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03/29/2006, City Univ1 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 Joint work with Qiya Hu (CAS, Beijing)

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2 Outline of the Talk

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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

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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

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

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6 Preconditioned Uzawa Algorithm Given two preconditioners:

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

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8 Preconditioned inexact Uzawa algorithm Algorithm Question :

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

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

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

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

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

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

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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

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

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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 :

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18 Edge Element Method

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19 Additive Preconditioner Theory ● Additive Preconditioner Theory ● Given an SPD S, define an additive Preconditioner M :

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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)

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

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22 Interface Equation on

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24 Global Coarse Subspace

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25 Two Global Coarse Spaces

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27 Nonoverlapping DD Preconditioner I

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

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29 Variational Formulation

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30 Equivalent Saddle-point System can not apply Uzawa iteration

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

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32 DD Preconditioners Let Theorem

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33 DD Preconditioner II

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34 Local & Global Coarse Solvers

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35 Stable Decomposition of V H

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36 Condition Number Estimate The additive preconditioner Condition number estimate: Independent of jumps in coefficients

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37 Mortar Edge Element Methods

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

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

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40 Local Multiplier Spaces : crucial !

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41 Near Optimal Convergence

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

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43 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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