(i) Consider the Poisson equation

$$\nabla^{2} u=f, \quad-1 \leqslant x, y \leqslant 1$$

with the periodic boundary conditions

and the normalization condition

$$\int_{-1}^{1} \int_{-1}^{1} u(x, y) d x d y=0$$

Moreover, $f$ is analytic and obeys the periodic boundary conditions $f(-1, y)=$ $f(1, y), f(x,-1)=f(x, 1),-1 \leqslant x, y \leqslant 1 .$

Derive an explicit expression of the approximation of a solution $u$ by means of a spectral method. Explain the term convergence with spectral speed and state its validity for the approximation of $u$.

(ii) Consider the second-order linear elliptic partial differential equation

$$\nabla \cdot(a \nabla u)=f, \quad-1 \leqslant x, y \leqslant 1$$

with the periodic boundary conditions and normalization condition specified in (i). Moreover, $a$ and $f$ are given by

$$a(x, y)=\cos (\pi x)+\cos (\pi y)+3, \quad f(x, y)=\sin (\pi x)+\sin (\pi y)$$

[Note that $a$ is a positive analytic periodic function.]

Construct explicitly the linear algebraic system that arises from the implementation of a spectral method to the above equation.

$$\begin{aligned} & u(-1, y)=u(1, y), \quad u_{x}(-1, y)=u_{x}(1, y), \quad-1 \leqslant y \leqslant 1, \\ & u(x,-1)=u(x, 1), \quad u_{y}(x,-1)=u_{y}(x, 1), \quad-1 \leqslant x \leqslant 1 \end{aligned}$$