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

## TL;DR

This paper introduces an efficient discontinuous Galerkin discretization for elliptic problems that leverages summation by parts operators to achieve improved convergence properties, especially for gradients, blending hyperbolic and elliptic perspectives.

## Contribution

It demonstrates how to implement a DG method with SBP operators for elliptic problems, enhancing convergence and computational efficiency.

## Key findings

- High order convergence of gradients achieved
- Efficient implementation using SBP operators
- Improved error estimates for elliptic solutions

## Abstract

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.