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. To preserve the mathematical structure of the evolutionary equation on the fully discrete level, suitable generalizations of the distribution gradient and divergence operators on broken polynomial spaces on which the discontinuous Galerkin approach is built on are defined. Well-posedness of the fully discrete problem and error estimates for the discontinuous Galerkin approximation in space and time are proved.

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