  The majority of the most common physical phenomena can be described using
partial differential equations (PDEs). However, they are very often
characterized by strong nonlinearities. Such features lead to the coexistence
of multiple solutions studied by the bifurcation theory. Unfortunately, in
practical scenarios, one has to exploit numerical methods to compute the
solutions of systems of PDEs, even if the classical techniques are usually able
to compute only a single solution for any value of a parameter when more
branches exist. In this work we implemented an elaborated deflated continuation
method, that relies on the spectral element method (SEM) and on the reduced
basis (RB) one, to efficiently compute bifurcation diagrams with more
parameters and more bifurcation points. The deflated continuation method can be
obtained combining the classical continuation method and the deflation one: the
former is used to entirely track each known branch of the diagram, while the
latter is exploited to discover the new ones. Finally, when more than one
parameter is considered, the efficiency of the computation is ensured by the
fact that the diagrams can be computed during the online phase while, during
the offline one, one only has to compute one-dimensional diagrams. In this
work, after a more detailed description of the method, we will show the results
that can be obtained using it to compute a bifurcation diagram associated with
a problem governed by the Navier-Stokes equations.
