A posteriori existence for the Keller-Segel model via a finite volume - finite element scheme

Abstract

We derive two forms of conditional a posteriori error estimates for a finite volume scheme approximating the parabolic-elliptic Keller-Segel system. The estimates control the error in the $L^{\infty }(0,T,L^2(\hat{I}\text{c◯}))$- and $L^2(0,T;H^1(\hat{I}\text{c◯}))$-norm and exhibit linear convergence in the mesh size, as observed in numerical experiments. Crucially, we show that, as long as the condition of the error estimate is satisfied, a weak solution exists. This means, as long as the numerical solution has good properties, we can rigorously infer existence of an exact solution.