Paper

[PetrKryslUCSD]

Line 21: uniquely -- is it used to indicate "solely"?
Line 70: patch-based -- not sure what it means. It does not appear to be "standard" terminology.
Line 82, 83: Shur -> Schur
Line 84: it's spectral -> its spectral
Line 88: The mass matrix is approximated by a Conjugate Gradient (CG) solver with... This sentence needs some work. I am not sure what you mean.
Line 91: it is setup -> it is set up
Line 128 - 131: h(div) -> H(div), h(curl) -> H(curl)

[JordiManyer]

I will address your semantic/grammar related comments.
Concerning the rest:

Line 70: patch-based -- not sure what it means. It does not appear to be "standard" terminology.

I agree, it is not the most standard terminology. We are basically addressing the same smoothers described here.
"Patch-based" refers to topologically-generated patches of cells/faces/edges/nodes where we build overlapping functional spaces to create smoothing operators.
Would their terminology of topological construction of space decompositions for multigrid relaxation methods be suitable? How can I make this clearer?

Line 88: The mass matrix is approximated by a Conjugate Gradient (CG) solver with... This sentence needs some work. I am not sure what you mean.

In this sentence, approximated is usually used in the literature as a way of saying solve using an inexact solver instead of a direct method. But I agree it's not the clearest. The idea is that our block-preconditioner has two diagonal blocks we somehow need to invert/solve at each iteration. The pressure block we solve using a CG+Jacobi, whereas the velocity block we solve using a GMG.

Maybe we could rephrase it as:

"""
The diagonal blocks have to be solved at each iteration. The mass matrix is solved by a Conjugate Gradient (CG) solver with Jacobi preconditioner. The eliminated velocity block is solved by a 2-level V-cycle GMG solver, where the coarsest level is inverted exactly in a single processor.
"""

[PetrKryslUCSD]

Patch based: maybe you could just add that explanation of what those patches are (essentially what you are saying above)?

The blocked system: maybe one could refer to solves with coefficient matrices (a) mass matrix and (b) solve with a velocity matrix, the first one with CG+Jacobi, the second with GMG?

[JordiManyer]

I've hopefully addressed everything. You can find the new compiled version of the pdf here. Let me know if there is anything that needs to be looked at.

[Leticia-maria]

Hello, I have some points about your paper:

• Review reference syntax

🟡 SKIP DOIs

- No DOI given, and none found for title: PETSc/TAO Users Manual
- No DOI given, and none found for title: MPI: A Message-Passing Interface Standard Version ...
- No DOI given, and none found for title: GridapPETSc
- No DOI given, and none found for title: GridapP4est
- No DOI given, and none found for title: PartitionedArrays
- No DOI given, and none found for title: The Trilinos Project Website
- No DOI given, and none found for title: \sl hypre: High Performance Preconditioners

(I think it is okay, just double check to make sure you have not missed any DOIs)

• More details on how the benchmark was performed: details about which benchmark method was used. It was not clear if the algorithm is compute-bound or memory-bound, maybe expand a little bit more on that.
• Overall, good propositions, I think it fits into the journal.

[JordiManyer]

Those skipped dois are all repos/codes. I tried to find dois for them, but to my best knowledge there is none.

[Leticia-maria]

Those skipped dois are all repos/codes. I tried to find dois for them, but to my best knowledge there is none.

That sounds good (I actually assumed so, I just wanted to make sure you haven't missed anything)