  Density functional theory plus $U$ (DFT+$U$) is one of the most efficient
first-principles methods to simulate the cold pressure properties of
strongly-correlated materials. However, the applicability of DFT+$U$ at
ultra-high pressure is not sufficiently studied, especially in the widely-used
augmented schemes [such as projector augmented wave (PAW) and linearized
augmented plane wave (LAPW)]. This work has systematically investigated the
performance of DFT+$U$ in PAW and LAPW at the pressure up to several hundred
GPa for the lanthanide metals, which is a typical strongly-correlated series.
We found DFT+$U$ simulation in PAW exhibits an unphysical underestimating of
force at high pressure. By delicate analysis and comparison with
local-orbital-independent hybrid functional results, we have demonstrated that
this unphysical behavior is related to a normalization problem on the local
density matrix caused by the overlap of local orbitals in PAW under high
pressure. Additionally, we observed a slight softening of force in DFT+$U$ in
heavy lanthanides (Tm, Yb and Lu) at high pressure comparing with the DFT
results without the influence of local orbital overlap, and it might be related
to the enhancement of bonding effect in correlation correction methods at high
pressure. Our work reveals the underestimating of high pressure in DFT+$U$
simulation, analyses two sources of this unusual behavior and proposes their
mechanism. Most importantly, our investigation highlights the breakdown of
DFT+$U$ for high pressure simulation in VASP package based on PAW framework.
