Multigrid Methods for Solving the Convection-Diffusion Equations

Chin-Tien Wu

National Center for Theoretical SciencesMathematics DivisionTsing-Hua University

Motivation

1. Incompressible Navier-Stokes equation models many flows in practices.

0 in [0, ]

0 in [0, ],

uu u v u p T

tu T

0

( , ) ( , ) on [0,T]

u(x,0)=u ( ) in

u x t g x t

x

11 1 1

1 1

0

0 and on

m mm m m m

m m

u uv u u u p

t

u u g

0 0T

F B u f

B p

Linearized backward Euler

Discretization

Oseen Problem

Saddle Point Problem

2. A preconditioning technique proposed by Silvester, Elman, Kay and Wathen for solving linearized incompressible Naiver-Stokes has been shown to be very efficient. (J. Comput. and Appl. Math. 2001 No 128 P. 261-279) .

3. For achieving efficiency, a robust Poisson solver and a scalar convection-diffusion solver are needed.

-11

0 where

0 0

TT

T

F B I F BS BF B

B BF I S

1 1

1

00

0 00 0

T

T

F B IF I B

B SI I

1 is a scaled Laplace operator where

M is the mass matrix

AS AF M

4. Mulrigrid is efficient for solving Poisson problem. Can we have a robust multigrid solver for the convection-diffusion problems?

Convection-Diffusion Equation

1 D

2 N

,

on ,

on

u b u cu f

u g

ug

n

The problem we consider here is

where for some constant c0 and b, c and f are

sufficiently smooth.

0

10

2c b c

What are the foundations for building an

efficient and accurate solver?

I. Stable discretization methods for FEM. II. A good mesh.

III. Reliable error estimation.

IV. Fast iterative linear solver.

Challenges in solving the convection-diffusion problem

1. When the convection is dominant, <<1, the solution has sharp gradients due to the presents of Dirichlet outflow boundaries or discontinuity in inflow boundaries.

2. The associated linear system is not symmetric.

I. Discretization

• Galerkin FEM

• Streamline diffusion upwinding FEM by Hughes

and Brooks 1979

• Discontinuous Galerkin FEM by Reed and Hill 1973

• Edge-averaged FEM by Xu and Zikatanov 1999

(the linear system form this discretization is an M-matrix)

Example 1: Characteristic and downstream layers

0

1 if (y=0 x>0) or x=1, |

0 otherwise,

where =[-1,1] [-1,1].

uu

y

u

0 1.

1

1

1

Solution from SDFEM discretization on 32x32 grid

Solution from Galerkin discretization on 32x32 grid

II. Methods for producing a good mesh?

1. Delaunay triangulation (DT): DT maximizes the minimal angle of the triangulation.

2. Advancing Front algorithm (AF): controls the element shape through control variables such as element stretch ratio (by Löhner 1988).

3. Advancing front Delaunay triangulation: combination of DT and AF (by Mavriplis 1993, Müller 1993 and Marcum 1995).

4. Longest side bisection: produces nested grids and guarantee the minimal angle on the fine grid is greater than or equal to half of the minimal angle on the coarse grid (by Rivara 1984).

5. Mesh relaxation includes moving mesh by equidistribution, MMPDE (by Huang, Ren and Russell 1994), and moving mesh FEM (by Carlson and Miller 1994, and Baines 1994).

III. A posteriori error indicators are used to pinpoint where the errors are large.

• A posteriori error estimation based on residual is proposed by Babuška and Rheinboldt 1978

• A posteriori error estimation based on solving a local problem is proposed by Bank and Weiser 1985

• A posteriori error estimation based on recovered gradient is proposed by Zienkiewicz and Zhu 1987

• A posteriori error estimations for the convection-diffusion problems are proposed by Verfürth 1998, Kay and Silvester 2001.

Adaptive Solution Strategies

Strategy I :

*

*Element T is refined if max , for a prescribed threshold .TT T

In mesh refinement, we employee the maximum marking strategy:

E0

T T T ET E TR ,( e , ) R , ,

2vv v

0 0T T T T T h

0 2 0T

for all consisting of edge and interior bubble functions, where R = ( f-b u ) ,

is the L projection onto the space P (T) and

v V Q B

hh,

E EE h,D E

hh,N

E

uE E

nR = 0 for all E E and R

u-2 E E

n

Solve Compute error indicator Refine meshi ii

i

ii

(Kay and Silvester 2001)

Improve Solution Accuracy by Mesh Refinement

Adaptive meshes with threshold 0.01 in the maximum marking strategy

Contour plots of solutions on adaptive meshes

Problem 1: 3( 10 and =0)

IV. Solving Linear System Ax=b by Iterative Methods

Methods:

• Stationary Methods: Jacobi, Gauss Seidel (GS), SOR.

• Krylov Subspace Methods: GMRES, by Saad and Schultz 1986, and MINRES, by Paige and Saunders 1975.

• Multigrid Methods: Geometric multigrid (MG), by Fedorenko 1961, and algebraic multigrid (AMG), by Ruge and Stüben 1985.

Why Multigrid?

Numerical Scheme Operations for Square/Cube

2-D 3-D

Gaussian elimination O(N2) O(N7/3)

Jacobi O(N2) O(N5/3)

Conjugate Gradient O(N3/2) O(N4/3)

Multigrid O(Nlog(N)) O(Nlog(N))

Multigrid convergence is independent with problem size.

Why multigrid works?

1. Relaxation methods converge slowly but smooth the error quickly. Consider

-1 12finite difference

- 2'' k k k

k

u u uu u u

h

2k 2

4Eigenvalues = sin and eigenvectors sin

h 2( 1) 1kj

k kj

N N

Richardson relaxation:1( ) where A=tridiag[-1 2 -1].RE I A

Fourier analysis shows that

2m

1 1

e 1 1 sin2( 1)

mmN Nk kk

k k

k

N

, after m relaxation and choosing 2

4.

h

2. Smooth error modes are more oscillatory on coarse grids. Smooth errors can be better corrected by relaxation on coarser grids.

Multigrid (I)

k k

-1k k k k k k

k-1k k k kk

i k-1 i-1 0k

kk-1 mm

k k m

1. x =w

2. (pre-smoothing) x =w +M g -A x

3. (restriction) g =I g -A x

4. (correction) q =MG (q ,g ) for 1 i m, m=1 or 2 and q =0

5. (prolongation) q =I q

6. set x =x +q

7. (post-sm

-1k k k k k k

k k k k

oothing) x =x +M g -A x

8. set MG (w ,g )=x

Multigrid (MG) Algorithm:

1 1 1h H h Hmg h H H h h s s H H h hE A I A I A E E I I A I A

MG Error reduction operator:

Pre-smoothing only Post-smoothing only

Multigrid (II)A

dapt

ive

refi

nem

ent

MG V-cycle

Restriction

Prolongation

AMG V-cycle

AM

G

Coarsening

Restriction

Prolongation

k-1kI

kk-1I

Multigrid (GMG) and Algebraic Multigrid (AMG)

MG AMG

1. A priori generated coarse grids are needed. Coarse grids need to be generated based on geometric information of the domain.

2. Interpolation operators are defined independent with coarsening process.

3. Smoother is not always fixed.

1. A priori generated coarse grids are not needed! Coarse grids are generated by algebraic coarsening from matrix on fine grid.

2. Interpolation operators are defined dynamically in coarsening process.

3. Smoother is fixed.

Multigrid Components

• Smoothing operator Es:

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

For

war

d H

-lin

e G

S

Back

ward

H-lin

e G

S

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

Forward V-line GS

Backward V-line GS

: linear interpolation

R/S coarseningAMG

Discretization on VHGMG

Coarse grid operatorRestrictionProlongationhHI TH h

h HI I

H hh h HI A I TH h

h HI I

GMG Convergence

, for all 0 and 0.l l lAS m A m l Smoothing property:

Approximation property:

Convergence needs robust smoothers together with semi-coarsening and operator dependent prolongation for the convection-diffusion equation (Reusken 2002).

Choices of interploations:

1 2 11

2 4 216

1 2 1

r

1 2 11

2 4 24

1 2 1

p

• Linear interpolation:

• Operator-dependent interpolation: De Zeeuw 1990

Recent results:

11 11 , for all 0.H H

l h l h A lA I A I C A l

GMG Convergence for Problem 1

h Theorem 1: Horizontal line Gauss-Seidel (HGS) converges for

Theorem 2.: Geometric multigrid with HGS smoother and bilinear interpolation

converge and

2/3for h 3/ 2mgE

h

9 8 6 464x64

8 7 5 432x32

8 6 4 316x16

10-2 10-3 10-4 10-5mesh

5 4 2 264x64

4 3 2 132x32

4 2 2 116x16

10-2 10-3 10-4 10-5mesh

HGS convergence on rectangular mesh MG convergence on rectangular mesh

2

3 3min ,1s

hE

h

1.

2

3 3(1 min ,1 )s

hAE

h

2.

Auxiliary inequalities:

AMG convergence

-1h0 1 2

,here v = Dv,v , v = Av,v , v = D Av,Av , for v V

2 2 2 2 2

11 1 2 11 1s c c c cE E E e E e E e e

2

1 2 0 2 1,c c c h c c h c c

H H H HE e AE e E e I e E e E e I e E e E e

Smoothing assumption:2 2 2

1 21 >0 for all s

hE e e e e V

Approximation assumption:2 2

10min where is independent with .

H

hH He

e I e e e

AMG works when A is a symmetric positive definite M-matrix.

s s1 1e esE • Smooth error is characterized by

2 1/ 2 1/ 2

1 2 0 1 0e D Ae D e e e e e

2 2 2, , ,

,

2 2,

,

2

,

2,

1, ( )

2

1

2

2

i j i j i j i i i ii j i j i

i j i j ii ii j

i j i j

j i i i i

Ae e a e e a e a e

a e e a e

a e e

a e

• Smoother errors vary slowly in the direction of strong connection, from ei to ej

, where are large.,

,

i j

i i

a

a

• AMG coarsening should be done in the direction of the strong connections.

• In the coarsening process, interpolation weights are computed so that the approximation assumption is satisfied. (detail see Ruge and Stüben 1985)

AMG works for Problem 1

Smooth error varies slowly along strong connected direction when h>>2/3.

• R/S AMG coarsening and interpolation work.

Smooth error es satisfies

s s s s s sAe ,e AE e ,E e -1/2 -1/2s s s s

2

s s3 2

AE D E D De ,e

3 3 3 3min ,1 (1 min ,1 ) De ,e

h h

h h

2

s s3De ,e

h

3/ 2

1 if , where =

2 if

amgA

hE c

h h h

Assuming the Galerkin coarse grid correction satisfied the approximation property, inequalities in the auxiliary lemmas and the approximation assumption imply

• AMG converges more rapidly than GMG as long as 1

.A

hC

2

2 2

,

2 2

Approximation assumption

1( )( )

2

Since ( ) (1 )

ii i ik k ij i j ij ii F k C i j i j

ii i ik k ii ik i k i ii F k C i F k C

a e w e a e e a e

a e w e a w e e s e

2 2

2 2

,

( ) (1 )

, here 0 is the interpolation weight from node k to node i, and 1,

clearly, if

( ) ( )( )2

ii ik i k i ii F k C

ik i ikk C

ii ik i k ij i ji F k C i j

a w e e s e

w s w

a w e e a e e

2 2

(1 ) ,

the approximation assumption holds. For to hold, we can simply require

0 and 0 1 .

ii i i ij ii F i j

ii ik ik ii i ijj

a s e a e

a w a a s a

AMG Coarsening Algorithm (I)

AMG Coarsening Algorithm (II)

AMG Coarsening

AMG coarsening with strong connection parameter /h << << 0.25

C-point

C-point

3 3 32 8

6 3 3 3 6 32

6 3 3 3 6 3

h h h

h h h

SDFEM discretization of Problem 1

with h

2T has matrix stencil:

AMG Coarsening on Uniform Meshes

Problem 1: b=(0,1)

Flow field Coarse grids from AMG coarseningCoarse grids from GMG coarsening

Problem 2: circulating flow b=(2y(1-x2),2x(1-y2))

Coarse grids from AMG coarseningFlow field Coarse grids from GMG coarsening

GMG and AMG as a Solver (I)

On the uniform mesh:

On the adaptive mesh:

Problem 1:

571

672

883

694

AMGGMGLevel

7171

8182

9243

8224

AMGGMGLevel

7341

8472

10573

14594

AMGGMGLevel

=10-2 =10-3 =10-4

=10-2 =10-3 =10-4

6121

6132

7133

AMGGMGlevel

6161

7262

8273

AMGGMGlevel

6171

8352

11513

AMGGMGlevel

GMG and AMG as a Solver (II)Problem 2:

On the uniform mesh:

On the adaptive mesh:

GMG and AMG as a Preconditioner of GMRES

Problem 1:

Problem 2:

1288GMRES-AMG

282014GMRES-GMG

433226GMRES-HGS

947558GMRES

1188AMG

512713GMG

10-410-310-2

1484GMRES-AMG

36165GMRES-GMG

593111GMRES-HGS

-14665GMRES

1486AMG

59229GMG

10-410-310-2

Iterative steps on uniform mesh Iterative steps on adaptive mesh

332411GMRES-AMG

453213GMRES-GMG

775937GMRES-HGS

---GMRES

-25629AMG

-18726GMG

10-410-310-2

161610GMRES-AMG

14128GMRES-GMG

404234GMRES-HGS

---GMRES

24214223AMG

13198GMG

10-410-310-2

Iterative steps on uniform mesh Iterative steps on adaptive mesh

Stopping Criterion for Iterative Solver

u : the weak solution of the convection-diffusion equation,

uh : the finite element solution

uh,n : the approximate iterative solution of uh.

Given a prescribed tolerance . If || u-uh|| < 0.5, clearly, we only have to ask

|| uh,n - uh || < 0.5 for large enough n.

• From a posteriori estimation || u-uh || < c, where c is some constant, it is natural

to acquire the iterative solution satisfies a stopping tolerance such that

C1. || uh,n - uh || < c,

• For adaptive mesh refinement, it is also desirable that

C2. n (n is the error indicator computed from uh,n )

Question :

1. What stopping criteria should be imposed for iterative solutions?

2. What solver requires least iterative steps to satisfy the stopping criteria?

Basic Ideas on Deriving the Stopping Criteria

T T

T

T T

T

T

nh h ω ω h,T

n nh,T h,T h h ω

nω h,T h,T ω h,T

nω h,T h,T h,T

n nh h ω r h rT

1. Assume ||| u u ||| c

2. Estimate | | c ||| u u |||

3. (1-2c c ) (1+2c c )

1 1 34. c

4c 2 2

5. ||| u u ||| c r , where c max{

h

p h

nh h,TT

r

T h,Th

h T

h,1}.

6. In order to satisfy 1, iterative steps n need to be large enough such that

1 r

4c c

max||| u u ||| 17. Use assumptions ,

||| u u ||| max2 2

p pp

p p

h ,T

h,T h ,T

0, and a

posteriori error bounds to replace by .

With the assumptions we prove1,

1,

| |||| ||| 1 1 and ,

||| ||| | |2 2 2p p

hh

h h

u uu u

u u u u

, when Kay and Silvester’s error indicator is used for mesh refniement.

hp

1/2

n 2h ,

Tmax

max h

If the number of iterations, n, is large enough such that

1 r ,

8 b h

where h is the maximum diameter of elements in , then

Theorem 5.2.5

p ph T

h

1/ 2

n 2h h ,

T

u u c

where c is a constant independent with h and .

h T

Stopping criterion for satisfying C1.

With the assumption we prove,

,,

max0 ,

maxh

h p pp

T h T

T h T

3/ 2nh , , h

, , ,

Given threshold in the maximum marking strategy, if

r max for all T4

then

1 3 , for any mar

2 2

Theorem 5.2.6

p pp hp

h TT Tp

nh T h T h T

h

,h,T

h,T ,

ked element T.

Moreover, element T, satisfying max , will not be marked by the same 4

marking strategy with replaced by .

hh T

T

nh T

, when Kay and Silvester’s error indicator is used for mesh refinement.

Stopping criterion for satisfying C2.

Verification of assumption of Theorem 5.2.6

Kay and Silvester’s indicator:

Problem 1: Characteristic and downstream layers (ii)

Number of points in refined meshes

Problem 2: Flow with Closed Characteristics (ii)

Kay and Silvester’s indicator:

Verification of assumption of Theorem 5.2.6

Number of points in refined meshes

Problem 1: Characteristic and downstream layers (i)

Verfürth’s indicator:

Verification of assumption of Theorem 5.1.6

Number of points in refined meshes

Problem 2: Flow with Closed Characteristics (i)

Verfürth’s indicator:

Verification of assumption of Theorem 5.1.6

Number of points in refined meshes

Recent Numerical Results of Multigrid Methods on Solving Convection-Diffusion Equations

• GMRES and BiCGSTAB accelerated W-cycle MG with alternative zebra line GS smoother and upwind prolongation achieves h-independent convergence on problems with closed characteristics in which upwind discretization is used (by Oosterlee and Washio 1998).

• BICGSTAB accelerated V- and F- cycle AMG with symmetric GS smoother shows very slightly h-dependent convergence for problems with closed characteristics in which upwind discretization is used (Trottenberg, Oosterlee and Schüller: Multigrid p.519 2001).

• GMRES accelerated V-cycle MG with line GS smoother and bilinear, upwind or matrix-dependent prolongation achieves h-independent convergence for the model problems in which SDFEM discretization is used (by Ramage 1999).

• By GMRES acceleration, improvement on convergence of MG and AMG is obtained on both uniform and adaptive meshes. GMRES accelerated AMG is an attractive black-box solver for the SDFEM discretized convection-diffusion equation.

Thank You

Conclusion

1. SDFEM discretization is more stable than Galerkin discretization.2. Our error-adapted mesh refinement is able to produce a good mesh

for resolving the boundary layers. 3. On adaptive meshes, MG is a robust solver only for problems with

only characteristics and AMG is robust for problems with only exponential layers. Both MG and AMG are good preconditioner for GMRES.

4. Fewer iterative steps are required for the MG solver to satisfy our stopping criteria ( in Theorem 5.1.6 and Theorem 5.2.6) than to satisfy the heuristic tolerance (residual less than 10-6). No such saving can be seen if GMRES is used. The total saving of computation works is significant (can be more than half of the total works with heuristic tolerance).

Furture works

1. Investigate the performance of different linear solvers from EAFEM.

2. Deriving a posteriori error estimations for EAFEM.3. Numerical studies on the a posteriori error estimation by

Kunert on anisotropic mesh generated by error-adapted refinement process.

4. Extend our stopping criteria to different problems.5. Solve more difficult problems such as Navier-Stokes

equations by more accurate and efficient methods.

• Discretization :

Solutions from Galerkin and SDFEM discretizations.

• Error estimator :

Reliability of a posteriori error estimators, based on residual and based on solving a local problem.

• Mesh improvement:

Moving mesh and error-adapted mesh refinement strategy

• Linear Solver :

1. Introduction of geometric multigrid and algebraic multigrid2. Convergence of line Gauss-Seidel and multigrid with line Gauss-Seidel

smoother when 3. Comparison of geometric multigrid and algebraic multigrid methods as a

solver and as a preconditioner of GMRES.4. Stopping criteria for iterative solvers based on a posteriori error bounds.

In this talk, we will discuss:

.h

Galerking and SDFEM

h 0 1h h D h T 1 h

h h h 0b h

h hb

Find u V { V : 0 on } with V { H ( ) : v| P (T), T } such that

( u , ) u , c u , f, , for all V ,

where, u b u .

v v v

v v v v v

Galerkin method :

SDFEM method :

h

h

h 0h

h h hT b b T b T b T b

h h h h h hb T b T b b T b T bT T T

T

h h h h h hb T b b T bT

T

Find u V such that

u , u , c u , f,

u , u , c u , u , u , cu , f,

u , u , c u , u u cu , f, f, ,

where,

v v v v v v v v

v v v v v v v v

v v v v v v

T

T

T

T e T,TT e

e

h , if P 1 b h= is the stablization parameter and the Peclet number P .

0 , if P 1 2

0

Problem 1: Downstream boundary layers:

Consider is a solution of the

convection-diffusion equation: 1 2( , ) 0u u

1 2

1 2

/ /

/ /

1 1( , )

1 1

x ye eu x y

e e

Dirichlet boundary condition given by where [0,1] [0,1].

with

|u

Solution from Galerkin discretization on 32x32 grid

Solution from SDFEM discretization on 32x32 grid

Mesh for compuitngerror

A Posteriori Error Estimation Based on Residual Proposed by Verfürth

Upper Bound:

Local lower Bound:

where

T T

1

2

1

2

TN

T

1/22

T E0;ωh h 0;EE T ωR,T TL (ω ) L (ω ) h || f-f || g-g 1 ||c|| ||b|| ||| u-u |||

12

1 1/2/2

h h h,N

2 2

T Eh h0;T 0;ET E

2Rh

E,T

T||| u-u ||| f-f g-g

and

1 1

2 2

E EN

2 222 2n nR,T T E Eh h h h h h h0;T EE T E T0;E 0;E

1 f u b u cu u g u2

1 1

2 2T T E E min h ,1 , min h ,1

A Posteriori Error Estimation Base on Solving a Local Problem Proposed by Kay and Silvester

Upper Bound:

Local lower Bound:

hh,

E EE h,D E

hh,N

E

uE E

nR = 0 for all E E and R

u-2 E E

n

T T T

0 0 0 2 0T h T T T T

for all consisting of edge and interior bubble functions respectively, where

R f-b u , R = R , is the L projection onto the space P (T) and

v Q B

E0

T T T ET E TR ,( e , ) R , ,

2vv v

T T THere, = e and e is a solution of the following local problem

2

02 21/ 2

Chh

h T TT

TT T

T

hRe R

0CT T

T T

TT TT h h T

T

TT

T

hR R

he b e

Comparison of VR and KS Error Indicators (i)

Problem 1:

1

1024

Comparison of VR and KS Error Indicators (ii)

14.0419.9628.6241.491/4096

6.9669.56213.5819.681/1024

4.5555.4157.13710.011/256

5.0644.8254.6735.7641/64

64x6432x3216x168x8

2.8674.0755.8428.4701/4096

1.4221.9522.7724.0161/1024

0.9291.1051.4572.0441/256

1.0220.9790.9511.1561/64

64x6432x3216x168x8

E of VR indicator E of KS indicator

Comparison of the global effectivity indices

2 1/ 2,( )

E||| |||

h

h TT

hu u

h

90.54181.0362.0724.11/4096

22.6745.2690.51181.01/1024

8.62712.4622.6945.261/256

7.7418.6018.62012.431/64

64x6432x3216x168x8

18.4836.9573.90147.81/4096

4.6299.23918.4836.951/1024

1.7502.5364.6379.2421/256

1.5571.6871.7142.5041/64

64x6432x3216x168x8

ET of VR indicator ET of KS indicator

Comparison of the local effectivity indices,

TE max||| |||h

h T

Th Tu u

h

Moving Mesh and Error-Adapted Mesh Refinement

Why mesh movement?

• A heuristic strategy for increasing the accuracy of numerical solution.

• Mesh movement tends to cluster nodes in the area with sharp gradient.

We move meshes by following the equidistribution principle where Kay and Silvester’s error indicator is employed as a monitor function.

Why error-adapted mesh refinement?

• To resolve boundary layers, regular mesh refinement may take too many steps and generate too many nodes.

• Error-adapted mesh refinement is able to cluster nodes to the locations where sharp gradients appear.

• Moving mesh destroy the nested grid structure from adaptive refinement process. Error-adapted refinement maintain nested grid structure.

Mesh Moving by Equidistributionix

jx

jdx

Soluiton on refinemnet mesh Soluiton on movement + refinement mesh

Moving Mesh (I)

Problem 2

Moving Mesh (II)

Solution on refinement mesh

Solution on movement + refinement mesh

1 2

2 21 2

, 0 on =[-1,1] [0,1] ,

u(x)=1 (-0.5 x<0 y=0) or x=1,

u(BC) 0 0 x<1 y=0,

n0 otherwise,

and , (2 (1 ) , 2 (1 ))

u u

y x x y

Problem 3: IAHR/CEGB:

Error-Adapted Mesh Refinement

1. Compute recovered error indicator for every node

2. Compute external force for each edge

3. Modify external force to

and

4.

i=1 ni

i

i

i TT

ii

T

T

T

i,ji,j i

i,j

eF =( - )

|e |j i,je .

*

*

,*

, i,j j,

i,j

,*

, i,j j,

| |1 if e and ,

| |

F| |

1 if e and ,| |

h

h

i je E

i j h ii j

i je E

j i h ii j

min Fe E

F

min Fe E

F

* *i,j i,jF 0 for e , where is the set of edges in marked elements.h hE E

i ,*i,j

xFor each edge e , place new node at .

2j i j

h

x FE

Choice of error sensitivity parameter

0.05 and =1

0.25 and =1/3

0.25 and =1

0.25 and =0

Problem 2:

Error-Adapted Mesh Refinement v.s. Regular Mesh Refinement (I)Problem 2 with ε=10-4

Number of node

Regular Refinement

Error-adapted

Refinement

7001 2729

1. 16 refinement steps are performed.2 Threshold = 0.5 in maximum marking strategy.

Variant IAHR/CEGB problem

1 2

2 21 2

, 0 on =[-1,1] [0,1] ,

u(x)=1 -0.5 x<0 y=0,

u(BC) 0 0 x<1 y=0,

n0 otherwise,

and , ( (4 (1 ) ), 2(1 )(1 ))

u u

y x x y

Flow Field:

Error-Adapted Mesh Refinement v.s. Regular Mesh Refinement (II)variant IAHR/CEGB problem with ε=10-4

Number of node

Regular Refinement

Error-adapted

Refinement

2858 2749

1. 8 refinement steps are performed.2 Threshold = 0.25 in maximum marking strategy.

Driven Cavity Flow for Re=100

GMRES

FEM Discretization