Dave, I'm thinking you are wanting to attach the "L" to the EDGE of the drywall, not the face, right? If that's the case, no, you cannot do this, and anchors will not work.
If the edge you're wanting to join up to is perpendicular to the joists, attach a 1X2 or similar wood trimmer to the joists at the end of the drywall, then attach your "L" to the trimmer. You could also make a tee-shaped trim which nailed to the joists and covered the edge of the drywall and creates and edge for the tiles and abandon the "L" altogether.
If the joists are parallel with the end of the drywall, and it ends on a joist, this will also work, but you may need to plate the side of the joist to get a surface to attach to.