  We study the numerical approximation of a coupled hyperbolic-parabolic system
by a family of discontinuous Galerkin space-time finite element methods. The
model is rewritten as a first-order evolutionary problem that is treated by the
unified abstract solution theory of R. Picard. For the discretization in space,
generalizations of the distribution gradient and divergence operators on broken
polynomial spaces are defined. Since their skew-selfadjointness is perturbed by
boundary surface integrals, adjustments are introduced such that the
skew-selfadjointness of the first-order differential operator in space is
recovered. Well-posedness of the fully discrete problem and error estimates for
the discontinuous Galerkin approximation in space and time are proved.
