In this work we show that the flexibility of the discontinuous Galerkin (dG) discretization can be fruitfully exploited to implement numerical solution strategies based on the use of arbitrarily shaped elements. Specifically, we propose and investigate a new h-adaptive technique based on agglomeration coarsening of a fine mesh. The main building block of our dG method consists of defining discrete polyno- mial spaces on arbitrarily shaped elements. For this purpose we orthonormalize with respect to the L2-product a set of monomials relocated in a specific element frame. This procedure provides high-order hierarchical physical space basis functions that are also optimal from the point of view of conservation property. To complete the dG formulation for second order problems, two extensions of the BRMPS scheme to arbitrary polyhedral grids, including a sharp estimate of the stabilization parameter ensuring the coercivity property, are here proposed. The freedom in defining the mesh topology leads to a new, agglomeration-based, mesh adaptivity approach, which is validated on a Poisson problem. The possibility to enhance the error distribution over the computational domain is investigated with the goal of obtaining a mesh independent discretization. The grid is considered as a degree of freedom of the computation and the nodes connectivity is decided on the fly as is usually done in mesh-free implementations. Finally, we propose an easy way to reduce the cost related to numerical integration on agglomerated meshes.
