How to determine the saddle-point nature of PDEs like those in Step-20, 21?

[Krishnakumar Gopalakrishnan]

In Step-21 tutorial, we have a statement that starts with the following (emphasis is mine):

"Given the saddle point structure of the first two equations and their similarity to the mixed Laplace formulation we have introduced in step-20"

It would be helpful if someone could explain the saddle-point structure of the problem (and/or point to some easily readable online resources).  Before coming to this Step-21 tutorial, I have gone through Step-20 tutorial wherein again the saddle-point nature of the problem is not explained.  In fact, it is entirely glossed over:

"It is a well-known fact stated in almost every book on finite element theory that if one chooses discrete finite element spaces for the approximation of u,p inappropriately, then the resulting discrete saddle-point problem is instable and the discrete solution will not converge to the exact solution."

I acknowledge that the deal.II tutorials is not intended to educate the user on such theory.  However, it is generally helpful to have one or two sentences explaining the saddle-point nature of the problem and or point to accessible online resource(s).

I have also finished watching Lecture 33.25 a couple of times which discusses saddle-point problems, but even this feels a bit too high-level to me i.e. all the important details are skipped. For example, the PDE is posed as an energy minimization problem without explaining why (the specific words being "where exactly this step from here to here comes from is not terribly important, but if you will believe me that I can rewrite this equation in this form, now....."), and the rest of the lecture (the inf/sup LBB condition) is also a bit too mathematical for me.

I apologise upfront and would like to clarify that I mean no disrespect to either the tutorial authors or Prof Bangerth. Together you have created a mountain of phenomenal quality work which I am thankful for.  However, I'd really appreciate if there was some "simple test" or practical advice to determine whether our own PDEs and DAEs belong to the saddle-point category or not, i.e. how to detect the presence of saddle-point nature of the PDEs, just simulate with Qp elements and look for a checkerboard pattern in the results? 

Regards,

Krishna

[Wolfgang Bangerth]

On 3/11/20 7:57 AM, Krishnakumar Gopalakrishnan wrote:
> In Step-21 tutorial, we have a statement that starts with the following
> (emphasis is mine):
>

> /_"Given the saddle point structure_ of the first two equations and

> their similarity to the mixed Laplace formulation we have introduced in
> step-20

> <https://nam01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fdealii.org%2Fdeveloper%2Fdoxygen%2Fdeal.II%2Fstep_20.html&data=02%7C01%7CWolfgang.Bangerth%40colostate.edu%7Ccd1596e29c4a427d467808d7c5c41aa8%7Cafb58802ff7a4bb1ab21367ff2ecfc8b%7C0%7C1%7C637195318357612315&sdata=1Y%2Fp8SQc%2F2YETGls%2FU5J8XPtkz5JLEZW%2FYGdLEnM9rc%3D&reserved=0>"/
> /
> /
> //

> It would be helpful if someone could explain the saddle-point structure
> of the problem (and/or point to some easily readable online resources).

> [...]

> However, I'd really appreciate if there was some "simple test" or
> practical advice to determine whether our own PDEs and DAEs belong to
> the saddle-point category or not, i.e. how to detect the presence of
> saddle-point nature of the PDEs, just simulate with Qp elements and look
> for a checkerboard pattern in the results?

I thought we had addressed this a while ago already:
https://github.com/dealii/dealii/pull/9470/files

Is that not enough? Or is the issue that the text added there just says
"indefinite", whereas you are looking for the term "saddle point problem"?

I think a good approximation is that
saddle point problem = indefinite + symmetric

We could presumably add this there.

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Krishnakumar Gopalakrishnan]

Dear Prof Bangerth.

Thank you for your reply.

I had seen that github issue/pull-request, but didn't realise that "saddle-point problems = (indefinite + symmetric)". These three words do not appear in close-proximity to each other anywhere in the slides, tutorials or video lectures. While I acknowledge the importance of mathematical terminology, perhaps it would be quite useful to de-jargonise this for the newcomers?

Now, my question becomes "how to test for definiteness and symmetric nature of my DAEs?", i.e. simply by looking at the weak form of the equation which consists of only symbolic notations, can one somehow infer the definiteness/symmetry of the matrix?

Regards,

Krishna

[Wolfgang Bangerth]

> I had seen that github issue/pull-request, but didn't realise that
> "saddle-point problems = (indefinite + symmetric)". These three words do
> not appear in close-proximity to each other anywhere in the slides,
> tutorials or video lectures. While I acknowledge the importance of
> mathematical terminology, perhaps it would be quite useful to
> de-jargonise this for the newcomers?

Or just link to wikipedia:

https://en.wikipedia.org/wiki/Ladyzhenskaya%E2%80%93Babu%C5%A1ka%E2%80%93Brezzi_condition
That's what I'm doing here:
https://github.com/dealii/dealii/pull/9651/files

> Now, my question becomes "how to test for definiteness and symmetric
> nature of my DAEs?", i.e. simply by looking at the weak form of the
> equation which consists of only symbolic notations, can one somehow
> infer the definiteness/symmetry of the matrix?

Yes. If your bilinear form satisfies
a(u,v)=a(v,u)
then your matrix is symmetric. If in addition
a(u,u) >= c \|u\|
with some norm of u on the right hand side, then the matrix is also
positive definite. Both is the case for Laplace equation, for example.

If the second condition only applies to the top left block, but the rest
of the problem has the structure shown in the wikipedia article above,
then you have a saddle point problem.