FEM, Space embedding doubt

In summary: This means that every function in H^{-1} can be expressed as a composition of a function in L^2 and a function in H^1_0, and every function in L^2 can be expressed as a composition of a function in H^1_0 and a function in H^1_0.
  • #1
SqueeSpleen
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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 [tex]\Phi[/tex] and [tex]\Xi[/tex]
Then it says:
"If we suppose that there exists Hilbert Spaces [tex]H_{\Phi}[/tex] and [tex]H_{\Xi}[/tex] such that the following dense and continuous embeddings hold in a compatible way
[tex]\Phi \subset H_{\Phi} \simeq H_{\Phi} ' \subset \Phi'[/tex]
[tex]\Xi \subset H_{\Xi} \simeq H_{\Xi} ' \subset \Xi'[/tex]
Here [tex]\subset[/tex] 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 [tex]\Phi[/tex] with [tex]\Phi'[/tex].

I have found an advice against identifying a space which is not [tex]L^{2}[/tex] with it's dual, because otherwise in constructions like this, if [tex]H_{\Phi}=L^{2}[/tex] and [tex]\Phi=H^{1}[/tex] we would end up identifying the four spaces [tex]\Phi \equiv H_{\Phi} \equiv H_{\Phi} ' \equiv \Phi'[/tex] 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.
 
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  • #3
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 .
 
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  • #4
When working with PDEs and variational methods it is not advised to identify [itex]H^1_0[/itex] with its dual [itex]H^{-1}[/itex], because you lose some subtleties. The space [itex]L^2[/itex] is sometimes called the pivot space and is sort of in the middle in terms of regularity, i.e. functions in [itex]L^2[/itex] are first derivatives of functions in [itex]H^1_0[/itex] and the elements of [itex]H^{-1}[/itex] are first derivatives of functions in [itex]L^2[/itex]. We can use this to characterize the dual space [itex]H^{-1}[/itex] in the following way:

For every [itex]\mathcal{l}\in H^{-1}[/itex] there exist [itex]v_0,v_1,\ldots,v_d[/itex] such that
[tex]\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}[/tex]
 
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What is FEM?

FEM stands for Finite Element Method. It is a numerical technique used to solve problems in engineering and physics by dividing a complex system into smaller, simpler elements and solving them individually.

How does FEM work?

FEM works by discretizing a continuous system into finite elements, which are connected at discrete points called nodes. The equations governing the behavior of the system are then solved at each node, and the solutions are combined to give an overall solution for the entire system.

What is space embedding doubt in FEM?

Space embedding doubt is a term used in FEM to describe uncertainties in the values assigned to the nodes of a finite element model. These uncertainties can arise due to errors in measurement, modeling assumptions, or numerical approximations.

How is space embedding doubt handled in FEM?

Space embedding doubt is typically accounted for in FEM by incorporating probabilistic methods, such as Monte Carlo simulations, to analyze the effects of uncertainties on the overall solution. This allows for more accurate and reliable results, as the uncertainties are taken into consideration.

What are the advantages of using FEM?

Some of the advantages of FEM include its ability to handle complex geometries, its versatility in solving a wide range of engineering and physics problems, and its efficiency in computing solutions for large systems. Additionally, FEM allows for the incorporation of material properties and boundary conditions, making it a powerful tool for analysis and design.

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