Continuum and discrete initial-boundary value problems and the Einstein field equations

In this talk we outline some of the theory necessary to understand continuum and discrete initial-boundary value problems arising from hyperbolic partial differential equations, and discuss applications to numerical relativity. In particular, we present a well posed initial and initial-boundary value formulations for Einstein's equations, and discuss multi-domain high order finite difference techniques and spectral methods to discretize them. The talk is a very brief outline of the contents of an upcoming Living Review in Relativity, ``Continuum and discrete initial boundary value problems and the Einstein field equations,'' by Olivier Sarbach and Manuel Tiglio.