# 1D Finite element method: Function contineously differentiable?

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?

• 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. – player100 Jan 6 at 16:15
• 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... – Jan SE Jan 6 at 17:01
• Are you coming to this from a theoretical or a computational mindset? – player100 Jan 6 at 18:26
• In the end I of course want to compute something, but at the moment I try to understand the mathematics. – Jan SE Jan 6 at 23:41
• 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 – player100 Jan 7 at 1:44