Convergence of a finite-volume scheme for aggregation-diffusion equations with saturation

In [Bailo, Carrillo, Hu. SIAM J. Appl. Math. 2023] the authors introduce a finite-volume method for aggregation-diffusion equations with non-linear mobility. In this paper we prove convergence of this method using an Aubin–Simons compactness theorem due to Gallouët and Latché. We use suitable discrete $H^1$ and $W^{-1,1}$ discrete norms. We provide two convergence results. A first result shows convergence with general entropies ($U$) (including singular and degenerate) if the initial datum does not have free boundaries, the mobility is Lipschitz, and the confinement ($V$) and aggregation ($K$) potentials are $W_0^{2,\mathrm{\infty }}$. A second result shows convergence when the initial datum has free boundaries, mobility is just continuous, and $V$ and $K$ are $W^{1,\mathrm{\infty }}$, but under the assumption that the entropy $U$ is $C^1$ and strictly convex.