Improved $L^2$ estimate for gradient schemes, and super-convergence of the TPFA finite volume scheme

TL;DR

This paper improves the $L^2$ error estimate for gradient schemes, demonstrates super-convergence of the HMM scheme, and explores the super-convergence of TPFA finite volume schemes, advancing understanding of diffusion approximation accuracy.

Contribution

It provides an improved $L^2$ error estimate for gradient schemes and establishes super-convergence results for HMM and TPFA schemes.

Findings

Achieved an $ ext{O}(h^2)$ super-convergence rate in $L^2$ norm for HMM schemes.

Designed a modified HMM method with super-convergence on general meshes.

Linked HMM and TPFA schemes to partially confirm a long-standing super-convergence conjecture.

Abstract

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.