On 4/18/21 12:24 PM, John Smith wrote:
Thank you for your reply. It is a bit difficult to read formulas in text. So I
have put a few questions I have in a pdf file. Formulas are better there. It
is attached to this message. May I ask you to have a look at it?
John:
If I understand your first question right, then you are given a vector field J
and you are looking for a vector field T so that
curl T = J
I don't really have anything to offer to this kind of question. There are many
vector fields T that satisfy this because of the large null space of curl. You
have a number of condition you would like to "approximate" but there are many
ways to "approximate" something. In essence, you have two goals: To satisfy
the equation above and to approximate something. You have to come up with a
way to weigh these two goals. For example, you could look to minimize
F(T) = ||curl T - J||^2 + alpha ||T-T_desired||^2
where T_desired is the right hand side of (0.0.5) and you have to determine
what alpha should be.
As for your other questions: (0.0.9) is indeed called "interpolation"
The issue with the Nedelec element (as opposed to a Q(k)^dim) field is that
for the Nedelec element,
\vec phi(x_j)
is not independent of
\vec phi(x_k)
and so you can't choose the matrix in (0.0.8) as the identity matrix. You
realize this in (0.0.13). The point I was trying to make is that (0.0.13)
cannot be exactly satisfied if T(x_j) is not a vector parallel to \hat e_j,
which in general it will not be. You have to come up with a different notion
of what it means to solve (0.0.13) because you cannot expect the left and
right hand side to be equal.
Best
W>
--
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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