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Math 228C Computational Assignment 1 Spring 1999

Due: Monday, April 26, 1999

Discretize the 1D model problem

$$\begin{eqnarray*}\Delta u &=&f=0 \\ u&=&0 \qquad \mbox{ on } \partial \Omega \end{eqnarray*}$$

for f=0 with the standard three-point discretization of the Laplacian. In 1D, the domain $\Omega$ is the unit interal [0,1], and the boundary of $\Omega$ is the set containing the points 0 and 1.
On a grid with N gridpoints,use Weighted Jacobi (WJ) with $\omega=2/3$, regular Gauss-Seidel(GS) and Red-Black Gauss-Seidel(GSRB) with initial guesses v_k for k=1,3,6 and N/2-1 on grids of N=64,128,256. Here v_k denotes the k Fourier mode, which has components

$$\begin{eqnarray*}(\textbf v_k)_j&=&\rm {sin}(\frac {jk\pi}{N}) \end{eqnarray*}$$

For each of these computations, plot the max norm of the error $\vert\vert\textbf e\vert\vert _{\infty}$ against the iterations number for 100 iterations.

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