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Exercise where

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Exercise

where

Discretize the problem as usual on square grid ofpoints (including boundaries). Definegand fsuch that the solution to thedifferential equationis

ImplementV(2,1)cycles usingnlevels, with a random initial guess. Use full-weighted restriction, bi-linear interpolation, and the following relaxation methods:

- Damped Jacobi,
- Gauss-Seidel in natural order;
- Gauss-Seidel in Red-Black order.

Perform manyV-cycles (until convergence slows down and stops due to round-off errors). Compute theL2norm (root-mean-square) of the residual after each cycle. Plot the residual reduction rate per cycle:

where is the iteration number. (EachV-cycle is one iteration). Use your code to answer the following questions. Explain all the results. Use at leastn = 5(but use at leastn = 6for Question 3).

1. For which is the best relaxation method? Which is the worst? What is the optimal for Jacobi relaxation found in practice? How do the results of damped Jacobi and Gauss-Seidel in natural order compare with the predictions ofsmoothing analysis?

2. For , what is the influence of (only positive) . Compare

3. For compare between the convergence behavior of the residual norm and that of the “differential error”, how are these behaviors influenced by the resolution? Give quantitative answers and explain if they match our expectations.

4. For what is the influence of (positive, negative, big, small). Starting with , reduce gradually until the two-grid convergence deteriorates. Use the IR/ICG two-grid analysis to determine the cause of the problem. Explain.

5. For discretize the problem using the skew Laplacian:

Test the residual convergence. Use the IR/ICG two-grid analysis to determine the cause of the problem. Explain.