Agglomeration based discontinuous Galerkin discretization of the Euler and Navier-Stokes equations

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 orthonormal 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.