This abstract focuses on the computation of a typical periodic cell problem that arises in homogenization. This model problem is intended to be discretized on a Fourier-basis and solved using a Fast Fourier Transform (FFT)-based iterative scheme. Such a method, which makes use of a uniform grid and global basis functions, has inherent limitations for correctly capturing any localized features of the solution, such as singularities or discontinuities. An adaptive method is proposed here to overcome these shortcomings, by allowing computations on a non-uniform grid.