Improved $L^2$ estimate for gradient schemes, and super-convergence of the TPFA finite volume scheme
Provides theoretical error bounds for numerical schemes solving diffusion equations, relevant to computational scientists and engineers.
The paper establishes an improved $L^2$ error estimate for gradient schemes, achieving $\mathcal O(h^2)$ super-convergence for Hybrid Mimetic Mixed schemes under local compensations, and partially resolves a long-standing conjecture on super-convergence of Two-Point Flux Approximation finite volume schemes.
The gradient discretisation method is a generic framework that is applicable to a number of schemes for diffusion equations, and provides in particular generic error estimates in $L^2$ and $H^1$-like norms. In this paper, we establish an improved $L^2$ 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 $\mathcal O(h^2)$ super-convergence rate in $L^2$ 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.