l41 - Lecture 41 Multigrid Methods 1 Last Time Looked at...

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1 Lecture 41: Multigrid Methods
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2 Last Time… Looked at the relationship between mesh size and the wave number content of error » Deduced that involving coarse meshes would help reduce low-k errors
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3 This Time … We will Analyze the Jacobi scheme to get a deeper understanding of the connection between mesh size and error Discuss the basics of the multigrid method Look particularly at the geometric multigrid method
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4 Poisson Equation Consider 1-D Poisson equation: Boundary conditions:   2 2 0 sx x 
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5 Analysis of Jacobi Scheme Look at Jacobi (instead of Gauss-Seidel) » More convenient to analyze and gives good insights Discretize this on a mesh of N equal control volumes of size h Look at slightly different discretization: Equispaced mesh of N interior nodes – no half- spacing at boundaries h
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6 Linear System Discretization yields Note: “2”, not “3” Can find eigenvalues analytically for this matrix
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7 Error Reduction Recall iteration matrix For Jacobi: P = D -1 (L+U) Can show that error at n th iteration is related to initial error by
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8 Error Reduction (cont’d)     1 1 10 0 = P since ; Also, g Applying for n iterations starting from k=0, we have k k k k k k exact exact kk exact exact exact k k k k exact exact nn x Px g x x x Px g x x x Px g e Pe x Px g g e P e 
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9 Error Reduction (Cont’d) For the error to reduce in successive iterations, the spectral radius , which is the largest eigenvalue of the iteration matrix, must be less than unity The rate of error reduction depends on how small the spectral radius is
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10 Eigenvalues of Iteration Matrix Can find eigenvalues of P to be: Eigenvectors turn out to be Fourier modes. The jth component of eigenvector corresponding to k is:
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11 Error and Eigenvalues of P If initial error is written in terms of Fourier modes, it can be expressed in terms of eigenvectors With a little manipulation, can show that:
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12 Errors and Eigenvalues of P (cont’d) 0 10 2 1 2 From the definition of the eigenvalue problem, we have Therefore, ... kk k k k k k k k k k k k k k k nn k k k ew Pw w e Pe Pw w e Pe Pw w    
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13 Error and Eigenvalues (Cont’d) Error reduction of k th mode at n th iteration is proportional to n k Since eigenvalues are given by: highest eigenvalue is for k=1 Lowest k modes are therefore the slowest to converge Also, magnitude of largest eigenvalue increases with N » Fine meshes slow to converge
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l41 - Lecture 41 Multigrid Methods 1 Last Time Looked at...

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