Adaptive Multigrid FE Methods Adaptive Multigrid FE Methods -- -- An optimal way to solve PDEsAn optimal way to solve PDEs

Zhiming Chen

Institute of Computational Mathematics

Chinese Academy of Sciences

Beijing 100080

Adaptive Concept

Refine: more nodes around singularities

Coarsen: less nodes in smooth region

x

)(xf

An “optimal mesh” is the mesh on which the error is approximately the same on each element. This motivates the error equi-distribution strategy.

The adaptive FEM based on a posteriori error estimates provides a systematic way to refine or coarsen the mesh according to the local a posteriori error estimators on each element.

The adaptive method finds the solution of given tolerance on a self-generated mesh according to the properties of the solution (singularities,oscillations).

A Linear Elliptic ProblemA Linear Elliptic Problem

Elliptic problem with piecewise constant coefficients:

Variational problem:

on 0 ,in )( ufuxa

)( )( 10

Hvdxfvdxvuxa

Discrete Problem: find such that

A priori error estimate:

hh Vu

hhhhh Vvdxfvdxvuxa )(

10 ,)(max)( 1

HEh uChuu

dxxahhEKK h

22

)(max | |)( ,max

nodes ofNumber

1 :mesh Uniform max h

A posteriori error estimate (Babuska & Miller, 1987)

The error indicator

where

2

1

2

)(

hKKEh Cuu

2

)(

2

12

)(

2

2

2

eLKeeeKLKK Jhfh

eKhKhe uxauxaJ 21

|))((|))((

1K

2K

*

2*

2

)(22

1

2

)()(

KTTLTTKKEh ffhCCuu

. , ||

1h

T

T TdxfT

f

Theorem (Verfürth, 1992): We have

where K

*K

Adaptive AlgorithmAdaptive Algorithm

Solve → Estimate → Refine/Coarsen

Error equi-distribution strategy

where tolerance, constant ,

number of elements in

hK KM

refine

If

: : )1( :M h

Numerical ExperimentsNumerical Experiments

where and

Exact solution (Kellogg)

)1,1()1,1(

in 0)div(a(x) u

45.1611 a 12 a

smooth 0.1, ),( ru

1.0 ),( 1 Hu

1a

1a

2a

2a

The exact solution for .1.0

FEM with uniform mesh

128x128 mesh:

512x512 mesh:

1024x1024 mesh:

8547.0)(

Ehuu

6954.0)(

Ehuu

A priori error analysis implies that one must introduce nodes in each space direction to bring the energy error under 0.1.

1110

7981.0)(

Ehuu

Convergence rate: 08.0maxh

The surface plot of the relative error

The maximum of the relative error is 0.2368.

||max/)( )1024 uuu

The adaptive mesh of 2673 nodes. The energy error is 0.07451.

The surface plot of the adaptive solution and the relative error . The maximum of the relative error is0.0188.

2673u||max/)( 2673 uuu

Definition

Let be the sequence of FE solutions generated by the adaptive algorithm. The meshes and the associated numerical complexity are called quasi-optimal if

are valid asymptotically. DOFs(k) is the number of degree of freedoms of the mesh .

ku

2

1

)()(DOFs

kCuu

Ek

h

1

)()(DOFs

kCuu

Lk

y)(Optimalit

Quasi-optimality of the estimators. The quasi-optimal decay is

indicated by the dotted line of slope –1/2.

Gauss-Seidel Iteration MethodGauss-Seidel Iteration Method

ibxaxaxa

bydefinedisxxxxGiven

Nibxa

bAx

iij

kjij

kiii

kj

ijij

kkN

kk

ij

N

jij

),,...,(

,...,1

)()1()1(

)1()()(1

)(

1

ijjijijijiij fhuuuuu 21,1,,1,1 ][4

0 , onuinfu

5-point finite difference scheme

Multigrid V-cycle AlgorithmMultigrid V-cycle AlgorithmspacesFEXXX J ... 21

mmiyAfRyy

yAfQByy

miyAfRyy

givenXy

beyfBletXfjAB

byyrecursiveldefinedisXXBmGiven

ijjii

mjjjmm

ijjii

j

mjj

jjj

2,...,1 ),(

)(

,...,1 ),(

,,1,

: ,1

11

11

11

0

121

11

)(

)()()1( k

JJJJkJ

kJ

JJJ

uAfBuu

methoditerativethebyfuAsystemtheSolve

Adaptive Multigrid MethodAdaptive Multigrid Method

◆ Local relaxation: Gauss-Seidel relaxation performed only on new nodes and their immediate neighboring nodes

◆ Each multigrid iteration requires only O(N)

number of operations

◆ Theorem (Wu and Chen): We have

◆ Numerical Example

JandoftindependenforABI JJ X 1 j

ApplicationsApplications

Continuous casting problem

Chen, Nochetto and Schmidt (2000)Wave scattering by periodic structure

Chen and Haijun Wu (2002)Convection diffusion problem

Chen and Guanghua Ji (2003)

Continuous Casting problemContinuous Casting problem

.in )()0,(

),,0(on 0)( ),,0(on

,in )(

,in 0)(

0

0

xuxu

TpTg

Qu

Qutvu

NextD

T

Tzt

s

例子：振荡铸钢速度系数例子：振荡铸钢速度系数

.225,1),10(|)(| :

]/)[00175.0sin(*005.00175.0)(

][75.2][000,10 ],)[25,21.0(

2

vOu

smttv

hsTm

无量纲化常数

变化铸钢速度 : 速度 v(t), 单元个数 , 时间步长

t=0.05 和 t=0.07 时的网格和温度。

周期结构上电磁波的散射问题

0i

0i

EH

HE

.)()(

0)(

1

)(

1

)0,,0( :onPolarizati TM

22

32

312

1

T

xxk

ux

u

xkxx

u

xkx

uH

其中

221 i) 71.622.0( ,1

A Linear Convection Diffusion Problem

Rotating Cylinder problem:

Convergence Rate

Epsilon=10e-3 Epsilon=10e-5

Thank you !Thank you !