The paper with Víctor Navarro-Fernández on Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients got published at ESAIM: Mathematical Modelling and Numerical Analysis (M2AN). In the revision (also on arXiv:2201.10411), we arrived at uniform errors estimate in the diffusion constant also in the limit of vanishing diffusion.

Abstract: We study the implicit upwind finite volume scheme for numerically approximating the advection-diffusion equation with a vector field in the low regularity DiPerna-Lions setting. That is, we are concerned with advecting velocity fields that are spatially Sobolev regular and data that are merely integrable. We study the implicit upwind finite volume scheme for numerically approximating the advection-diffusion equation with a vector field in the low regularity DiPerna-Lions setting. We prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique solution of the continuous model is at least one. The numerical error is estimated in terms of logarithmic Kantorovich-Rubinstein distances and provides thus a bound on the rate of weak convergence.