Extended analysis of a pseudo-spectral approach to the vortex patch problem Mattias Bertolino www.diva-portal.org/smash/get/diva2:1221196/FULLTEXT01.pdf
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Of course and as usual, this is not a real question, it is here, only a reference added to my project, that RG doesnt permit to add otherwise (at the time), eg here on its "suggestions" etc.
The spectrum of a series is estimated by its periodogram. This involves Fourier expansion of its autocorrelation function. Spectral analysis is the frequency domain approach to time series analysis. It involves breaking up the series into movements of various frequencies.
There are many benefits to solving PDEs in the frequency domain. From the paper you have attached, I see the usual tesselation approach to "divide and conquer" the problem. Lastly, after generating a matrix of coefficients (from all the geometric breakage), the author is stuck with solving this matrix problem by employing a pseudo-spectral approach, which is, specifically, the use of some orthogonal basis set, like Chebyshev, Fourier and similar series, to solve for the unknown coefficients. A laborious approach that does not take into account discretization errors and other errors. With that said, pseudo-spectral methods are generally more accurate than the provincial Finite-Difference Time Domain method.
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