Growing an LVM disk takes a few steps. The recurring trap is forgetting the filesystem resize after the logical volume one.
The disk is full, the hypervisor has already allocated more space, but the partition does not grow. LVM adds a useful layer of abstraction, as long as you know its three tiers.
PV, VG, LV: three layers to align
The physical volume (PV) is the raw disk or partition handed to LVM. The volume group (VG) is the aggregated pool of space. The logical volume (LV) is the slice presented to the system as a partition.
Growing follows the order of the layers. You extend the PV so it sees the new disk space. The VG picks up that capacity. You then extend the LV so it takes the free space in the VG.
The forgotten step: the filesystem
At this point the logical volume is larger, but df still shows the old size. The filesystem on top has not moved. It must be resized so it fills the grown LV.
One flag avoids the miss: extend the LV while requesting the filesystem resize in the same command. A single operation, LV and filesystem grow together. That is the step most often skipped when done in two passes.
The principle: three layers, one order. The disk is useless until the filesystem on the top tier has been extended.