Post on 04-Feb-2018
7/21/2019 Talk Multiscale
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Multiscale Finite Elements
Basic methodology and theory for periodic coefficients for second-orderelliptic equations
Markus Kollmann
October 18th, 2011
7/21/2019 Talk Multiscale
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Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
Outline
1 Motivation
2 Introduction to MsFEM
3 Analysis in 2D
4 Reducing boundary effects
5 Generalization of MsFEM
Markus Kollmann Multiscale Finite Elements
d l d b d ff l f
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Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
Outline
1 Motivation
2 Introduction to MsFEM
3 Analysis in 2D
4 Reducing boundary effects
5 Generalization of MsFEM
Markus Kollmann Multiscale Finite Elements
i i d i l i i d i b d ff li i f
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Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
Motivation
Many scientific and engineering problems involve multiple scales,particularly multiple spatial and (or) temporal scales (e.g. compositematerials, porous media,...)
Difficulty of direct numerical solution: size of the computationFrom an application perspective: sufficient to predict the macroscopicpropertiesof the multiscale systems
Multiscale modeling: calculation of material properties or systembehaviour on the macroscopic level using information or models frommicroscopic levels (capture the small scale effect on the large scale,without resolving the small-scale features)
Markus Kollmann Multiscale Finite Elements
M i i I d i M FEM A l i i 2D R d i b d ff G li i f M FEM
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Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
Outline
1 Motivation
2 Introduction to MsFEM
3 Analysis in 2D
4 Reducing boundary effects
5 Generalization of MsFEM
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
Introduction
Capture the multiscale structure of the solution via localized basisfunctions
Basis functions contain information about the scales that are smaller thanthe local numerical scale (multiscale information)
Basis functions are coupled through a global formulation to provide afaithful approximation of the solution
Two main ingredients of MsFEM:Global formulation of the method
Construction of basis functions
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
Problem Formulation
Consider the linear elliptic equation
Lu=f in ,
u=0 on ,(1)
where
Lu:=div(k(x)u) .
... domain in Rd (d=2, 3)
k(x) ... heterogeneous field varying over multiple scales
Additionally assume:k(x) = (kij(x)) is symmetric
||2 kijij ||2 Rd (0< < )
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
Problem Formulation, contd.
Variational formulation of(1):
Find uH10 () such thata(u, v) =f, v, v H10 (),
where
a(u, v) =
ku vdx and f, v=
fvdx.
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
Basis Functions
LetTh be a partition of into finite elements K(coarse grid which can beresolved by a fine grid).Letxibe the interior nodes ofTh and 0i be the nodal basis of the standardfinite element space Wh =span{0i}.
Definition of multiscale basis functions i:Li =0 in K, i =
0i on K, K Th, K Si, (2)
where Si =supp(0i).
Denote by Vh the finite element space spanned by i
Vh =span(i).
((2)is solved on the fine grid)
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
10/31
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
Basis Functions, contd.
Computational regions smaller than Kare used if one can use smaller regions(Kloc) to characterize the local heterogeneities within the coarse-grid block(e.g. periodic heterogeneities). Such regions are called Representative Volume
Elements (RVE).
Definition of multiscale basis functions i:
Li =0 in Kloc, i =0i on Kloc, Kloc Th, Kloc Si;
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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y g y
Source: [Y. Efendiev and T.Y. Hou, Multiscale Finite Element Methods:Theory and Applications, Springer, New York, 2009]
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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y g y
Global Formulation
The representation of the fine-scale solution via multiscale basis functionsallows reducing the dimension of the computation. When the approximation ofthe solution uh =
iuii is substituted into the fine-scale equation, the
resulting system is projected onto the coarse-dimensional space to find ui.
The MsFEM reads:
Find uhVh such that:
KK
k
uh
vhdx=
fvhdx
vh
Vh (3)
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
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Global Formulation, contd.
E.g. (3) is equivalent to
Aunodal=b, (4)
where A= (aij) with
aij=K
K
ki jdx,
unodal= (u1, ..., ui,...) are the nodal values of the coarse-scale solution andb= (bi) with
bi =
fi.
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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Outline
1 Motivation
2 Introduction to MsFEM
3 Analysis in 2D
4 Reducing boundary effects
5 Generalization of MsFEM
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
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Repetition
Consider (periodic case)
Lu=f in , u=0 on , (5)
where
Lu:=div (k(x/)u) ,with kij(y), y=x/ smooth periodic in y in a unit square Y ( is a smallparameter), fL2() and a convex polygonal domain.Looking for expansion:
u=u0(x, x/) +u1(x, x/) +...
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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Repetition, contd.
u0 =u0(x) satisfies the homogenized equation:
L0u0 :=div (ku0) =f in , u0 =0 on , (6)where
k
ij =
1
|Y| Y kil(y)ljj
yl
dy,
and j is the periodic solution of
divy
k(y)yj
=
yikij(y) in Y,
Y
j(y)dy=0.
In addition we have
u1(x, y) =j(y) u0xj
(x). (7)
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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Repetition, contd.
Note that
u0(x) +u1(x, y)=u on ,therefore we introduce a first order correction term :
L =0 in , =u1(x, y) on , (8)
then u0(x) +(u1(x, y) ) satisfies the boundary condition ofu.Now we have the following homogenization result:
Lemma 1
Let u0
H2() be the solution of (6),
H1() be the solution of (8) and
u1 be given by (7). Then there exists a constant C independent of u0, andsuch that
u u0 (u1 )1,Cu02,. (9)
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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Convergence for h <
Multiscale method and standard linear finite element method are closely related.First we have Cas lemma
Lemma 2
Let u and uh be the solutions of (1) and (3) respectively. Then
u uh1,C infvhVh u vh1,, (10)
and the regularity estimate
|u|2, Cf0, (11)
(1/ is due to small-scale oscillations in u).
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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Convergence for h <
Lagrange interpolation operator:
h :C()Wh
hu(x) :=J
j=1
u(xj)0j(x)
Interpolation operator defined through multiscale basis functions:
Ih :C()Vh
Ihu(x) :=J
j=1
u(xj)j(x)
From (2) we have
L(Ihu) =0 in K, Ihu= hu on K, K Th (12)
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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Convergence for h <
Lemma 3
Let uH2() be the solution of(1). Then there exist constants C1 >0 andC2 >0, independent of h and, such that
u Ihu0,C1 h2
f0,,
u Ihu1,C2h
f0,.(13)
Theorem 4
Let uH2() and uh be the solutions of (1)and (3)respectively. Then thereexists a constant C, independent of h and, such that
u uh1,Chf0,. (14)
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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Convergence for h >
Convergence result uniform in as tends to zero (main feature of the MsFEMover the traditional FEM).Main result:
Theorem 5
Let uH2() and uh be the solutions of (1)and (3)respectively. Then thereexist constants C1, C2, independent of h and, such that
u uh1,C1(h+)f0,+C2
h1/2
. (15)
Lemma 6
Let uH2() be the solution of(1) and uI =Ihu0Vh. Then there existconstants C1, C2, independent of h and, such that
u uI1,C1(h+)f0,+C2
h
1/2, (16)
where u0H2() W1,() is the solution of the homogenized equation (6).
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
f h
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Convergence for h >
From (12)
LuI =0 in K, uI = hu0 on K, K Th.Thus (in K)
uI =uI0+uI1 I+...where
L0uI0 =0 in K, uI0 = hu0 on K,
uI1(x, y) =j(y) uI0xj
(x),
LI =0 in K, I =uI1(x, y) on K.
Note that
uI0 = hu0 in K, (17)
becausehu0 is linear on K.
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
C f h
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Convergence for h >
Lemma 7
There exists a constant C, independent of h and, such that
uI uI0 uI1+I1,Cf0,. (18)
proof:
By standard approximation theory
uI02,K uI0 u02,K+ u02,K Cu02,K.Using (9)
uI uI0 uI1+I1,K CuI02,K Cu02,K.Summing over Kand using the regularity estimate
u02,Cf0, (19)finishes the proof.
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
C f h >
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Convergence for h >
Now we have
u uI1,u (u0+(u1 ))1,+ (uI0+(uI1 I)) uI1,+ (u0+(u1 )) (uI0+(uI1 I))1,.
Using (9), the last Lemma, the regularity estimate (19)and the triangleinequality we get:
u uI1,Cf0,+ u0 uI01,+ (u1 uI1)1,+ ( I)1,.
(20)
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
Convergence for h >
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Convergence for h >
Lemma 8
We haveu0 uI01,Chf0,,
(u1 uI1)1,C(h+)f0,.(21)
Lemma 9
We have
1,C
. (22)
Lemma 10
We have
I1,C
h
1/2. (23)
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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Outline
1 Motivation
2 Introduction to MsFEM
3 Analysis in 2D
4 Reducing boundary effects
5 Generalization of MsFEM
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
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Motivation
Boundary conditions for basis functions play a crucial role
If local boundary conditions do not reflect the nature of the underlyingheterogeneities, MsFEM can have large errorsh
term in the convergence rate is large when h
(resonance effect)
h term comes from I
Remember: I term is due to the mismatch between the fine-scalesolution and MsFEM solution along the boundaries of the coarse-gridblock (MsFEM solution is linear there)
Mismatch propagates into the interior of the coarse-grid block
But boundary layer is thin (oscillations decay quickly as we move awayfrom the boundaries)
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
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Oversampling
Sample in a domain with size larger than h and use only the interiorsampled information to construct the basis functions
Doing this, the influence of the boundary layer in the larger sample domainon the basis functions is reduced
LetEj satisfying
LEj =0 in KE K, Ej =0j on KE,
then we form i by
i =
j
cijEj
where cijare determined by i(xj) =ij for nodal points xj.
= Nonconforming MsFEM
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
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Source: [Y. Efendiev and T.Y. Hou, Multiscale Finite Element Methods:Theory and Applications, Springer, New York, 2009]
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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Outline
1 Motivation
2 Introduction to MsFEM
3 Analysis in 2D
4 Reducing boundary effects
5 Generalization of MsFEM
Markus Kollmann Multiscale Finite Elements
Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM
7/21/2019 Talk Multiscale
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General Framework of MsFEM
Lu=f (24)
where L: X Y is an operator.Multiscale basis functions are replaced by multiscale maps EMsFEM :WhVh.For each vhWh, vr,h =E
MsFEM
vh is defined as:Lmapvr,h =0 in K.
Lmap captures the small scales.Solving (24):
Find ur,hVh such that:Lglobal
ur,h, vr,h=f, vr,h, vr,hVh.Correct choices ofLmap and Lglobal are the essential part of MsFEM (can bedifferent) and guarantee the convergence of the method.
Markus Kollmann Multiscale Finite Elements