Preconditioners based on Calderon's formulae for FMMs in periodic wave problems

Wednesday, August 4, 2010 - 1:30pm - 2:00pm
Keller 3-180
Naoshi Nishimura (Kyoto University)
Wave problems in periodic domains have many interesting applications
such as photonic crystals and metamaterials in Maxwell's equations and
phononic crystals in elastodynamics, etc. Fast multipole methods are
considered to be effective as solvers of such problems, particularly
when the problems under consideration are of scattering nature. In
view of this, we have developed periodic FMMs for various wave
problems. We are now interested in further enhancing the performance
of the periodic FMMs with the help of effective preconditioners. In
this presentation we shall discuss our recent efforts on the use of
preconditioners based on Calderon's formulae in periodic transmission
problems in Helmholtz' equation and elastodynamics. In Helmholtz, we
shall show that the matrix of the discretised integral equations
itself serves as an effective preconditioner. This fact leads to a
very simple preconditioning scheme as one uses GMRES as the solver. We
shall also see that a similar approach is possible in elastodynamics,
but either with some restrictions on the material constants or with
more complicated formulations. We shall present a number of numerical
examples to test the performance of the proposed approaches.