  We show how the basic idea of parabolic Jacobi relaxation can be modified to
obtain a new class of hyperbolic relaxation schemes that are suitable for the
solution of elliptic equations. Some of the analytic and numerical properties
of hyperbolic relaxation are examined. We describe its implementation as a
first order system in a pseudospectral evolution code, demonstrating that
certain elliptic equations can be solved within a framework for hyperbolic
evolution systems. Applications include various initial data problems in
numerical general relativity. In particular we generate initial data for the
evolution of a massless scalar field, a single neutron star, and binary neutron
star systems.
