In this paper we assess the neutron sensitivity of Graphics Processing Units (GPUs) when executing a Fast Fourier Transform (FFT) algorithm, and propose specific software-based hardening strategies to reduce its failure rate. Our research is motivated by experimental results with an unhardened FFT that demonstrate a majority of multiple errors in the output in the case of failures, which are caused by data dependencies. In addition, the use of the built-in error-correction code (ECC) showed a large overhead, and proved to be insufficient to provide high reliability. Experimental results with the hardened algorithm show a two orders of magnitude failure rate improvement over the original algorithm (one order of magnitude over ECC) and an overhead 64% smaller than ECC