Deflationaccelerated preconditioning of the PoissonNeumann Schur problem on long domains with a highorder discontinuous elementbased collocation method
Abstract
A combination of blockJacobi and deflation preconditioning is used to solve a highorder discontinuous elementbased collocation discretization of the Schur complement of the PoissonNeumann system as arises in the operator splitting of the incompressible NavierStokes equations. The preconditioners and deflation vectors are chosen to mitigate the effects of illconditioning due to highlyelongated domains typical of simulations of strongly nonhydrostatic environmental flows, and to achieve Generalized Minimum RESidual method (GMRES) convergence independent of the size of the number of elements in the long direction. The illposedness of the PoissonNeumann system manifests as an inconsistency of the Schur complement problem, but it is shown that this can be accounted for with appropriate projections out of the null space of the Schur complement matrix without affecting the accuracy of the solution. The blockJacobi preconditioner is shown to yield GMRES convergence independent of the polynomial order and only weakly dependent on the number of elements within a subdomain in the decomposition. The combined deflation and blockJacobi preconditioning is compared with twolevel nonoverlapping blockJacobi preconditioning of the Schur problem, and while both methods achieve convergence independent of the grid size, deflation is shown to require half as many GMRES iterations and 25% less wallclock time for a variety of grid sizes and domain aspect ratios. The deflation methods shown to be effective for the twodimensional PoissonNeumann problem are extensible to the threedimensional problem assuming a Fourier discretization in the third dimension. A Fourier discretization results in a twodimensional Helmholtz problem for each Fourier component that is solved using deflated blockJacobi preconditioning on its Schur complement. Here again deflation is shown to be superior to twolevel nonoverlapping blockJacobi preconditioning, requiring about half as many GMRES iterations and 15% less time.
 Publication:

Journal of Computational Physics
 Pub Date:
 May 2016
 DOI:
 10.1016/j.jcp.2016.02.033
 arXiv:
 arXiv:1512.01756
 Bibcode:
 2016JCoPh.313..209J
 Keywords:

 Poisson equation;
 Spectral element;
 Deflation;
 Preconditioning;
 Schur complement;
 Domain decomposition;
 Mathematics  Numerical Analysis;
 Physics  Computational Physics
 EPrint:
 doi:10.1016/j.jcp.2016.02.033