Abstract
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.
Highlights
Introduction and motivationUsually, when one wants to numerically compute a bifurcation diagram, one has to combine many numerical methods in order to obtain it
The offline phase is very expensive, but its cost is reduced exploiting the fact that the solutions obtained with the spectral element method (SEM) are characterized by a lower number of degrees of freedom when compared to their counterpart computed with the standard finite element method (FEM)
We showed that the eigenvalues decay of the proper orthogonal decomposition (POD) matrix is the expected one and that, exploiting the continuation and the deflation method, it is possible to construct a bifurcation diagram in the offline phase and to reconstruct it accurately and efficiently during the online one
Summary
When one wants to numerically compute a bifurcation diagram, one has to combine many numerical methods in order to obtain it. We rely on the reduced basis method [25] This is important because, after a very expensive offline phase, the computation of a solution in the online one is, in a repetitive computational environment, very efficient. We decided to use the spectral element method (SEM) [28] to compute the snapshots in the offline phase This is important because numerous solutions have to be computed to obtain a reduced space able to capture all the branches in the online phase. The offline phase is very expensive, but its cost is reduced exploiting the fact that the solutions obtained with the SEM are characterized by a lower number of degrees of freedom when compared to their counterpart computed with the standard finite element method (FEM). It is important to highlight that the computational cost of
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