An Incidence-Based Edge-Averaged Method for Convection-Diffusion Problems on Polytopal Meshes
We develop an incidence-based edge-averaged method for scalar convection-diffusion problems on convex polytopal meshes in arbitrary dimensions. The proposed method extends the classical edge-averaged finite-element scheme through a graph-theoretic formulation based on vertex-edge incidence matrices. We first establish this formulation on simplices in arbitrary dimensions and then extend it to convex polytopes by introducing complementary edges and generalized barycentric coordinates. A key ingredient is a matrix identity that relates the generalized Whitney edge mass matrix to a diagonal representation of the Poisson stiffness matrix, enabling an exponentially-fitted discretization expressed entirely in terms of nodal values and edge weights. We prove that the resulting stiffness matrix is a nonsingular $M$-matrix under suitable positivity and connectivity conditions, ensuring monotonicity and the discrete maximum principle. Numerical experiments on two- and three-dimensional simplicial and nonsimplicial meshes demonstrate first-order convergence in the $H^1$ semi-norm and stable, nonoscillatory solutions in convection-dominated regimes.