Differential Equations Solutions 182

Differential Equations Solutions 182 - fastest iterative...

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192 Chapter 32. Solutions: Multigrid Methods: Managing Massive Meshes CHALLENGE 32.2. The right-hand sides have 15 + 7 + 3 + 1 = 16(1 + 1 / 2+1 / 4+1 / 8) 4 , elements, which is less than twice the storage necessary for the right-hand side for the Fnest grid. The same is true for the solution vectors. Similarly, each matrix A h has at most half of the number of nonzeros of the one for the next Fner grid, so the total matrix storage is less than 2 times that for A 1 / 16 . The matrices P h can be stored as sparse matrices or, since we only need to form their products with vectors, we can just write a function to perform multiplication without explicitly storing them. CHALLENGE 32.3. In Chapter 27 we saw that the fastest algorithm for the Fnest grid for myproblem=1 was the AMD-Cholesky algorithm, which, on my com- puter, took about 0.2 seconds and storage about 5 times that for the matrix. The
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Unformatted text preview: fastest iterative method, conjugate gradients with an incomplete Cholesky precon-ditioner, took 0.9 seconds. My implementation of multigrid for this problem took 4 iterations and 8.2 seconds. The virtue of multigrid, though, is if we want a Fner grid, we will probably still get convergence in about 4 iterations, while the number of iterations of the other algorithms increases with h , so eventually multigrid will win. CHALLENGE 32.4. The number of iterations remained 4 for κ = 10, 100, and − 10, but for κ = − 100, multigrid failed to converge. As noted in the challenge, a more complicated algorithm is necessary. Note that the function smooth.m is a much faster implementation of Gauss– Seidel than that given in the solution to Challenge 28.6 in Chapter 28....
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