You Do Need power plug adapter while traveling from Luxembourg to Faroe Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Faroe Islands are C, E, F and K and in Luxembourg are C and F. This is how they look like.

Luxembourg

In Luxembourg, outlets of type C and F are in use.

Type C

Type F

Faroe Islands

In Faroe Islands, outlets of type C, E, F and K are in use.

Type C

Type E

Type F

Type K