# On the Necessity of Superparametric Geometry Representation for Discontinuous Galerkin Methods on Domains with Curved Boundaries

## TL;DR

This paper investigates the importance of superparametric geometry representation in Discontinuous Galerkin methods for curved domains, showing it is necessary for Euler equations but not for Navier-Stokes equations to achieve optimal convergence.

## Contribution

It provides numerical evidence clarifying when superparametric versus isoparametric geometry representations are needed for optimal convergence in DG methods.

## Key findings

- Superparametric geometry is necessary for optimal convergence with Euler equations.
- Isoparametric geometry suffices for Navier-Stokes equations in the tested cases.
- The study offers guidance on geometry representation choices for DG methods on curved domains.

## Abstract

We provide numerical evidence demonstrating the necessity of employing a superparametric geometry representation in order to obtain optimal convergence orders on two-dimensional domains with curved boundaries when solving the Euler equations using Discontinuous Galerkin methods. However, concerning the obtention of optimal convergence orders for the Navier-Stokes equations, we demonstrate numerically that the use of isoparametric geometry representation is sufficient for the case considered here.