On reproducing uniform flow exactly on general hexahedral cells using one degree of freedom per surface

The solution of second order elliptic partial differential equations typically arising for flow problems, e.g., in porous media, is frequently expressed as face fluxes of a flow field at hexahedral cells. In this letter, we show by example that the following properties are incompatible: (1) A local velocity reconstruction, (2) reproduction of uniform flow for general hexahedral cells, and (3) a flow field in H(div). This is in particular relevant to mixed finite element methods approximating H(div), and to other methods trying to reconstruct a flow field in a hexahedron using only local face fluxes, e.g., streamline tracing methods for reservoir simulation.