New transparent boundary conditions for Schrödinger, advection-diffusion and wave equation are derived, that are highly accurate but local in time and space.
To this end the exterior of some convex polygon/polyhedron is discretized by infinite trapezes in two space dimensions or infinite prisms in three space dimensions. This exterior discretisation yields a generalised ``radial'' variable. Using the pole condition in the radial variable one is able to discriminate incoming/physical and outgoing/unphysical solutions. The pole condition can be reformulated in such a way that outgoing solutions in the transformed variable must belong to the Hardy space of the unit ball , whereas incoming solutions must not belong to . Approximating outgoing solutions by a power series expansion and matching moments an efficient algorithm is derived, that is local in time and space. In numerical experiments these Hardyspace infinite elements show super-algebraic convergence in the number of coefficients used in the power series expansion.