Attached is an apple-to-apple comparison among Spectral Element Method (SEM), Finite Difference Method (FDM) and Finite Element Method (FEM). The accuracy achieved by FEM or FDM with 100 nodes can be achieved by SEM with 10 nodes.
In my opinion, researchers don't consider SEM just because of complexity in coding when compared with FEM or FDM. But when it comes to accuracy, no way.
Note: here I mean the low order FEM
Nothing of strange in this, what about your specific question? You should consider the field of application, SEM can be computational expensive and somehow not easily stable for non-linear problems.
What are the orders of the discretization schemes used in your formulations based on these three methods?
Most likely, it will be something like SEM >> FEM ≈ FDM. In that case, it would be interesting to look at the computational time not just the error as the number of nodes increases.
I am no expert, but I think the problem with SEM is computational time (as is the case with many very accurate higher order methods) rather than just coding complexity.
SEM is a special case of FEM with higher-order piecewise polynomials as the basis functions. Fundamentally there is no difference between FEM and SEM in terms of weak form. Higher-order FE schemes are used extensively.
What is the difference between the SEM and the FEM considered for your comparison?
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