Workshop on

numerical methods for multiscale problems

 

     
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  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 efficiently.
Impressum
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