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FEM Is Not a Local Method (and It Isn't Global Either)

Ajit R. Jadhav's picture

In the literature, FEM has sometimes been characterized as a local approach, but IMO this needs to be corrected.

The piecewise continuous trial-functions of FEM can be looked at from two different viewpoints:

(i) If FEM is seen as an expansion method making use of basis functions, then naturally the comparison is with the Fourier-theoretic approaches (and all the derived, consequent or similar ones). The basis functions for the latter are global in the sense they have supports all over the domain. This, indeed, is unlike the limited (piecewise) support of the FEM trial-functions.

(ii) Yet, on the other hand, if FEM is seen an integral approach, i.e. an approach which is based either on functionals or on domain integrals of the residuals, then notice that both such expressions enjoy support over a finite region—not infinitesimal. Consequently, FEM cannot really be said to be a local approach. Compare it, for instance, with FDM. Or, with the ideas of the differential calculus per say.

I therefore object to characterizations of FEM as a local approach. Also for its characterization as a global approach. I believe that FEM is “midway,” really speaking.

As a relevant aside, also consider here that, in structural FEM for instance, the potential energy (and its stationarity) *is* globally defined, and yet, the displacements are defined only in a piecewise manner—i.e. over only sub-intervals of the overall domain.

I, therefore, suggest that a term like “sub-global” (or "supra-local") might be used to characterize FEM.  And also, other methods like FEM.

In between the two candidates considered here, the first (”sub-global”) appears a more complete description as compared to the second (”supra-local”.) The first also sounds more honest and less pseudo-intellectual. (A third candidate, viz. “ultra-local,” seems to imply exactly the opposite of the intended meaning: it seems to doubly emphasize the local nature. Hence, it is unsuitable.)

I, thus, vote for the “sub-global” term.

Other methods where this term becomes relevant and applicable include, for example, BEM. Also MD, wherein, despite the use of point-particles, the potential itself is not a point-phenomenon (just the way it also does not have infinite support due to the cut-off). As such, MD, too, too should be characterized as a “sub-global” method.

BTW, this issue is neither obscurely academic nor pedantic. … Here, ask yourself why quantum entanglement is at all considered to be so “counter-intuitive” or “dramatic” (or gets so hotly debated/discussed).

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Well thought-out corrections to my position would be welcome, as also any really relevant complementary observation(s).

Thanks in advance.

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