% (c) 2004/2005 Universitaet Stuttgart % Chair 'Numerische Mathematik fuer Hoechstleistungsrechner' % http://www.ians.uni-stuttgart.de/nmh % % ---------------------------------------------------------- % Permission to use this exercise for non-profit educational % purposes is hereby granted provided that this copyright % notice is included in all copies or substantial portions % of the LaTeX files. % ---------------------------------------------------------- \begin{exercise}{Jacobi and SOR method - example}{tutorial}{10}{indir03\_eng} The discretization of the Poisson equation $-\Delta u = f$ over the unit square with simple finite elements (or finite differences) and a uniform grid gives a sparse $n^2 \times n^2$ matrix $A$ that can be generated in Matlab using the command \texttt{gallery('poisson',n)}. \begin{itemize} \item[a)] Examine the number of required iterations for the Jacobi method and the Gauss-Seidel method when solving $Au=f$ for the right-hand side $f = \frac{1}{(n+1)^2} \begin{pmatrix}1 \\ 1 \\ \vdots \\ 1 \end{pmatrix}.$ Use the methods of exercise \texttt{indir02\_eng} to do this. As the termination condition, use a tolerance \texttt{tol} of $10^{-5}$ for the relative residuum (in the Euclidean norm). Use the zero vector as start vector of the iteration. Generate a plot showing the number of iterations vs. the dimension $n$, for $4 \leq n \leq 18$ (set the maximum number of iterations to perform to 1000). \item[b)] Solve the system $Bu=f$ with $B=A+2I$ ($I$: identity matrix) under the same conditions as in a). How does the number of required iterations change? Explain the observed behavior. \end{itemize} \end{exercise}