  For a Keller-Segel model for chemotaxis in two spatial dimensions we consider
a modification of a positivity preserving fully discrete scheme using a local
extremum diminishing flux limiter. We discretize space using piecewise linear
finite elements on an quasiuniform triangulation of acute type and time by the
backward Euler method. We assume that initial data are sufficiently small in
order not to have a blow-up of the solution. Under appropriate assumptions on
the regularity of the exact solution and the time step parameter we show
existence of the fully discrete approximation and derive error bounds in $L^2$
for the cell density and $H^1$ for the chemical concentration. We also present
numerical experiments to illustrate the theoretical results.
