Comparison of hierarchical and non-hierarchical error indicators for adaptive mesh refinement for the Euler equations

Abstract

This article deals with two automatic mesh refinement methods for the Euler equations discretized by the discontinuous Galerkin finite element method of lowest order. The main purpose is the comparison of an hierarchical error indicator with a non-hierarchical one. The underlying mesh refinements are based on quadrilateral meshes with hanging nodes. The hierarchical indicator is based on the comparison of the solutions of two consecutive globally refined meshes. It therefore includes information of the nonlinear problem on the fine mesh. In contrast to this, the non-hierarchical indicator employs only information on the given mesh in form of jumps of the solution over internal edges. As often done in practice, in both cases, we chose a certain physical quantity such as the density or the Mach number as a sensor variable to which the indicator is applied. Three different standard test problems are considered: •A scramjet test case representing an internal supersonic flow, •A cylinder test case representing an external supersonic flow, •A Naca wing case representing an external transonic flow.