Further progress of fundamental particle physics requires
high intensity and high brightness of accelerated proton and ion
beams. This goal is essential for the FAIR hadron beams at GSI, for
the neutrino production at the facilities such as Fermilab and
JPARC, and for the Large Hadron Collider luminosity at CERN. One of
the most formidable obstacles toward that goal is the beam's own
space charge, whose forces cause beam emittance growth, losses and
lifetime degradation. Typically, such effects become intolerable
when the space charge tune-shift parameter Δ QSC exceeds
∼ - (0.25–0.5). To reduce these detrimental effects, it
was suggested to use electron lenses to compensate the space charge
forces. This paper reports on detailed particle-in-cell space charge
and electron lens compensation simulations for extremely intense
proton bunches whose space charge tune-shift parameter exceeds
-1.0. Different scenarios were evaluated based on reduction in the
emittance growth and particle loss at a 4σ aperture. We
investigate phenomena and issues related to the focusing lattice
errors, importance of the transverse and longitudinal matching of
the electron beam profiles to the proton ones, and vary the strength
and the number of the electron lenses distributed around a circular
machine to optimize the reduction of harmful space charge effects.