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{\large DELFT UNIVERSITY OF TECHNOLOGY}\\
{\bf Faculty of Information Technology and Systems}\\
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{\bf 
Numerical Analysis CII (wi4014TU)\\
Numerical Methods PDE (wi3001)\\
Fourth Series\\
Numerical Mathematics SS (wi4012)\\
Second series\\
2000/01}
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\begin {enumerate}
\item
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The Poisson equation 
\beq
-\Delta u = f
\eeq
is solved on the 3 regions above by use of the standard
5-point difference molecule. Describe for which region(s) a band method may be
used and for which region(s) a profile method has to be used.
Give the optimal numbering strategy for the band method.

\item
A possible algorithm to solve
$A{\bf x} = {\bf b}$
is Jacobi's method. Let $A = D - L - U$, in which $D$
diagonal, $L$ lower triangle, $U$ upper triangle; Jacobi's method is given by
\beq
{\bf x}^{n+1} = D^{-1} (L+U) {\bf x}^n + D^{-1} {\bf b}
\eeq
\begin {enumerate}
\item
Show that this process gives the solution of
$A{\bf x} = {\bf b}$, whenever it converges.
\item
Show that, when
\beq
|a_{ii}| > \sum_{\stackrel {j=1} {j \neq i}}^n |a_{ij}|, \quad i=1, 2, \dots
, n
\eeq
is satisfied, Jacobi's method converges.
(Hint: use theorem a2.8)
\end {enumerate}
\item
Let $A$ be diagonal-blocktridiagonal. Show that, if $\omega$ is chosen larger
than $\omega_{\rm opt}$, it follows, that all eigenvalues of $Q_\omega$
are complex and that
\beq
|\lambda_i| = \omega - 1, \quad i=1,2, \dots, n.
\eeq
holds. (Use theorem 11.4). Why can't you use rule of thumb 11.2 in this
case?
\item
The Heat Equation in two dimensions is solved by use of ADI:
\bea
\frac {u^* - u^n}{\Delta t} = \frac12 (A_x u^* + A_y u^n) + \frac12
f^n\\
\frac {u^{n+1} - u^*}{\Delta t} = \frac12 (A_x u^* + A_y u^{n+1}) +
\frac12 f^{n+1}
\eea
The vectors $u^n, u^{n+1}, f^n, f^{n+1}$  contain unknowns and source terms on
two different time levels and the matrices $A_x$ and
$A_y$ represent the second divided differences in x- and y-directions. 
Compare this method to Crank-Nicolson's method:
\beq
\frac {u^{n+1} - u^n}{\Delta t} = \frac12 (A_x + A_y) (u^{n+1} + u^n) +
\frac12 (f^{n+1} + f^n)
\eeq
Show by elimination of the intermediate step
$u^*$, that the order of the {\em difference} between both methods (in $\Delta t$)
equals the order of Crank-Nicolson's local truncation error. So what is 
the order of ADI's local truncation error?
\item Excercise 16.5.
\end {enumerate}
\medskip
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{\bf wi4014 to be submitted  before April 1, 2001}
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