Spectral methods

• How does a spectral method differ from a psuedo-spectral method?
• Write a point on a circle in the complex plane in Cartesian ($x,y$) and Polar ($r,\theta$) coordinates.
• What type of boundary conditions are appropriate for spectral methods?
• If you have $u$ at $N$ grid points and compute the discrete Fourier trasform to compute $\hat{u}$, the $\hat{u}$ are complex numbers with real and imaginary parts, giving $2N$ degrees of freedom. How are the $N$ $u$ and the $2N$ $\hat{u}$ reconciled?
• For real $u$, what is the constraint on the values of the complex $\hat{u}$?
• What is one of the reasons that spectral methods are so accurate?
• What is the psuedo-spectral representation of a derivative $du/dx$?
• Write psuedo-spectral formulation of a term like $\frac{d}{dx}(c(x)\frac{dy}{dx})$.