University of Sussex

We propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities.

We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, convergence of the algorithm is shown by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented.

In dynamical systems given by an ODE, one is interested in the basin

of attraction of invariant sets, such as equilibria or periodic

orbits. The basin of attraction consists of solutions which converge

towards the invariant set. To determine the basin of attraction, one

can use a solution of a certain linear PDE which can be approximated

by meshless collocation.

The basin of attraction of an equilibrium can be determined through

sublevel sets of a Lyapunov function, i.e. a scalar-valued function

which is decreasing along solutions of the dynamical system. One

method to construct such a Lyapunov function is to solve a certain

linear PDE approximately using Meshless Collocation. Error estimates

ensure that the approximation is a Lyapunov function.

The basin of attraction of a periodic orbit can be analysed by Borg’s

criterion measuring the time evolution of the distance between

adjacent trajectories with respect to a certain Riemannian metric.

The sufficiency and necessity of this criterion will be discussed,

and methods how to compute a suitable Riemannian metric using

Meshless Collocation will be presented in this talk.

Meshfree methods become more and more important for the numerical simulation of complex real-world processes. Compared to classical, mesh-based methods they have the advantage of being more flexible, in particular for higher dimensional problems and for problems, where the underlying geometry is changing. However, often, they are also combined with classical methods to form hybrid methods.

In this talk, I will discuss meshfree, kernel based methods. After a short introduction along the lines of optimal recovery, I will concentrate on results concerning convergence orders and stability. After that I will address efficient numerical algorithms. Finally, I will present some examples, including one from fluid-structure-interaction, which will demonstrate why these methods are currently becoming Airbus's preferred solution in Aeroelasticity.