Numerical relativity initial data for neutron star binaries and the hyperbolic relaxation method Numerical Relativity Initial Data for Neutron Star Binaries and the Hyperbolic Relaxation Method. A new class of relaxation schemes for the solution of systems of elliptic partial differential equations is derived from a modification of the parabolic Jacobi relaxation scheme. The new scheme can be viewed as a formal embedding of the elliptic partial differential equations into a system of hyperbolic partial differential equations with damping. The hyperbolicity of the hyperbolic relaxation equations is examined and the implementation of the method in the hyperbolic relaxation code bamps is discussed. Furthermore the convergence properties investigated for simple model problems in terms of an analytical mode analysis and an experimental comparison to other numerical solution methods for elliptic partial differential equations. Subsequently the hyperbolic relaxation scheme is applied to the initial data problem in numerical relativity for the case of massless scalar fields single neutron stars and binary neutron stars. For binary neutron stars a scheme avoiding adapted coordinates by formally extending the elliptic partial differential equations to regions outside the neutron stars is investigated. To improve the physical accuracy of initial data for neutron star binaries with spin an attempt is made to take into account terms that are neglected in current state of the art formalisms. By investigation of additional constraints originating from the neglected terms that the current formalism would actually pose unphysical requirements on the solution. Consequently some inconsistencies in the current formalism are pointed out and a modified formalism fixing these inconsistencies is discussed.