Hi Wolfgang,
thanks for your quick response and yes, I tried to indeed ask for the most
elegant and deal.ii-native-way to rewrite
(f*g, \phi),
if f and g are finite element approximations of the functions I am trying
to find and \phi (for all \phi) are the test functions.
To clarify: In Tutorial-23 (u, \phi) was rewritten into M U, with M_{i,j} =
(\phi_i , \phi_j) being the Mass Matrix and U the parameter vector of u.
Now I would love to similarly pull the parameter vector of f and g out from
(f*g, \phi) and still somehow have a linear system, if possible?
Maybe even something similar to MatrixCreator::create_mass_matrix ?
Writing it out (sloppy to not overextend this question) :
(f g , \phi) = (\phi F \phi G, \phi), where F,G are the parameter vectors,
i.e. f = \phi F, g = \phi G.
=M G, with M_{i,j} = (\phi F \phi_i, \phi_j) G, (like a mass matrix)
Can I now somehow put this into a 3-D tensor / block-matrix such that
M_{i,j}= (\phi F \phi_i, \phi_j)
= (\phi_i \phi F, \phi_j)
= N_{i,j} F, with N_{i,j,k} = (\phi_i \phi_k, \phi_j)
? Or do I simply have to use a Newton solver for (f g, \phi) for f and g?
Also, evaluating this term for a past time-step (f, g resp F,G are known)
seems more complex?
I hope this time I expressed the question better but also didn't overwhelm
you with hard-to-read notation.
Thanks a lot for your help and best,
Jost
Wolfgang Bangerth schrieb am Mittwoch, 21. Juni 2023 um 23:51:06 UTC+2:
> On 6/21/23 08:07, 'Jost Arndt' via deal.II User Group wrote:
> > Linearizing the weak formulation of a product of two functions, i.e. in
> the
> > original equation the term f*g appears, both f,g are real-valued
> functions.
> > The weak formulation looks therefore somehow like (f*g, \phi).
> > This term appears implicitly and explicitly (i.e. f,g are sometimes
> known,
> > sometimes both unknown).
> > I was wondering if there is a tidy version to linearize this in kind of
> a 3D
> > Tensor?
> > As an example in Tutorial-23 (f,\phi) gets linearized into A * b, A
> being a
> > Matrix of the products of base functions and b only the parameter vector
> of f.
>
> Having read over this a number of times, I must admit that I still don't
> quite
> understand what you want to do. Can you be more concrete what you want to
> do,
> and how? Are you asking what happens when both f and g are finite element
> functions, and whether one can write
> (f*g, \phi_i)
> in a more elegant way?
>
> Best
> W.
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bang...@colostate.edu
> www: http://www.math.colostate.edu/~bangerth/
>
>
>
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