Agglomeration-based physical frame DG discretizations for high-order accurate CFD

One of the major advantages of finite element discretizations over low-order methods is the possibility to achieve high-order accurate discretizations on unstructured hybrid meshes featuring elements of general shape. Since this comes at the price of an increased computational effort, the geometrical flexibility and increased accuracy need to be fruitfully exploited to stay competitive. In this work we propose agglomeration based physical frame discontinuous Galerkin methods (abdG) as an effective way to retain optimal approximation properties on general meshes and further improve the flexibility in the domain discretization. On one hand the convergence rates of physical frame discretizations are not influenced by non-linear reference-to-physical frame mappings and, on the other hand, physical frame polynomial spaces can be defined on very general elements, well beyond the standardized reference frame polygons. This suggests the possibility to decouple the domain discretization from the solution approximation using agglomerated elements. A fine grid, whose cardinality is chosen according to the domain discretization requirements, is first obtained using standard mesh generator. Thus the agglomerated elements are generated clustering together the cells of the fine grid. The mesh density can be decided at will. The coarser agglomerated mesh retains the boundary discretization capabilities of the fine grid and allows to exploit the convergence rates of high-order discretizations while keeping the number of degrees of freedom under control.