Download a stabilised finite element method for fictitious domain problems

introduction
Block 1
Block 2
Conclusions and perspectives
A STABILISED FINITE ELEMENT
METHOD FOR FICTITIOUS DOMAIN
PROBLEMS
Gabriel R. Barrenechea & Cheherazada Gonzalez Aguayo
University of Strathclyde
27th of April, 2017
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introduction
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Block 2
Conclusions and perspectives
introduction
BLOCK 1
BLOCK 2
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Introduction
Stability
Convergence
Numerical results
Conclusions
introduction
Block 1
Block 2
Conclusions and perspectives
BLOCK 1: Introduction
Suppose the following Time-Dependent problem:
For f˜ given, find ũ such that
∂t ũ − ∆ũ = f˜ in
ũ = g
on
ũ(x, 0)= ũ0 (x)
ω×(0, T ),
γ×(0, T ),
in
ω,
where ω is a very complicated domain.Then, a larger ( simpler) domain Ω is
introduced, taking an extension f of f˜ to Ω, and the problem to solve becomes:
1
Find (u, λ) ∈ L2 (0, T ; H01 (Ω)) × L∞ (0, T ; H − 2 (γ)) such that
(∂t u, v )Ω + (∇u, ∇v )Ω − hλ, v iγ = (f , v )Ω
hµ, uiγ = hµ, g iγ
1
∀v ∈ H01 (Ω), µ ∈ H − 2 (γ),
where λ : γ −→ R is a Lagrange multiplier.
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Conclusions and perspectives
Introduction
Stability
Convergence
Numerical results
Conclusions
Meshes ( option 1):
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Introduction
Stability
Convergence
Numerical results
Conclusions
Fully Discrete Problem
The fully discrete problem is given by:
Given a suitable approximation of uh0 ∈ Vh of ũ0 , for n = 0, 1, 2... , find
(uhn+1 , λn+1
) ∈ Wh̃ such that
h̃
1 n+1
(u
− uhn , vh )Ω + (∇uhn+1 , ∇vh )Ω − hλn+1
, vh iγ = (f , vh )Ω
h̃
δt h
hµh̃ , uhn+1 iγ = hµh̃ , g iγ , ∀(vh , µh̃ ) ∈ Wh̃ .
where
Wh̃ = Vh × Λh̃
Vh = {vh ∈ C 0 (Ω̄) ∩ H01 (Ω) : vh |h ∈ P1 (K ), ∀K ∈ Th }
Λh̃ = {µh̃ ∈ L2 (γ) : µh̃ |ẽ ∈ P0 (ẽ), ∀ẽ ∈ γh̃ }
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Introduction
Stability
Convergence
Numerical results
Conclusions
Lemma. If |ẽ| > 3h, there exist β > 0 such that
sup
vh ∈Vh \{0}
b(µh̃ , vh )
> βkµh̃ k− 1 ,γ
2
|vh |1,Ω
∀µh̃ ∈ Λh̃ .
( V. Girault and R. Glowinski. Error analysis of a fictitious domain method applied to a Dirichlet problem. Japan J. Indust. Appl. Math.,
12(3) : 487 − 514, 1995.)
Lemma. The fully discrete inf-sup stable problem is well-posed.
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Introduction
Stability
Convergence
Numerical results
Conclusions
Meshes ( option 2):
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Conclusions and perspectives
Introduction
Stability
Convergence
Numerical results
Conclusions
Stabilised Method
Stabilised problem:
For f given, find (uhn+1 , λhn+1 ) ∈ Wh such that
1 n+1
(u
− uhn , vh )Ω + (∇uhn+1 , ∇vh )Ω − hλn+1
h , vh iγ = (f , vh )Ω
δt h
hµh , uhn+1 iγ +j(λn+1
h , µh ) = hµh , g iγ , ∀(vh , µh ) ∈ Wh ,
where Wh = Vh × Λh and Λh = {µh ∈ L2 (γ) : µh |e ∈ P0 (e), ∀e ∈ γh }.
In order to stabilise the Lagrange multiplier, we introduce a bilinear form
j : Λh × Λh −→ R defined as
X
j(λh , µh ) =
|ẽ|(λh − Πλh , µh − Πµh )ẽ
ẽ∈γh̃
where Π : L2 (γ) −→ Λh̃ is defined as (Πµh )|ẽ = |ẽ|−1 (µh , 1)ẽ .
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Introduction
Stability
Convergence
Numerical results
Conclusions
Stabilised Method
Lemma. There exist two constants C , β > 0, independent of h, such that
sup
vh ∈Vh \{0}
1
|b(λh , vh )|
+ C j(λh , λh ) 2 > βkλh k− 1 ,γ
2
|vh |1,Ω
∀λh ∈ Λh .
(Gabriel R. Barrenechea and Franz Chouly. A local projection stabilised method for fictitious domains. Appl. Math.
Lett.,25(12) : 2071 − 2076, 2012.)
Lemma. The fully discrete stabilised problem is well-posed.
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Conclusions and perspectives
Introduction
Stability
Convergence
Numerical results
Conclusions
Stability
Lemma. Let uh0 be a given H01 -stable approximation of u0 in Vh and let
{(uhn , λnh )}Nn=1 be the solution of the fully discrete stabilised problem. The
following estimate holds for 1 ≤ n ≤ N:
kuhn k20,Ω +
n−1
X
m=0
n−1
2 X
) +
δt (uhm+1 , λm+1
δtkλm+1
k2− 1 ,γ
h
h
h
2
m=0
≤C (f , u0 , g )+
n−1
X
δt −1 kuhm+1 − uhm k20,Ω ,
m=0
where |||(vh , µh )|||2h = |vh |21,Ω + j(µh , µh ).
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Introduction
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Convergence
Numerical results
Conclusions
introduction
Block 1
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Conclusions and perspectives
Stability
1
The Ritz-projection operator. For each (w , ξ) ∈ H01 (Ω) × H − 2 (γ), the
projection Sh (w , ξ) = (Ph (w , ξ), Rh (w , ξ)) ∈ Wh is defined as the unique
solution of
(∇Ph (w , ξ), ∇vh ) − hRh (w , ξ), vh iγ = (∇w , ∇vh ) − hξ, vh iγ
hµh , Ph (w , ξ)iγ +j(Rh (w , ξ), µh ) = hµh , w iγ
∀(vh , µh ) ∈ Wh .
Theorem. Let {(uhn , λnh )}Nn=1 be the solution of the fully discrete problem. If
u0 ∈ H 1 (Ω) and uh0 = Ph (u0 , 0), the following estimate holds for 1 ≤ n ≤ N:
kuhn k20,Ω +
n−1
X
m=0
n−1
2 X
δt (uhm+1 , λm+1
) +
δtkλm+1
k2− 1 ,γ ≤C (f , u0 , g )
h
h
h
m=0
2
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Introduction
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Conclusions
Convergence
Theorem. Assume that u ∈ H 1 (0, T ; H 2 (Ω)) ∩ H 2 (0, T ; L2 (Ω)) and
1
λ ∈ C 0 (0, T ; H 2 (γ)), and uh0 := Ph (u0 , 0). Then the following estimate holds
for 1 ≤ n ≤ N:
keun k20,Ω +
n−1
X
2
δt |||(eum , eλm )|||h + keλm k2− 1 ,γ ≤ C (h2 + δt 2 ),
2
m=0
where
eun := u(tn ) − uhn
eλn := λ(tn ) − λnh
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Introduction
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Conclusions
Numerical results
We consider a regular solution. To measure convergence for the stabilised
method, we have computed the norms
k(u − uh,δt , λ − λh,δt )k2+ :=kuhN − u(tN )k20,Ω
N−1
2
X m+1
+
− λ(tm+1 )) ,
δt (uh − u(tm+1 ), λm+1
h
h
m=0
and
kλ − λh,δt k2++ :=
N−1
X
δtkλm+1
− λ(tm+1 )k20,γ .
h
m=0
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Numerical results
Conclusions
Recall the error estimates kerror k ≤ C (h + δt). So, to balance both terms, we
have chosen δt = h in the experiments.
Figure: Error of u and λ for stabilised method for nonhomogeneous BCs.
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Conclusions
Recall the error estimates kerror k ≤ C (h + δt). If we take δt −→ 0, fixing
n = 1,
Figure: Lambda behavior for stabilised method for nonhomogeneous BCs.
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Stability
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Conclusions
Conclusions
-Stability and convergence analysis of both inf-sup stable and stabilised
methods for the transient problem.
- Numerical results confirm the theory.
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Conclusions and perspectives
Introduction
Stability
Convergence
Numerical results
BLOCK 2: Introduction
The purpose of this work is to approximate numerically an elliptic partial
differential equation posed on domains with small perforations (or inclusions),
for instance,
where
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Conclusions and perspectives
Introduction
Stability
Convergence
Numerical results
The problem of interest reads as follows:
Find ũ : ω −→ R such that

 −∆ũ =
ũ =

ũ =
f˜ in ω,
gi on γi , i = 1, ..., N ,
0 on ∂Ω,
1
where f˜ ∈ L2 (ω) and gi ∈ H 2 (γi ) for all i = 1, ..., N.
Applying fictitious domain method, the mixed problem is:
1
−2
Find (u, λ) ∈ W := H01 (Ω) × ΠN
(γi ), where λ = (λi )N
i=1 H
i=1 , such that
(∇u, ∇v )Ω −
N
X
hλi , v iγi = (f , v )Ω ,
i=1
N
N
X
X
hµi , uiγi =
hµi , gi iγi ,
i=1
for all (v , µ) ∈ W , µ =
i=1
(µi )N
i=1 .
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First Idea:
We use λ = [[∂n u]]γ and propose to add a stabilised term of the form:
N
X
hh[[∂n uh ]] − λi,h̃ , [[∂n vh ]] + µi,h̃ iγi
i=1
Problem:
This modifies all the entries of the matrix, and leads to technical complications.
Thus, we propose a simplified version.
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Conclusions and perspectives
Introduction
Stability
Convergence
Numerical results
BLOCK 2: Stability
We now propose the stabilised finite element method considered in this work:
Find (uh , λh̃ ) ∈ Wh := Vh × Λh̃ such that
B ((uh , λh̃ ), (vh , µh̃ )) = (f , vh )Ω −
N
X
hµi,h̃ , gi iγi
∀(vh , µh̃ ) ∈ Wh ,
i=1
where
B ((uh , λh̃ ), (vh , µh̃ )) = (∇uh , ∇vh )Ω −
N
X
hλi,h̃ , vh iγi
i=1
N
N
X
X
−
hµi,h̃ , uh iγi −
hhλi,h̃ , µi,h̃ iγi .
i=1
i=1
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Numerical results
Theorem
For all (vh , µh̃ ) ∈ Wh , the following holds
B ((vh , µh̃ ), (vh , −µh̃ )) = k(vh , µh̃ )k2W
h
.
Then, the stabilised problem is well-posed.
The next result states the consistency of the method for smooth solutions.
Lemma
Let (u, λ) and (uh , λh̃ ) ∈ Wh be the solutions. Then
B ((u − uh , λ − λh̃ ), (vh , µh̃ )) = −
N
X
hhλi , µi,h̃ iγi ,
i=1
3
for all (vh , µh̃ ) ∈ Wh . Moreover, if u ∈ H 2 + (Ω) for some > 0, then
B ((u − uh , λ − λh̃ ), (vh , µh̃ )) = 0
∀ (vh , µh̃ ) ∈ Wh .
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BLOCK 2: Convergence
We start making various assumptions on the meshes and inclusions.
where dist(xj , γij ) = 12 h.
Lemma
For every v ∈ H 1 (Ω), the following local trace inequality holds
kv k20,γ j ≤ 8 h−1 kv k20,T j + kv k0,T j k∇v k0,T j .
i
i
i
i
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Hypothesis:
#{Bi : Bi ∩ K 6= 0} ≤ M
∀ K.
(1)
Theorem
Let us suppose that u ∈ H 1+s (Ω), for s ∈ (0, 1], and λ ∈
N
Q
H δ (γi ), for
i=1
δ ∈ [0, 12 ]. Then, there exists a constant C > 0, independent of h and h̃, such
that

! 12 
N
X
√
1 +δ
2
u
λ
s
.
kλi kδ,γi
(2)
k(e , e )kWh ≤ C (1 + M)h |u|1+s,Ω + h 2
i=1
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CASE 1: Smooth analytical solution
n
u(x, y ) = sin(x) sin(y ) in Ω,
3
21
P
ku − uh k1,Ω order
hkλ − λh k20,γi
order
i=1
1
2
3
4
8
12
16
3.5800
1.7993
1.1986
0.8989
0.4489
0.3001
0.2244
0.99
1.00
1.00
1.00
0.99
1.00
0.1986
0.0595
0.0256
0.0156
0.0038
0.0020
0.0011
1.73
2.08
1.72
2.03
1.58
2.07
Table: Finite element errors for the smooth example and r = 0.1.
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CASE 2:The exact solution is not known
To estimate the error commited, we have computed a reference solution in ω
using a mesh containing P2 elements and we consider the approximation of the
following boundary value problem:
−∆u = 0
u = 10
n
1
2
3
4
8
12
16
on
γi , i = 1, 2, 3,
kuref − uh k0,ω
17.9261
12.4341
9.5128
7.9418
4.5119
3.1452
2.3856
order
0.52
0.66
0.62
0.81
0.88
0.96
in
ω
u = 0
on
kuref − uh k1,ω
28.069
20.0713
15.7798
13.0286
7.9581
5.9118
4.7040
∂Ω ,
order
0.48
0.59
0.66
0.71
0.73
0.79
Table: Errors kuref − uh k0,ω and kuref − uh k1,Ω for r = 0.2.
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CASE 2:The exact solution is not known
Figure: Reference solution uref (top) and approximate solution uh for n = 10, for
r = 0.2.
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CASE 3: More inclusions closer together
Considering
The cross-sections of uref and uh at y = 9.2 for different values of n are
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n=1
n=2
n=3
n=4
n=8
n = 12
n = 16
kuref − uh k0,ω
14.782
9.7017
7.2801
5.9660
3.3513
2.3579
1.7774
Introduction
Stability
Convergence
Numerical results
order
0.60
0.70
0.69
0.83
0.86
0.98
kuref − uh k1,ω
24.374
18.4522
14.9537
12.447
8.1707
6.1067
5.0732
order
0.40
0.51
0.63
0.60
0.71
0.64
Table: Numerical results for 9 inclusions when d = 1.
n=1
n=2
n=3
n=4
n=8
n = 12
n = 16
kuref − uh k0,ω
5.2804
3.3580
2.6033
2.1365
1.2823
0.9562
0.7457
order
0.65
0.62
0.68
0.73
0.72
0.86
kuref − uh k1,ω
14.8082
11.6816
9.8871
8.8501
6.679
5.4516
4.8062
order
0.34
0.41
0.38
0.40
0.50
0.43
Table: Numerical results for 9 inclusions when d = 0.1.
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Conclusions and perspectives
- We have proposed a simple stabilised finite element method, a fictitious
domain method enhanced with a stabilisation term, to approximate the solution
of partial differential equations posed in domains containing a moderate
amount of small perforations.
- A simple alternative to previously existing references when we consider a
domain with the number of perforations remains moderate.
FUTURE WORK
- When the domain ω is moving.
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Thank you for your attention
[email protected]
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