# On The Weak Consistency of Finite Volumes Schemes for Conservation Laws on General Meshes

## TL;DR

This paper develops tools to establish weak consistency of finite volume schemes for multi-dimensional conservation laws on general meshes, extending classical results under minimal regularity assumptions.

## Contribution

It introduces a discrete integration by parts approach to prove weak consistency, generalizing the Lax-Wendroff theorem to complex meshes and minimal regularity.

## Key findings

- Establishes weak consistency of finite volume schemes on general meshes
- Extends Lax-Wendroff theorem to multi-dimensional, irregular meshes
- Demonstrates convergence of discrete gradients in L-infinity weak topology

## Abstract

The aim of this paper is to develop some tools in order to obtain the weak consistency of (in other words, analogues of the Lax-Wendroff theorem for) finite volume schemes for balance laws in the multi-dimensional case and under minimal regularity assumptions for the mesh. As in the seminal Lax-Wendroff paper, our approach relies on a discrete integration by parts of the weak formulation of the scheme. This makes a discrete gradient of the test function appear, and the central argument for the scheme consistency is to remark that this discrete gradient is convergent in L $\infty$ weak .