adaptive multigrid fe methods -- an optimal way to solve pdes
DESCRIPTION
Adaptive Multigrid FE Methods -- An 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. - PowerPoint PPT PresentationTRANSCRIPT
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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
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Adaptive Concept
Refine: more nodes around singularities
Coarsen: less nodes in smooth region
x
)(xf
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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).
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A Linear Elliptic ProblemA Linear Elliptic Problem
Elliptic problem with piecewise constant coefficients:
Variational problem:
on 0 ,in )( ufuxa
)( )( 10
Hvdxfvdxvuxa
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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
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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
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*
2*
2
)(22
1
2
)()(
KTTLTTKKEh ffhCCuu
. , ||
1h
T
T TdxfT
f
Theorem (Verfürth, 1992): We have
where K
*K
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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
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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
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The exact solution for .1.0
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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
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The surface plot of the relative error
The maximum of the relative error is 0.2368.
||max/)( )1024 uuu
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The adaptive mesh of 2673 nodes. The energy error is 0.07451.
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The surface plot of the adaptive solution and the relative error . The maximum of the relative error is0.0188.
2673u||max/)( 2673 uuu
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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
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Quasi-optimality of the estimators. The quasi-optimal decay is
indicated by the dotted line of slope –1/2.
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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
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ijjijijijiij fhuuuuu 21,1,,1,1 ][4
0 , onuinfu
5-point finite difference scheme
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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
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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
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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)
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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
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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
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t=0.05 和 t=0.07 时的网格和温度。
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周期结构上电磁波的散射问题
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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
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221 i) 71.622.0( ,1
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A Linear Convection Diffusion Problem
Rotating Cylinder problem:
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Convergence Rate
Epsilon=10e-3 Epsilon=10e-5
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Thank you !Thank you !