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I often hear the lecturer making comparisons between continuous and discrete setting in FEM (topics related to Poisson equation Mixed FEM). My vague understanding is that it is impossible to find a solution in the continuous setting, hence we introduce the discrete setting followed by interpolation and other stuff. My current idea about the whole thing is:

  1. We develop a variational formulation (test function and trial functions are elements of $V$, an infinite dimensional space or an arbitrary space which we claim to have the solution)

  2. Say that we will find a solution in $V_h ⊂ V$. ($V_h$ a finite dimensional subspace of $H_1$, $H(\mathrm{div})$, …)

  3. Define/Find basis function to span the space(is it the interpolation space?)

  4. Formulation of mass matrix and other processes to find the solution.

I couldn't see any references online to these specific keywords "discrete setting" and " continuous setting". If one cannot find a solution in the continuous setting, why studying it?

Can someone give me a clear understanding on these concepts?

nicoguaro
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Jobin Raju
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  • We can find analytic solutions for the strong version of the equations for simples cases. – nicoguaro Jun 17 '19 at 19:52
  • @nicoguaro can you tell me what exactly is " continuous setting" and "discrete setting" ? – Jobin Raju Jun 18 '19 at 20:18
  • I can't, I haven't been in your classes with you. But I would say that the "continuous setting" refers to looking for solutions in the infinite dimensional space $V$ and the "discrete setting" refers to looking for "solutions" in the finite dimensional subspace $V_h \subset V$. – nicoguaro Jun 18 '19 at 20:29
  • Thank you. That's all I wanted ! – Jobin Raju Jun 20 '19 at 22:18
  • In that case, let me rewrite the comment as an answer. – nicoguaro Jun 20 '19 at 22:21

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I haven't been in your classes with you, so I can't tell for sure what your lecturer means. Probably, the best option is to ask her directly.

That being said, I would say that the "continuous setting" refers to looking for solutions in the infinite dimensional space $V$ and the "discrete setting" refers to looking for "solutions" in the finite dimensional subspace $V_h \subset V$. In the Finite Element Method, the finite dimensional space is generated from the interpolation functions defined over your elements.

nicoguaro
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