Thank you for the clear answer, I understand my mistake now, it is much
simpler like that.
On a side note (irrelevant now), when I was talking about orthonormal
basis, I was refering to something like \int w_i(x) w_j(x)=
kronecker_{i,j}, which should be doable and in which basis T_n^3 would look
like *T_{n}^3 = sum([T_{n}]_i^3 * w_i).*
I was also considering to mimic tutorial 15 with the Newton method if this
strategy (linearising T^4) was not performing well, but with your reply now
I will stick to my method.
By the way, Isn't there a mistake in tutorial 15 when it says F'(u_n, δu_n)
= -F(u_n) (just after "...use a damping parameter αn to get better global
convergence behavior: "). Shouldn't we read F'(u_n, δu_n) * δu_n = -F(u_n)
otherwise it is not homogeneous ?
Best,
Sylvain
El viernes, 2 de julio de 2021 a la(s) 17:49:20 UTC+2, Wolfgang Bangerth
escribió:
>
> > So I applied the Crank Nicolson discretisation first before multiplying
> by
> > test functions and integrating. But then I am however confused how to
> handle a
> > term like *(T_{n}^3 * T_{n+1} , w_i)_omega*
> > (where *(A,B)_omega* is the usual notation for the integral over the
> spatial
> > region omega of the product A*B and *w_i* is my test function).
> >
> > If I write *T_{n}* and *T_{n+1}* as a finite linear combinaison of w_i
> such
> > that (*T_{n} = sum_i([T_{n}]_i * w_i)*) then *T_{n}^3* is monstruous
> unless I
>
> You're thinking of T_n^3 as a polynomial of particular high degree, but we
> don't actually care about that. We just need to evaluate it at quadrature
> points, and to do that you evaluate
> T_n(x_q) = \sum [T_n]_j \varphi_j(x_q)
> and then take whatever value that gives you to the third power. You never
> need
> T_n^3 as a *function*, you just need to be able to evaluate it at
> quadrature
> points, and so questions such as whether functions are orthogonal don't
> actually matter.
>
>
> > 1- Is this even mathematically sound ? I should be able to choose my
> basis
> > orthogonal as far as I can think. Can I then do that with deal.II ?
>
> Think about it this way: You want to compute an integral
> \int c(x) w_i(x) w_j(x)
> where c(x) is a coefficient that in your case involves T_n^3. You won't in
> general be able to find orthogonal basis functions w_k for these kind of
> weights c(x). You just have to approximate the integral by quadrature.
>
>
> > 2 - Is this the approach I should be following for handling such term or
> > should I rethink the time discretisation scheme ?
>
> No. Just think of the old solution as a coefficient. You probably want to
> take
> a look at step-15, which shares many of the characteristics of what you
> want
> to do.
>
> Best
> W.
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: [email protected]
> www: http://www.math.colostate.edu/~bangerth/
>
>
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