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I can't see a difference. 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.

References:

[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.org/10.1016/j.cma.2018.10.019,

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