Numerical Analysis Seminar
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Transcript of Numerical Analysis Seminar
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8/3/2019 Numerical Analysis Seminar
1/27
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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8/3/2019 Numerical Analysis Seminar
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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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8/3/2019 Numerical Analysis Seminar
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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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8/3/2019 Numerical Analysis Seminar
4/27
Outline
Motivation
The abstract framework
The advection-diffusion model problem
Projection and localization
Amos Otasowie Egonmwan 2 / 24
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8/3/2019 Numerical Analysis Seminar
5/27
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8/3/2019 Numerical Analysis Seminar
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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
Amos Otasowie Egonmwan 3 / 24
Th b f k
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The abstract framework
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
Amos Otasowie Egonmwan 4 / 24
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
Amos Otasowie Egonmwan 5 / 24
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
Amos Otasowie Egonmwan 6 / 24
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)
Amos Otasowie Egonmwan 7 / 24
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
Amos Otasowie Egonmwan 8 / 24
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 (, )
Amos Otasowie Egonmwan 9 / 24
The advection diffusion model problem
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The advection-diffusion model problem
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
Amos Otasowie Egonmwan 10 / 24
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
Amos Otasowie Egonmwan 11 / 24
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)
Amos Otasowie Egonmwan 12 / 24
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
Amos Otasowie Egonmwan 13 / 24
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.
Amos Otasowie Egonmwan 14 / 24
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)
Amos Otasowie Egonmwan 15 / 24
Linear elements and H10 -optimality in 1D contd
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Linear elements and H0 optimality in 1D, contd.
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),
Amos Otasowie Egonmwan 16 / 24
Linear elements and H10 -optimality in 1D, contd.
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Linear elements and H0 optimality in 1D, contd.
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
Amos Otasowie Egonmwan 17 / 24
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)
Amos Otasowie Egonmwan 18 / 24
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)
Amos Otasowie Egonmwan 19 / 24
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
Amos Otasowie Egonmwan 20 / 24
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
Amos Otasowie Egonmwan 21 / 24
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
Amos Otasowie Egonmwan 22 / 24
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.
Amos Otasowie Egonmwan 23 / 24
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Thank You!
Amos Otasowie Egonmwan 24 / 24