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.
- two-point flux approximation finite volumes
- hybrid mimetic mixed methods
- gradient schemes