  We numerically investigate the interplay of disorder and electron-electron
interactions in the integer quantum Hall effect. In particular, we focus on the
behaviour of the electronic compressibility as a function of magnetic field and
electron density. We find manifestations of non-linear screening and charging
effects around integer filling factors, consistent with recent imaging
experiments. Our calculations exhibit $g$-factor enhancement as well as strong
overscreening in the centre of the Landau bands. Even though the critical
behaviour appears mostly unaffected by interactions, important implications for
the phase diagram arise. Our results are in very good agreement with the
experimental findings and strongly support the relevance of electron-electron
interactions for understanding integer quantum Hall physics.
