Stefan Sauter: Lattice Equations
In many practical applications such as lightweight constructions or
large crystals, the physical problem is formulated on a lattice,
i.e., on a set of nodes which are connected by some rods. In my talk,
I will propose solution methods for solving and analysing abstract
lattice equations. We will consider two types of approaches.
a) Fourier analysis
Various properties of periodic lattices can be analysed
and understood by considering, as a model, an infinite cell-periodic
lattice with possibly complicated micro-structures via Fourier analysis.
We will discuss mathematical problems
i) to determine the connectivity of the infinite lattice in finite time,
ii) to formulate algebraic properties on the equations such that they
are elliptic and
iii) to approximate the equations by means of homogenisation.
b) Multigrid methods for problems on finite lattices
We will present a methodology to assign an elliptic boundary value
problem to a lattice equation. The finite element discretisation of this
elliptic PDE serves as an optimal preconditioner for the lattice equation
and multigrid methods can be employed to realise the preconditioner