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{\large Hisham bin Zubair \\
{\bf Multigrid preconditioning for the Indefinite Helmholtz equation on locally refined grids}}

Mekelweg 4 \\ 2628 CD \\ Delft \\ The Netherlands
\\
{\tt h.binzubair@tudelft.nl}\end{center}

Efficient solution of the indefinite Helmholtz equation remains a
challenge for multigrid based numerical solution techniques. Multigrid
preconditioning techniques based on the complex shifted Helmholtz
operator exist, which we exploit in this work. We deal with specific
Helmholtz equations, arising out of applications from particle physics.
Accuracy requirements constraint the maximum mesh size to obey $kh <
0.625$ limit. In our model Helmholtz equations, the wave number has
spatial dependence, which allows us to restrict the maximum mesh size
limit to certain spatially chosen areas within the domain; elsewhere we
discretize with larger mesh sizes. In this talk we demonstrate multigrid
preconditioning based solution of these equations. The emphasis is on the
L-shaped coarsening techniques developed for such locally refined grids,
which allow standard coarsening throughout the domain, in contrast with
existing MLAT techniques, which (usually) only coarsen in the refined
subdomain. This multigrid method is based on the simplest possible
components, and the resulting convergence is good. We present the
obtained results in this talk.


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