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

This repository contains information and code to reproduce the results presented in the article

@article{ranocha2023discontinuous,
  title={A discontinuous {G}alerkin discretization of elliptic problems with
         improved convergence properties using summation by parts operators},
  author={Ranocha, Hendrik},
  journal={Journal of Computational Physics},
  year={2023},
  month={07},
  doi={10.1016/j.jcp.2023.112367},
  eprint={2302.12488},
  eprinttype={arxiv},
  eprintclass={math.NA}
}

If you find these results useful, please cite the article mentioned above. If you use the implementations provided here, please also cite this repository as

@misc{ranocha2023discontinuousRepro,
  title={Reproducibility repository for
         "{A} discontinuous {G}alerkin discretization of elliptic problems with
         improved convergence properties using summation by parts operators"},
  author={Ranocha, Hendrik},
  year={2023},
  howpublished={\url{https://github.com/ranocha/2023_elliptic}},
  doi={10.5281/zenodo.7672744}
}

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, Guzmán, 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.

Numerical experiments

To reproduce the numerical experiments presented in this article, you need to install Julia. The numerical experiments presented in this article were performed using Julia v1.8.3.

First, you need to download this repository, e.g., by cloning it with git or by downloading an archive via the GitHub interface. Then, you need to start Julia in this directory and execute the following commands in the Julia REPL.

julia> include("code.jl")

julia> convergence_tests_1d()

julia> convergence_tests_2d()

This will show the results in the REPL. If you want to display LaTeX code for the convergence tables, add the keyword argument latex = true to the function calls shown above.

Authors

• Hendrik Ranocha (University of Hamburg, Germany)

Disclaimer

Everything is provided as is and without warranty. Use at your own risk!