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Multigrid Methods Shijie Zhong Dept. of Physics University of Colorado Boulder, Colorado

Multigrid Methods Shijie Zhong Dept. of Physics University of Colorado Boulder, Colorado Workshop for Advancing Numerical Modeling of Mantle Convection and Lithospheric Dynamics July 2008, UC-Davis. Numerical modeling. A scientific problem . Partial differential equations

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Multigrid Methods Shijie Zhong Dept. of Physics University of Colorado Boulder, Colorado

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  1. Multigrid Methods Shijie Zhong Dept. of Physics University of Colorado Boulder, Colorado Workshop for Advancing Numerical Modeling of Mantle Convection and Lithospheric Dynamics July 2008, UC-Davis

  2. Numerical modeling A scientific problem  Partial differential equations within a domain  Discretize PDE using FE, FD, FV, … on a certain grid  a matrix equation: Kd=F f=ma

  3. A toy problem: 1-D heat conduction 0 1 x

  4. e x=0 x=1 Discretize with FE

  5. e 1 0

  6. Kd=F

  7. Iterative Solvers A matrix equation: Kd=F Iterative solvers: • memory usage ~ N (# of unknowns in d), • # of flops ~ N (e.g., for multigrid solver), • suitable for parallel computing.

  8. Jacobi and Gauss-Seidel methods Matrix Equation: Rewrite matrix K: Jacobi method: Start with a guessed solution d(0), then update d iteratively to get d(1), … until residual e=||Kd(n)-F|| is less than some tolerance. Gauss-Seidel method:

  9. Jacobi method

  10. Gauss-Seidel Method

  11. The idea behind multigrid Gauss-Seidel

  12. A road map

  13. A road map continued but reversed

  14. Different cycles W-cycle V-cycle n n-1 1

  15. THE method for elliptic equations (i.e., “diffusion” like problems)

  16. Execution Time vs Grid Size N for Multi-grid Solvers in Citcom t ~ N-1 FMG: Zhong et al. 2000 MG: Moresi and Solomatov, 1995

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