course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
II |
106 |
II |
2011 |
Numerical Analysis |
|
Paper 4, Section II, A |
2011 |
(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.
$$\begin{aligned}
& u(-1, y)=u(1, y), \quad u_{x}(-1, y)=u_{x}(1, y), \quad-1 \leqslant y \leqslant 1, \\
& u(x,-1)=u(x, 1), \quad u_{y}(x,-1)=u_{y}(x, 1), \quad-1 \leqslant x \leqslant 1
\end{aligned}$$