Iterative Methods with Inexact Preconditioners and Applications to

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) 03/29/2006, City Univ 1

Outline of the Talk 2

Inexact Uzawa Methods for SPPs • Linear saddle-point problem: where A, C : SPD matrices ; B: nxm (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 3

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 4

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

Preconditioned Uzawa Algorithm Given two preconditioners: 6

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

Preconditioned inexact Uzawa algorithm • Algorithm • Question : 8

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

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

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

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

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

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

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

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

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 : 17

Edge Element Method 18

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

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

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

Interface Equation on 22

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Global Coarse Subspace 24

Two Global Coarse Spaces 25

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Nonoverlapping DD Preconditioner I 27

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

Variational Formulation 29

Equivalent Saddle-point System can not apply Uzawa iteration 30

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

DD Preconditioners Let Theorem 32

DD Preconditioner II 33

Local & Global Coarse Solvers 34

Stable Decomposition of VH 35

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

Mortar Edge Element Methods 37

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

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

Local Multiplier Spaces: crucial ! 40

Near Optimal Convergence 41

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

Optimal DD and MG Preconditioners • Edge element of 1 st family for discretization • Edge element of 2 nd family for preconditioning • Mesh-independent condition number • Extension to elliptic and parabolic equations Thank You ! 43