Numerical Analysis Seminar

8/3/2019 Numerical Analysis Seminar

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Variational Multiscale Analysis:The fine-scale Greens function, projection, optimization,

and localization

Amos Otasowie Egonmwan

Supervisor: a.Univ.-Prof. Dipl.-Ing. Dr. Walter Zulehner

14th November 2011


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Outline

Motivation

The abstract framework

The advection-diffusion model problem

Projection and localization

Amos Otasowie Egonmwan 2 / 24


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Outline

Motivation






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Outline

Motivation






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Motivation

Introduced as a framework for incorporating missing fine-scale effects intonumerical problems governing coarse-scale behaviour

Provided a rationale for stabilized methods and for development of robustmethods

Direct application of Galerkins methods with standard bases (such as FE) in thepresence of multiscale phenomena leads to wrong solution

Multiscale phenomena is ubiquitous in science and engineering applications


Th b f k


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The abstract problem

Let V Hilbert space, V norm, (, )V scalar product, V dual of V,

, VV duality pairing, L : V V linear isomorphism.

Given f V, find u V such that

Lu = f (1)

VF: find u V such that

Lu, vVV = f, vVV v V (2)

Solution of (1) u = Gf, G : V V, where G := L1


Th i ti l lti l f l ti


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The variational multiscale formulation

Let V closed subspace of V, P : V V linear projection

R(P) = V, define Ker(P) = V

and let P continuous in V

V = VV

(3)

= v V, v = v + v

where v V, v

V

also, u = u+ u

V space of computable coarse scale, V

space of unresolved fine scale

Aim of VMS: Find u = Pu


Th i ti l lti l f l ti td


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The variational multiscale formulation, contd.

Procedure : VF (2) splits;

Lu, vVV + Lu

, vVV = f, vVV v V (4)

Lu, v

VV + Lu

, v

VV = f, v

VV v

V

(5)

Assume (4) and (5) are well posed for u and u

respectively.

Associate with (5); G

: V V, which yields u

= G

(f Lu)

Having G

, eliminate u

from (4) = VMS for u

Lu, vVV LG

Lu, vVV = f, vVVLG

f, vVV v V (6)

admits a unique solution u = Pu


Th fi l G t


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The fine-scale Greens operator

P

T

:V

V adjoint of

P, i.e

PT, vVV = , PvVV V

Theorem

Under the assumption of the abstract problem and VMS formulation, we have

G

= G GP T(PGPT)1PG (7)

G

PT = 0, and PG

= 0 (8)


The fine scale Greens operator contd


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The fine-scale Green s operator, contd.

If dim(V) = N, we can find a set of functionals : V R, {i}i=1, ,N

such that, for all v V

i

, vV

V= 0 i = 1, , N Pv = 0 (9)

In this case, (8) is equivalent to:

G

i = 0 i = 1, , N (10)

and

i, G

VV = 0 V i = 1, , N (11)

Introduce (V)N, GT VN,

GT =

1, G1VV 1, GNVV

.

.

.. . .

.

.

.

N, G1V

V N, GNV

V

RRR V

and vector of functionals; i.e G : (V) RN, then (7) is equivalent to:

G

= G GT(GT)1G (12)

since {i}

i=1, ,Nis a basis for the image of PT


Orthogonal projectors and optimization


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Orthogonal projectors and optimization

Given scalar product (, ) defined on V V, the related orthogonal projector

P is defined by(Pw, v) = (w, v) w V, v V

In this case, V

and V are orthogonal complements with respect to (, ), and the

VMS formulation thus provide optimal approximation u V of u, with respect

to the induced by the scalar product (, )


The advection diffusion model problem


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Model problem:

Lu = u+ u = f in u| = 0 (13)

> 0 (scalar diffusivity)

: Rd, div() = 0

Rd

(d = 1, 2); regular domainf L2()

VF: V = H10 H10 (), V

= H1

Representation: Greens operator G through the Greens function g : R

such that:

u(y) =

g(x, y)f(x)dx a.e y


The advection diffusion model problem contd


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The advection-diffusion model problem, contd.

Similar representation: fine-scale Greens operator G

through the fine-scale Greens

function g

: R gives the fine scale component u

of u from

r = f Lu :

u

(y) =

g

(x, y)r(x)dx (14)

where the space V

and function g

depends on the underlying projector P


The advection-diffusion model problem contd


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The advection-diffusion model problem, contd.

Having {i}i,N such that:

i(x)v(x)dx = 0 i, , N Pv = 0

Then, g

is obtained by (12) as:

g(x, y) = g(x, y)

g(x, y)1(x)dx

g(x, y)N(x)dx

g(x, y)1(x)1(y)dxdy . . .

g(x, y)N(x)1(y)dxdy...

. . ....

g(x, y)1(x)N(y)dxdy . . .

g(x, y)N(x)N(y)dxdy

1

g(x, y)1(y)dy

..

. g(x, y)N(y)dy

(15)


VMS for advection-diffusion model problem


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VMS for advection-diffusion model problem

The property (10) and (11) = for all x, y and for all i = 1, , N

g

(x, y)i(x)dx = 0 and

g

(x, y)i(y)dy = 0 (16)

In this context, the VMS formulation (6) reads: Find u V such that

(u(x) u) v(x)dx

Lu(x)g

(x, y)Lv(y)dxdy

=

f(x)v dx

f(x)g

(x, y)Lv(y)dxdy

v V


Linear elements and H1-optimality in 1D


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Linear elements and H0 optimality in 1D

Take: d = 1, = (0, L), and consider: 0 = x0 < x1 < < xel1 < xel = L

Subdivision: (0, L) into nel elements (xi1, xi), i =, , nel

For the H10

-projector P = PH10

the VMS provides a nodally exact approximation u of

the exact solution u.

In this case, i = (x xi) and the property (16) in the theorem is equivalent to:

g

(x, xi) = g

(xi, y) = 0 i = 1, , N, 0 x, y L; (17)

i.e., g

vanishes if one of its two arguments is a node of the grid.


Linear elements and H10 -optimality in 1D contd


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Linear elements and H0 optimality in 1D, contd.

As a result, the expression (16) for the fine-scale Greens function in this casebecomes:

g(x, y) = g(x, y)

g(x1, y) g(xN, y)

g(x1, x1) . . . g(xN, x1)

..

.. . .

..

.g(x1, xN) . . . g(xN, xN)

1

g(x, x1)

.

.

.

g(x, xN)


Linear elements and H10 -optimality in 1D contd


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Since g(x, y)

= 0 only when x and y belong to the same element, (14) can be

localized within each element:

u

(y) =

xixi1

g

(x, y)r(x)dx y (xi1, xi)

V

is the space of bubbles:

V

=

i=1, ,nel

H10 (xi1, xi),


Linear elements and H10 -optimality in 1D, contd.


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If we assume piecewise-constant coefficients , and source term f , the fine-scaleVE:

L0

L0

Lv(y)g

(x, y)r(x)dxdy =

neli=1

xixi1

xixi1

Lv(y)g

(x, y)r(x)dxdy

=

neli=1

xixi

1xix

i

1

g

(x, y)dxdy

xi xi1

xixi1

r(x)Lv(x)dx,

which is recognized as a classical stabilization term depending on the parameter

1 1,(xi1,xi) =

xixi1

xixi1

g

(x, y)dxdy

xi xi1 =

h

2

coth()

1

where = h2

is the mesh Peclet number, h = xi xi1 is the local mesh-size


Higher order elements and H10 -optimality in 1D


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Higher order elements and H0 optimality in 1D

Consider higher-order piecewise-polynomial coarse: 0 = x0 < x1 < < xnel1 = L

and setV = {v H10 (0, L) such that v|(xi1,xi) Pk, 1 i nel}

V

is a strict subset of bubbles:

V

=

i=1, ,nel

H10 (xi1, xi), (18)


Higher order elements and H10 -optimality in 1D, contd.


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Higher order elements and H0 optimality in 1D, contd.

Consider: V

i= V

|(xi1,xi); as fine-scale of bubbles,

Vi = V|(xi1,xi) H10

(xi1, xi) space of bubbles of coarse-scale

Vi = H10

(xi1, xi) = Vi

V

ispace of unconstrained bubbles

In this case, the Greens function of the unconstrained bubble is the element Greens

function gel. From (12), we get g

in terms of gel : on (0, h) (0, h)

g(x, y) = gel(x, y)

h0

gel(x, y)dx

h0

xk2gel(x, y)dx

h0

h0

gel(x, y)dxdy h

0

h0

xk2gel(x, y)dxdy...

. . ....

h

0 h

0yk2gel(x, y)dxdy . . .

h0

h0

xk2yk2gel(x, y)dxdy

1

h0 gel(x, y)dy

.

.

.h0

yk2gel(x, y)dy

(19)


Higher order elements and H10 -optimality in 1D, contd.


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g 0 p y ,

For k = 2, (21) yields:

g(x, y) = gel(x, y)

h0

gel(x, y)dxh

0gel(x, y)dyh

0

h0

gel(x, y)dxdy

and for k = 3, (21) becomes:

g(x, y) = gel(x, y) h

0gel(x, y)dx

h0

xgel(x, y)dx

h0

h0

gel(x, y)dxdyh

0

h0

xgel(x, y)dxdyh0

h0

y gel(x, y)dxdyh

0

h0

xy gel(x, y)dxdy

1

h

0 gel(x, y)dyh0

yk2gel(x, y)dy


Expression for element Greens function


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p

Element Greens function : gel(x, y)

Lgel(x, y) = (x y) x

gel(x, y) = 0 x

e = 1, , nel (20)

where L = u , and solution of (20) [T.J.R. Hughes, et al 1998]

gel(x, y) =

C1(y)(1 e

2x/h) x yC2(y)(e

2(x/h) e2) x y

where

C1 = 1 e2(1(y/h))

(1 e2), C2 = e

2(y/h)

1(1 e2)

where = h2

is the mesh Peclet number, h = xi xi1 is the local mesh-size


L2-optimality in 1D and localization of g


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p y g

In 1D, P = PH1

0fine-scale Greens function g

is fully localized within each element

= conveniet evaluation of the fine-scale effects in VMS formulation.

This feature is not guaranteed for other projectors, e.g P = PL2

Remark:

Selection of the projection is crucial in the development of a multiscale method


Linear element in 2D


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In 2D:

There is difficulty in getting an analytical expression for g and g

through (18)

Remedy: Numerically compute g and g

on a fine mesh of 524,288 elements(by standard Galerkin method) which is able to resolve the fine-scale effects

In general (for finer mesh), g

when P = PH10

is better than P = PL2 because itsfully localized = coarse-scale approximation u for PH1

0is better than PL2

However, if coarse scales are piecewise-polynomials, then fine-scales are notlocalized within each element for any projector (including PH1

0) = difficulty in

evaluating the fine-scale effects in VMS formulation.



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Thank You!


Numerical Analysis Seminar

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