In this article we study discontinuous Galerkin finite element discretizations of linear
second-order elliptic partial differential equations with Dirac delta right-hand side. In
particular, assuming that the underlying computational mesh is quasi-uniform, we derive an
a priori bound on the error measured in terms of the
L2-norm. Additionally, we develop residual-based a
posteriori error estimators that can be used within an adaptive mesh refinement
framework. Numerical examples for the symmetric interior penalty scheme are presented
which confirm the theoretical results.