Methods of Weighted Residuals (MWR) [1] usually include Galerkin, collocation, method of moments, least-squares and subdomain methods.

Spectral methods [2] usually include Galerkin, tau and pseudospectral methods. Pseudospectral is another name for collocation and the tau method is the same as the method of moments [3].

I can't see a difference. If there is no difference between them, why the different names? I'm writing a monograph about these methods and want to make sure I'm not missing something. If you'd like to see my unfinished monograph, it is at http://tildentechnologies.com/Numerics


[1] Finlayson, B.A., The Method of Weighted Residuals and Variational Principles, Academic Press, New York, NY (1972), SIAM Classics in Applied Mathematics (2014)

[2] Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A., Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin (1988)

[3] Young, L.C., “Orthogonal Collocation Revisited,” Comp. Methods in Appl. Mech. and Engr. 345 (1) 1033-1076 (Mar. 2019) (doi: 10.1016/j.cma.2018.10.019).

I tried this over at the Mathematics SE but got no response.


1 Answer 1


To my knowledge these are the same things. However, this type of thing is common. For example, the proper orthogonal decomposition also has field-specific names. Others call it principal component analysis, the Karhunen--Loeve expansion, or empirical orthogonal functions. It is also no different than an autoencoder with linear activation function. I'm sure this has roots in when and how the method was introduced in the literature of that field.

  • $\begingroup$ Yes, it is common to have "field specific" names, but in this case the names are beyond a specific field. MWR seems like a perfectly good descriptive name, which dates back to observations of Courant in 1924. The name "Spectral Method" seems to have originated with atmospheric modeling in the 1960s. It started as a field specific name and somehow has all but supplanted the name MWR. I am sure I'm not the only one to be confused. It took me quite a while to figure out that the Tau method (in the spectral literature) is just a moments method. $\endgroup$
    – L. Young
    Dec 22, 2019 at 19:16

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