Here are the Postscript file:homework2.ps and the LaTeX file:homework2.tex.

Math 228C Computational Assignment 2 Spring 1999

Due: Monday, May 3, 1999

1.

Solve the Neumann problem for 1D Poisson equation with GSRB (Red Black Gauss-Seidel)

$$\Delta {\textbf u}$$ (1) = 0

$$\frac {\partial {\textbf u}} {\partial {\textbf n}} \big \vert _{\partial \Omega}$$ (2) = 0

on $\Omega$= [0,1]. For N=64,128 and 256,plot

• The error $\vert\vert{\textbf e}\vert\vert _{\infty}$ versus iteration for 1000 iterations.
• The error $\vert\vert{\textbf e}\vert\vert _{\infty}$ versus x after 10,50,100,500 and 1000 iterations.

Experiment with different initial guesses v⁰, and determine whether all initial guesses will result in a sequence of iterates that converge to a correct solution. If not, try to find a method for picking v⁰ such that the iterates are certain to converge to a correct solution.

2.

Use a Multigrid V-Cycle with Weighted Jacobi ($\omega$=2/3) to solve

$$\Delta {\textbf u}$$ (3) = 0

$$0 \qquad \mbox { on } \partial \Omega$$ (4) u =

in 1D with N=48 and initital guess:

$$\frac 12 [\rm {\sin}(\frac {12 j \pi}N) +\rm {\sin} (\frac {30 j \pi} N)]$$ (5) v_j =

• Plot the initial guess v⁰.
• Plot the error $\vert\vert{\textbf e}\vert\vert _{\infty}$ versus x in [0,1] after 1 fine grid relaxation.
• Plot the error $\vert\vert{\textbf e}\vert\vert _{\infty}$ versus x in [0,1] after 3 fine grid relaxation.
• Transfer the fine grid residual after 3 fine grid relaxations to the coarse grid using full weighting and relax once on the coarse grid. Plot the error $\vert\vert{\textbf e}\vert\vert _{\infty}$ on the fine grid relaxation.
• Do the same after 3 relaxation on the coarse grid

Compare these plots to Figures (18a-e) in the Multigrid Tutorial book.

3.

Plot the error $\vert\vert{\textbf e}\vert\vert _{\infty}$ versus x in [0,1] after 1,2,5 and 10 Full Multigrid V-Cycles applied to Problem 2.

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