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Introduction to Scientific Computing II

From Gaussian Elimination to Multigrid – A Recapitulation. Dr. Miriam Mehl. Introduction to Scientific Computing II. Tasks – SLE. ???. Tasks – Molecular Dynamics. Prerequisites. discretisation of PDEs linear algebra Gaussian elimination basics on iterative solvers

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Introduction to Scientific Computing II

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  1. From Gaussian Elimination to Multigrid – A Recapitulation Dr. Miriam Mehl Introduction to Scientific Computing II

  2. Tasks – SLE ???

  3. Tasks – Molecular Dynamics

  4. Prerequisites • discretisation of PDEs • linear algebra • Gaussian elimination • basics on iterative solvers • Jacobi, Gauss-Seidel, SOR, MG • matlab

  5. Organization • lecture (90 min/week) • theory • methods • simple examples • tutorials (45 min/week) • more examples • make your own experiences

  6. What Determines the Grading? • written exam at the end of the semester • no weighting of tutorials !!!! solving tutorials is essential !!!! • for understanding and remembering subjects • for your success in the exam

  7. Materials • slides (short, only headwords) • exercise sheets • make your own lecture notes! • find your own solutions! • solutions presented in the tutorials

  8. Rules • for questions ask or fix a date per emailDr. Miriam Mehl: mehl@in.tum.deMartin Buchholz:buchholm@in.tum.de

  9. Introduction to Scientific Computing II From Gaussian Elimination to Multigrid – A Recapitulation Dr. Miriam Mehl

  10. What’s the Problem to be Solved? Application Scenario Partial Differential Equations Modelling Scientific Computing I Finite Elements Finite Differences (Finite Volumes) Scientific Computing I Numerical Programming II Systems of linear equations LU, Richardson, Jacobi, Gauss-Seidel, SOR, MG Scientific Computing I, Scientific Computing Lab, Numerical Programming I More on this!!!

  11. Example Equation v v v v v v v v v v v v v v v two-dimensional Poisson equation • heat equation • diffusion • membranes • … grid + finite differences

  12. Typical SLE sparse band structure

  13. Example

  14. Gaussian Elimination (LU)

  15. Gaussian Elimination (LU)

  16. Gaussian Elimination (LU)

  17. Gaussian Elimination (LU)

  18. Gaussian Elimination (LU)

  19. Gaussian Elimination (LU)

  20. Gaussian Elimination (LU)

  21. Gaussian Elimination (LU)

  22. Gaussian Elimination (LU)

  23. Gaussian Elimination – Costs 2D: O(N4) 3D: O(N7)

  24. Gaussian Elimination – Costs 2D hallo

  25. Gaussian Elimination – Costs 3D hallo

  26. Iterative Solvers – Principle series of approximations • costs per iteration? • convergence? • stopping criterion?

  27. Relaxation Methods problem: order an amount of peas on a straight line (corresponds to solving uxx=0)

  28. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  29. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  30. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  31. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  32. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  33. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  34. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  35. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  36. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  37. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  38. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  39. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  40. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  41. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  42. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  43. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  44. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  45. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  46. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  47. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  48. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

  49. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours • we get a smooth curve instead of a straight line • global error is locally (almost) invisible

  50. Relaxation Methods problem: order an amount of peas on a straight line (corresponds to solving uxx=0)

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