0
C Interior Penalty Methods
Domain Decomposition
Current Trends in Computational Methods for PDEs
CIMPA-NPDE-NBHM Research School
Bangalore, July 2013
Friday, July 19, 2013
Outline
�
Preconditioned Conjugate Gradients
�
Additive Schwarz Preconditioners
�
An Overlapping Domain Decomposition Algorithm
Friday, July 19, 2013
General References
• Smith, Bjørstad and Gropp, Domain Decomposition
Cambridge University Press, 1996
• Toselli and Widlund, Domain Decomposition Methods - Al-
gorithms and Theory
Springer-Verlag, 2005
• Mathew, Domain Decomposition Methods for the Numeri-
cal Solution of Partial Differential Equations
Springer-Verlag, 2008
• PETSc (Portable, Extensible Toolkit for Scientific Compu-
tation)
http://www.mcs.anl.gov/petsc
Friday, July 19, 2013
Preconditioned Conjugate Gradients
Friday, July 19, 2013
Conjugate Gradients
SPD Problem
Ax = b
�
A ∈ Rn×n , x, b ∈ Rn
�
A is symmetric positive definite (and sparse).
�
Exact solution is denoted by x∗ .
Friday, July 19, 2013
Conjugate Gradients
SPD Problem
Ax = b
�
A ∈ Rn×n , x, b ∈ Rn
�
A is symmetric positive definite (and sparse).
�
Exact solution is denoted by x∗ .
Equivalent Formulation
Find
where
Friday, July 19, 2013
x∗ = argmin E(x)
x∈Rn
1 T
E(x) = x Ax − bT x
2
Conjugate Gradients
Conjugate Gradient Algorithm with Initial Guess x0
r0 = b − Ax0
d0 = r0
For k = 0, 1, 2, . . .
αk = (rTk rk )/(dTk Adk )
xk+1 = xk + αk dk
rk+1 = rk − αk Adk
βk = (rTk+1 rk+1 )/(rTk rk )
dk+1 = rk+1 + βk dk
Friday, July 19, 2013
Conjugate Gradients
Conjugate Gradient Algorithm with Initial Guess x0
r0 = b − Ax0
d0 = r0
For k = 0, 1, 2, . . .
αk = (rTk rk )/(dTk Adk )
xk+1 = xk + αk dk
rk+1 = rk − αk Adk
βk = (rTk+1 rk+1 )/(rTk rk )
dk+1 = rk+1 + βk dk
�
Friday, July 19, 2013
Reach exact solution x∗ within n steps.
Conjugate Gradients
Error Estimate after k Steps (k � n)
�
� κ(A) − 1 �k
�x∗ − xk �A ≤ 2 �
�x∗ − x0 �A
κ(A) + 1
where
so that
Energy Norm
Friday, July 19, 2013
λmax (A)
κ(A) =
λmin (A)
(≥ 1)
�
κ(A) − 1
0≤ �
<1
κ(A) + 1
�
�1
�v�A = v Av 2
T
Conjugate Gradients
Error Estimate after k Steps (k � n)
�
� κ(A) − 1 �k
�x∗ − xk �A ≤ 2 �
�x∗ − x0 �A
κ(A) + 1
where
so that
λmax (A)
κ(A) =
λmin (A)
(≥ 1)
�
κ(A) − 1
0≤ �
<1
κ(A) + 1
If κ(A) � 1, i.e., A is ill-conditioned, then
�
κ(A) − 1
�
≈1
κ(A) + 1
and the convergence will be slow.
Friday, July 19, 2013
Preconditioning
Choose an SPD matrix B. Instead of solving
Ax = b
we can solve the equivalent SPD problem (with x = B1/2 x� )
(B1/2 AB1/2 )x� = B1/2 b
more efficiently provided
κ(B1/2 AB1/2 ) � κ(A)
Moreover this can be accomplished without using B1/2 .
Friday, July 19, 2013
Preconditioning
Choose an SPD matrix B. Instead of solving
Ax = b
we can solve the equivalent SPD problem (with x = B1/2 x� )
(B1/2 AB1/2 )x� = B1/2 b
more efficiently provided
κ(B1/2 AB1/2 ) � κ(A)
Moreover this can be accomplished without using B1/2 .
Note that BA = B1/2 (B1/2 AB1/2 )B−1/2 and hence
1/2
1/2
λ
(B
AB
)
λmax (BA)
max
1/2
1/2
κ(B AB ) =
=
= κ(BA)
1/2
1/2
λmin (B AB )
λmin (BA)
Friday, July 19, 2013
Preconditioned Conjugate Gradient
r0 = b − Ax0
d0 = Br0
For k = 0, 1, 2, . . .
αk = (rTk Brk )/(dTk Adk )
xk+1 = xk + αk dk
rk+1 = rk − αk Adk
βk = (rTk+1 Brk+1 )/(rTk Brk )
dk+1 = Brk+1 + βk dk
Friday, July 19, 2013
Preconditioned Conjugate Gradient
r0 = b − Ax0
d0 = Br0
For k = 0, 1, 2, . . .
αk = (rTk Brk )/(dTk Adk )
xk+1 = xk + αk dk
rk+1 = rk − αk Adk
βk = (rTk+1 Brk+1 )/(rTk Brk )
dk+1 = Brk+1 + βk dk
Error Estimate
�
�
�x∗ − xk �A ≤ 2 �
Friday, July 19, 2013
κ(BA) − 1 �k
κ(BA) + 1
�x∗ − x0 �A
Preconditioned Conjugate Gradient
CG
r0 = b − Ax0 , d0 = r0
αk = (rTk rk )/(dTk Adk )
xk+1 = xk + αk dk
rk+1 = rk − αk Adk
βk = (rTk+1 rk+1 )/(rTk rk )
dk+1 = rk+1 + βk dk
PCG
r0 = b − Ax0 , d0 = Br0
αk = (rTk Brk )/(dTk Adk )
xk+1 = xk + αk dk
rk+1 = rk − αk Adk
βk = (rTk+1 Brk+1 )/(rTk Brk )
dk+1 = Brk+1 + βk dk
Friday, July 19, 2013
Two Requirements for B
�
κ(BA) � κ(A)
�
The evaluation of Bv (v ∈ Rn ) can be carried out efficiently.
Friday, July 19, 2013
Additive Schwarz Preconditioners
Friday, July 19, 2013
Set-Up
V is a finite dimensional vector space.
V � is the dual space.
�·, ·� is the canonical bilinear form on V � × V.
A : V −→ V � is a linear operator that is SPD, i.e.,
�Av, w� = �Aw, v�
�Av, v� > 0
Friday, July 19, 2013
∀ v, w ∈ V
∀ v ∈ V \ {0}
Set-Up
V is a finite dimensional vector space.
V � is the dual space.
�·, ·� is the canonical bilinear form on V � × V.
A : V −→ V � is a linear operator that is SPD, i.e.,
�Av, w� = �Aw, v�
∀ v, w ∈ V
�Av, v� > 0
∀ v ∈ V \ {0}
Remark If we choose a basis v1 , . . . , vn for V and equip V � with
the dual basis, then the matrix A representing A is given by
A(i, j) = �Avi , vj �
Hence A is SPD iff A is SPD.
Friday, July 19, 2013
for
1 ≤ i, j ≤ n
Set-Up
V is a finite dimensional vector space.
V � is the dual space.
�·, ·� is the canonical bilinear form on V � × V.
A : V −→ V � is a linear operator that is SPD, i.e.,
�Av, w� = �Aw, v�
�Av, v� > 0
∀ v, w ∈ V
∀ v ∈ V \ {0}
Additive Schwarz preconditioners are preconditioners for the
operator A that generalize classical diagonal (Jacobi) preconditioners.
Friday, July 19, 2013
Ingredients for Additive Schwarz Preconditioners
V1 , V2 , . . . , VJ are finite dimensional vector spaces.
Ij : Vj −→ V is a linear operator for 1 ≤ j ≤ J.
V=
J
�
Ij Vj
j=1
Ijt : V � −→ Vj� is the transpose of Ij with respect to the canonical
bilinear forms.
�Ijt β, v� = �β, Ij v�
∀ β ∈ V � , v ∈ Vj
Aj : Vj −→ Vj� is a linear SPD operator.
� −→ V is therefore also SPD, i.e.,
A−1
:
V
j
j
−1
�β, A−1
γ�
=
�γ,
A
j
j β�
�β, A−1
j β� > 0
Friday, July 19, 2013
∀ β, γ ∈ Vj�
∀ β ∈ Vj� \ {0}
Additive Schwarz Preconditioners
B : V � −→ V
B=
J
�
−1 t
Ij Aj Ij
j=1
Goal
Characterize the condition number of the operator
BA : V −→ V
Friday, July 19, 2013
Additive Schwarz Preconditioners
B : V � −→ V
B=
J
�
−1 t
Ij Aj Ij
j=1
Goal
Characterize the condition number of the operator
BA : V −→ V
Remark A−1
j represents either a subdomain solve or a coarse
grid solve in domain decomposition.
Friday, July 19, 2013
Additive Schwarz Preconditioners
B : V � −→ V
B=
J
�
−1 t
Ij Aj Ij
j=1
Goal
Characterize the condition number of the operator
BA : V −→ V
Remark A−1
j represents either a subdomain solve or a coarse
grid solve in domain decomposition.
Remark If we choose bases for V and Vj (1 ≤ j ≤ J) and equip
V � and Vj� with the dual bases, then the matrix B representing
the additive Schwarz preconditioner has the same form:
B=
J
�
j=1
Friday, July 19, 2013
T
Ij A−1
I
j
j
Additive Schwarz Preconditioners
Example
V = Rn
A : V −→ V � is given by
�Av, w� = vt Aw
∀ v, w ∈ Rn
where A ∈ Rn×n is SPD.
Vj is the one dimensional subspace of Rn spanned by êj .
Ij : Vj −→ V is the natural injection.
Aj : Vj −→ Vj� is defined by �Aj êj , êj � = A(j, j).
B=
J
�
t
Ij A−1
I
j
j
j=1
is then represented by the matrix B = D−1 with respect to the
canonical basis of Rn , where D is the diagonal part of A.
Friday, July 19, 2013
Property of B
Lemma
Proof.
B=
The operator B : V � −→ V is SPD.
�J
−1 t
I
A
j=1 j j Ij is clearly symmetric positive semi-definite.
If β ∈ V � and �β, Bβ� = 0, then
0 = �β,
implies
and hence
Friday, July 19, 2013
J
�
t
Ij A−1
I
j β� =
j
j=1
j=1
t
�Ijt β, A−1
I
j β� = 0
j
t
Ij β
=0
J
�
for
for
t
�Ijt β, A−1
I
j β�
j
1≤j≤J
1 ≤ j ≤ J.
Property of B
Therefore
�β,
J
�
j=1
Ij vj � =
J
�
j=1
�Ijt β, vj � = 0
which implies
β=0
since
V=
J
�
j=1
Friday, July 19, 2013
Ij Vj
∀ vj ∈ Vj , 1 ≤ j ≤ J
Property of B
Therefore
�β,
J
�
j=1
Ij vj � =
J
�
j=1
�Ijt β, vj � = 0
which implies
∀ vj ∈ Vj , 1 ≤ j ≤ J
β=0
since
V=
J
�
Ij Vj
j=1
Corollary
Friday, July 19, 2013
The operator B−1 : V −→ V � is SPD.
Fundamental Lemma for ASPs
�B−1 v, v� =
Friday, July 19, 2013
v=
min
�
J
j=1 Ij vj
vj ∈Vj
J
�
j=1
�Aj vj , vj �
∀v ∈ V
Fundamental Lemma for ASPs
�B−1 v, v� =
Step 1
If v =
J
�
v=
min
�
J
j=1 Ij vj
vj ∈Vj
J
�
j=1
∀v ∈ V
Ij vj , then
j=1
Step 2
�Aj vj , vj �
�B−1 v, v� ≤
J
�
j=1
�Aj vj , vj �
For a particular decomposition v =
J
�
j=1
�B
Friday, July 19, 2013
−1
v, v� =
J
�
j=1
∗ ∗
�Aj vj , vj �
Ij v∗j , we have
Step 1
�B−1 v, v� = �B−1 v,
=
J
�
j=1
=
J
�
j=1
≤
=
J
�
j=1
Ij v j )
j=1
�Ijt B−1 v, vj �
�Ijt B−1 v, A−1
j Aj vj �
J
��
j=1
J
��
j=1
J
�1 � �
�1
2
2
−1 t −1
−1
t −1
�Ij B v, Aj Ij B v�
�Aj vj , Aj Aj vj �
j=1
J
�1 � �
�1
2
2
−1 t −1
−1
�B v, Ij Aj Ij B v�
�Aj vj , vj �
j=1
J
�
�
1
= �B−1 v, v� 2
Friday, July 19, 2013
(v =
Ij vj �
J
�
j=1
�1
2
�Aj vj , vj �
Step 2
t B−1 v.
Let v∗j = A−1
I
j
j
J
�
Ij v∗j =
j=1
J
�
j=1
J
�
t −1
−1
Ij A−1
I
B
v
=
BB
v=v
j
j
J
�
�Ijt B−1 v, v∗j �
j=1
�Aj v∗j , v∗j � =
=
j=1
J
�
j=1
�B−1 v, Ij v∗j �
= �B−1 v,
J
�
j=1
= �B−1 v, v�
Friday, July 19, 2013
Ij v∗j �
Characterizations of λmax (BA) and λmin (BA)
The operator BA : V −→ V is symmetric with respect to the
inner product
((·, ·)) = �B−1 ·, ·�
((BAv, w)) = �B−1 (BA)v, w�
= �Av, w� = �Aw, v� = ((BAw, v))
Friday, July 19, 2013
∀ v, w ∈ V
Characterizations of λmax (BA) and λmin (BA)
The operator BA : V −→ V is symmetric with respect to the
inner product
((·, ·)) = �B−1 ·, ·�
((BAv, w)) = �B−1 (BA)v, w�
= �Av, w� = �Aw, v� = ((BAw, v))
∀ v, w ∈ V
Rayleigh Quotient Formulas
�Av, v�
((BAv, v))
λmin (BA) = min
= min −1
v∈V �B v, v�
v∈V ((v, v))
((BAv, v))
�Av, v�
λmax (BA) = max
= max −1
v∈V ((v, v))
v∈V �B v, v�
Friday, July 19, 2013
Characterizations of λmax (BA) and λmin (BA)
λmin (BA) = min
v∈V
v=
�Av, v�
J
�
min
�Aj vj , vj �
�
J
j=1 Ij vj
vj ∈Vj
λmax (BA) = max
v∈V
v=
Friday, July 19, 2013
j=1
�Av, v�
J
�
min
�A
v
,
v
�
j
j
j
�
J
j=1 Ij vj
vj ∈Vj
j=1
An Overlapping
Domain Decomposition Algorithm
Friday, July 19, 2013
Overlapping Domain Decomposition
Th
TH
Ωj
�
given a triangulation Th
�
create a coarse grid TH
�
enlarge the subdomains of the coarse grid to construct an
overlapping domain decomposition
Friday, July 19, 2013
Overlapping Domain Decomposition
Th
TH
Ωj
Vh ⊂ H01 (Ω) = Qk (k ≥ 2) finite element space associated with
Th
VH ⊂ H01 (Ω) = Q1 or Q2 finite element space associated with
TH
Vj ⊂ H01 (Ωj ) = Qk (k ≥ 2) finite element space associated with
Th (1 ≤ j ≤ J)
Friday, July 19, 2013
Discrete Operator for Clamped Plates
Ah : Vh −→ Vh�
�Ah v1 , v2 � = ah (v1 , v2 )
ah (w, v) =
�
�
R∈Th
R
∀ v1 , v2 ∈ Vh
D2 w : D2 v dx
� ∂2w �
��� ∂v �� �
� ∂2v �
��� ∂w ���
� � ��
+
+
ds
2
2
∂n
∂n
∂n
∂n
e
e∈Eh
� σ � �� ∂w �� �� ∂v ��
+
ds
(clamped plates)
|e| e ∂n
∂n
e∈Eh
Ah is the SPD operator to be preconditioned.
Friday, July 19, 2013
Ingredients for the Preconditioner
Friday, July 19, 2013
Ingredients for the Preconditioner
A0 : VH −→ VH�
�A0 v1 , v2 � = aH (v1 , v2 )
aH (w, v) =
��
R∈TH
R
∀ v1 , v2 ∈ VH
D2 w : D2 v dx
� ∂2w �
��� ∂v �� �
� ∂2v �
��� ∂w ���
� � ��
+
+
ds
2
2
∂n
∂n
∂n
∂n
e
e∈EH
�
�� ∂w �� �� ∂v ��
� σ
+
ds
|e| e ∂n
∂n
e∈EH
Friday, July 19, 2013
Ingredients for the Preconditioner
A0 : VH −→ VH�
�A0 v1 , v2 � = aH (v1 , v2 )
∀ v1 , v2 ∈ VH
I0 : VH −→ Vh
I0 = Πh ◦ EH
where EH : VH −→ H02 (Ω) is the enriching operator from VH
to a Bogner-Fox-Schmit space associated with TH and Πh :
H02 (Ω) −→ Vh is the nodal interpolation operator.
Friday, July 19, 2013
Ingredients for the Preconditioner
A0 : VH −→ VH�
�A0 v1 , v2 � = aH (v1 , v2 )
∀ v1 , v2 ∈ VH
I0 : VH −→ Vh
I0 = Πh ◦ EH
where EH : VH −→ H02 (Ω) is the enriching operator from VH
to a Bogner-Fox-Schmit space associated with TH and Πh :
H02 (Ω) −→ Vh is the nodal interpolation operator.
Remark The correct choice of the connection operator I0 is
crucial for the good performance of the preconditioner.
Friday, July 19, 2013
Ingredients for the Preconditioner
Aj : Vj −→ Vj�
�Aj v1 , v2 � = aj (v1 , v2 )
aj (w, v) =
��
R∈Th
R⊂Ωj
R
∀ v1 , v2 ∈ Vj
D2 w : D2 v dx
� ∂2w �
��� ∂v �� �
� ∂2v �
��� ∂w ���
� � ��
+
ds
+
2
2
∂n
∂n
∂n
∂n
e
e∈Eh
e⊂Ω̄j
� σ � �� ∂w �� �� ∂v ��
+
ds
|e| e ∂n
∂n
e∈Eh
e⊂Ω̄j
Friday, July 19, 2013
Ingredients for the Preconditioner
Aj : Vj −→ Vj�
�Aj v1 , v2 � = aj (v1 , v2 )
∀ v1 , v2 ∈ Vj
Ij : Vj −→ Vh
Ij = natural injection
Friday, July 19, 2013
Two-Level Additive Schwarz Preconditioner
BTL : Vh� −→ Vh
BTL =
J
�
t
Ij A−1
I
j
j
j=0
�Ijt α, vj � = �α, Ij vj �
Friday, July 19, 2013
Dryja and Widlund (1987)
∀ α ∈ V � , vj ∈ Vj
Two-Level Additive Schwarz Preconditioner
BTL : Vh� −→ Vh
BTL =
J
�
t
Ij A−1
I
j
j
j=0
�Ijt α, vj � = �α, Ij vj �
Dryja and Widlund (1987)
∀ α ∈ V � , vj ∈ Vj
�
The coarse grid solve
and the subdomain solves
(1 ≤ j ≤ J) can be carried out in parallel.
�
The coarse grid solve provides global communication
among the subdomains so that this preconditioner is scalable.
Friday, July 19, 2013
−1
A0
−1
Aj
Condition Number Estimates
�
λmax (BTL Ah )
H �3
κ(BTL Ah ) =
≤C 1+
λmin (BTL Ah )
δ
where δ measures the overlap among the subdomains and the
positive constant C is independent of h, H, δ and J.
Friday, July 19, 2013
Condition Number Estimates
�
λmax (BTL Ah )
H �3
κ(BTL Ah ) =
≤C 1+
λmin (BTL Ah )
δ
where δ measures the overlap among the subdomains and the
positive constant C is independent of h, H, δ and J.
�
κ(Ah ) ≈ h−4
�
κ(BTL Ah ) ≤ C if H/δ is bounded
(optimal preconditioner in the case of generous overlap)
�
Friday, July 19, 2013
This is a scalable estimate.
Condition Number Estimates
Characterization of λmax (BTL Ah )
λmax (BTL Ah ) = max
v∈Vh
v=
�Ah v, v�
J
�
min
�Aj vj , vj �
�J
j=0 Ij vj
j=0
vj ∈Vj
Characterization of λmin (BTL Ah )
λmin (BTL Ah ) = min
v∈Vh
v=
�Ah v, v�
J
�
�Aj vj , vj �
min
�J
j=0 Ij vj
vj ∈Vj
Friday, July 19, 2013
j=0
Upper Bound for λmax (BTL Ah )
�Ah v, v�
≤ C∗
J
�
min
�Aj vj , vj �
�
λmax (BAh ) = max
v∈Vh
v= Jj=0 vj
j=0
vj ∈Vj
because
�Av, v� = �A
J
�
Ij vj ,
j=0
for any decomposition v =
J
�
k=0
�J
Ik vk � ≤ C∗
j=0 Ij vj
j=0
(vj ∈ Vj ).
(C∗ is independent of h, H, δ and J.)
Friday, July 19, 2013
J
�
�Aj vj , vj �
Stability of I0
The stability estimate
�Ah I0 vH , I0 vH � ≤ C�AH vH , vH �
∀ vH ∈ V0 = VH
for the operator I0 = Πh ◦ EH plays a key role in the derivation
of the upper bound for λmax (BTL Ah ).
Friday, July 19, 2013
Stability of I0
The stability estimate
�Ah I0 vH , I0 vH � ≤ C�AH vH , vH �
∀ vH ∈ V0 = VH
for the operator I0 = Πh ◦ EH plays a key role in the derivation
of the upper bound for λmax (BTL Ah ).
Recall
�Ah v, v� ≈
�AH v, v� ≈
Friday, July 19, 2013
�
R∈Th
�
R∈TH
�� ∂v ���2
� σ�
�
�
2
|v|H 2 (T) +
�
�
|e| ∂n L2 (e)
e∈Eh
�� ∂v ���2
� σ�
�
�
2
|v|H 2 (T) +
�
�
|e| ∂n L2 (e)
e∈EH
Stability of I0
The stability estimate
�Ah I0 vH , I0 vH � ≤ C�AH vH , vH �
∀ vH ∈ V0 = VH
Since the magnitudes of the scaling factor 1/|e| are different
for the fine and coarse meshes, if we just use the natural injection as I0 , then the constant in the stability estimate for I0
and consequently the upper bound for λmax (BTL Ah ) will depend
adversely on the ratio between the mesh sizes.
Friday, July 19, 2013
Stability of I0
The stability estimate
�Ah I0 vH , I0 vH � ≤ C�AH vH , vH �
∀ vH ∈ V0 = VH
Since the magnitudes of the scaling factor 1/|e| are different
for the fine and coarse meshes, if we just use the natural injection as I0 , then the constant in the stability estimate for I0
and consequently the upper bound for λmax (BTL Ah ) will depend
adversely on the ratio between the mesh sizes.
We avoid this problem by including the enriching operator in the
definition of I0 :
I0 = Πh ◦ EH
Because functions in the Bogner-Fox-Schmit spaces are C1
functions, the sum involving the jumps of the normal derivatives
disappears.
Friday, July 19, 2013
Lower Bound for λmin (BTL Ah )
It only remains to find a lower bound for
λmin (BAh ) = min
v∈Vh
v=
Friday, July 19, 2013
�Ah v, v�
J
�
min
�Aj vj , vj �
�
J
j=0 Ij vj
vj ∈Vj
j=0
Lower Bound for λmin (BTL Ah )
It only remains to find a lower bound for
λmin (BAh ) = min
v∈Vh
v=
�Ah v, v�
J
�
min
�Aj vj , vj �
�
J
j=0 Ij vj
vj ∈Vj
j=0
For any v ∈ Vh and any decomposition
v=
J
�
j=0
we have
v=
Friday, July 19, 2013
†
Ij vj
†
(vj
∈ Vj )
�Ah v, v�
�Ah v, v�
≥ J
J
�
�
† †
�Aj vj , vj �
�Aj vj , vj �
min
�
J
j=0 Ij vj
vj ∈Vj
j=0
j=0
Lower Bound for λmin (BTL Ah )
The key is to find, for any v ∈ Vh , a special decomposition
v=
J
�
†
Ij vj
†
(vj
j=0
such that
∈ Vj )
�Ah v, v�
J
�
† †
�Aj vj , vj �
j=0
is as large as possible, or equivalently
J
�
j=0
† †
�Aj vj , vj �
≤ C2 �Ah v, v�
for some positive constant C2 that is as small as possible.
Friday, July 19, 2013
Lower Bound for λmin (BTL Ah )
Analytical Tools
Partition of unity
θj ∈ C∞ (Ω̄) (1 ≤ j ≤ J)
θj ≥ 0 and θj = 0 on Ω \ Ωj
�J
j=1 θj = 1 on Ω
C
C
2
�∇θj �L∞ (Ω) ≤
�D θj �L∞ (Ω) ≤ 2
δ
δ
(δ = measure of the overlap among the subdomains)
Friday, July 19, 2013
Lower Bound for λmin (BTL Ah )
Analytical Tools
Partition of unity
θj ∈ C∞ (Ω̄) (1 ≤ j ≤ J)
θj ≥ 0 and θj = 0 on Ω \ Ωj
�J
j=1 θj = 1 on Ω
C
C
2
�∇θj �L∞ (Ω) ≤
�D θj �L∞ (Ω) ≤ 2
δ
δ
(δ = measure of the overlap among the subdomains)
Nodal Interpolation Operators
Enriching Operator
Friday, July 19, 2013
EH
Πh and ΠH
Construction of the Special Decomposition
Given any v ∈ Vh we take
†
v0
and
Friday, July 19, 2013
†
vj
= ΠH v ∈ VH = V0
= Πh (θj (v −
†
I0 v0 ))
∈ Vj
(I0 = Πh ◦ EH )
Construction of the Special Decomposition
Given any v ∈ Vh we take
†
v0
and
†
vj
J
�
†
Ij vj
= ΠH v ∈ VH = V0
= Πh (θj (v −
=
†
I0 v0
j=0
+
J
�
j=1
=
†
I0 v0
+ Πh
†
I0 v0 ))
∈ Vj
Πh (θj (v −
J
��
j=1
=
=v
Friday, July 19, 2013
+ (v −
†
I0 v0 ))
�
†
θj (v − I0 v0 )
= I0 v†0 + Πh (v − I0 v†0 )
†
I0 v0
(I0 = Πh ◦ EH )
†
I0 v0 )
Property of the Special Decomposition
From
�∇θj �L∞ (Ω)
C
≤
δ
2
�D θj �L∞ (Ω)
C
≤ 2
δ
properties of the nodal interpolation operator
properties of the enriching operator
we have
J
�
j=0
�A0 v†j , v†j � = �Aj v†0 , v†0 ) +
�
J
�
j=1
�Ah Πh (θj (v − I0 v†0 )), Πh (θj (v − I0 v†0 )�
H3 �
≤ C† 1 + 3 �Ah v, v�
δ
where C† is independent of h, H, δ and J.
Friday, July 19, 2013
Lower Bound for λmin (BTL Ah )
The estimate
J
�
j=0
v=
† †
�A0 vj , vj �
�
≤ C† 1 +
�
3
H
δ3
�Ah v, v� implies
� �
�Ah v, v�
a(v, v)
H �3 �−1
≥
≥
C
1
+
†
J
J
δ
�
�
† †
min
�A
v
,
v
�
�A
v
j j j
j j , vj �
�
J
j=0 Ij vj
vj ∈Vj
j=0
j=0
and hence
λmin (BTL Ah ) = min
v∈Vh
v=
Friday, July 19, 2013
�
�
�Ah v, v�
J
�
min
�Aj vj , vj �
�
J
j=0 Ij vj
vj ∈Vj
j=0
H �3 �−1
≥ C† 1 +
δ
Estimate for κ(BTL Ah )
Theorem
�
H �3
κ(BTL Ah ) ≤ C 1 +
δ
where C is independent of h, H, δ and J.
Proof.
λmax (BTL Ah )
κ(BTL Ah ) =
λmin (BTL Ah )
λmax (BTL Ah ) ≤ C∗
�
�
λmin (BTL Ah ) ≥ C† 1 +
Friday, July 19, 2013
�
�
3
−1
H
δ
Numerical Results
biharmonic equation with the boundary conditions of
clamped plates on the unit square
Vh = Q2 finite element space
VH = Q2 /Q1 finite element space
σ=5
Friday, July 19, 2013
Numerical Results
Number of PCG iterations needed for reducing the error in the
energy norm (not the residual error) by a factor of 10−6
h
G
2
2
2
2
-2
-3
-4
-5
2
-2
13
2
-3
2
-4
2
-5
2
-6
2
-7
13
13
12
12
11
14
14
13
13
12
18
18
18
16
31
31
29
4 subdomains, H = 1/2, Q2 coarse space
Friday, July 19, 2013
Numerical Results
Number of PCG iterations needed for reducing the error in the
energy norm by a factor of 10−6
G
2
2
2
h
-3
-4
-5
2
-3
16
2
-4
2
-5
2
-6
2
-7
16
16
15
14
15
14
14
13
22
20
19
16 subdomains, H = 1/4, Q2 coarse space
Friday, July 19, 2013
Numerical Results
Number of PCG iterations needed for reducing the error in the
energy norm by a factor of 10−6
G
2
2
h
-4
-5
2
-4
17
2
-5
2
-6
2
-7
16
16
15
16
15
14
64 subdomains, H = 1/8, Q2 coarse space
Friday, July 19, 2013
Numerical Results
Number of PCG iterations needed for reducing the error in the
energy norm by a factor of 10−6
J
4
16
64
256
2
12
15
16
16
4
13
14
15
16
8
18
20
22
HG
h = 2−6 , Q2 coarse space
(scalable)
Friday, July 19, 2013
Numerical Results
H/δ
frag replacements
λmax
λmin
κ(BTL Ah )
4
4.83
4.30 x 10 -1
1.12 x 10 1
8
4.80
1.04 x 10 -1
4.62 x10 1
16
4.75
1.69 x 10 -2
2.81 x 10 2
32
4.66
2.30 x 10 -3
2.04 x 10 3
64
4.59
2.95 x 10 -4
1.56 x 10 4
4 subdomains, H = 1/2, h = 2−7
Q1 coarse space
κ(Ah ) ≈ 2 × 109
Friday, July 19, 2013
�
κ(BTL Ah ) ≤ C 1 +
�
H 3
δ
Effect of the Enriching Operator
Number of PCG iterations needed for reducing the error
(in the energy norm) by a factor of 10−6
G
2
2
2
h
-3
-4
-5
2
-3
19
2
-4
2
-5
2
-6
2
-7
19
17
17
15
22
21
20
18
41
39
35
16 subdomains, H = 1/4, Q1 coarse space
(with enriching operator)
Friday, July 19, 2013
Effect of the Enriching Operator
Number of PCG iterations needed for reducing the error
(in the energy norm) by a factor of 10−6
G
2
2
2
h
-3
-4
-5
2
-3
24
2
-4
2
-5
2
-6
2
-7
28
30
33
37
35
42
49
58
60
70
84
16 subdomains, H = 1/4, Q1 coarse space
(without enriching operator)
Friday, July 19, 2013
References
• Brenner and Wang, Two-level additive Schwarz precondi-
tioners for C0 interior penalty methods
Numer. Math., 2005
• Brenner and Wang, An iterative substructuring algorithm
for a C0 interior penalty method
Elect. Trans. Numer. Anal., 2012
• Antonietti and Ayuso de Dios, Schwarz domain decompo-
sition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case
M2AN Math. Model. Numer. Anal., 2007
• Antonietti and Ayuso de Dios, Two-level Schwarz precondi-
tioners for super penalty discontinuous Galerkin methods
Commun. Comput. Phys., 2009
Friday, July 19, 2013