I have several relatives who "divorced" in the late 1800's and early 1900's, but none of them bothered with filing any paperwork.
Up until some point in Louisiana, not sure when the law changed, but I've heard that there were two ways to get a divorce; one was to get an act of the state legislature and the other was to desert your spouse and move out of state for a period of time.
From my own geneaology, it appears that there was a 3rd method - just declare yourself "divorced" or "widowed" even if your spouse was still alive and kicking and living in the next parish. I have seen this technique used on the 1940 census. Luckily, all the cases in the 1940's census were just living apart - no one was married to a 3rd party.