Improved L² estimate for gradient schemes and super-convergence of the TPFA finite volume scheme

The gradient discretization method is a generic framework that is applicable to a number of schemes for diffusion equations, and provides in particular generic error estimates in L² and H1-like norms. In this article, we establish an improved L² error estimate for gradient schemes. This estimate is applied to a family of gradient schemes, namely the hybrid mimetic mixed (HMM) schemes, and yields an O(h²) superconvergence rate in L² norm, provided local compensations occur between the cell points used to define the scheme and the centers of mass of the cells. To establish this result, a modified HMM method is designed by just changing the quadrature of the source term; this modified HMM enjoys a super-convergence result even on meshes without local compensations. Finally, the link between HMM and two-point flux approximation (TPFA) finite volume schemes is exploited to partially answer a long-standing conjecture on the super-convergence of TPFA schemes.