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新表述为线性双曲系统的稳态问题。该方法对椭圆方程的解及其梯度均能获得最优误差估计，但阻碍了椭圆问题常用求解器的应用。我们展示了该方法与Cockburn、Guzmán和Wang（2009）分析的间断伽辽金（DG）方法之间的联系——后者在一般情形下极难实现。随后，我们证明了如何利用求和分（SBP）算子高效实现该方法，特别是在SBP DG方法（如DG谱元法DGSEM）的框架下。所构建的格式融合了双曲视角与椭圆视角的良好特性，尤其是梯度具备比椭圆问题DG方法常规预期高一阶的超收敛收敛阶。