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

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