Active Flux and Polygons: why a stabilisation and how to implement boundary conditions

In a recent paper, the authors developed an Active Flux-type algorithm for polygonal meshes (see \cite{BP_Pampa_VEM}). The main difference between Active Flux methods on polygonal meshes and those on Cartesian or simplex meshes is that the basis functions are not explicitly known. These functions describe the solution using point-value approximations on the polygon boundaries and the average value within the polygons. In \cite{BP_Pampa_VEM}, it was shown that directly transposing the algorithm from simplexes to polygons leads to a mild instability, and a remedy was proposed without a rigorous explanation. In the present paper, we clarify this issue and demonstrate that the solution presented in \cite{BP_Pampa_VEM} is indeed correct. Furthermore, another challenge-the implementation of boundary conditions-was previously solved heuristically. Building on \cite{pampa_dg}, we show how to properly implement these boundary conditions. Finally, the method is illustrated through several examples with hybrid meshes.