I submitted the talk A Discretization of Elliptic Terms with Improved Convergence Properties Using Summation by Parts Operators to the SIAM CSE Conference 2023 in Amsterdam (The Netherlands) scheduled on Tuesday, 2023-02-28, 14:15 CEST.
Nishikawa (2007) proposed to reformulate the classical Poisson equation as a steady-state problem for a linear hyperbolic system. This enables a unified discretization based on hyperbolic PDE solvers, e.g., in the context of coupled elliptic-hyperbolic systems such as the Euler equations with self-gravity studied by Schlottke-Lakemper, Winters, Ranocha, and Gassner (2021). It also results in optimal error estimates for the solution of the elliptic equation and its gradients, which are of primary interest in self-gravity. However, it prevents the application of well-known solvers for elliptic problems such as the (preconditioned) conjugate gradient method. We show connections to a discontinuous Galerkin (DG) method analyzed by Castillo, Cockburn, Perugia, and Schötzau (2000) 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. 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.