> How small is the Reynolds number? There is a difference between 0.1 and 100, > although both may be laminar. It's more like in the range of 100.
> I assume the XFEM basis is just resolving the jump across the interface. Does > this mean the size of the matrix is changing as the interface moves? In any > case, it looks like you can't amortize setup costs between time steps, so we > need a solution short of a direct solve. Yes, exactly the pressure approximation space is enriched with discontinuous functions. Due to the XFEM basis I then get an additional degree of freedom at the nodes of elements cut by the interface. > Unfortunately, multigrid methods for XFEM are a recent topic. Perhaps the > best results I have seen (at conferences) use some geometric information in > an otherwise algebraic framework. For this problem (unlike many fracture > problems), the influence of the extended basis functions may be local enough > that you can build a coarse level using the conventional part of the problem. > The first thing I would try is probably to see if a direct solve with the > conventional part makes an effective coarse level. If that works, I would see > if ML or Hypre can do a reasonable job with that part of the problem. > > I have no great confidence that this will work, it's highly dependent on how > local the influence of the extended basis functions is. Perhaps you have > enough experience with the method to hypothesize. Unfortunately, I don't have any practical experience with multigrid methods, but I'm afraid that the conventional part won't make a good coarse level. Depending on the test case (large density ratios between the phases) the standard approximation won't do well. Anyway, I'll take a closer look at the multigrid topic. > Note: for the conventional part of the problem, it is still incompressible > flow. It sounds like you are using equal-order elements (Q1-Q1 stabilized; > PSPG or Dohrmann&Bochev?). Q1-Q1 SUPG, PSPG stabilized. > For those elements, you will want to interlace the velocity and pressure > degrees of freedom, then use a smoother that respects the block size. PETSc's > ML and Hypre interfaces forward this information if you set the block size on > the matrix. When using ML, you'll probably have to make the smoothers > stronger. There are also some "energy minimization" options that may help. Best, Henning
