A comparison of refinement indicators for p-adaptive discontinuous Galerkin methods for the Euler and Navier-Stokes equations

Abstract

This paper presents an analysis of refinement indicators for the numerical solution of the Euler and the Navier-Stokes equations using a p-adaptive discontinuous Galerkin (DG) method. Residual error, discretization error and feature-based indicators are studied in the context of p-adaptive DG simulations of two inviscid flow configurations, namely, the flow over a Gaussian bump and the flow past a circular cylinder. As expected, the superiority of local p-refinement as compared to uniform p-refinement is clearly observed. The results obtained highlight the efficiency of the considered refinement indicators with respect to an indicator based on the local entropy error. In particular, they identify regions where numerical errors are being produced and not those where errors are being convected. Finally the p-adaptive algorithm is applied to the simulation of the laminar flow past a circular cylinder at Re = 40. The great potential of the spectral decay 1 and VMS 2 indicators, which combine accuracy and computational efficiency, is also demonstrated.

Local Multiwavelet-Based Adaptation within a Discontinuous Galerkin Framework

“…An important decrease in the number of degrees of freedom (#DOFs) is observed when the adaptive process is activated. A comparison with the modal error estimators developed in [4,5] shows that, for our proposed estimator, fewer elements are adapted for a similar global level of the error. Interestingly, the effectivity index of the proposed method is close to unity.…”

“…A further analysis of the effectivity index for the initial conditions IC-smooth and = 3 is performed in Fig. (4). This example shows no difference between uniform and adapted mesh due to the regularity of the solution.…”

“…Finally, in our study two error estimators extensively tested in [4,5] are employed as reference estimators: the small-scale energy density (SSED) [29] and the spectral decay (SD) [30] error estimators. The SSED measures the energy associated with the highest-order modes.…”

“…With this in mind, the research in [4,5] shows the benefits of p-adaptation, and how its use can considerably reduce the number of DOFs for a prescribed level of accuracy, compared to the use of a uniform approximation order.…”

A p-multigrid strategy with anisotropic p-adaptation based on truncation errors for high-order discontinuous Galerkin methods

“…This has been done historically using three different approaches: the feature-based adaptation, the adjoint-based adaptation, and the local error-based adaptation. A comparison of these three approaches was performed by Fraysse et al [24] in the context of finite volume approximations and by Kompenhans et al [25] and Naddei et al [26] for high-order DG methods. The key ideas behind the adaptation approaches are:…”