An agglomeration‐based adaptive discontinuous Galerkin method for compressible flows

Abstract

Summary In this work, we exploit the possibility to devise discontinuous Galerkin discretizations over polytopic grids to perform grid adaptation strategies on the basis of agglomeration coarsening of a fine grid obtained via standard unstructured mesh generators. The adaptive agglomeration process is here driven by an adjoint‐based error estimator. We investigate several strategies for converting the error field estimated solving the adjoint problem into an agglomeration factor field that is an indication of the number of elements of the fine grid that should be clustered together to form an agglomerated element. As a result the size of agglomerated elements is optimized for the achievement of the best accuracy for given grid size with respect to the target quantities. To demonstrate the potential of this strategy we consider problem‐specific outputs of interest typical of aerodynamics, eg, the lift and drag coefficients in the context of inviscid and viscous flows test cases.

Anisotropic a posteriori error estimate for the Virtual Element Method

“…However, while (isotropic and anisotropic) error estimates and a posteriori error estimates and adaptive finite element methods (AFEMs) have been intensively investigated during the last decades (see, e.g., for the isotropic case the monographs [32,31] and the references therein and for the anisotropic case [4,21,22,23,24,25,26] and the references therein), the corresponding study of a posteriori error estimates and adaptivity for polytopal methods is still in its infancy. See, for example, [5,8,2] for the study of a posteriori error estimates in the context of Mimetic Finite Differences, [9,13,15,29,10,19,16] for the Virtual Element Method, [34,35,38,37] for polygonal BEMbased FEM, [39] for the polygonal Discontinuous Galerkin method, [20] for the Mixed High Order method, [30] for the Weak Galerkin method and [33] for lowest-order locally conservative methods on polytopal meshes. Moreover, despite the great flexibility provided by polytopal meshes, the above works focused on the isotropic case, only.…”