  The nature of the charge order in the charge frustrated compound
LuFe$_2$O$_4$ and its effect on magnetocapacitance were examined on the basis
of first-principles electronic structure calculations and Monte Carlo
simulations of electrostatic energy. Our work shows that two different types of
charge order of almost equal stability (i.e., $\sqrt{3}\times \sqrt{3}$ and
chain types) occur in the W-layers of LuFe$_2$O$_4$, and that the ground state
of LuFe$_2$O$_4$ has a ferrielectric arrangement of the W-layers with
$\sqrt{3}\times \sqrt{3}$ charge order. The giant magnetocapacitance effect of
LuFe$_2$O$_4$ at room temperature is accounted for in terms of charge
fluctuations arising from the interconversion between the two types of charge
order, that becomes hindered by an applied magnetic field.
