Modeling chemotaxis and cell aggregation by a parabolic system of PDEs, the Keller-Segel system has played a pivotal role in mathematical biology for over 50 years. Besides its possible blowup in finite time and various modeling applications, e.g., in tumor progression, a mathematical aspect of interest is its numerical discretization, which has been a challenge due to localized high concentrations appearing in the solutions. In this talk we review the model and its prominent properties and introduce new a posteriori estimates that lay the foundation of adaptive mesh refinement schemes. The results are based on stability estimates and suitable reconstructions of the numerical solution. We also discuss implications on well-posedness, provide estimates for modified cell migration models and elucidate the behavior of the error estimator in numerical experiments.

* This is joint work with Jan Giesselmann and Kiwoong Kwon from the Technical University of Darmstadt.