Lawrence Mitchell <[email protected]> writes: > So my approach was to transfer using projection and then use riesz > representation to get the residual from the dual space back into the > primal space so I can apply the operator at the next level. Is there > an obvious reason why this is worse than the transpose of > prolongation? The analysis I've read typically approaches from an FD > angle, which doesn't always obviously (at least to me) map into FE > speak. In particular, moving forward, I'm interested in GMG for > H(div) problems, for which the analysis I'm aware of uses projections > to move residuals between levels (for example, Arnold, Falk and > Winther Num Math 85:197 (2000)).
FE multigrid usually talks about embedding of spaces. If I have a continuous function f and want to evaluate \int v_i f for each coarse basis function, I can do it by quadrature directly or I can choose a large space containing v_i, integrate for all basis functions w_j in that larger space, then sum up those fine-grid contributions. This amounts to the transpose of interpolation, but is exact in the sense that it is equivalent to integrating against v_i directly using a suitable quadrature. >> Integration is the transpose of your prolongation. We can do a precise >> analysis, but I expect that you'll see slightly better convergence if >> you modify this operator. > > So I coded this up, I only have the matrix-free application of the > prolongation, so it was not just taking the transpose. I get > marginally better convergence for this problem, although it's almost a > wash. Good to hear it's at least as good. Now to make FMG work... >> I would also recommend trying FMG soon. You should be able to converge >> in one F-cycle, and that will be faster than V-cycles with Krylov. > > This is just -pc_mg_type full, right? Given that I only have > matrix-free transfer operators do I need to explicitly set do > PCMGSetRscale such that the state is restricted correctly? If you want FAS, you should implement state restriction as nodal injection. State restriction should be as accurate as possible on low-frequency modes, but does not require damping aliased high-frequency modes. > If I just select -pc_mg_type full, I take more than one Krylov > iteration to converge. You have to compare to discretization error. For example, HPGMG-FE computes errors against an analytic solution, and delivers the design third-order accuracy using 3,1 cycles.
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