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A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems A Priori Error Analysis of Fully Discrete Elliptic model problem First convergence results FE-HMM Error Analysis of the Fully

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SLIDE 1

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

A Priori Error Analysis of Fully Discrete FE-HMM

Monika Wolfmayr 5th December 2011

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SLIDE 2

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Outline

Introduction FE-HMM for Elliptic Problems Elliptic model problem First convergence results Error Analysis of the Fully Discrete FE-HMM Fully discrete FE problem Convergence results for the macrosolution

H1-error L2-error L2-projection of uε

Convergence results for the fully discrete solution

H1-error

Conclusions

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SLIDE 3

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Introduction

Heterogeneous Multiscale Method (HMM) introduced by E and Engquist in 2003 the name heterogeneous was used to emphasize that the models at different scales may be of very different nature Difference between traditional MM and HMM: MM: general purpose are microscale solvers, i.e. to resolve the details of the solutions of the microscale model HMM: objective is to capture the macroscale behavior of the system with a cost that is much less than the cost of full microscale solvers

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SLIDE 4

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Introduction

HMM: physical problem is directly discretized by a macroscopic finite element method model (coarse scale) microproblems are either unit-cell problems or problems on a patch with a fixed number of unit cells (fine scale) study of accuracy properties in HMM:

◮ first approach: assumption that the microproblems are

analytically given; macro- and microerrors often separately estimated

◮ further approach: combination of microscopic and

macroscopic models; microproblems are solved numerically as well; estimates for the errors transmitted

  • n 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

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SLIDE 5

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Notation

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

1 ...∂rn n ;

H1(Ω) = {u ∈ L2(Ω); Dru ∈ L2(Ω), |r| ≤ 1}, uH1(Ω) =  

|r|≤1

Dru2

L2(Ω)

 

1/2

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

per(Y ) = {v ∈ H1 per(Y );

  • Y

v dx = 0}; H1

0(Ω) = C ∞ 0 (Ω) ·H1, H1 per(Y ) = C ∞ per(Y ) ·H1, Y = (0, 1)n

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SLIDE 6

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Elliptic Model Problem

−∇ · (aε∇uε) = f in Ω ⊂ Rn uε = 0

  • n ∂Ω,

(1) ε ... length scale, Ω ... convex polygon, f ∈ L2(Ω), aε(x) = a(x, x

ε) = 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 Ω u0 = 0

  • n ∂Ω,

(2) a0 ... smooth matrix with coefficients a0

ij(x) =

  • Y (aij(x, y) + n

k=1 aik(x, y) ∂χj ∂yk (x, y))dy,

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

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SLIDE 7

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Elliptic Model Problem

Macro FE space: S1

0(Ω, TH) = {uH ∈ H1 0(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 1

per(Kε) and

  • Kε

∇u a(xk, x/ε)(∇z)Tdx = 0 ∀z ∈ W 1

per(Kε).

(5)

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SLIDE 8

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Elliptic Model Problem

It can be shown that u = uH + ε

n

  • 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

eT

j a(xk, y)(∇z)Tdy

(7) for all z ∈ W 1

per(Y ); {ej}n j=1 ... standard basis in Rn.

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SLIDE 9

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Elliptic Model Problem

Variational problem for the macrosolution: Find uH ∈ S1

0(Ω, TH) such that

B(uH, vH) = f , vH ∀vH ∈ S1

0(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 variational problem for the macrosolution (8) is of semidiscrete nature.

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SLIDE 10

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

First convergence estimates for the macrospace

Assumption: H2-regularity for u0, exact microsolution uε − u0L2(Ω) ≤ C ε f L2(Ω), (10) uε − uHH1(Ω) ≤ C (H/ε) f L2(Ω), (11) u0 − uHH1(Ω) ≤ C H f L2(Ω), (12) uε − uε

p¯ H1(Ω) ≤ C (√ε + H) f L2(Ω),

(13) uε

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

periodically extended on each K, can be discontinuous across K, hence ¯ H1-norm is mesh-dependent; Puε − uHH1(Ω) ≤ C (ε/H + H) f L2(Ω), (14) Puε ... L2-projection of the solution

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SLIDE 11

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Micro FE problem

Sampling domain: Kε = xk + ε[−1/2, 1/2]n Micro FE space: S1

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

(15) Th ... quasi-uniform triangulation of Kε with meshsize h, S1

per(Kε, Th) ⊂ W 1 per(Kε)

Discrete microproblem: For uH ∈ S1

0(Ω, TH) find uh such that

(uh − uH) ∈ S1

per(Kε, Th) and

  • Kε

∇uh a(xk, x/ε)(∇zh)Tdx = 0 ∀zh ∈ S1

per(Kε, Th).

(16)

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SLIDE 12

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Fully discrete FE problem

It can be shown that uh = uH + ε

n

  • j=1

χj,h(xk, x/ε)∂uH(xk) ∂xj , (17) χj,h(xk, y) ... unique solutions of the cell problems in S1

per(Kε, Th)

Fully discrete macrobilinear form: ¯ B(uH, vH) =

  • K∈TH

|K| |Kε|

  • Kε

∇uh a(xk, x/ε)(∇vh)Tdx (18) Variational problem for the fully discrete macrosolution: Find ¯ uH ∈ S1

0(Ω, TH) such that

¯ B(¯ uH, vH) = f , vH ∀vH ∈ S1

0(Ω, TH).

(19)

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SLIDE 13

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Fully discrete FE problem

Proposition

The problem (19) has a unique solution which satisfies ¯ uHH1(Ω) ≤ Cf 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 Dα

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

(21)

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SLIDE 14

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Fully discrete FE problem

Lemma

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

  • ¯

B(v H, w H) − B(v H, w H)

  • ≤ C

h ε 2 ∇v HL2(Ω)∇w HL2(Ω), (22)

where vH, wH ∈ S1

0(Ω, TH), h is the mesh size of the micro

FE space S1

per(Kε, Th).

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SLIDE 15

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Proof of the Lemma - Part I

  • B(v H, w H) − ¯

B(v H, w H)

  • =
  • K∈TH

|K| |Kε|

  • Kε

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

  • Inserting +/ − ∇v h a(xk, x/ε)(∇w)T yields

=

  • K∈TH

|K| |Kε|[

  • Kε

∇ (v − v h)

  • ∈W 1

per(Kε)

a(xk, x/ε) (∇w)T

∈W 1

per (Kε)

dx

  • =(5)0

−

  • Kε

∇v h a(xk, x/ε)(∇(w h − w))Tdx]

  • .
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SLIDE 16

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Proof of the Lemma - Part II

Due to (5), we have

  • Kε ∇v a(xk, x/ε)(∇(w h − w))Tdx = 0.

Together with the boundedness of the bilinear form, we obtain

  • B(v H, w H) − ¯

B(v H, w H)

  • =
  • K∈TH

|K| |Kε|

  • Kε

∇(v h − v) a(xk, x/ε)(∇(w h − w))Tdx]

  • ≤ C
  • K∈TH

|K| |Kε|∇(v h − v)L2(Kε)∇(w h − w)L2(Kε). Moreover, ∇(v h − v)L2(Kε) =

  • ε

n

  • j=1

∇(χj,h(xk, x/ε) − χj(xk, x/ε))∂v H(xk) ∂xj

  • L2(Kε)

≤

  • ε max

j

∇(χj,h − χj)

n

  • j=1

∂v H(xk) ∂xj

  • L2(Kε)

.

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SLIDE 17

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Proof of the Lemma - Part III

Since ∇v H is constant, ∇(v h − v)L2(Kε) ≤

  • ε max

j

∇(χj,h − χj)

  • L2(Kε)

n

  • j=1

∂v H ∂xj ≤ C ε max

j

  • ∇(χj,h − χj)
  • L2(Kε)(

n

  • j=1

(∂v H ∂xj )2)

1 2

≤ C εh

  • |Kε| max

j

  • χj
  • W 2,∞(Kε)

√ ∇v H∇v H ≤(21) C εh

  • |Kε|Cε−2√

∇v H∇v H = C h ε |Kε| √ ∇v H∇v H.

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SLIDE 18

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Proof of the Lemma - Part IV

Altogether, we obtain

  • B(v H, w H) − ¯

B(v H, w H)

  • ≤ C
  • K∈TH

|K| |Kε|∇(v h − v)L2(Kε)∇(w h − w)L2(Kε) ≤ C

  • K∈TH

|K| |Kε| h ε |Kε| √ ∇v H∇v H h ε |Kε| √ ∇w H∇w H = C h ε 2

K∈TH

  • |K|
  • |Kε|
  • |Kε|

√ ∇v H∇v H

  • |K|
  • |Kε|
  • |Kε|

√ ∇w H∇w H = C h ε 2

K∈TH

  • |K|

√ ∇v H∇v H |K| √ ∇w H∇w H. Since ∇v H and ∇w H are constant, = C h ε 2

K∈TH

∇v HL2(K)∇w HL2(K). Summing up over K finally yields (22).

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SLIDE 19

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

H1-error between the discrete and the fully discrete macrosolution

Remark: If M = dim S1

per(Kε), then h ≃ εM− 1

n with |Kε| = εn.

= ⇒ h/ε independent of ε

Proposition

Let uH, ¯ uH be the solutions of the variational problems for the macrosolution and the fully discrete macrosolution, respectively, and suppose that the assumptions of the Lemma hold. Then uH − ¯ uHH1(Ω) ≤ C M− 2

n f L2(Ω).

(23)

Proof.

Use coercivity of the bilinear forms, the Lemma before and the Proposition about existence and uniqueness of the fully discrete solution.

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SLIDE 20

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

H1-error between the homogenized and the fully discrete macrosolution

Theorem

Assume the solution of the homogenized problem u0 is H2-regular and ¯ uH is the fully discrete solution. Let the assumptions of the Lemma hold. Then u0 − ¯ uHH1(Ω) ≤ C(H + M− 2

n )f L2(Ω).

(24)

Proof.

Use triangle inequality, u0 − uHH1(Ω) ≤ C H f L2(Ω) and (23) from the Proposition before.

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SLIDE 21

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

L2-error between the homogenized/exact solution and the fully discrete macrosolution

Corollary

Suppose that the assumptions of the Theorem before hold. Then u0 − ¯ uHL2(Ω) ≤ C(H2 + M− 2

n )f L2(Ω),

(25) uε − ¯ uHL2(Ω) ≤ C(H2 + ε + M− 2

n )f L2(Ω).

(26)

Proof.

Use triangle inequality, u0 − uHL2(Ω) ≤ C H2 f L2(Ω), uε − u0L2(Ω) ≤ C ε f L2(Ω) and (23) from the Proposition before.

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SLIDE 22

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

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

For u ∈ H1(Ω), we define Pu ∈ S1

0(Ω, TH) as unique solution

  • f

Pu, vH = u, vH ∀vH ∈ S1

0(Ω, TH).

(27)

Theorem

Let Puε be the solution projected on S1

0(Ω, TH) and ¯

uH be the fully discrete solution. Suppose that the assumptions of the Lemma hold and that u0 is H2-regular. Then Puε − ¯ uHH1(Ω) ≤ C( ε H + H + M− 2

n )f L2(Ω),

(28) Puε − ¯ uHL2(Ω) ≤ C(ε + H2 + M− 2

n )f L2(Ω).

(29)

Proof.

Use triangle inequality, Puε − uHH1(Ω) ≤ C (ε/H + H) f L2(Ω), Puε − uHL2(Ω) ≤ C (ε + H2) f L2(Ω) and (23).

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SLIDE 23

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Procedure to retrieve the microscopic information

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

p(x) = ¯

uH(x) + (uh(x) − ¯ uH(x))|P

K

for x ∈ K ∈ TH, (30) where |P

K denotes the periodic extension of the fine-scale

solution (uh − ¯ uH), 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. ¯ uε

p can be discontinuous across K. Hence, we define the

following broken H1-seminorm: |u| ¯

H1(Ω) :=

 

K∈TH

∇u2

L2(K)

 

1 2

. (31)

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SLIDE 24

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

H1-error between the exact and the fully discrete solution

Theorem

Suppose that the assumptions of the Lemma hold. Then the error between the exact and the fully discrete solution can be estimated by |uε − ¯ uε

p| ¯ H1(Ω) ≤ C (√ε + H + M− 1

n )f L2(Ω).

(32)

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SLIDE 25

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Proof of the Theorem - Part I

|uε − ¯ uε

p| ¯ H1(Ω) ≤

|uε − uε

p| ¯ H1(Ω)

  • ≤C (√ε+H) f L2(Ω)

+|uε

p − ¯

uε

p| ¯ H1(Ω)

|uε

p − ¯

uε

p|2 ¯ H1(Ω) =

  • K∈TH

∇(uε

p − ¯

uε

p)2 L2(K)

=

  • K∈TH

∇(uH + (u − uH)|P

K) − ∇(¯

uH + (uh − ¯ uH)|P

K)2 L2(K)

=

  • K∈TH

∇uH +

=∇(u−uH)

  • n
  • j=1

∇(εχj(xk, x/ε))∂uH ∂xj − ∇¯ uH −

=∇(uh−¯ uH)

  • n
  • j=1

∇(εχj,h(xk, x/ε))∂¯ uH ∂xj 2

L2(K)

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SLIDE 26

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Proof of the Theorem - Part II

|uε

p − ¯

uε

p|2 ¯ H1(Ω) ≤ C

  • K∈TH

∇uH − ∇¯ uH2

L2(K)

  • =(I)

+ C

  • K∈TH
  • n
  • j=1

∇(εχj) ∂uH ∂xj − ∂¯ uH ∂xj

  • 2

L2(K)

  • =(II)

+ C

  • K∈TH
  • n
  • j=1

∇(ε(χj − χj,h))∂¯ uH ∂xj 2

L2(K)

  • =(III)

(I) ≤ uH − ¯ uH2

H1(K) ≤(23) (CM− 2

n f L2(K))2

slide-27
SLIDE 27

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Proof of the Theorem - Part III

(II) =

n

  • j=1

∇(εχj) ∂uH ∂xj − ∂¯ uH ∂xj

  • 2

L2(K)

≤ ε max

j

∇(χj)

n

  • j=1

∂uH ∂xj − ∂¯ uH ∂xj

  • 2

L2(K)

≤ ε2 max

j

∇(χj)2

L2(K)∇uH − ∇¯

uH2

L2(K)

≤(21) ε2C(ε−1)2uH − ¯ uH2

H1(K) ≤(23) (CM− 2

n f L2(K))2

(III) =

n

  • j=1

∇(ε(χj − χj,h))∂¯ uH ∂xj 2

L2(K)

≤Pr.Lem.III C 2( √ ∇¯ uH∇¯ uH)2ε2 max

j

∇(χj − χj,h)2

L2(K)

≤ C∇¯ uH∇¯ uHε2h2|K| max

j

  • χj
  • W 2,∞(K)
  • ≤(21)(Cε−2)2

≤ Ch2ε−2|K|∇¯ uH∇¯ uH ≤ C(εM− 1

n )2ε−2|K|∇¯

uH∇¯ uH

slide-28
SLIDE 28

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Proof of the Theorem - Part IV

Since ∇¯ uH is constant, (III) ≤ CM− 2

n ∇¯

uH2

L2(K) ≤ CM− 2

n ¯

uH2

H1(K) ≤ CM− 2

n f 2

L2(K).

Altogether, we obtain the estimate |uε

p − ¯

uε

p|2 ¯ H1(Ω) ≤ C

  • K∈TH

2(CM− 2

n f L2(K))2 + C

  • K∈TH

CM− 2

n f 2

L2(K)

= C2(M− 2

n )2f 2

L2(Ω) + CM− 2

n f 2

L2(Ω)

≤ CM− 2

n f 2

L2(Ω).

So, |uε

p − ¯

uε

p|2 ¯ H1(Ω) ≤ C (√ε + H) f L2(Ω) + CM− 1

n f L2(Ω)

= C (√ε + H + M− 1

n ) f L2(Ω).

slide-29
SLIDE 29

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems

Elliptic model problem First convergence results

Error Analysis of the Fully Discrete FE-HMM

Fully discrete FE problem Convergence results for the macrosolution H1-error L2-error L2-projection of uε Convergence results for the fully discrete solution H1-error

Conclusions

Conclusions

The numerical results presented in Abdulle, On A Priori Error Analysis of Fully Discrete Heterogeneous Multiscale FEM, Multiscale Model. Simul.

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

show that the theoretical bounds are sharp. Thanks for your attention!