Accurate knowledge of fission fragment yields is an essential ingredient of numerous applications ranging from the formation of elements in the r-process to fuel cycle optimization in nuclear energy. The need for a predictive theory applicable where no data is available, together with the variety of potential applications, is an incentive to develop a fully microscopic approach to fission dynamics. One of the most promising theoretical frameworks is the time-dependent generator coordinate method (TDGCM) applied under the Gaussian overlap approximation (GOA). Previous studies reported promising results by numerically solving the TDGCM+GOA equation with a finite difference technique. However, the computational cost of this method makes it difficult to properly control numerical errors. In addition, it prevents one from performing calculations with more than two collective variables. To overcome these limitations, we developed the new code FELIX-1.0 that solves the TDGCM+GOA equation based on the Galerkin finite element method. In this article, we briefly illustrate the capabilities of the solver FELIX-1.0, in particular its validation for n+239Pu low energy induced fission. This work is the result of a collaboration between CEA,DAM,DIF and LLNL on nuclear fission theory.