Goal-oriented anisotropic $hp$-adaptive discontinuous Galerkin method for the Euler equations

TL;DR

This paper develops a goal-oriented anisotropic hp-adaptive discontinuous Galerkin method for solving the Euler equations, emphasizing error estimation, adjoint consistency, and mesh adaptation, supported by theoretical analysis and numerical experiments.

Contribution

It introduces a novel goal-oriented anisotropic hp-adaptive DG scheme for Euler equations, with a new approach to adjoint consistency and error estimation.

Findings

The method achieves accurate solutions with efficient mesh adaptation.

Numerical experiments validate the theoretical error estimates.

The approach improves computational efficiency for Euler equations.

Abstract

We deal with the numerical solution of the compressible Euler equations with the aid of the discontinuous Galerkin (DG) method with focus on the goal-oriented error estimates and adaptivity. We analyze the adjoint consistency of the DG scheme where the dual problem is not formulated by the differentiation of the DG form and the target functional but using a suitable linearization of the nonlinear forms. Further, we present the goal-oriented anisotropic $hp$-mesh adaptation technique for the Euler equations. The theoretical results are supported by numerical experiments.