Efficient Computation of Bifurcation Diagrams with Spectral Element Method and Reduced Order Models

Many physical phenomena can be described using partial differential equations (PDEs), they usually are non-linear and, due to the non-linearity, multiple different solutions can exist. Understanding this property and the relation between the different solutions and some parameters is crucial to predict the evolution of a system when the parameters can vary. Even if these phenomena can be represented with bifurcation diagrams, obtaining them analytically or theoretically is impossible for almost every interesting problem. For this reason, one would like to compute them numerically, but, due to the complexity of the task, several different techniques must be used together to perform it in an accurate and efficient way. Firstly, one has to be able to compute a solution of the problem for many values of the parameters, therefore the solver should be as accurate and fast as possible because it will be used many times. Secondly, a suitable continuation method is needed to efficiently follow a single branch of the bifurcation diagram and, lastly, it is important to use another method to detect the bifurcation points or discover new branches. In particular, the deflation method has been chosen to find them. The problem is that, even if all these elements are available and optimized, the computation of a bifurcation diagram is a very expensive operation; indeed the described methods have to be performed a huge number of times and the associated computational cost can easily become prohibitive if more than one parameter are involved. In this work an efficient way to compute bifurcation diagrams with one or more parameters is proposed and all the different required techniques are explained with a particular focus on their implementation and the obtained results. We highlight that the described method can be used in several different scenarios, in fact the spectral element method (SEM) has been used to compute the full order solutions required to generate the reduced space with the reduced basis (RB) method. These two techniques are very general because they simply consider the variational formulation of an arbitrary equation. Moreover, we implemented an advanced deflated continuation method to efficiently compute the set of solutions that discretizes the bifurcation diagram. In such a technique we alternate the continuation method, implemented in two different ways in order to be able to accurately track all the branches during each phase of the computation, and the deflation one, that we paired with a novel heuristic to increase its effectiveness. Since a reduced order model (ROM) is used, the discussed methods can be divided in two phases: the first one, named offline phase, is the most expensive one and is responsible for the computation of the so called full order solutions or snapshots. Subsequently, in the online phase, the solution of the discrete problem is sought in a low-dimensional space and all the matrices and vectors are simply assembled with objects created during the offline phase and, for this reason, the computation is much faster.