Yes, I'm solving a transient wave equation. Should the behavior between M + dt*K and M + dt^2*K be similar when solving the linear system? I can only understand them as positive helmholtz equation, the latter more dominated by mass matrix.
I agree with the trade-off between iteration cost vs iteration numbers. I plan to increase the dofs to 100 million or even 1 billion, so multigrid (geometric or algebraic) might be the final option. I guess the growing iterations are mainly contributed by the stiffness matrix? 在2024年10月4日星期五 UTC+8 22:57:10<Wolfgang Bangerth> 写道： > On 10/4/24 08:10, 'yy.wayne' via deal.II User Group wrote: > > > > I've considered multigrid preconditioners as the iterations are > relatively > > constant, but isn't it suits diffusion problems more, > > as mentioned here (A former discussion > > < > https://groups.google.com/g/dealii/c/det9e4HWGrk/m/q0oj-yQlBAAJ#:~:text=-%20lumping%20of%20mass%20matrices%0A-%20AMG/GMG%20for%20laplace-like%20operators%0A-%20ILU/Jacobi%20for%20mass%20matrix-like%20operators%0A-%20spectrally-equivalent%20matrices%20to%20approximate%20Schur%20complements%0A-%20and%20more > >)? > > Multigrid works well for diffusion problems. It doesn't work well for > advection problems. > > But if your matrix really is M + dt^2 K (i.e., with dt^2 instead of dt), > then > you must be solving a problem with second time derivatives -- say the wave > equation. That's not a diffusion problem either. > > > > I assume multigrid works well for M+dt^2K type problem, but it's > expensive for > > such 'easy' problem(when dt is small). The only drawback > > for jacobi preconditioner is the iteration number scales with problem > size ... > > Well, that's the trade-off you have. Your simple method takes too many > iterations. You need to invest in a more complicated method that perhaps > per > iteration is more expensive, but at least results in a smaller number of > iterations. The total time to solution is (time per iteration) * (number > of > iterations). Your current approach results in (cheap) * (growing number) > whereas multigrid might be (expensive) * (constant). At some point, the > latter > will be better than the former. > > Best > W. > > -- > ------------------------------------------------------------------------ > Wolfgang Bangerth email: [email protected] > www: http://www.math.colostate.edu/~bangerth/ > > > -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/dealii/38a24339-034e-42a0-a140-222653b33899n%40googlegroups.com.
