A Why Are Hilbert Space Embeddings Used in FEM Eigenvalue Approximation?

[SqueeSpleen]

Hi. I'm studying Finite Elements Method, I was readding a paper written by Danielle Boffi and in a part dedicated to the approximation of eigenvalues in mixed form, it's about approximating eigenvalues in the Hilbert Spaces \Phi and \Xi
Then it says:
"If we suppose that there exists Hilbert Spaces H_{\Phi} and H_{\Xi} such that the following dense and continuous embeddings hold in a compatible way
\Phi \subset H_{\Phi} \simeq H_{\Phi} ' \subset \Phi'
\Xi \subset H_{\Xi} \simeq H_{\Xi} ' \subset \Xi'
Here \subset means dense and continously embedded.

I'm not sure why it does that, I have read more than one time that they work with a space identified with it's dual space, as they're Hilbert Spaces I know you can do it using Riesz Representation theorem, but I don't exacly see why they're doing this.

I didn't know why they don't simply identify \Phi with \Phi'.

I have found an advice against identifying a space which is not L^{2} with it's dual, because otherwise in constructions like this, if H_{\Phi}=L^{2} and \Phi=H^{1} we would end up identifying the four spaces \Phi \equiv H_{\Phi} \equiv H_{\Phi} ' \equiv \Phi' and it would mean that you're identifying a function with it's laplacian which is (and here I'm quoting a book) ""the beggining of the end""

I'm new in this of variational formulations and I don't have a strong background on partial differential equations, I have more background on functional and real analysis, I'm studying this subject to make my Licenciatura's thesis in this but I'm really lost sometimes, this is as clear as I could make the question so feel free to ask me to clarify something if I wasn't clear enough.

PD: I don't know how to write in latex without making a new line.

 

[akeleti8]

Hi there.
Φ≡HΦ≡H′Φ≡Φ'
does not make sense. i do not think i am negligent with my work in saying a function can't be equal to it's derivative. unless it's e^x. albeit Danielle boffi might have a different finding.

This comes from a guy who would enjoys going to a Lollapalooza wearing a rippling red scirt. This site helps me with your problem http://math.stackexchange.com/questions/644879/function-is-equal-to-its-own-derivative .

 

[Xiuh]

When working with PDEs and variational methods it is not advised to identify H^1_0 with its dual H^{-1}, because you lose some subtleties. The space L^2 is sometimes called the pivot space and is sort of in the middle in terms of regularity, i.e. functions in L^2 are first derivatives of functions in H^1_0 and the elements of H^{-1} are first derivatives of functions in L^2. We can use this to characterize the dual space H^{-1} in the following way:

For every \mathcal{l}\in H^{-1} there exist v_0,v_1,\ldots,v_d such that
\langle\mathcal{l},u\rangle_{H^{-1},H^1_0} = (v_o,u)_{L^2} + \sum_{i=1}^d (v_i, u_{x_i})_{L^2}