The optimal choice of the iteration parameter depends on explicit knowledge of ß = p(GJ) 2 1+ 1-3² Here GJ is the reduct

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The Optimal Choice Of The Iteration Parameter Depends On Explicit Knowledge Of Ss P Gj 2 1 1 3 Here Gj Is The Reduct 1 (117.04 KiB) Viewed 67 times

Two websites to help you:
https://en.wikipedia.org/wiki/Five-point_stencil
https://en.wikipedia.org/wiki/Laplace%27s_equation
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The optimal choice of the iteration parameter depends on explicit knowledge of ß = p(GJ) 2 1+ 1-3² Here GJ is the reduction matrix of the Jacobi method. Unfortunately, p(G₁) is not ana- lytically known for the five-point stencil on a triangular domain. However, the eigenvalues and eigenvectors of the five-point operator are known when the domain is a square. ordering function k(i, j) are given by Wopt The corresponding eigenvalue is = Using the node the eigenvectors of A5 (on the square) (Ok(j1,j2))k(l1,l2) = sin(j₁l₁/N) sin(Tj₂l₂/N). Ak(12)=4-2 cos(j1/N) - 2 cos(j2/N).
Here 11, 12, 11, 12 run from 1 to N - 1 so that the dimension of A5 on the square is (N − 1)². Using the fact that D 41 for A5, compute the eigenvalues of GJ (as a function of j1, 12 and N) corresponding to A5 on the square. Use this to compute ß = p(GJ) and compute Wopt (N) for A5 on the square as a function of N, N = 4, 8, 16, 32, 64. Wopt = =