Abstarct: In this thesis we focus on adaptive procedures for the time-discontinuous Galerkin space–time finite element method (DGFEM) including high-order accurate nonreflecting boundary conditions (NRBC) forunbounded wave problems are developed. Sparse multi-level iterative scheme baxsed on the Gauss–Seidel method are developed to solve the resulting fully-discrete system equations for the interior hyperbolic equations. Due to the local nature of wave propagation, the iterative strategy requires only a few iterations per time step. An h-adaptive space–time strategy is employed baxsed on the Zienkiewicz–Zhu spatial error estimate, together with a temporal error estimate arising from the discontinuous jump between time steps of both the interior field solutions and auxiliary boundary functions.