  We present a general abstract framework for the systematic numerical
approximation of dissipative evolution problems. The approach is based on
rewriting the evolution problem in a particular form that complies with an
underlying energy or entropy structure. Based on the variational
characterization of smooth solutions, we are then able to show that the
approximation by Galerkin methods in space and discontinuous Galerkin methods
in time automatically leads to numerical schemes that inherit the dissipative
behavior of the evolution problem. The proposed framework is rather general and
can be applied to a wide range of applications. This is demonstrated by a
detailed discussion of a variety examples ranging from diffusive partial
differential equations to Hamiltonian and gradient systems.
