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.
Best
W.
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Wolfgang Bangerth email: [email protected]
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