# Numerical analysis of a nonlinear free-energy diminishing Discrete Duality Finite Volume scheme for convection diffusion equations

## TL;DR

This paper introduces a nonlinear Discrete Duality Finite Volume scheme for convection diffusion equations that preserves energy dissipation at the discrete level, ensuring accurate long-term behavior and convergence to weak solutions.

## Contribution

The paper develops a novel nonlinear scheme that maintains energy relations on distorted meshes and proves its convergence and positivity properties.

## Key findings

- Scheme preserves energy dissipation even on distorted meshes
- Convergence of the scheme to weak solutions is established
- Numerical experiments confirm good long-time behavior

## Abstract

We propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. We establish the existence of positive solutions to the scheme. Based on compactness arguments, the convergence of the approximate solution towards a weak solution is established. Finally, we provide numerical evidences of the good behavior of the scheme when the discretization parameters tend to 0 and when time goes to infinity.