Thank you for the discussion.

Are we agreed then that the derivatives of the natural coordinates are required for the described approach? If so, is this something PETSc can currently do within the point-wise residual functions?

Matt - Thank you for the command line option for the 2nd derivatives. Those will be needed to implement the discussed approach. Specifically in the stabilization and shock capture parameters. (Ref.: B. Kirk's Thesis). What is a good reference for the usual SUPG method you are referencing? I've been looking through my textbooks but haven't found a good reference.

Jed - Thank you for the link. I will review the information on it. 

Sorry about the attachment. I will upload it to this thread later (I'm at work right now and I can't do it from here).
From: Jed Brown <jed@jedbrown.org>
Sent: Wednesday, October 11, 2023 1:38 PM
To: Matthew Knepley <knepley@gmail.com>
Cc: Brandon Denton <bldenton@buffalo.edu>; petsc-users <petsc-users@mcs.anl.gov>
Subject: Re: [petsc-users] FEM Implementation of NS with SUPG Stabilization
 
Matthew Knepley <knepley@gmail.com> writes:

> On Wed, Oct 11, 2023 at 1:03 PM Jed Brown <jed@jedbrown.org> wrote:
>
>> I don't see an attachment, but his thesis used conservative variables and
>> defined an effective length scale in a way that seemed to assume constant
>> shape function gradients. I'm not aware of systematic literature comparing
>> the covariant and contravariant length measures on anisotropic meshes, but
>> I believe most people working in the Shakib/Hughes approach use the
>> covariant measure. Our docs have a brief discussion of this choice.
>>
>> https://nam12.safelinks.protection.outlook.com/?url=https%3A%2F%2Flibceed.org%2Fen%2Flatest%2Fexamples%2Ffluids%2F%23equation-eq-peclet&data=05%7C01%7Cbldenton%40buffalo.edu%7Cd9372f934b26455371a708dbca80dc8e%7C96464a8af8ed40b199e25f6b50a20250%7C0%7C0%7C638326427028053956%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=skMsKDmpBxiaXtBSqhsyckvVpTOkGqDsNJIYo22Ywps%3D&reserved=0
>>
>> Matt, I don't understand how the second derivative comes into play as a
>> length measure on anistropic meshes -- the second derivatives can be
>> uniformly zero and yet you still need a length measure.
>>
>
> I was talking about the usual SUPG where we just penalize the true residual.

I think you're focused on computing the strong diffusive flux (which can be done using second derivatives or by a projection; the latter produces somewhat better results). But you still need a length scale and that's most naturally computed using the derivative of reference coordinates with respect to physical (or equivalently, the associated metric tensor).