Abstract
The subject of this work is an optimal and scalable parallel geometric multigrid solver for elliptic problems on the sphere, crucial to the forecasting and the data assimilation tools used at the U.K. Met office. The optimality of multilevel techniques for elliptic problems makes them a suitable choice for these applications. The Met office uses spherical polar grids which, although structured, have the drawback of creating strong anisotropies near the poles. Moreover, a higher resolution in the radial direction introduces further anisotropies, and so modifications to the standard multigrid relaxation and the coarsening procedures are necessary to retain optimal efficiency. As the strength of anisotropy varies, we propose a nonuniform strategy, coarsening the grid only in regions that are sufficiently isotropic. This is combined with line relaxation in the radial direction. The success of nonuniform coarsening strategies has been demonstrated with algebraic multigrid (AMG) methods. Without the large setup costs required by AMG, however, we aim to surpass them with the geometric approach. We demonstrate the advantages of the method with experiments on model problems, both sequentially and in parallel, and show robustness and optimal efficiency of the method with constant convergence factors of less than 0.1. It substantially outperforms Krylov subspace methods with onelevel preconditioners and the BoomerAMG implementation of AMG on typical grid resolutions. The parallel implementation scales almost optimally on up to 256 processors, so that a global solve of the quasigeostrophic omegaequation with a maximum horizontal resolution of about 10 km and 3 x 109 unknowns takes about 60s.
Original language  English 

Pages (fromto)  325342 
Number of pages  18 
Journal  Numerical Linear Algebra with Applications 
Volume  17 
Issue number  23 
Early online date  11 Mar 2010 
DOIs  
Publication status  Published  1 Apr 2010 
Keywords
 anisotropic multigrid
 spherical polar grid
 line relaxation
 quasigeostrophic omegaequation
 conditional semicoarsening
 anisotropy
 geometric multigrid
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