A vertex-based scheme on polyhedral meshes for advection–reaction equations with sub-mesh stabilization

We devise and analyze vertex-based schemes on polyhedral meshes to approximate advection–reaction equations. Error estimates of order O(h^3/2) are established in the discrete inf–sup stability norm which includes the mesh-dependent weighted advective derivative. The two key ingredients are a local polyhedral reconstruction map leaving affine polynomials invariant, and a local design of stabilization whereby gradient jumps are only penalized across some subfaces in the interior of each mesh cell. Numerical results are presented on three-dimensional polyhedral meshes.