I m currently reading the book "The finite element method in electromagnetics" written by Jian-Ming Jin.

On page 50 and 51 which I attach as a screenshot below they show for an example functional, that minimizing the functional will lead to the solution of the pde in 3.12.

My question is: In the step 3.8 to 3.9 one did an integration by parts. For this step $\alpha \frac{d\phi}{dx}$ has to be differentiable? If I assume that $\alpha$ to be differentiable, then $\phi$ has to be differentiable twice.

When I minimize the functinal in practice I discretize space and assume in all discretized space intervals for example linear functions (linear element functions). With these linear functions I stick together my $\phi$. But this $\phi$ is then of course only contineous, but not twice differentiable!

Does anyone know a solution to this? Maybe it is okay if the discretization is fine enough?

Many thanks in advance. 1 2

  • $\begingroup$ Yes, the idea is that in the limit of discretization the solution of the weak (variational) form approaches the solution to the strong (PDE) form. There are many assumptions of course. $\endgroup$ – player100 Jan 6 at 16:15
  • $\begingroup$ Do you have a recommendation where I can read about this in a little bit more detail? It seems that this book goes over these details without even mentioning them... $\endgroup$ – Jan SE Jan 6 at 17:01
  • $\begingroup$ Are you coming to this from a theoretical or a computational mindset? $\endgroup$ – player100 Jan 6 at 18:26
  • $\begingroup$ In the end I of course want to compute something, but at the moment I try to understand the mathematics. $\endgroup$ – Jan SE Jan 6 at 23:41
  • $\begingroup$ There are many online resources. Try web.stanford.edu/class/energy281/FiniteElementMethod.pdf There is the much more formal, such as folk.uio.no/kent-and/sommerskole/material/book_20april.pdf $\endgroup$ – player100 Jan 7 at 1:44

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