a priori error analysis of fully discrete fe-hmm...a priori error analysis of fully discrete fe-hmm...

A Priori ErrorAnalysis of Fully

Discrete FE-HMM

Monika Wolfmayr

Introduction

FE-HMM forElliptic Problems

Elliptic model problem

First convergenceresults

Error Analysis ofthe Fully DiscreteFE-HMM

Fully discrete FEproblem

Convergence resultsfor the macrosolution

H1-error

L2-error

L2-projection of uε

Convergence resultsfor the fully discretesolution

H1-error

Conclusions

A Priori Error Analysis of Fully DiscreteFE-HMM

Monika Wolfmayr

5th December 2011


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Outline

Introduction

FE-HMM for Elliptic ProblemsElliptic model problemFirst convergence results

Error Analysis of the Fully Discrete FE-HMMFully discrete FE problemConvergence results for the macrosolution

H1-errorL2-errorL2-projection of uε

Convergence results for the fully discrete solutionH1-error

Conclusions


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Introduction

Heterogeneous Multiscale Method (HMM) introduced by Eand Engquist in 2003

the name heterogeneous was used to emphasize that themodels at different scales may be of very different nature

Difference between traditional MM and HMM:

MM: general purpose are microscale solvers, i.e. to resolvethe details of the solutions of the microscale model

HMM: objective is to capture the macroscale behavior of thesystem with a cost that is much less than the cost of fullmicroscale solvers


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Introduction

HMM:physical problem is directly discretized by a macroscopicfinite element method model (coarse scale)microproblems are either unit-cell problems or problems on apatch with a fixed number of unit cells (fine scale)

study of accuracy properties in HMM:

I first approach: assumption that the microproblems areanalytically given; macro- and microerrors oftenseparately estimated

I further approach: combination of microscopic andmacroscopic models; microproblems are solvednumerically as well; estimates for the errors transmittedon the macroscale by discretizing the fine scale(Abdulle, 2005)

analysis for piecewise linear continuous FEMs in the micro-and macrospaces and for the periodic case


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Notation

r = (r1, ..., rn) ∈ Nn, |r | = r1 + ...+ rn, Dr = ∂r11 ...∂rnn ;

H1(Ω) = {u ∈ L2(Ω); Dru ∈ L2(Ω), |r | ≤ 1},

‖u‖H1(Ω) =

∑|r |≤1

‖Dru‖2L2(Ω)

1/2 ;W l ,∞(Ω) = {u ∈ L∞(Ω); Dru ∈ L∞(Ω), |r | ≤ l};

W 1per (Y ) = {v ∈ H1per (Y );∫Y

v dx = 0};

H10 (Ω) = C∞0 (Ω)

‖·‖H1 , H1per (Y ) = C∞per (Y )

‖·‖H1 , Y = (0, 1)n


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Elliptic Model Problem

−∇ · (aε∇uε) = f in Ω ⊂ Rn

uε = 0 on ∂Ω,(1)

ε ... length scale, Ω ... convex polygon, f ∈ L2(Ω),aε(x) = a(x , xε ) = a(x , y) ... symmetric and coercive tensor,periodic with respect to each component in Y = (0, 1)n,

aij(x , ·) ∈ L∞(Rn)

uε converges weakly to a homogenized solution u0 of

−∇ · (a0∇u0) = f in Ωu0 = 0 on ∂Ω,

(2)

a0 ... smooth matrix with coefficientsa0ij(x) =

∫Y (aij(x , y) +

∑nk=1 aik(x , y)

∂χj

∂yk(x , y))dy ,

χj(x , ·) ... solution of the cell problems


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Elliptic Model ProblemMacro FE space:

S10 (Ω, TH) = {uH ∈ H10 (Ω); uH |K ∈ P1(K ) ∀K ∈ TH}, (3)

P1(K ) ... space of linear polynomials on the triangle K ,TH ... quasi-uniform triangulation of Ω of shape regular K ,H ... size of triangulation

Macrobilinear form:

B(uH , vH) =∑K∈TH

|K ||Kε|

∫Kε

∇u a(xk , x/ε)(∇v)Tdx , (4)

Kε = xk + ε[−1/2, 1/2]n ... sampling subdomain

u is the solution of the exact microproblem:Find u such that (u − uH) ∈W 1per (Kε) and∫

Kε

∇u a(xk , x/ε)(∇z)Tdx = 0 ∀z ∈W 1per (Kε). (5)


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions


It can be shown that

u = uH + εn∑

j=1

χj(xk , x/ε)∂uH(xk)

∂xj, (6)

χj(xk , y) ... unique solutions of the cell problems:∫Y∇χja(xk , y)(∇z)Tdy = −

∫Y

eTj a(xk , y)(∇z)Tdy (7)

for all z ∈W 1per (Y ); {ej}nj=1 ... standard basis in Rn.


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions


Variational problem for the macrosolution:Find uH ∈ S10 (Ω, TH) such that

B(uH , vH) = 〈f , vH〉 ∀vH ∈ S10 (Ω, TH). (8)

B(·, ·) elliptic, bounded ⇒ unique solution of (8)

It can be shown that

B(uH , vH) =∑K∈TH

∫K∇uH a0(xk)(∇vH)Tdx . (9)

Remark:Since we assume an exact microsolver, so far the variationalproblem for the macrosolution (8) is of semidiscrete nature.


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

First convergence estimates for the macrospace

Assumption: H2-regularity for u0, exact microsolution

‖uε − u0‖L2(Ω) ≤ C ε ‖f ‖L2(Ω), (10)‖uε − uH‖H1(Ω) ≤ C (H/ε) ‖f ‖L2(Ω), (11)‖u0 − uH‖H1(Ω) ≤ C H ‖f ‖L2(Ω), (12)‖uε − uεp‖H̄1(Ω) ≤ C (

√ε+ H) ‖f ‖L2(Ω), (13)

uεp ... reconstructed solution from uH with (u − uH)

periodically extended on each K , can be discontinuousacross K , hence H̄1-norm is mesh-dependent;

‖Puε − uH‖H1(Ω) ≤ C (ε/H + H) ‖f ‖L2(Ω), (14)

Puε ... L2-projection of the solution


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Micro FE problemSampling domain: Kε = xk + ε[−1/2, 1/2]n

Micro FE space:

S1per (Kε, Th) = {uh ∈W 1per (Kε); uh|T ∈ P1(T ), T ∈ Th},(15)

Th ... quasi-uniform triangulation of Kε with meshsize h,S1per (Kε, Th) ⊂W 1per (Kε)

Discrete microproblem:For uH ∈ S10 (Ω, TH) find uh such that(uh − uH) ∈ S1per (Kε, Th) and∫

Kε

∇uh a(xk , x/ε)(∇zh)Tdx = 0 ∀zh ∈ S1per (Kε, Th).

(16)


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Fully discrete FE problemIt can be shown that

uh = uH + εn∑

j=1

χj ,h(xk , x/ε)∂uH(xk)

∂xj, (17)

χj ,h(xk , y) ... unique solutions of the cell problems inS1per (Kε, Th)

Fully discrete macrobilinear form:

B̄(uH , vH) =∑K∈TH

|K ||Kε|

∫Kε

∇uh a(xk , x/ε)(∇vh)Tdx (18)

Variational problem for the fully discrete macrosolution:Find ūH ∈ S10 (Ω, TH) such that

B̄(ūH , vH) = 〈f , vH〉 ∀vH ∈ S10 (Ω, TH). (19)


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Fully discrete FE problem

Proposition

The problem (19) has a unique solution which satisfies

‖ūH‖H1(Ω) ≤ C‖f ‖L2(Ω). (20)

Proof.Verify the assumptions of the Lax-Milgram Theorem.

Assumption: the solutions χj of the cell problems satisfy

χj(xk , ·) ∈W 2,∞(Y ).

If χj(xk , y) = χj(xk , x/ε), then

‖Dαx (χj(xk , x/ε))‖L∞(Kε) ≤ Cε−|α|, |α| ≤ 2, α ∈ Nn. (21)


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Fully discrete FE problem

LemmaSuppose that the solutions χj of the cell problems satisfy(21). Then the following estimation holds:

∣∣B̄(vH ,wH)− B(vH ,wH)∣∣ ≤ C (hε

)2‖∇vH‖L2(Ω)‖∇wH‖L2(Ω),

(22)

where vH ,wH ∈ S10 (Ω, TH), h is the mesh size of the microFE space S1per (Kε, Th).


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Proof of the Lemma - Part I

∣∣B(vH ,wH)− B̄(vH ,wH)∣∣=

∣∣∣∣ ∑K∈TH

|K ||Kε|

∫Kε

[∇v a(xk , x/ε)(∇w)T −∇vh a(xk , x/ε)(∇wh)T ]dx∣∣∣∣

Inserting +/−∇vh a(xk , x/ε)(∇w)T yields

=

∣∣∣∣ ∑K∈TH

|K ||Kε|

[

∫Kε

∇ (v − vh)︸︷︷︸∈W 1per (Kε)

a(xk , x/ε) (∇w)T︸︷︷︸∈W 1per (Kε)

dx

︸︷︷︸=(5)0

−∫Kε

∇vh a(xk , x/ε)(∇(wh − w))Tdx ]∣∣∣∣.


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Proof of the Lemma - Part IIDue to (5), we have

∫Kε∇v a(xk , x/ε)(∇(wh − w))Tdx = 0.

Together with the boundedness of the bilinear form, we obtain∣∣B(vH ,wH)− B̄(vH ,wH)∣∣=

∣∣∣∣ ∑K∈TH

|K ||Kε|

∫Kε

∇(vh − v) a(xk , x/ε)(∇(wh − w))Tdx ]∣∣∣∣

≤ C∑K∈TH

|K ||Kε|‖∇(vh − v)‖L2(Kε)‖∇(w

h − w)‖L2(Kε).

Moreover,

‖∇(vh − v)‖L2(Kε)

=

∥∥∥∥ε n∑j=1

∇(χj,h(xk , x/ε)− χj(xk , x/ε))∂vH(xk)

∂xj

∥∥∥∥L2(Kε)

≤∥∥∥∥εmaxj ∇(χj,h − χj)

n∑j=1

∂vH(xk)

∂xj

∥∥∥∥L2(Kε)

.


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Proof of the Lemma - Part III

Since ∇vH is constant,

‖∇(vh − v)‖L2(Kε)

≤∥∥εmax

j∇(χj,h − χj)

∥∥L2(Kε)

n∑j=1

∂vH

∂xj

≤ C εmaxj

∥∥∇(χj,h − χj)∥∥L2(Kε)

(n∑

j=1

(∂vH

∂xj)2)

12

≤ C εh√|Kε|max

j

∣∣χj ∣∣W 2,∞(Kε)

√∇vH∇vH

≤(21) C εh√|Kε|Cε−2

√∇vH∇vH

= C

(h

ε

)√|Kε|√∇vH∇vH .


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Proof of the Lemma - Part IVAltogether, we obtain∣∣B(vH ,wH)− B̄(vH ,wH)∣∣≤ C

∑K∈TH

|K ||Kε|‖∇(vh − v)‖L2(Kε)‖∇(w

h − w)‖L2(Kε)

≤ C∑K∈TH

|K ||Kε|

(h

ε

)√|Kε|√∇vH∇vH

(h

ε

)√|Kε|√∇wH∇wH

= C

(h

ε

)2 ∑K∈TH

√|K |√|Kε|

√|Kε|√∇vH∇vH

√|K |√|Kε|

√|Kε|√∇wH∇wH

= C

(h

ε

)2 ∑K∈TH

√|K |√∇vH∇vH

√|K |√∇wH∇wH .

Since ∇vH and ∇wH are constant,

= C

(h

ε

)2 ∑K∈TH

‖∇vH‖L2(K)‖∇wH‖L2(K).

Summing up over K finally yields (22). �


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

H1-error between the discrete and the fullydiscrete macrosolutionRemark:If M = dim S1per (Kε), then h ' εM−

1n with |Kε| = εn.

=⇒ h/ε independent of ε

Proposition

Let uH , ūH be the solutions of the variational problems forthe macrosolution and the fully discrete macrosolution,respectively, and suppose that the assumptions of theLemma hold. Then

‖uH − ūH‖H1(Ω) ≤ C M−2n ‖f ‖L2(Ω). (23)

Proof.Use coercivity of the bilinear forms, the Lemma before andthe Proposition about existence and uniqueness of the fullydiscrete solution.


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

H1-error between the homogenized and the fullydiscrete macrosolution

TheoremAssume the solution of the homogenized problem u0 isH2-regular and ūH is the fully discrete solution. Let theassumptions of the Lemma hold. Then

‖u0 − ūH‖H1(Ω) ≤ C (H + M−2n )‖f ‖L2(Ω). (24)

Proof.Use triangle inequality, ‖u0 − uH‖H1(Ω) ≤ C H ‖f ‖L2(Ω) and(23) from the Proposition before.


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

L2-error between the homogenized/exact solutionand the fully discrete macrosolution

Corollary

Suppose that the assumptions of the Theorem before hold.Then

‖u0 − ūH‖L2(Ω) ≤ C (H2 + M−2n )‖f ‖L2(Ω), (25)

‖uε − ūH‖L2(Ω) ≤ C (H2 + ε+ M−2n )‖f ‖L2(Ω). (26)

Proof.Use triangle inequality, ‖u0 − uH‖L2(Ω) ≤ C H2 ‖f ‖L2(Ω),‖uε − u0‖L2(Ω) ≤ C ε ‖f ‖L2(Ω) and (23) from the Propositionbefore.


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

H1- and L2-error between the L2-projection of thesolution and the fully discrete macrosolution

For u ∈ H1(Ω), we define Pu ∈ S10 (Ω, TH) as unique solutionof

〈Pu, vH〉 = 〈u, vH〉 ∀vH ∈ S10 (Ω, TH). (27)

TheoremLet Puε be the solution projected on S10 (Ω, TH) and ūH bethe fully discrete solution. Suppose that the assumptions ofthe Lemma hold and that u0 is H2-regular. Then

‖Puε − ūH‖H1(Ω) ≤ C (ε

H+ H + M−

2n )‖f ‖L2(Ω), (28)

‖Puε − ūH‖L2(Ω) ≤ C (ε+ H2 + M−2n )‖f ‖L2(Ω). (29)

Proof.Use triangle inequality, ‖Puε − uH‖H1(Ω) ≤ C (ε/H + H) ‖f ‖L2(Ω),‖Puε − uH‖L2(Ω) ≤ C (ε+ H2) ‖f ‖L2(Ω) and (23).


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Procedure to retrieve the microscopic information

We define the fully discrete fine-scale approximation of uε by

ūεp(x) = ūH(x) + (uh(x)− ūH(x))|PK for x ∈ K ∈ TH ,

(30)

where |PK denotes the periodic extension of the fine-scalesolution (uh − ūH), available in Kε on each K .Extension is defined for w ∈ H1(Kε):

wp(x + εl) = w(x) ∀l ∈ Zn ∀x ∈ Kε s. t. x + εl ∈ K .

ūεp can be discontinuous across K . Hence, we define thefollowing broken H1-seminorm:

|u|H̄1(Ω) :=

∑K∈TH

‖∇u‖2L2(K)

12 . (31)


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

H1-error between the exact and the fully discretesolution

TheoremSuppose that the assumptions of the Lemma hold. Then theerror between the exact and the fully discrete solution can beestimated by

|uε − ūεp|H̄1(Ω) ≤ C (√ε+ H + M−

1n )‖f ‖L2(Ω). (32)


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Proof of the Theorem - Part II

|uεp − ūεp |2H̄1(Ω) ≤ C∑K∈TH

‖∇uH −∇ūH‖2L2(K)︸︷︷︸=(I )

+ C∑K∈TH

‖n∑

j=1

∇(εχj)(∂uH

∂xj− ∂ū

H

∂xj

)‖2L2(K)︸︷︷︸

=(II )

+ C∑K∈TH

‖n∑

j=1

∇(ε(χj − χj,h))∂ūH

∂xj‖2L2(K)︸︷︷︸

=(III )

(I ) ≤ ‖uH − ūH‖2H1(K) ≤(23) (CM− 2n ‖f ‖L2(K))2


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Proof of the Theorem - Part III

(II ) = ‖n∑

j=1

∇(εχj)(∂uH

∂xj− ∂ū

H

∂xj

)‖2L2(K)

≤ ‖εmaxj∇(χj)

n∑j=1

(∂uH

∂xj− ∂ū

H

∂xj

)‖2L2(K)

≤ ε2‖maxj∇(χj)‖2L2(K)‖∇u

H −∇ūH‖2L2(K)

≤(21) ε2C (ε−1)2‖uH − ūH‖2H1(K) ≤(23) (CM− 2n ‖f ‖L2(K))2

(III ) = ‖n∑

j=1

∇(ε(χj − χj,h))∂ūH

∂xj‖2L2(K)

≤Pr.Lem.III C2(√∇ūH∇ūH)2ε2 max

j‖∇(χj − χj,h)‖2L2(K)

≤ C∇ūH∇ūHε2h2|K |maxj

∣∣χj ∣∣W 2,∞(K)︸︷︷︸

≤(21)(Cε−2)2

≤ Ch2ε−2|K |∇ūH∇ūH

≤ C (εM− 1n )2ε−2|K |∇ūH∇ūH


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Proof of the Theorem - Part IV

Since ∇ūH is constant,

(III ) ≤ CM− 2n ‖∇ūH‖2L2(K) ≤ CM− 2n ‖ūH‖2H1(K) ≤ CM

− 2n ‖f ‖2L2(K).

Altogether, we obtain the estimate

|uεp − ūεp |2H̄1(Ω) ≤ C∑K∈TH

2(CM−2n ‖f ‖L2(K))2 + C

∑K∈TH

CM−2n ‖f ‖2L2(K)

= C 2(M−2n )2‖f ‖2L2(Ω) + CM

− 2n ‖f ‖2L2(Ω)≤ CM− 2n ‖f ‖2L2(Ω).

So,

|uεp − ūεp |2H̄1(Ω) ≤ C (√ε+ H) ‖f ‖L2(Ω) + CM−

1n ‖f ‖L2(Ω)

= C (√ε+ H + M−

1n ) ‖f ‖L2(Ω). �


Discrete FE-HMM

Monika Wolfmayr

Introduction







H1-error

L2-error



H1-error

Conclusions

Conclusions

The numerical results presented in

Abdulle, On A Priori Error Analysis of Fully DiscreteHeterogeneous Multiscale FEM, Multiscale Model. Simul.

Vol. 4, No. 2, pp. 447 - 459, 2005

show that the theoretical bounds are sharp.

Thanks for your attention!

IntroductionFE-HMM for Elliptic ProblemsElliptic model problemFirst convergence results

Error Analysis of the Fully Discrete FE-HMMFully discrete FE problemConvergence results for the macrosolutionConvergence results for the fully discrete solution

Conclusions

a priori error analysis of fully discrete fe-hmm...a priori error analysis of fully discrete fe-hmm...

Documents