Hi, if your problem is related to polynomial interpolation I would suggest to use cubic spline instead of Chebyshev:

http://www.geos.ed.ac.uk/~yliu23/docs/lect_spline.pdf

Thank you Richard Epenoy 

Hello,a few years ago I solved stability of a rotating disc flow(6th order eigenvalue DE similar to Orr Sommerfeld equation) using both approaches. I started with spectral FEM using Chebyshev polynomial on a whole interval but I soon found out that I would have needed very high degree of the polynomials  to capture the shape of the eigenfunction. Splitting the original interval into 5 subintervals and using quite low degree Ch. polynomials led to much smaller systems of linear eq. (having the same accuracy as the high degree spectral method on a single interval).

Thank you Jiří Falta . It was very prefitable

Hi all, 

Increasing the degree of Chebyshev approximation leads to more accurate results, but to badly conditioned matrices (preconditioning can, of course, help).

Decomposing the interval into subdomains can help in general but you have to be careful how you couple the solutions on adjacent intervals: which level of smoothness do you want? If your exact solution is smooth enough, you may want to preserve this property for the approximation and the only points where you can loose the smoothness are those separating the sub-intervals.

In fact, I think that this decomposition makes really sense if you have regions where the gradients are changing rapidly (like in the case of boundary layers or interior layers). We tried it long time ago and it worked fine, and this is basically also the idea of adaptive discretization methods. Then I would consider such regions decomposed into small sub-intervals wherein a Chebyshev discretization, not necessary lower order but possibly smoothed to avoid oscillations, can be applied. Regions with small variations in the solution gradient can be still approximated on larger sub-intervals. There is no need to consider small sub-domains for such regions. 

Good luck,

Sorin

Dear Iuliu Sorin Pop,

Thank you for your answer. it was very beneficial.

Also, if your problem has very smooth solution by predictions, then I suggest to use the second 2) choice. If the problem is supposed to be non-smooth or even discontinuous, then the first 1) choice is better. In the latter case, you need to choose cutting points as well.

Thank you Zhang. your answer is so useful