Unexpected performance of different preconditioners

[Wayne901]

Unexpected performance of different preconditioners

[jandrej]

When choosing a preconditioner you have to think about the matrix structure. In an implicit time dependent formulation your most dominant eigenvalues are determined by a mass-type matrix with a scaled diagonal proportional to the time step chosen. As a result you see that the inverse diagonal (Jacobi) is an efficient preconditioner.

[najlkin]

Yes, choice of the preconditioner very much depends on the problem. AMS or similar preconditioners are mainly useful for diffusive cases or hyperbolic with long steps (which get diffusive as well due to numerical effects). For hyperbolic problems with short steps, your solution is just the same profile, but slightly shifted, so there is nothing much to precondition other than a plain integral projection and a cheap Jacobi makes most sense intuitively 😉

[Wayne901]

Thank you jandrej and najlkin, I've checked an article Wathen and it seems Jacobi preconditioner is the appropriate one for mass matrix.
I mostly dealt with Poisson and Helmholtz (definite and indefinite) equations before, where Jacobi behaves poorly. Is there a more efficient preconditioner than Jacobi for a mass-dominated matrix? For example, element-wise Schwarz domain decomposition? I'm interested in the case when applying high-order basis functions.

[najlkin]

Even with Jacobi there is a lot to play with, there are multiple parameters of DSmoother or you may use different HypreSmoother options for Jacobi (or Gauss-Seidel). Feel free to experiment! 😉

[Wayne901]

Thank you @najlkin! I'll compare the performance of different Jacobi settings then.