Boundary and Finite Element Methods for Dirichlet Boundary

Institut für Numerische Mathematik
Boundary and Finite Element Methods
for Dirichlet Boundary Control Problems
G. Of, T. X. Phan, O. Steinbach
Institut für Numerische Mathematik, Technische Universität Graz
SFB: Mathematical Optimization
and Applications in Biomedical Sciences
Subproject: Fast Finite and Boundary Element Methods
for Optimality Systems (FEMBEM)
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
1 / 29
Institut für Numerische Mathematik
Outline
Dirichlet boundary control problem
Boundary integral equations
Bi-Laplace boundary integral operators
Boundary element methods
Error estimates
Finite element methods
Numerical results
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
2 / 29
Institut für Numerische Mathematik
Dirichlet boundary control problem
Dirichlet boundary control problem
Z
1
J(u, z) =
[u(x) − u(x)]2 dx +
2
to minimize
1
αkzk2A
2
for (u, z) ∈ H 1 (Ω) × H 1/2 (Γ)
Ω
subject to
−∆u(x) = f (x)
for x ∈ Ω,
u(x) = z(x)
for x ∈ Γ = ∂Ω
where the operator A
A : H 1/2 (Γ) −→ H −1/2 (Γ),
kzk2A ≃ hAz, ziΓ ≃ kzk2H 1/2(Γ)
e modified hypersingular operator
Example: A = D
e v iΓ := hDu, v iΓ + hu, 1iΓ hv , 1iΓ
hDu,
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
3 / 29
Institut für Numerische Mathematik
Dirichlet boundary control problem
Primal Dirichlet boundary value problem
−∆u(x) = f (x) for x ∈ Ω,
u(x) = z(x) for x ∈ Γ
Adjoint Dirichlet boundary value problem
−∆p(x) = u(x) − u(x) for x ∈ Ω,
p(x) = 0 for x ∈ Γ
Optimality condition
αAz =
∂
p(x)
∂n
for x ∈ Γ
Laplace fundamental solution

1

−
log |x − y |
∗
2π
U (x, y ) =
1
1


4π |x − y |
T. X. Phan
for d = 2
for d = 3
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
4 / 29
Institut für Numerische Mathematik
Boundary integral equation for primal problem
Primal Dirichlet boundary value problem
−∆u(x) = f (x) for x ∈ Ω,
Boundary integral equation
1
I + K z − N0 f ,
Vt =
2
where
u(x) = z(x) for x ∈ Γ
t=
∂
u ∈ H −1/2 (Γ)
∂n
(Vt)(x) :=
Z
U ∗ (x, y )t(y ) dsy
for x ∈ Γ,
(Kz)(x) :=
Z
∂ ∗
U (x, y )z(y ) dsy
∂ny
for x ∈ Γ,
Z
U ∗ (x, y )f (y ) dy
for x ∈ Γ.
Γ
Γ
(N0 f )(x) :=
Ω
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
5 / 29
Institut für Numerische Mathematik
Boundary integral equation for adjoint problem
Adjoint Dirichlet boundary value problem
−∆p(x) = u(x) − u(x)
for x ∈ Ω,
p(x) = 0 for x ∈ Γ
Representation formula for x ∈ Ω
Z
Z
∂
∗
p(x) = U (x, y ) p(y ) dsy + U ∗ (x, y )[u(y ) − u(y )] dy
∂n
Γ
Ω
Boundary integral equation
Vq =
1
I +K
2
where
q=
T. X. Phan
p − N0 (u − u)
∂
p ∈ H −1/2 (Γ)
∂n
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
6 / 29
Institut für Numerische Mathematik
Boundary integral equation for adjoint problem
Due to p = 0 on Γ, we obtain
Z
Z
U ∗ (x, y )q(y ) dsy = − U ∗ (x, y )[u(y ) − u(y )] dy
Γ
for x ∈ Γ
Ω
U ∗ (x, y ) = ∆x V ∗ (x, y ) = ∆y V ∗ (x, y )
Bi-Laplace fundamental solution V ∗ (x, y )


− 1 |x − y |2 (log |x − y | − 1)
∗
8π
V (x, y ) =
1

 |x − y |
8π
T. X. Phan
for d = 2
for d = 3
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
7 / 29
Institut für Numerische Mathematik
Boundary integral equation for adjoint problem
Newton potential
Z
Ω
T. X. Phan
∗
U (x, y )u(y ) dy =
Z
∆y V ∗ (x, y )u(y ) dy
Ω
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
8 / 29
Institut für Numerische Mathematik
Boundary integral equation for adjoint problem
Newton potential
Z
∗
U (x, y )u(y ) dy =
Ω
=
Z
Γ
T. X. Phan
Z
∆y V ∗ (x, y )u(y ) dy
Ω
∂ ∗
V (x, y )u(y ) dsy −
∂ny
Z
Γ
V ∗ (x, y )
∂
u(y ) dsy +
∂n
Z
V ∗ (x, y )∆u(y ) dy
Ω
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
8 / 29
Institut für Numerische Mathematik
Boundary integral equation for adjoint problem
Newton potential
Z
∗
U (x, y )u(y ) dy =
Ω
=
∆y V ∗ (x, y )u(y ) dy
Ω
Z
∂ ∗
V (x, y )u(y ) dsy −
∂ny
=
Z
Z
V ∗ (x, y )
∂
u(y ) dsy +
∂n
Γ
Γ
Γ
T. X. Phan
Z
∂ ∗
V (x, y )z(y ) dsy −
∂ny
Z
V ∗ (x, y )∆u(y ) dy
Ω
Z
Γ
V ∗ (x, y )t(y ) dsy −
Z
V ∗ (x, y )f (y ) dy
Ω
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
8 / 29
Institut für Numerische Mathematik
Boundary integral equation for adjoint problem
Representation formula for x ∈ Ω
Z
Z
∗
p(x) = U (x, y )q(y ) dsy +
Γ
−
Z
Γ
∗
∂ ∗
V (x, y )z(y ) dsy
∂ny
Γ
V (x, y )t(y ) dsy −
Z
∗
V (x, y )f (y ) dy −
Ω
Z
U ∗ (x, y )u(y ) dy
Ω
Boundary integral equation for x ∈ Γ
Vq = V1 t − K1 z + N0 u + M0 f
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
9 / 29
Institut für Numerische Mathematik
Boundary integral equation for adjoint problem
Representation formula for x ∈ Ω
Z
Z
∗
p(x) = U (x, y )q(y ) dsy +
Γ
−
Z
Γ
∗
∂ ∗
V (x, y )z(y ) dsy
∂ny
Γ
V (x, y )t(y ) dsy −
Z
∗
V (x, y )f (y ) dy −
Ω
Z
U ∗ (x, y )u(y ) dy
Ω
Boundary integral equation for x ∈ Γ
Vq = V1 t − K1 z + N0 u + M0 f
Normal derivative for x ∈ Γ
1
′
q=
I + K q − D1 z − K1′ t − N1 u − M1 f = αAz
2
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
9 / 29
Institut für Numerische Mathematik
Boundary integral equations
We end up with a system (Nonsymmetric formulation)

1

Vt = 2 I + K z − N0 f
Vq = V1 t − K1 z + N0 u + M0 f


αAz = q
where
t, q ∈ H −1/2 (Γ),
z ∈ H 1/2 (Γ).
The Schur complement system
Tα z = V −1 N0 u + V −1 M0 f − V −1 V1 V −1 N0 f
where the operator Tα is defined by
1
Tα := V −1 K1 − V −1 V1 V −1 ( I + K ) + αA
2
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
10 / 29
Institut für Numerische Mathematik
Bi-Laplace boundary integral operators
Consider Bi-Laplace equation
−∆2 u(x) = 0
for
x ∈Ω
Representation formula for xe ∈ Ω
Z
Z
∂ ∗
∂
U (e
x , y )u(y ) dsy
u(e
x) =
U ∗ (e
x , y ) u(y ) dsy −
∂n
∂ny
Γ
Γ
Z
Z
∂
∂ ∗
∗
x , y ) ∆u(y ) dsy −
x , y )∆u(y ) dsy
V (e
+ V (e
∂n
∂ny
Γ
T. X. Phan
Γ
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
11 / 29
Institut für Numerische Mathematik
Boundary integral equations
Taking the limit Ω ∋ e
x → x ∈ Γ and using the following notations
v (x) = γ0int u(x),
r (x) = γ0int ∆u(x),
w (x) = γ1int u(x),
s(x) = γ1int ∆u(x)
we obtain a system, including the so-called Calderón projection C ,
 
  1
V
−K1
V1
v
v
2I − K
1
′
′



w   D
I
+
K
D
K
w
1
1
2
 
 =
1
r 
r   0
0
I
−
K
V
2
1
′
s
s
0
0
D
2I + K
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
12 / 29
Institut für Numerische Mathematik
Mapping properties
Lemma
The Calderón operator C is a projection, i.e., C = C 2 .
Lemma
For all boundary integral operators there hold the relations
VK ′ = KV ,
K ′ D = DK ,
VD =
K1 V − V1 K ′ = VK1′ − KV1 ,
DV1 + D1 V + K ′ K1′ + K1′ K ′ = 0,
T. X. Phan
1
I − K 2,
4
DV =
1
I − K ′2 ,
4
K ′ D1 − D1 K = DK1 − K1′ D,
VD1 + V1 D + KK1 + K1 K = 0.
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
13 / 29
Institut für Numerische Mathematik
Mapping properties
Lemma
For any q ∈ H −1/2 (Γ) there holds the equality
e qk2 2 = hK1 Vq, qiΓ − hV1 ( 1 + K ′ )q, qiΓ .
kV
L (Ω)
2
Theorem
The boundary integral operator
1
Tα := V −1 K1 − V −1 V1 V −1 ( I + K ) + αA : H 1/2 (Γ) → H −1/2 (Γ)
2
is self-adjoint, bounded and H 1/2 (Γ)-elliptic.
The Schur complement system
Tα z = V −1 N0 u + V −1 M0 f − V −1 V1 V −1 N0 f
admits a unique solution z ∈ H 1/2 (Γ).
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
14 / 29
Institut für Numerische Mathematik
Boundary element method
We have to find t, q ∈ H −1/2 (Γ), z ∈ H 1/2 (Γ) such that


−V1 t + Vq + K1 z = N0 u + M0 f
Vt − 21 I + K z = −N0 f


q − αAz = 0
Let
−1/2
Sh0 (Γ) = span{ϕ0k }N
(Γ),
k=1 ⊂ H
1/2
SH1 (Γ) = span{ϕ1k }M
(Γ)
k=1 ⊂ H
be some boundary element spaces of piecewise constant and piecewise linear
basis functions.
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
15 / 29
Institut für Numerische Mathematik
Galerkin variational formulation
The Galerkin boundary element formulation then reads to find (th , qh , zH ) ∈
Sh0 (Γ) × Sh0 (Γ) × SH1 (Γ) such that


i + hK1 zH , wh iΓ = hN0 u + M0 f , wh iΓ ,
−hV1 th , wh iΓ + hVqh , w
h Γ
1
hVth , τh iΓ − h 2 I + K zH , τh iΓ = −hN0 f , τh iΓ ,


−hqh , vH iΓ + αhAzH , vH iΓ = 0
is satisfied for all (wh , τh , vH ) ∈ Sh0 (Γ) × Sh0 (Γ) × SH1 (Γ).
Linear system of algebraic equations,
   

t
f1
−V1,h
Vh
K1,h
 Vh
− 21 Mh − Kh  q  = f 2 
−MhT
αAH
0
z
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
16 / 29
Institut für Numerische Mathematik
Unique solvability
The Galerkin matrix Vh is positive definite and therefore invertible. Hence we
obtain the Schur complement system
1
αAH − MhT Vh−1 V1,h Vh−1 ( Mh + Kh ) + MhT Vh−1 K1,h z
2
= MhT Vh−1 f 1 + V1,h Vh−1 f 2
where
eα,H := αAH − M T V −1 V1,h V −1 ( 1 Mh + Kh ) + M T V −1 K1,h
T
h h
h h
h
2
defines Galerkin boundary element approximation of the Schur complement
boundary integral operator Tα .
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
17 / 29
Institut für Numerische Mathematik
Unique solvability
Theorem
eα,H is positive definite, i.e.
The approximate Schur complement T
eα,H z, z) >
(T
1 T
c kzH k2H 1/2 (Γ)
2 1
for all z ∈ RM ↔ zH ∈ SH1 (Γ), if
h 6 c0 H
for some constant c0 > 0.
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
18 / 29
Institut für Numerische Mathematik
Error estimates
Theorem
Let t, z be the unique solution of system of boundary integral equations
minimizing the reduced cost functional J(u, z). Let th , zH be the unique
solution of the Galerkin variational problem. When assuming z ∈ H 2 (Γ)
there hold the error estimates
kz − zH kH 1/2 (Γ) 6 C H 3/2 |z|H 2 (Γ) + h3/2 kzkH 2(Γ) + h3/2 ,
kt − th kH −1/2 (Γ) 6 C H 3/2 |z|H 2 (Γ) + h3/2 kzkH 2(Γ) + h3/2 .
Remark: In L2 (Γ) norm
kz − zH kL2 (Γ) = O(h2 + H 2 ),
kt − th kL2 (Γ) = O(h + H).
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
19 / 29
Institut für Numerische Mathematik
Symmetric formulation
Boundary integral equations


  
−V1
V
K1
t
N0 u + M 0 f
 V
0
− 21 I − K  q  =  −N0 f 
1
′
′
z
N1 u + M 1 f
−K1 2 I + K −(αA + D1 )
The Schur complement system
h
1
1
K1′ V −1 ( I + K ) + ( I + K ′ )V −1 K1
2
2
i
1
1
′
−1
− ( I + K )V V1 V −1 ( I + K ) + (αA + D1 ) z
2
2
1
′ −1
′
=K1 V N0 f − ( I + K )V −1 V1 V −1 N0 f
2
1
1
′
−1
+ ( I + K )V N0 u + ( I + K ′ )V −1 M0 f − N1 u − M1 f
2
2
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
20 / 29
Institut für Numerische Mathematik
Unique solvability for symmetric formulation
Theorem
The boundary integral operator
1
1
1
Tα := K1′ V −1 ( I + K ) − ( I + K ′ )V −1 V1 V −1 ( I + K )
2
2
2
1
′
−1
1/2
+( I + K )V K1 + (αA + D1 ) : H (Γ) → H −1/2 (Γ)
2
is self-adjoint, bounded and H 1/2 (Γ)-elliptic.
Linear system of algebraic equations,
   

−V1,h
Vh
K1,h
t
a
 Vh
− 12 Mh − Kh  q  = b
T
K1,h
− 21 MhT − KhT αAH + D1,H
c
z
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
21 / 29
Institut für Numerische Mathematik
Finite Element Methods
Primal Dirichlet boundary value problem
−∆u(x) = f (x) for x ∈ Ω,
u(x) = z(x) for x ∈ Γ
u = u0 + Ez, u0 ∈ H01 (Ω):
Z
Z
∇[u0 (x) + Ez(x)]∇v (x) dx = f (x)v (x) dx
Ω
for all
v ∈ H01 (Ω)
Ω
Adjoint Dirichlet boundary value problem
−∆p(x) = u(x) − u(x) for x ∈ Ω,
p(x) = 0 for x ∈ Γ
p ∈ H01 (Ω) is the weak solution of the variational problem
Z
Z
∇p(x)∇v (x) dx = [u(x) − u(x)]v (x) dx for all v ∈ H01 (Ω)
Ω
T. X. Phan
Ω
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
22 / 29
Institut für Numerische Mathematik
Finite Element Methods
Optimality condition
α(Az)(x) = q(x) =
∂
p(x) for x ∈ Γ
∂n
w ∈ H 1/2 (Γ)
αhAz, w iΓ
=
Z
=
Z
∂
p(x)w (x) dsx
∂nx
Γ
∇p(x)∇Ew (x) dx −
Ω
Finite element discretization

−MII
KII
 KII
MIC −KIC
T. X. Phan
Z
[u(x) − u(x)]Ew (x) dx
Ω

  
uI
−f 1
−MCI
p  =  f 2 
KCI
f3
(MCC + αAh )
z
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
23 / 29
Institut für Numerische Mathematik
Numerical results (nonsymmetric formulation)
In our numerical example, we consider the square Ω = (0, 0.5)2 , h = H, and
1
8
u(x) = − 4 +
[x1 (1 − 2x1 ) + x2 (1 − 2x2 )] , f (x) = − , α = 0.01
α
α
e:
A=D
e v iΓ := hDu, v iΓ + hu, 1iΓ hv , 1iΓ
hDu,
with non zero control z(x), (see paper of S. May, R. Rannacher, B. Vexler).
Mesh
16
32
64
128
256
512
1024
expected
T. X. Phan
kth − th9 kL2 (Γ)
Error
Order
33.5093
24.0337 0.48
14.7956 0.70
8.4434
0.81
4.5915
0.88
2.4033
0.93
1.1718
1.04
1.00
kzh − zH9 kL2 (Γ)
Error
Order
0.45219
0.12773
1.82
0.03536
1.85
9.026e-03 1.97
2.175e-03 2.05
5.158e-04 2.08
1.389e-04 1.89
2.00
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
24 / 29
Institut für Numerische Mathematik
Numerical results (symmetric formulation)
We consider the square Ω = (0, 0.5)2 , h = H, and
1
8
u(x) = − 4 +
[x1 (1 − 2x1 ) + x2 (1 − 2x2 )] , f (x) = − ,
α
α
e:
A=D
α = 0.01
e v iΓ := hDu, v iΓ + hu, 1iΓ hv , 1iΓ
hDu,
with non zero control z(x).
Mesh
16
32
64
128
256
512
1024
expected
T. X. Phan
kth − th9 kL2 (Γ)
Error
Order
39.8736
28.6702 0.48
16.2201 0.82
8.8797
0.87
4.7272
0.91
2.4459
0.95
1.1853
1.05
1.00
kzh − zH9 kL2 (Γ)
Error
Order
2.24551
0.46550
2.27
0.08836
2.39
0.01631
2.44
3.019e-03 2.43
5.731e-04 2.40
1.243e-04 2.20
2.00
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
25 / 29
Institut für Numerische Mathematik
Compare numerical results: BEM and FEM
Consider the square Ω = (0, 0.5)2 , h = H and
1
[x1 (1 − 2x1 ) + x2 (1 − 2x2 )] ,
u(x) = − 4 +
α
e:
A=D
8
f (x) = − ,
α
α = 0.01
e v iΓ := hDu, v iΓ + hu, 1iΓ hv , 1iΓ
hDu,
with non zero control z(x).
Level
2
3
4
5
6
7
8
T. X. Phan
Symmetric BEM
FEM
kL2 (Γ)
kzH − zHBEM
9
order
kL2 (Γ)
kzH − zHFEM
9
order
kL2 (Γ)
kzH − zHBEM
9
2.24551
0.46550
0.08836
0.01630
3.019e-03
5.731e-04
1.243e-04
2.27
2.39
2.44
2.43
2.40
2.20
0.38934
0.10745
0.02809
7.281e-03
1.872e-03
4.691e-04
1.060e-04
1.86
1.93
1.95
1.96
2.00
2.15
0.38936
0.10747
0.02811
7.298e-03
1.889e-03
4.883e-04
1.337e-04
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
26 / 29
Institut für Numerische Mathematik
Linear systems: BEM and FEM
Finite element system

−MII
KII
 KII
MIC −KIC
  

uI
−MCI
−f 1
p  =  f 2 
KCI
(MCC + αAh )
f3
z
Boundary element system

−V1,h
Vh
 Vh
T
K1,h
− 21 MhT − KhT
   
K1,h
t
a
− 12 Mh − Kh  q  = b
αAH + D1,H
c
z
• Preconditioned iterative system solvers (saddle point structure)
• Construction of preconditioners (BEM and FEM)
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
27 / 29
Institut für Numerische Mathematik
Outlook/Plan
• Stability analysis for symmetric formulation: h = H
• Mixed boundary control problems
• Box boundary control problems
• Parabolic control problems
• BIE for time dependent problems
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
28 / 29
Institut für Numerische Mathematik
Thanks for your attention!
T. X. Phan
Boundary and Finite Element Methods for Dirichlet Boundary Control Problems
SFB Status Seminar, 07/11/2008
29 / 29