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Nishikawa (2007) proposed to reformulate the classical Poisson equation as asteady state problem for a linear hyperbolic system. This results in optimalerror estimates for both the solution of the elliptic equation and itsgradient. However, it prevents the application of well-known solvers forelliptic problems. We show connections to a discontinuous Galerkin (DG) methodanalyzed by Cockburn, Guzm\'an, and Wang (2009) that is very difficult toimplement in general. Next, we demonstrate how this method can be implementedefficiently using summation by parts (SBP) operators, in particular in thecontext of SBP DG methods such as the DG spectral element method (DGSEM). Theresulting scheme combines nice properties of both the hyperbolic and theelliptic point of view, in particular a high order of convergence of thegradients, which is one order higher than what one would usually expect from DGmethods for elliptic problems.

A discontinuous Galerkin discretization of elliptic problems with improved convergence properties using summation by parts operators