Attached to this post is the Blender file containing my attempt in making a groin or cross vault.
Since it appears that vertices are not automatically created when edges intersect, creating the groined vault was a challenge. I started with 2 perpendicularly intersected cylinders, then removed the lower halves to create 2 barrel vaults. One at a time, extended a vertex from an end arch towards the middle while constrained to the proper axis. The snap-to-edge was enabled in order to properly connect the extended vertex to corresponding perpendicular edge. Also I had to delete the same axis edge before extruding the vertex, this way I didnt have double edges on the same axis/location. This process allowed me to build the vault along one axis. The next axis was almost the same process except I used snap-to-vertex. This process creates lots of double vertices and will have to be removed. I added edges as required in order to create four sided faces.
The Blender model file for download 2013-08-10-blender-groin-vault.zip 1.1MB