Choosing preconditioner for unsymmetric pressure-velocity coupled system

Question

I'm working with pressure-velocity coupled systems. It means that instead of solving 4 different linear systems in segregated approach (1 for pressure and 3 for Ux, Uy, Uz), we can solve only one coupled system. Assuming simple 3D rectangular mesh resulting matrix appears to be 7-diagonal with each element is a block 4x4 of arrow form \begin{pmatrix} a_{11} & 0 & 0 & a_{14} \\ 0 & a_{22} & 0 & a_{24} \\ 0 & 0 & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix}

I would like to make efficient GPU-based BiCGStab solver for it, but it seems, that I have problem with preconditioning. On CPU BiCGStab + ILU or AMG works just fine. But on GPU, I can't use ILU family, because they are not suitable. I've tried AINV preconditioner, but it's either too slow and big or not robust enough. Diagonal preconditioner isn't effective at all. Now I'm thinking of polynomial, though I expect it to have similar problems as AINV.

Are there any other options that could possibly work with my approach, or the whole idea of GPU BiCGStab is a dead end in the first place?

Answer 1

Both ILU and diagonal scaling are not efficient preconditioners for "real" problems, i.e., if you let the size of your discrete problem become large. In other words, if you insist on timing GPUs vs CPUs with these preconditioners, you are only considering academic problems that are not really of interest -- if the problem is small enough that the use of these preconditioners make sense, then the problem is so small that nobody actually cares about the run time in practice.

If you do care about run time (because run time is an obstacle to doing actual simulations, not just because you want to do an academic timing study without practical implications), then you need to use preconditioners that make sense for large problems -- and these are typically based on the physical structure of the underlying problem, not just on the matrix values. An excellent example is the one by Silvester and Wathen. We implement and discuss its use in step-22 of the deal.II tutorial (see https://www.dealii.org/developer/doxygen/deal.II/step_22.html); pay particular attention to the discussion in the "Possibilities for extensions" section at the end of the "Results" section.