Method of averaged element splittings for diffusion terms discretization in vertex-centered framework

•Method of averaged element splittings to approximate diffusion terms.•Coincides with the P1-Galerkin method on simplicial meshes.•Yields a 5-point (2D) or 7-point (3D) approximation for the Laplacian operator on Cartesian meshes.•Convergence with the order 3/2 in 2D and with the first order in 3D is proved for elliptic problems.•Well suitable for an implicit time integration based on the Newton method.