High-order complete flux schemes for convection-diffusion equations on arbitrary subdivisions

We develop a complete flux finite volume method for convection-diffusion equations that works on arbitrary meshes in two and three dimensions and for discrete spaces of any polynomial degree. Unlike standard finite volume discretizations, where the numerical flux is directly approximated from the flux definition, we derive the exact normal flux across each control volume edge/face from the underlying PDE. This exact flux splits naturally into a homogeneous part (the classical Scharfetter-Gummel flux) and an inhomogeneous part expressed via a Green's function that incorporates the tangential flux and the source term. The resulting formulation is exactly equivalent to the continuous equation and, once the discrete space is chosen, yields high-order schemes without any correction or stabilization. For piecewise linear spaces, the scheme achieves optimal second-order accuracy in convection-dominated regimes and can preserve positivity on moderately coarse meshes. For quadratic spaces, standard finite volume methods, based on the Lagrange elements or B-splines, fail to attain optimal $L^2$ convergence unless the control volume mesh is specially designed. The proposed complete flux scheme, however, always achieves optimal $L^2$ convergence independently of the control volume mesh. Numerical experiments in two and three dimensions confirm the robustness and optimal accuracy of the approach.