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

TL;DR

This paper introduces an efficient deflated continuation method combined with spectral element and reduced basis techniques to compute complex bifurcation diagrams of PDEs with multiple parameters and solutions.

Contribution

It presents a novel approach integrating deflation, spectral element, and reduced basis methods for efficient bifurcation analysis of nonlinear PDEs.

Findings

Successfully computed bifurcation diagrams for Navier-Stokes equations.

Achieved online phase computation of multi-parameter bifurcation diagrams.

Enhanced efficiency by reducing offline computations to one-dimensional diagrams.

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.