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The majority of the most common physical phenomena can be described usingpartial differential equations (PDEs). However, they are very oftencharacterized by strong nonlinearities. Such features lead to the coexistenceof multiple solutions studied by the bifurcation theory. Unfortunately, inpractical scenarios, one has to exploit numerical methods to compute thesolutions of systems of PDEs, even if the classical techniques are usually ableto compute only a single solution for any value of a parameter when morebranches exist. In this work we implemented an elaborated deflated continuationmethod, that relies on the spectral element method (SEM) and on the reducedbasis (RB) one, to efficiently compute bifurcation diagrams with moreparameters and more bifurcation points. The deflated continuation method can beobtained combining the classical continuation method and the deflation one: theformer is used to entirely track each known branch of the diagram, while thelatter is exploited to discover the new ones. Finally, when more than oneparameter is considered, the efficiency of the computation is ensured by thefact that the diagrams can be computed during the online phase while, duringthe offline one, one only has to compute one-dimensional diagrams. In thiswork, after a more detailed description of the method, we will show the resultsthat can be obtained using it to compute a bifurcation diagram associated witha problem governed by the Navier-Stokes equations.

Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method