This is a follow up to this question.
I have mapped from Chebyshev space to the physical space using the mapping $x = ((1-w)\xi^3 - x_p\xi^2 + w\xi + x_p + 1 )*(L/2)$.
When in Chebyshev space I have easily defined the spectral differentiation matrix as follows:
$$(D_N)_{00} = \frac{2N^2 +1}{6}$$ $$(D_N)_{NN} = -\frac{2N^2 +1}{6}$$ $$(D_N)_{jj} = -\frac{\xi_j}{2(1-\xi_j^2)}, \ \forall \ 1 \leq j \leq N-1$$ $$(D_N)_{ij} = \frac{c_i}{c_j} \frac{(-1)^{i+j}}{\xi_i - \xi_j}, \ \forall \ i \neq j$$
where
$ c_i= \begin{cases} 2,& \text{if } i = 0 \text{ or } N\\ 1, & \text{otherwise} \end{cases}$
$ c_j= \begin{cases} 2,& \text{if } j = 0 \text{ or } N\\ 1, & \text{otherwise} \end{cases}$
- How do I find the analogous differentiation matrix for the mapping
mentioned above? - Will this mapping affect the stability and convergence of collocation method?