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Just curious if someone knew this answer. I know for conformal finite element method that the basis functions and the hat functions are from the same function space.

I would conjecture that abstract would be that the test functions and basis functions do not have to exist in the same functional space, but I would like conformation on that.

Thanks

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The possibility of using different spaces for trial functions (or basis functions, where the solution is approximated) and test functions is certainly one generalization of finite element methods that has been investigated. Cf. Projection Methods With Different Trial and Test Spaces by M.S. Mock (Math. Comp., 1976).

However I do not think most authors would describe this as "abstract finite element method". Rather I suspect that phrase refers to developing the theory of finite element approximations based on the properties of a weak formulation/coercive bilinear form in a Hilbert space, specifically the Lax-Milgram Theorem.

A presentation of the finite element method starting from this "abstract" formulation essentially decouples the existence and uniqueness of approximate solutions from the approximation properties (and any geometric underpinnings) of the space chosen for the trial functions. Such a point of view is articulated by Larson and Bengzon (2012) in Chapter 7 of The Finite Element Method: Theory, Implementation, and Applications, Abstract Finite Element Analysis.

Once approximation theory supplies the accuracy of the trial functions to approach a (smooth) solution, the abstract analysis is immediately applicable to provide the error estimates for a chosen finite element method.


Extending points about terminology from earlier Comments: When different spaces for trial and test functions are used (as broached in the Question), then the Lax-Milgram Theorem is replaced by a functional analysis result by Babuška (1971) which works out what, instead of simple coercivity of the bilinear form (weak formulation) is sufficient to give quasi-optimal approximations.

A non-conformal scheme does concern the approximation spaces, but it signifies not that the trial and test spaces are different, but rather that the trial functions are not conforming (entirely within the Sobolev space of the continuous weak formulation). For example, one might approximate a solution with piecewise constant functions "rougher" than the $\mathscr{H}^1$ functions natural to the weak form. This is however generally connected to using a different test function space with additional "smoothness" in order to give (by further integration by parts) a modified bilinear form.

A search term that might be helpful here is collocation method, emphasizing point-evaluations of solutions and drawing close to finite difference methods in that respect.

For a broader class of methods, not necessarily tied to the geometric implications of "finite element" methods, but sharing the underlying approach of obtaining a discrete problem from a continuous one via weak formulations (e.g. of an elliptic PDE) see Galerkin method.

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  • $\begingroup$ Wow this was perfect. Thank you so much. $\endgroup$ – Vogtster Mar 19 '16 at 20:23
  • $\begingroup$ Just one question though. So in FEM once you pick your trial space, you assume those are the basis functions? Can you choose your trial space to be for example, hat functions, while your basis space are some other function that are not hat functions? $\endgroup$ – Vogtster Mar 19 '16 at 20:25
  • $\begingroup$ It is typical in the FEM to use equal trial space and test space, but as noted this can be generalized. In the abstract analysis one replaces the usual Lax-Milgram Thm. already noted with a 1971 generalization by Babuška. $\endgroup$ – hardmath Mar 19 '16 at 20:31
  • $\begingroup$ Thank you for the great information. $\endgroup$ – Vogtster Mar 19 '16 at 21:22

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