  Nishikawa (2007) proposed to reformulate the classical Poisson equation as a
steady state problem for a linear hyperbolic system. This results in optimal
error estimates for both the solution of the elliptic equation and its
gradient. However, it prevents the application of well-known solvers for
elliptic problems. We show connections to a discontinuous Galerkin (DG) method
analyzed by Cockburn, Guzm\'an, and Wang (2009) that is very difficult to
implement in general. Next, we demonstrate how this method can be implemented
efficiently using summation by parts (SBP) operators, in particular in the
context of SBP DG methods such as the DG spectral element method (DGSEM). The
resulting scheme combines nice properties of both the hyperbolic and the
elliptic point of view, in particular a high order of convergence of the
gradients, which is one order higher than what one would usually expect from DG
methods for elliptic problems.
