Agglomeration based discontinuous Galerkin discretization of the Euler and Navier–Stokes equations F. Bassi a , L. Botti a,b,⇑ , A. Colombo a , S. Rebay c a Università degli Studi di Bergamo, via Marconi 4, 24044 Dalmine, BG, Italy b Biomedical Engineering Department, Mario Negri Institute for Pharmacological Research, Villa Camozzi, via Camozzi 3, Ranica, BG, Italy c Università degli studi di Brescia, via Branze 38, 25123 Brescia, Italy article info Article history: Received 11 February 2011 Received in revised form 21 October 2011 Accepted 3 November 2011 Available online 18 November 2011 Keywords: Discontinuous Galerkin methods Compressible Euler and Navier–Stokes equations Polyhedral elements Curved boundaries abstract In this work we exploit the flexibility associated to discontinuous Galerkin methods to perform high- order discretizations of the Euler and Navier–Stokes equations on very general meshes obtained by means of agglomeration techniques. Agglomeration is here considered as an effective mean to decouple the geometry representation from the solution approximation being alternative to standard isoparametric or breakthrough isogeometric discretizations. The mesh elements can be composed of a set of standard sub-elements belonging to a fine low-order mesh whose cardinality can be freely chosen according to the domain discretization capabilities. The number of mesh sub-elements still have an impact on the cost of numerical integration. Since the agglomerated elements are arbitrarily shaped, physical space ortho- normal basis functions are here considered as a key ingredient to build a suitable discrete dG space. As a result we are allowed to perform high-order discretizations on top of the set of (possibly) subparametric sub-cells composing an agglomerated element. Our approach is validated on challenging viscous and inviscid test cases. We demonstrate the use of low-order meshes as a starting point to obtain high-order accurate solutions on coarse meshes. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction In recent years, the ability to deal with general unstructured polyhedral grids has received growing interest in the CFD commu- nity. Several Finite Volume (FV) schemes have been extended in this direction in order to improve grid quality and accuracy of numerical solutions for a given computational cost. In this context, the discontinuous Galerkin (dG) framework appears very well sui- ted to cope with the issue of higher-order approximations on arbi- trarily shaped elements. In a paper by Bassi et al. [6] the key ingredients of a physical frame dG formulation able to handle gen- eral meshes have been presented in detail and a new h-adaptivity strategy based on agglomeration coarsening of an underlying fine grid has been devised. Moreover, the BRMPS dG discretization of the Poisson equation has been extended to general meshes. In this paper we consider the compressible Euler and Navier–Stokes equa- tions and assess the benefits of agglomeration-based high-order dG discretizations in the context of inviscid and viscous compress- ible flows. In particular, we focus on the important issue of provid- ing an accurate representation of complex (possibly curved) domain boundaries by means of agglomeration techniques. An accurate treatment of curved boundaries has been recognized as mandatory for the application of finite element discretizations to real-life CFD computations. Indeed, the computational mesh of a do- main X results, in general, in a tessellation of an approximate do- main X h and boundary conditions are set on the approximate geometry. This introduces a consistency error with respect to the unknown exact solution that may dominate the approximation er- ror and degrade convergence properties. Clearly the higher is the approximation accuracy the higher must be the domain representa- tion accuracy provided by the computational mesh. In this context the possibility to increase the discretization accuracy without resorting to time consuming mesh generation techniques would be greatly appreciated by all the CFD practitioners. Isoparametric finite elements derived in the context of standard Galerkin discretizations has since long been considered the unique viable solution to perform high-order discretization on sufficiently coarse meshes. As a matter of fact, in the context of high-order dis- cretization, the possibility to improve the representation of X decreasing the mesh step size is suboptimal due to the significant increase in the number of degrees of freedom. A better approach on relatively coarser meshes is to improve the quality of the geomet- 0045-7930/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2011.11.002 ⇑ Corresponding author at: Università degli Studi di Bergamo, via Marconi 4, 24044 Dalmine, BG, Italy. E-mail addresses: francesco.bassi@unibg.it (F. Bassi), lorenzo.botti@unibg.it (L. Botti), alessandro.colombo@unibg.it (A. Colombo), stefano.rebay@ing.unibs.it (S. Rebay). Computers & Fluids 61 (2012) 77–85 Contents lists available at SciVerse ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid