  In this paper we analyze a homogeneous parabolic problem with initial data in
the space of regular Borel measures. The problem is discretized in time with a
discontinuous Galerkin scheme of arbitrary degree and in space with continuous
finite elements of orders one or two. We show parabolic smoothing results for
the continuous, semidiscrete and fully discrete problems. Our main results are
interior $L^\infty$ error estimates for the evaluation at the endtime, in cases
where the initial data is supported in a subdomain. In order to obtain these,
we additionally show interior $L^\infty$ error estimates for $L^2$ initial data
and quadratic finite elements, which extends the corresponding result
previously established by the authors for linear finite elements.
