simulation of black hole collisions and similar events, based on the numerical
solution of the Einstein eld equations. In this note we introduce the scienti c, mathe
matical, and computational problems and discuss the development of a computer code
to solve the initial data problem for colliding black holes, a nonlinear elliptic bound
ary value problem posed in an unbounded three dimensional domain which is a key
step in solving the full eld equations. The code is based on nite elements, adaptive
meshes, and a multigrid solution process. Here we will particularly emphasize the
mathematical and algorithmic issues arising in the generation of adaptive tetrahedral
meshes.
1 Introduction
In Einstein'stheory of general relativityspacetimeis representedas a fourdimensional
semiRiemannian manifold. The geodesics of this manifold are the paths of freely
