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.

翻译：Nishikawa (2007) 提出将经典泊松方程重构为线性双曲系统的稳态问题。该方法能同时获得椭圆方程解及其梯度的最优误差估计，但阻碍了椭圆问题已知求解器的应用。我们揭示了该方法与Cockburn、Guzmán及Wang (2009) 所分析的间断Galerkin (DG) 方法之间的关联——后者在一般情况下极难实现。进而，我们展示了如何利用分部求和 (SBP) 算子高效实现该方法，特别是在SBP DG方法（如DG谱元法DGSEM）框架下。最终格式融合了双曲与椭圆视角的优良特性，尤其能使梯度收敛阶达到比椭圆问题DG方法常规预期高出一阶的精度。