adaptive multigrid fe methods -- an optimal way to solve pdes zhiming chen institute of...
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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
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
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 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
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