talk multiscale

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


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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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Outline

1 Motivation


3 Analysis in 2D




i i d i l i i d i b d ff li i f


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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)


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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Outline

1 Motivation


3 Analysis in 2D






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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




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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< < )




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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.




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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)




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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;




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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]




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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)




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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.




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Outline

1 Motivation


3 Analysis in 2D






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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/) +...




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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)




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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)




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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).




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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)




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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)




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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).



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.



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.



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)



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)




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Outline

1 Motivation


3 Analysis in 2D






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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)




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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




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Source: [Y. Efendiev and T.Y. Hou, Multiscale Finite Element Methods:Theory and Applications, Springer, New York, 2009]




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Outline

1 Motivation


3 Analysis in 2D






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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.


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