Computational Mathematics and Scientific Computing Seminar

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Computing disconnected bifurcation diagrams of partial differential equations

Speaker: Patrick Farrell, University of Oxford

Location: Warren Weaver Hall online

Date: March 12, 2021, 10 a.m.

Synopsis:

Computing the distinct solutions $u$ of an equation $f(u, \lambda) = 0$ as a parameter $\lambda \in \mathbb{R}$ is varied is a central task in applied mathematics and engineering.  The solutions are captured in a bifurcation diagram, plotting (some functional of) $u$ as a function of $\lambda$. In this talk I will present a new algorithm, deflated continuation, for this task.

Deflated continuation has three advantages. First, it is capable of computing disconnected bifurcation diagrams; previous algorithms only aimed to compute that part of the bifurcation diagram continuously connected to the initial data. Second, its implementation is extremely simple: it only requires a minor modification to an existing Newton-based solver. Third, it can scale to very large discretisations if a good preconditioner is available; no auxiliary problems must be solved.

We will present applications to hyperelastic structures, liquid crystals, and Bose-Einstein condensates.