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Jun Zou Department of Mathematics The Chinese University of Hong Kong

Iterative Methods with Inexact Preconditioners and Applications to Saddle-point Systems & Electromagnetic Maxwell Systems. Jun Zou Department of Mathematics The Chinese University of Hong Kong http://www.math.cuhk.edu.hk/~zou Joint work with Qiya Hu (CAS, Beijing).

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Jun Zou Department of Mathematics The Chinese University of Hong Kong

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  1. Iterative Methods with Inexact Preconditioners and Applications to Saddle-point Systems & Electromagnetic Maxwell Systems Jun Zou Department of Mathematics The Chinese University of Hong Kong http://www.math.cuhk.edu.hk/~zou Joint work with Qiya Hu (CAS, Beijing)

  2. Outline of the Talk

  3. Inexact Uzawa Methods for SPPs • Linear saddle-point problem: where A, C: SPD matrices ; B : n x m ( n > m ) • Applications: Navier-Stokes eqns, Maxwell eqns, optimizations, purely algebraic systems , … … • Well-posedness : see Ciarlet-Huang-Zou, SIMAX 2003 • Much more difficult to solve than SPD systems • Ill-conditioned: need preconditionings, parallel type

  4. Why need preconditionings ? • When solving a linear system • A is often ill-conditioned if it arises from discretization of PDEs • If one finds a preconditioner B s.t. cond(BA) is small, then we solve • If B is optimal, i.e. cond (BA) is independent of h, then the number of iterations for solving a system of h=1/100 will be the same as for solving a system of h=1/10 • Possibly with a time difference of hours & days, or days & months, especially for time-dependent problems

  5. Schur Complement Approach A simple approach: first solve for p , Then solve for u , We need other more effective methods !

  6. Preconditioned Uzawa Algorithm Given two preconditioners:

  7. Preconditioned inexact Uzawa algorithm • Algorithm • Randy Bank, James Bramble, Gene Golub, ... ...

  8. Preconditioned inexact Uzawa algorithm • Algorithm • Question :

  9. Uzawa Alg. with Relaxation Parameters(Hu-Zou, SIAM J Maxtrix Anal, 2001) • Algorithm I • How to choose

  10. Uzawa Alg with Relaxation Parameters • Algorithm with relaxation parameters: • Implementation • Unfortunately, convergence guaranteed under But ensured for any preconditioner for C ; scaling invariant

  11. (Hu-Zou, Numer Math, 2001) • Algorithm with relaxation parameter • This works well only when both • This may not work well in the cases

  12. (Hu-Zou, Numer Math, 2001) • Algorithm with relaxation parameter • For the case : more efficient algorithm: • Convergence guaranteed if

  13. (Hu-Zou, SIAM J Optimization, 2005)

  14. Inexact Preconditioned Methods for NL SPPs • Nonlinear saddle-point problem: • Arise from NS eqns, or nonlinear optimiz :

  15. Time-dependent Maxwell System ● The curl-curl system: Find u such that ● Eliminating H to get the E - equation: ● Eliminating E to get the H - equation: ● Edge element methods (Nedelec’s elements) : see Ciarlet-Zou : Numer Math 1999; RAIRO Math Model & Numer Anal 1997

  16. Time-dependent Maxwell System ● The curl-curl system: Find u such that ● At each time step, we have to solve

  17. Non-overlapping DD Preconditioner I(Hu-Zou, SIAM J Numer Anal, 2003) ● The curl-curl system: Find u such that ● Weak formulation: Find ●Edge element of lowest order : ● Nodal finite element :

  18. Edge Element Method

  19. Additive Preconditioner Theory ●Given an SPD S, defineanadditive Preconditioner M : ● Additive Preconditioner Theory

  20. DDMs for Maxwell Equations • 2D, 3D overlapping DDMs: Toselli (00), Pasciak-Zhao (02), Gopalakrishnan-Pasciak (03) • 2D Nonoverlapping DDMs : Toselli-Klawonn (01), Toselli-Widlund-Wohlmuth (01) • 3D Nonoverlapping DDMs : Hu-Zou (2003), Hu-Zou (2004) • 3D FETI-DP: Toselli (2005)

  21. Nonoverlapping DD Preconditioner I(Hu-Zou, SIAM J Numer Anal, 2003)

  22. Interface Equation on

  23. Global Coarse Subspace

  24. Two Global Coarse Spaces

  25. Nonoverlapping DD Preconditioner I

  26. Nonoverlapping DD Preconditioner II(Hu-Zou, Math Comput, 2003)

  27. Variational Formulation

  28. Equivalent Saddle-point System can not apply Uzawa iteration

  29. Equivalent Saddle-point System Write the system into equivalent saddle-point system : Important : needed only once in Uzawa iter. Convergence rate depends on

  30. DD Preconditioners Let Theorem

  31. DD Preconditioner II

  32. Local & Global Coarse Solvers

  33. Stable Decomposition of VH

  34. Condition Number Estimate Theadditivepreconditioner Condition number estimate: Independent of jumps in coefficients

  35. Mortar Edge Element Methods

  36. Mortar Edge Element Methods See Ciarlet-Zou, Numer Math 99:

  37. Mortar Edge M with Optim Convergence(nested grids on interfaces)

  38. Local Multiplier Spaces: crucial !

  39. Near Optimal Convergence

  40. Auxiliary Subspace Preconditioner(Hiptmair-Zou, Numer Math, 2006) Solve the Maxwell system : by edge elements on unstructured meshes

  41. Optimal DD and MG Preconditioners • Edge element of 1st family for discretization • Edge element of 2nd family for preconditioning • Mesh-independent condition number • Extension to elliptic and parabolic equations Thank You !

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