03/29/2006, city univ1 iterative methods with inexact preconditioners and applications to...

03/29/2006, City Univ 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 withQiya 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

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)

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

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

● Additive Preconditioner Theory

● Given an SPD S, define an additive Preconditioner M :

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

24

Global Coarse Subspace

25

Two Global Coarse Spaces

26

27

Nonoverlapping DD Preconditioner I

28

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

29

Variational Formulation

30

Equivalent Saddle-point System

can not apply Uzawa iteration

31

Equivalent Saddle-point System

Write the system

into equivalent saddle-point system :

Convergence rate depends on

Important : needed only once in Uzawa iter.

32

DD Preconditioners

Let

Theorem

33

DD Preconditioner II

34

Local & Global Coarse Solvers

35

Stable Decomposition of VH

36

Condition Number Estimate

The additive preconditioner

Condition number estimate:

Independent of jumps in coefficients

37

Mortar Edge Element Methods

38

Mortar Edge Element Methods

See Ciarlet-Zou, Numer Math 99:

39

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

40

Local Multiplier Spaces: crucial !

41

Near Optimal Convergence

42

Auxiliary Subspace Preconditioner(Hiptmair-Zou, Numer Math, 2006)

Solve the Maxwell system :

by edge elements on unstructured meshes

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 !