Summary This paper deals with bifurcation analysis methods based on the asymptotic‐numerical method. It is used to investigate 3‐dimensional (3D) instabilities in a sudden expansion. To do so, high‐performance computing is implemented in ELMER, ie, an open‐source multiphysical software. In this work, velocity‐pressure mixed vectors are used with asymptotic‐numerical method–based methods, remarks are made for the branch‐switching method in the case of symmetry‐breaking bifurcation, and new 3D instability results are presented for the sudden expansion ratio, ie, E=3. Critical Reynolds numbers for primary bifurcations are studied with the evolution of a geometric parameter. New values are computed, which reveal new trends that complete a previous work. Several kinds of bifurcation are depicted and tracked with the evolution of the spanwise aspect ratio. One of these relies on a fully 3D effect as it breaks both spanwise and top‐bottom symmetries. This bifurcation is found for smaller aspect ratios than expected. Furthermore, a critical Reynolds number is found for the aspect ratio, ie, Ai=1, which was not previously reported. Finally, primary and secondary bifurcations are efficiently detected and all post‐bifurcated branches are followed. This makes it possible to plot a complete bifurcation diagram for this 3D case.