Question about the advice about regularity of the stokes' equations in Step-22

[Krishnakumar Gopalakrishnan]

Step-22 has a strong-sounding stub-like statement: "In practice, one wants to impose as little regularity on the pressure variable as possible".

The above one-liner (fairly enough) assumes domain knowledge. As a new entrant to deal.ii (haven't studied fluid mechanics before), may I ask the following?

1. Why do we impose as little regularity on the pressure as possible?
2. When I attempt to solve my own PDE (outside traditional areas such as fluid or structural mechanics), what properties should I know about the field variable to correctly apply the procedure and learnings from this tutorial, i.e. what exactly is the regularity condition mentioned here?

Regards,

Krishna

[Wolfgang Bangerth]

On 3/10/20 12:59 PM, Krishnakumar Gopalakrishnan wrote:
> Step-22 has a strong-sounding stub-like statement:/"In practice, one

> wants to impose as little regularity on the pressure variable as possible".

> /
> /
> /

> The above one-liner (fairly enough) assumes domain knowledge. As a new
> entrant to deal.ii (haven't studied fluid mechanics before), may I ask
> the following?
>

> 1. Why do we impose as little regularity on the pressure as possible?

We do the same on the velocity. That's why we integrate by parts the form
\int v (-Delta u)
to
\int \nabla v . \nabla u


> 2. When I attempt to solve my own PDE (outside traditional areas such

> as fluid or structural mechanics), what properties should I know
> about the field variable to correctly apply the procedure and
> learnings from this tutorial, i.e. what exactly is the regularity
> condition mentioned here?

You'll have to take a second course on PDEs, and most importantly learn
a bit about the theory of weak solutions. Here's a patch that explains
this in about as much detail as I would like to go into:
https://github.com/dealii/dealii/pull/9652/files

Best
W.


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

[Krishnakumar Gopalakrishnan]

"We do the same on the velocity. That's why we integrate by parts the form"

Thank you so much for clarifying. That explains it, and I now understand this fully.  The original sentence was a bit misleading. "In practice, one wants to impose as little regularity on the pressure variable as possible". For someone who hasn't studied fluid dynamics, it can potentially plant doubts in one's head to mean something like:

"Perhaps there exists some special mathematical considerations informed by the physics of the pressure variable that requires it be as less regular as possible, and therefore maybe some extra work and careful thought is needed specifically for this problem, which is somehow different from all others studied in tutorials so far". 

I shall now de-emphase this in my mind and take this sentence to mean as "In practice, one wants to impose as little regularity on the pressure variable as possible (just like every field variable solved by the Galerkin method in every tutorial thus far)".

Thanks a lot once again for the clarification. Apologies for asking such basic questions.

Regards,

Krishna

[Wolfgang Bangerth]

On 3/12/20 6:49 AM, Krishnakumar Gopalakrishnan wrote:
> /"We do the same on the velocity. That's why we integrate by parts the form"/

>
> Thank you so much for clarifying. That explains it, and I now understand this
> fully.  The original sentence was a bit misleading. "In practice, one wants to
> impose as little regularity on the pressure variable as possible". For someone
> who hasn't studied fluid dynamics, it can potentially plant doubts in one's
> head to mean something like:
>

> /"Perhaps there exists some special mathematical considerations informed by

> the physics of the pressure variable that requires it be as less regular as
> possible, and therefore maybe some extra work and careful thought is needed
> specifically for this problem, which is somehow different from all others

> studied in tutorials so far". /

Well yes. The mathematics rarely reflects anything that has no basis in
physical reality. Mathematics really just makes things precise, and in the
case of the Stokes equation, it is that the pressure is not as smooth a
quantity as the velocity is: It can go to infinity, for example, whereas the
velocity can not. Mathematically, that is reflected by the fact that the
pressure has only one derivative on it, whereas the velocity has two in the
strong formulation. In the weak formulation, we get rid of one derivative from
both, ending up with no derivatives at all on the pressure.


> I shall now de-emphase this in my mind and take this sentence to mean as/"In

> practice, one wants to impose as little regularity on the pressure variable as
> possible (just like every field variable solved by the Galerkin method in

> every tutorial thus far)"./

Correct. That is exactly the lesson of going from the strong to the weak
formulation of PDEs.