My PhD research has a negative outcome -- how can I still graduate with it? A former PhD student at our lab had this idea for an algorithm for solving an engineering problem that has many well known solutions. The student developed preliminary results but ended up applying the algorithm to some other simpler engineering problems instead and graduating. I 'inherited' the project and it has always been a project my advisor has had high ambitions for. My advisor actually ended up changing my research project to this one after 6 month even though I heavily objected. The project is not funded. I TA to pay the bills.

At this point I have already advanced to candidacy and I'm in what I was planning to be my final 6 month. My research has basically consisted of building other algorithms based on the one the other student came up with. All are for an application in a measurement system. I have derived all of the mathematical solutions that will be at the heart of my algorithm and I am in the process of validating them with new experimental data. The previous data I worked with was all provided by the first student to work on the project.

I have recently discovered there are some issues with the original algorithm that create errors in the solution. The errors are larger than those  of the previous (well-established) methods. This does not happen in all conditions, however, the experimental data I was left with all happened to be cases where this error does not show up (I'm not entirely sure this was by coincidence).

How can I salvage this project and graduate without delaying my graduation much. I have family obligations that require me to graduate soon and I'm devastated that I've discovered this so late. This is partly my fault for trusting the research from the previous student too much (it was easy to do since my advisor can't stop praising him).

I personally have always been of the opinion that research projects do not need to have positive outcomes and PhD students should be assured that things will be fine for them even if there research has bad outcomes. I feel that doing research to show that something has a bad outcome is still equally valuable. I also feel that because of pressure to have good outcomes a lot of researchers exaggerate their results out of fear of possible consequences and it ends up doing more bad than good when people try to use their publications.

Do you think I have a good case of being able to graduate given my situation? And how would you approach convincing my advisor?
Conditions Under-which Fancy Algorithm Has Inflated Error Properties by A. Student. Journal of Science and The Doing Of It. 2017.

This would be a perfectly acceptable paper, and one which should appear in the literature. If you can, correcting said algorithm so it does not do so would be a bonus, but not necessary. Also, it's slightly late for you, but one should always ask when embarking on a thesis/dissertation project "What happens if the answer is "No"?"