C 0 Interior Penalty Methods
Geometric Multigrid
Current Research in Finite Element Methods
CIMPA Summer School
Mumbai, July 2015
Outline
I
Ideas
I
Set-Up
I
Multigrid Algorithms
I
Error Propagation Operators
I
Smoothing and Approximation
I
Convergence of W -Cycle
I
Convergence of V -Cycle
I
Other Algorithms
General References
• Hackbusch, Multi-grid Methods and Applications, Springer-
Verlag, 1985.
• Bramble, Multigrid Methods, Longman Scientific & Techni-
cal, 1993.
• Bramble and Zhang, The Analysis of Multigrid Methods, in
Handbook of Numerical Analysis VII, North-Holland, 2000.
• Trottenberg, Oosterlee and Schüller, Multigrid, Academic
Press, 2001.
• Briggs, Henson and McCormick, A Multigrid Tutorial (Sec-
ond Edition), SIAM, 2000.
Ideas
Numerical Linear Algebra
Numerical Linear Algebra
Let A be a SPD matrix. Suppose we solve
(L)
Ax = b
by an iterative method (Richardson, Gauss-Seidel, etc.).
Numerical Linear Algebra
Let A be a SPD matrix. Suppose we solve
(L)
Ax = b
by an iterative method (Richardson, Gauss-Seidel, etc.).
After m iterations (starting with some initial guess), we obtain
an approximate solution xm . Then the error em = x∗ − xm
satisfies the residual equation
(RE)
Aem = rm
where rm = b − Axm is the (computable) residual.
Numerical Linear Algebra
Let A be a SPD matrix. Suppose we solve
(L)
Ax = b
by an iterative method (Richardson, Gauss-Seidel, etc.).
After m iterations (starting with some initial guess), we obtain
an approximate solution xm . Then the error em = x∗ − xm
satisfies the residual equation
(RE)
Aem = rm
where rm = b − Axm is the (computable) residual.
If we can solve (RE) exactly, then we can recover the exact
solution x∗ of (L) by the relation
(C)
x∗ = xm + (x∗ − xm ) = xm + em
Numerical Linear Algebra
In reality, we will only solve (RE) approximately to obtain an
approximation e0m of em . Then, hopefully, the correction
(C0 )
x0 = xm + e0m
will give a better approximation of x∗ .
Numerical Linear Algebra
In reality, we will only solve (RE) approximately to obtain an
approximation e0m of em . Then, hopefully, the correction
(C0 )
x0 = xm + e0m
will give a better approximation of x∗ .
In the context of finite element equations
(FEh )
Ah xh = fh
there is a natural way to carry out this idea.
Numerical Linear Algebra
In reality, we will only solve (RE) approximately to obtain an
approximation e0m of em . Then, hopefully, the correction
(C0 )
x0 = xm + e0m
will give a better approximation of x∗ .
In the context of finite element equations
(FEh )
Ah xh = fh
there is a natural way to carry out this idea.
Smoothing Step
Apply m iterations of a classical iterative
method to obtain an approximation xm,h of xh and the corresponding residual equation
(REh )
Ah em,h = rm,h
em,h = xh − xm,h , rm,h = fh − Ah xm,h
Numerical Linear Algebra
Correction Step
Instead of solving (REh ), we solve a related equation on a coarser grid T2h (assuming that Th is obtained from T2h by uniform refinement).
(RE2h )
A2h e2h = r2h
r2h = projection of rh,m onto the coarse grid space
A2h = stiffness matrix for the coarse grid
Numerical Linear Algebra
Correction Step
Instead of solving (REh ), we solve a related equation on a coarser grid T2h (assuming that Th is obtained from T2h by uniform refinement).
(RE2h )
A2h e2h = r2h
r2h = projection of rh,m onto the coarse grid space
A2h = stiffness matrix for the coarse grid
We then use a transfer operator Ih2h to move e2h to the fine grid
Th and obtain the final output
xm+1,h = xm,h + Ih2h e2h
This is known as the two-grid algorithm.
Numerical Linear Algebra
I
Smoothing steps will damp out the highly oscillatory part
of the error so that we can capture em,h accurately on the
coarser grid by the correction step. Together they produce
a good approximate solution of (FEh ).
Numerical Linear Algebra
I
Smoothing steps will damp out the highly oscillatory part
of the error so that we can capture em,h accurately on the
coarser grid by the correction step. Together they produce
a good approximate solution of (FEh ).
I
Moreover, it is cheaper to solve the coarse grid residual
equation (RE2h ).
Numerical Linear Algebra
I
Smoothing steps will damp out the highly oscillatory part
of the error so that we can capture em,h accurately on the
coarser grid by the correction step. Together they produce
a good approximate solution of (FEh ).
I
Moreover, it is cheaper to solve the coarse grid residual
equation (RE2h ).
Of course we do not have to solve (RE2h ) exactly. Instead we
can apply the same idea recursively to (RE2h ). The resulting
algorithm is a multigrid algorithm.
Set-Up
Model Problem
Find u ∈ H02 (Ω) such that
Z
a(u, v) =
f v dx
∀ v ∈ H02 (Ω)
Ω
Ω = bounded polygonal domain in R2
f ∈ L2 (Ω)
Z
a(w, v) =
D2 w : D2 v dx
Ω
(Biharmonic equation with the boundary conditions of clamped
plates.)
Discrete Problems
Let T0 be an initial triangulation of Ω and Tk (k ≥ 1) be obtained
from Tk−1 by uniform refinement.
Discrete Problems
Let T0 be an initial triangulation of Ω and Tk (k ≥ 1) be obtained
from Tk−1 by uniform refinement.
Vk (k ≥ 0) is a Pj /Qj (j ≥ 2) finite element space associated
with Tk for clamped plates.
V0 ⊂ V1 ⊂ · · · Vk ⊂ Vk+1 ⊂ · · ·
Discrete Problems
k-th Level Discrete Problem
Find uk ∈ Vk such that
Z
ak (uk , v) =
f v dx
∀ v ∈ Vk
(f ∈ L2 (Ω))
Ω
n ∂ 2 w oohh ∂v ii
XZ n
ds
ak (w, v) =
D w : D v dx +
∂n2
∂n
T ∈Tk T
e∈Ek e
n ∂2v o
ohh ∂w ii
XZ n
+
ds
∂n2
∂n
e∈Ek e
X σ Z hh ∂w ii hh ∂v ii
ds
+
|e| e ∂n
∂n
X Z
2
2
e∈Ek
(Ek is the set of the edges of Tk .)
Discrete Problems
Discrete Operator Ak : Vk −→ Vk0
hAk v, wi = ak (v, w)
∀ v, w ∈ Vk
h·, ·i = canonical bilinear form on Vk0 × Vk
Discrete Problems
Discrete Operator Ak : Vk −→ Vk0
hAk v, wi = ak (v, w)
∀ v, w ∈ Vk
h·, ·i = canonical bilinear form on Vk0 × Vk
Ak is SPD in the sense that
hAk v, wi = hAk w, vi
∀ v, w ∈ Vk
hAk v, vi > 0
∀ v ∈ Vk \ {0}.
Discrete Problems
Discrete Operator Ak : Vk −→ Vk0
hAk v, wi = ak (v, w)
∀ v, w ∈ Vk
h·, ·i = canonical bilinear form on Vk0 × Vk
Ak is SPD in the sense that
hAk v, wi = hAk w, vi
∀ v, w ∈ Vk
hAk v, vi > 0
∀ v ∈ Vk \ {0}.
The matrix representing Ak with respect to the natural nodal
basis of the finite element space Vk and the dual basis of Vk0 is
the stiffness matrix, whose condition number is O(h−4
k ).
Discrete Problems
Discrete Operator Ak : Vk −→ Vk0
hAk v, wi = ak (v, w)
∀ v, w ∈ Vk
h·, ·i = canonical bilinear form on Vk0 × Vk
k-th Level Discrete Problem
Find uk ∈ Vk such that
Ak uk = φk
Z
hφk , vi =
f v dx
Ω
∀ v ∈ Vk
Discrete Problems
Discrete Operator Ak : Vk −→ Vk0
hAk v, wi = ak (v, w)
∀ v, w ∈ Vk
h·, ·i = canonical bilinear form on Vk0 × Vk
k-th Level Discrete Problem
Find uk ∈ Vk such that
Ak uk = φk
Z
hφk , vi =
f v dx
∀ v ∈ Vk
Ω
Multigrid algorithms are optimal order iterative methods for
(∗)
where z ∈ Vk and ψ ∈ Vk0 .
Ak z = ψ
Multigrid Algorithms
Two Ingredients
Two Ingredients
We need intergrid transfer operators to move data between
grids.
We need a good smoothing scheme for the equation
Ak z = ψ
to damp out the highly oscillatory part of the error so
that the remaining error can be captured accurately on a
coarser grid.
Intergrid Transfer Operators
V0 ⊂ V1 ⊂ · · · Vk−1 ⊂ Vk ⊂ · · ·
Coarse-to-Fine Operator
k
Ik−1
: Vk−1 −→ Vk
k
Ik−1
= natural injection
Intergrid Transfer Operators
V0 ⊂ V1 ⊂ · · · Vk−1 ⊂ Vk ⊂ · · ·
Coarse-to-Fine Operator
k
Ik−1
: Vk−1 −→ Vk
k
Ik−1
= natural injection
Fine-to-Coarse Operator
0
Ikk−1 : Vk0 −→ Vk−1
k
hIkk−1 ψ, vi = hψ, Ik−1
vi
for all ψ ∈ Vk0 and v ∈ Vk−1
Smoother for Ak z = ψ
(z ∈ Vk , ψ ∈ Vk0 )
Smoother for Ak z = ψ
(z ∈ Vk , ψ ∈ Vk0 )
Preconditioned Iteration
(S)
znew = zold + ωk Bk−1 (ψ − Ak zold )
where Bk : Vk −→ Vk0 is SPD and ωk > 0 is a damping factor
chosen so that the spectral radius of the operator ωk Bk−1 Ak
satisfies
ρ(ωk Bk−1 Ak ) ≤ 1
Smoother for Ak z = ψ
(z ∈ Vk , ψ ∈ Vk0 )
Preconditioned Iteration
(S)
znew = zold + ωk Bk−1 (ψ − Ak zold )
where Bk : Vk −→ Vk0 is SPD and ωk > 0 is a damping factor
chosen so that the spectral radius of the operator ωk Bk−1 Ak
satisfies
ρ(ωk Bk−1 Ak ) ≤ 1
Example 1
(standard smoother)
hBk v, wi = h2k
X
v(p)w(p)
p∈Nk
where Nk is the set of the (interior) nodes of the finite element
space Vk .
Smoother for Ak z = ψ
(z ∈ Vk , ψ ∈ Vk0 )
Preconditioned Iteration
(S)
znew = zold + ωk Bk−1 (ψ − Ak zold )
where Bk : Vk −→ Vk0 is SPD and ωk > 0 is a damping factor
chosen so that the spectral radius of the operator ωk Bk−1 Ak
satisfies
ρ(ωk Bk−1 Ak ) ≤ 1
Example 1
(standard smoother)
hBk v, wi = h2k
X
v(p)w(p)
p∈Nk
where Nk is the set of the (interior) nodes of the finite element
space Vk .
hBk v, vi ≈ kvk2L2 (Ω)
∀ v ∈ Vk
Smoother for Ak z = ψ
Example 2
(z ∈ Vk , ψ ∈ Vk0 )
(nonstandard smoother)
Bk−1 : Vk0 −→ Vk is an SPD operator which is an approximate
inverse of the discrete Laplace operator Lk : Vk −→ Vk0 defined
by
Z
hLk v1 , v2 i =
∇v1 · ∇v2 dx
Ω
∀ v ∈ Vk
(z ∈ Vk , ψ ∈ Vk0 )
Smoother for Ak z = ψ
Example 2
(nonstandard smoother)
Bk−1 : Vk0 −→ Vk is an SPD operator which is an approximate
inverse of the discrete Laplace operator Lk : Vk −→ Vk0 defined
by
Z
hLk v1 , v2 i =
∇v1 · ∇v2 dx
∀ v ∈ Vk
Ω
hBk v, vi ≈ |v|2H 1 (Ω)
∀ v ∈ Vk
(z ∈ Vk , ψ ∈ Vk0 )
Smoother for Ak z = ψ
Example 2
(nonstandard smoother)
Bk−1 : Vk0 −→ Vk is an SPD operator which is an approximate
inverse of the discrete Laplace operator Lk : Vk −→ Vk0 defined
by
Z
hLk v1 , v2 i =
∇v1 · ∇v2 dx
∀ v ∈ Vk
Ω
hBk v, vi ≈ |v|2H 1 (Ω)
∀ v ∈ Vk
We can take Bk−1 to be a multigrid Poisson solve, which can be
easily implemented because the finite element spaces in the C 0
interior penalty methods are standard spaces for second order
problems.
V -Cycle Algorithm for Ak z = ψ
V -Cycle Algorithm for Ak z = ψ
Output = M GV (k, ψ, z0 , m)
z0 = initial guess
m = number of smoothing steps
V -Cycle Algorithm for Ak z = ψ
Output = M GV (k, ψ, z0 , m)
z0 = initial guess
m = number of smoothing steps
For k = 0, we solve A0 z = ψ exactly to obtain
M GV (0, ψ, z0 , m) = A−1
0 ψ
V -Cycle Algorithm for Ak z = ψ
Output = M GV (k, ψ, z0 , m)
z0 = initial guess
m = number of smoothing steps
For k = 0, we solve A0 z = ψ exactly to obtain
M GV (0, ψ, z0 , m) = A−1
0 ψ
For k ≥ 1, we compute the multigrid output recursively in 3
steps.
V -Cycle Algorithm for Ak z = ψ
Output = M GV (k, ψ, z0 , m)
z0 = initial guess
m = number of smoothing steps
For k = 0, we solve A0 z = ψ exactly to obtain
M GV (0, ψ, z0 , m) = A−1
0 ψ
For k ≥ 1, we compute the multigrid output recursively in 3
steps.
Pre-smoothing Step
For 1 ≤ k ≤ m, compute
zk = zk−1 + ωk Bk−1 (ψ − Ak zk−1 )
V -Cycle Algorithm for Ak z = ψ
Correction Step
Transfer the residual ψ − Ak zm ∈ Vk0 to the
k−1
coarse grid using Ik and solve the coarse grid residual equation
Ak−1 ek−1 = Ikk−1 (ψ − Ak zm )
by applying the (k − 1)st level algorithm using 0 as the initial
guess, i.e., we compute
q = M GV (k − 1, Ikk−1 (ψ − Ak zm ), 0, m)
as an approximation to ek−1 . Then we make the correction
k
zm+1 = zm + Ik−1
q
V -Cycle Algorithm for Ak z = ψ
Correction Step
Transfer the residual ψ − Ak zm ∈ Vk0 to the
k−1
coarse grid using Ik and solve the coarse grid residual equation
Ak−1 ek−1 = Ikk−1 (ψ − Ak zm )
by applying the (k − 1)st level algorithm using 0 as the initial
guess, i.e., we compute
q = M GV (k − 1, Ikk−1 (ψ − Ak zm ), 0, m)
as an approximation to ek−1 . Then we make the correction
k
zm+1 = zm + Ik−1
q
Post-smoothing Step
For m + 2 ≤ k ≤ 2m + 1, compute
zk = zk−1 + ωk Bk−1 (ψ − Ak zk−1 )
V -Cycle Algorithm for Ak z = ψ
Final Output
M GV (k, ψ, z0 , m) = z2m+1
V -Cycle Algorithm for Ak z = ψ
Final
Output
C0 Interior Penalty Methods for Fourth Order Problems
Multigrid Algorithms
Multigrid Algorithms
for Ak z =
ψ
M G (k, ψ, z , m)
=z
V
V-cycle Algorithm
replacements
0
2m+1
(z ∈ Vk , ψ ∈ Vk0 )
p=1
k=3
k=2
k=1
k=0
Scheduling
Diagram
of the
V-Cycle
Algorithm
scheduling
diagram
for the
V -cycle
algorithm
W -Cycle Algorithm for Ak z = ψ
W -Cycle Algorithm for Ak z = ψ
Output = M GW (k, ψ, z0 , m)
z0 = initial guess
m = number of smoothing steps
W -Cycle Algorithm for Ak z = ψ
Output = M GW (k, ψ, z0 , m)
z0 = initial guess
Correction Step
m = number of smoothing steps
(apply the coarse grid algorithm twice)
q 0 = M GW (k − 1, Ikk−1 (ψ − Ak zm ), 0, m)
q = M GW (k − 1, Ikk−1 (ψ − Ak zm ), q 0 , m)
W -Cycle Algorithm for Ak z = ψ
Output = M GW (k, ψ, z0 , m)
z0 = initial guess
m = number of smoothing steps
C0 Interior Penalty Methods for Fourth Order Problems
Multigrid Algorithms
Correction
Step
(apply
grid
algorithm
Multigrid
Algorithms
for A the
z =coarse
ψ
(z ∈
V , ψ ∈twice)
V0)
k
0
q = M GW (k −
W-cycle Algorithm
ments
p=2
q = M GW (k −
1, Ikk−1 (ψ
1, Ikk−1 (ψ
k
− Ak zm ), 0, m)
− Ak zm ), q 0 , m)
k=3
k=2
k=1
k=0
scheduling diagram for the W -cycle algorithm
Scheduling Diagram of the W-Cycle Algorithm
k
F -Cycle Algorithm for Ak z = ψ
F -Cycle Algorithm for Ak z = ψ
Output = M GF (k, ψ, z0 , m)
z0 = initial guess
m = number of smoothing steps
F -Cycle Algorithm for Ak z = ψ
Output = M GF (k, ψ, z0 , m)
z0 = initial guess
Correction Step
m = number of smoothing steps
(coarse grid algorithm followed by V -cycle)
q 0 = M GF (k − 1, Ikk−1 (ψ − Ak zm ), 0, m)
q = M GV (k − 1, Ikk−1 (ψ − Ak zm ), q 0 , m)
F -Cycle Algorithm for Ak z = ψ
Output = M GF (k, ψ, z0 , m)
z0 = initial guess
Correction Step
m = number of smoothing steps
(coarse grid algorithm followed by V -cycle)
q 0 = M GF (k − 1, Ikk−1 (ψ − Ak zm ), 0, m)
q = M GV (k − 1, Ikk−1 (ψ − Ak zm ), q 0 , m)
k=3
replacements
k=2
k=1
k=0
scheduling diagram for the F -cycle algorithm
Operation Counts
nk = dimVk
(nk ≈ 4k )
Wk = number of flops for the k th level multigrid algorithm
m = number of smoothing steps
p = 1 or 2
Operation Counts
nk = dimVk
(nk ≈ 4k )
Wk = number of flops for the k th level multigrid algorithm
m = number of smoothing steps
p = 1 or 2
Wk ≤ C∗ mnk + pWk−1
Operation Counts
nk = dimVk
(nk ≈ 4k )
Wk = number of flops for the k th level multigrid algorithm
m = number of smoothing steps
p = 1 or 2
Wk ≤ C∗ mnk + pWk−1
≤ C∗ mnk + p(C∗ mnk−1 ) + p2 (C∗ mnk−2 ) + · · · pk−1 (C∗ mn1 ) + pk W
Operation Counts
nk = dimVk
(nk ≈ 4k )
Wk = number of flops for the k th level multigrid algorithm
m = number of smoothing steps
p = 1 or 2
Wk ≤ C∗ mnk + pWk−1
≤ C∗ mnk + p(C∗ mnk−1 ) + p2 (C∗ mnk−2 ) + · · · pk−1 (C∗ mn1 ) + pk W
≤ C† m4k + pC† m4k−1 + p2 C† m4k−2 + · · · pk−1 (C† m4) + pk W0
Operation Counts
nk = dimVk
(nk ≈ 4k )
Wk = number of flops for the k th level multigrid algorithm
m = number of smoothing steps
p = 1 or 2
Wk ≤ C∗ mnk + pWk−1
≤ C∗ mnk + p(C∗ mnk−1 ) + p2 (C∗ mnk−2 ) + · · · pk−1 (C∗ mn1 ) + pk W
≤ C† m4k + pC† m4k−1 + p2 C† m4k−2 + · · · pk−1 (C† m4) + pk W0
p p2
pk−1 = C† m4k 1 + + 2 + . . . k−1 + pk W0
4 4
4
Operation Counts
nk = dimVk
(nk ≈ 4k )
Wk = number of flops for the k th level multigrid algorithm
m = number of smoothing steps
p = 1 or 2
Wk ≤ C∗ mnk + pWk−1
≤ C∗ mnk + p(C∗ mnk−1 ) + p2 (C∗ mnk−2 ) + · · · pk−1 (C∗ mn1 ) + pk W
≤ C† m4k + pC† m4k−1 + p2 C† m4k−2 + · · · pk−1 (C† m4) + pk W0
p p2
pk−1 = C† m4k 1 + + 2 + . . . k−1 + pk W0
4 4
4
C† m4k
≤
+ p k W0
1 − p/4
Operation Counts
nk = dimVk
(nk ≈ 4k )
Wk = number of flops for the k th level multigrid algorithm
m = number of smoothing steps
p = 1 or 2
Wk ≤ C∗ mnk + pWk−1
≤ C∗ mnk + p(C∗ mnk−1 ) + p2 (C∗ mnk−2 ) + · · · pk−1 (C∗ mn1 ) + pk W
≤ C† m4k + pC† m4k−1 + p2 C† m4k−2 + · · · pk−1 (C† m4) + pk W0
p p2
pk−1 = C† m4k 1 + + 2 + . . . k−1 + pk W0
4 4
4
C† m4k
≤
+ pk W0 ≤ C] m4k + W0 pk ≤ C\ m nk
1 − p/4
Error Propagation Operators
V -Cycle Algorithm
V -Cycle Algorithm
Let EkV : Vk −→ Vk be the error propagation operator that maps
the initial error z −z0 to the final error z −M GV (k, ψ, z0 , m). We
V .
want to develop a recursive relation between EkV and Ek−1
V -Cycle Algorithm
Let EkV : Vk −→ Vk be the error propagation operator that maps
the initial error z −z0 to the final error z −M GV (k, ψ, z0 , m). We
V .
want to develop a recursive relation between EkV and Ek−1
It follows from
(S)
znew = zold + ωk Bk−1 (ψ − Ak zold )
and Ak z = ψ that
z−znew = z−zold −ωk Bk−1 (ψ−Ak zold ) = (Idk −ωk Bk−1 Ak )(z−zold )
where Idk is the identity operator on Vk .
V -Cycle Algorithm
Let EkV : Vk −→ Vk be the error propagation operator that maps
the initial error z −z0 to the final error z −M GV (k, ψ, z0 , m). We
V .
want to develop a recursive relation between EkV and Ek−1
It follows from
(S)
znew = zold + ωk Bk−1 (ψ − Ak zold )
and Ak z = ψ that
z−znew = z−zold −ωk Bk−1 (ψ−Ak zold ) = (Idk −ωk Bk−1 Ak )(z−zold )
where Idk is the identity operator on Vk .
Therefore the effect of one smoothing step is measured by the
operator
Rk = Idk − ωk Bk−1 Ak
V -Cycle Algorithm
Let Pkk−1 : Vk −→ Vk−1 be the transpose of the coarse-to-fine
k
operator Ik−1
with respect to the variational forms, i.e.
k
ak−1 (Pkk−1 v, w) = ak (v, Ik−1
w)
∀ v ∈ Vk , w ∈ Vk−1
V -Cycle Algorithm
Let Pkk−1 : Vk −→ Vk−1 be the transpose of the coarse-to-fine
k
operator Ik−1
with respect to the variational forms, i.e.
k
ak−1 (Pkk−1 v, w) = ak (v, Ik−1
w)
∀ v ∈ Vk , w ∈ Vk−1
Recall the coarse grid residual equation
Ak−1 ek−1 = Ikk−1 (ψ − Ak zm )
V -Cycle Algorithm
Let Pkk−1 : Vk −→ Vk−1 be the transpose of the coarse-to-fine
k
operator Ik−1
with respect to the variational forms, i.e.
k
ak−1 (Pkk−1 v, w) = ak (v, Ik−1
w)
∀ v ∈ Vk , w ∈ Vk−1
Recall the coarse grid residual equation
Ak−1 ek−1 = Ikk−1 (ψ − Ak zm )
ak−1 (ek−1 , v) = hAk−1 ek−1 , vi
(for any v ∈ Vk )
V -Cycle Algorithm
Let Pkk−1 : Vk −→ Vk−1 be the transpose of the coarse-to-fine
k
operator Ik−1
with respect to the variational forms, i.e.
k
ak−1 (Pkk−1 v, w) = ak (v, Ik−1
w)
∀ v ∈ Vk , w ∈ Vk−1
Recall the coarse grid residual equation
Ak−1 ek−1 = Ikk−1 (ψ − Ak zm )
ak−1 (ek−1 , v) = hAk−1 ek−1 , vi
= hIkk−1 (ψ − Ak zm ), vi
(for any v ∈ Vk )
V -Cycle Algorithm
Let Pkk−1 : Vk −→ Vk−1 be the transpose of the coarse-to-fine
k
operator Ik−1
with respect to the variational forms, i.e.
k
ak−1 (Pkk−1 v, w) = ak (v, Ik−1
w)
∀ v ∈ Vk , w ∈ Vk−1
Recall the coarse grid residual equation
Ak−1 ek−1 = Ikk−1 (ψ − Ak zm )
ak−1 (ek−1 , v) = hAk−1 ek−1 , vi
= hIkk−1 (ψ − Ak zm ), vi
k
= h(ψ − Ak zm ), Ik−1
vi
(for any v ∈ Vk )
V -Cycle Algorithm
Let Pkk−1 : Vk −→ Vk−1 be the transpose of the coarse-to-fine
k
operator Ik−1
with respect to the variational forms, i.e.
k
ak−1 (Pkk−1 v, w) = ak (v, Ik−1
w)
∀ v ∈ Vk , w ∈ Vk−1
Recall the coarse grid residual equation
Ak−1 ek−1 = Ikk−1 (ψ − Ak zm )
ak−1 (ek−1 , v) = hAk−1 ek−1 , vi
= hIkk−1 (ψ − Ak zm ), vi
k
= h(ψ − Ak zm ), Ik−1
vi
k
= hAk (z − zm ), Ik−1
vi
(for any v ∈ Vk )
V -Cycle Algorithm
Let Pkk−1 : Vk −→ Vk−1 be the transpose of the coarse-to-fine
k
operator Ik−1
with respect to the variational forms, i.e.
k
ak−1 (Pkk−1 v, w) = ak (v, Ik−1
w)
∀ v ∈ Vk , w ∈ Vk−1
Recall the coarse grid residual equation
Ak−1 ek−1 = Ikk−1 (ψ − Ak zm )
ak−1 (ek−1 , v) = hAk−1 ek−1 , vi
= hIkk−1 (ψ − Ak zm ), vi
k
= h(ψ − Ak zm ), Ik−1
vi
k
= hAk (z − zm ), Ik−1
vi
k
= ak (z − zm , Ik−1
v)
(for any v ∈ Vk )
V -Cycle Algorithm
Let Pkk−1 : Vk −→ Vk−1 be the transpose of the coarse-to-fine
k
operator Ik−1
with respect to the variational forms, i.e.
k
ak−1 (Pkk−1 v, w) = ak (v, Ik−1
w)
∀ v ∈ Vk , w ∈ Vk−1
Recall the coarse grid residual equation
Ak−1 ek−1 = Ikk−1 (ψ − Ak zm )
ak−1 (ek−1 , v) = hAk−1 ek−1 , vi
(for any v ∈ Vk )
= hIkk−1 (ψ − Ak zm ), vi
k
= h(ψ − Ak zm ), Ik−1
vi
k
= hAk (z − zm ), Ik−1
vi
k
= ak (z − zm , Ik−1
v) = ak−1 (Pkk−1 (z − zm ), v)
V -Cycle Algorithm
Let Pkk−1 : Vk −→ Vk−1 be the transpose of the coarse-to-fine
k
operator Ik−1
with respect to the variational forms, i.e.
k
ak−1 (Pkk−1 v, w) = ak (v, Ik−1
w)
∀ v ∈ Vk , w ∈ Vk−1
Recall the coarse grid residual equation
Ak−1 ek−1 = Ikk−1 (ψ − Ak zm )
ak−1 (ek−1 , v) = hAk−1 ek−1 , vi
(for any v ∈ Vk )
= hIkk−1 (ψ − Ak zm ), vi
k
= h(ψ − Ak zm ), Ik−1
vi
k
= hAk (z − zm ), Ik−1
vi
k
= ak (z − zm , Ik−1
v) = ak−1 (Pkk−1 (z − zm ), v)
V -Cycle Algorithm
ek−1 = Pkk−1 (z − zm )
V -Cycle Algorithm
ek−1 = Pkk−1 (z − zm )
Recall q = M GV (k−1, Ikk−1 (ψ−Ak zm ), 0, m) is the approximate
solution of the coarse grid residual equation obtained by using
the (k − 1)st level V -cycle algorithm with initial guess 0.
V -Cycle Algorithm
ek−1 = Pkk−1 (z − zm )
Recall q = M GV (k−1, Ikk−1 (ψ−Ak zm ), 0, m) is the approximate
solution of the coarse grid residual equation obtained by using
the (k − 1)st level V -cycle algorithm with initial guess 0.
∴
ek−1 − q = Ek−1 (ek−1 − 0) =⇒ q = (Idk−1 − Ek−1 )ek−1
V -Cycle Algorithm
ek−1 = Pkk−1 (z − zm )
Recall q = M GV (k−1, Ikk−1 (ψ−Ak zm ), 0, m) is the approximate
solution of the coarse grid residual equation obtained by using
the (k − 1)st level V -cycle algorithm with initial guess 0.
∴
ek−1 − q = Ek−1 (ek−1 − 0) =⇒ q = (Idk−1 − Ek−1 )ek−1
k
z − zm+1 = z − (zm + Ik−1
q)
k q
zm+1 = zm + Ik−1
V -Cycle Algorithm
ek−1 = Pkk−1 (z − zm )
Recall q = M GV (k−1, Ikk−1 (ψ−Ak zm ), 0, m) is the approximate
solution of the coarse grid residual equation obtained by using
the (k − 1)st level V -cycle algorithm with initial guess 0.
∴
ek−1 − q = Ek−1 (ek−1 − 0) =⇒ q = (Idk−1 − Ek−1 )ek−1
k
z − zm+1 = z − (zm + Ik−1
q)
k
= z − zm − Ik−1
(Idk−1 − Ek−1 )ek−1
V -Cycle Algorithm
ek−1 = Pkk−1 (z − zm )
Recall q = M GV (k−1, Ikk−1 (ψ−Ak zm ), 0, m) is the approximate
solution of the coarse grid residual equation obtained by using
the (k − 1)st level V -cycle algorithm with initial guess 0.
∴
ek−1 − q = Ek−1 (ek−1 − 0) =⇒ q = (Idk−1 − Ek−1 )ek−1
k
z − zm+1 = z − (zm + Ik−1
q)
k
= z − zm − Ik−1
(Idk−1 − Ek−1 )ek−1
k
= z − zm − Ik−1
(Idk−1 − Ek−1 )Pkk−1 (z − zm )
V -Cycle Algorithm
ek−1 = Pkk−1 (z − zm )
Recall q = M GV (k−1, Ikk−1 (ψ−Ak zm ), 0, m) is the approximate
solution of the coarse grid residual equation obtained by using
the (k − 1)st level V -cycle algorithm with initial guess 0.
∴
ek−1 − q = Ek−1 (ek−1 − 0) =⇒ q = (Idk−1 − Ek−1 )ek−1
k
z − zm+1 = z − (zm + Ik−1
q)
k
= z − zm − Ik−1
(Idk−1 − Ek−1 )ek−1
k
= z − zm − Ik−1
(Idk−1 − Ek−1 )Pkk−1 (z − zm )
k
k
= (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1 Pkk−1 )(z − zm )
V -Cycle Algorithm
ek−1 = Pkk−1 (z − zm )
Recall q = M GV (k−1, Ikk−1 (ψ−Ak zm ), 0, m) is the approximate
solution of the coarse grid residual equation obtained by using
the (k − 1)st level V -cycle algorithm with initial guess 0.
∴
ek−1 − q = Ek−1 (ek−1 − 0) =⇒ q = (Idk−1 − Ek−1 )ek−1
k
z − zm+1 = z − (zm + Ik−1
q)
k
= z − zm − Ik−1
(Idk−1 − Ek−1 )ek−1
k
= z − zm − Ik−1
(Idk−1 − Ek−1 )Pkk−1 (z − zm )
k
k
= (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1 Pkk−1 )(z − zm )
k
k
= (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1 Pkk−1 )Rkm (z − z0 )
V -Cycle Algorithm
k
k
z − zm+1 = (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1 Pkk−1 )Rkm (z − z0 )
V -Cycle Algorithm
k
k
z − zm+1 = (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1 Pkk−1 )Rkm (z − z0 )
z − M GV (k, γ, z0 , m) = z − z2m+1
= Rkm (z − zm+1 )
k
k
V
= Rkm (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1
Pkk−1 )Rkm (z − z0 )
V -Cycle Algorithm
k
k
z − zm+1 = (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1 Pkk−1 )Rkm (z − z0 )
z − M GV (k, γ, z0 , m) = z − z2m+1
= Rkm (z − zm+1 )
k
k
V
= Rkm (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1
Pkk−1 )Rkm (z − z0 )
{z
}
|
V
= Ek
V -Cycle Algorithm
k
k
z − zm+1 = (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1 Pkk−1 )Rkm (z − z0 )
z − M GV (k, γ, z0 , m) = z − z2m+1
= Rkm (z − zm+1 )
k
k
V
= Rkm (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1
Pkk−1 )Rkm (z − z0 )
{z
}
|
V
= Ek
Recursive Relation for V -Cycle
k
k
V
EkV = Rkm (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1
Pkk−1 )Rkm
E0V = 0
W -Cycle Algorithm
Let EkW : Vk −→ Vk be the error propagation operator
that maps the initial error z − z0 to the final error z −
M GW (k, ψ, z0 , m).
k
k
W 2 k−1
EkW = Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
) Pk )Rkm
E0W = 0
F -Cycle Algorithm
Let EkF : Vk −→ Vk be the error propagation operator that maps
the initial error z − z0 to the final error z − M GF (k, ψ, z0 , m).
k
k
V
F
EkF = Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
Ek−1
)Pkk−1 )Rkm
E0F = 0
Two-Grid Algorithm
Two-Grid Algorithm
In the two grid algorithm the coarse grid residual equation is
solved exactly. The error propagation operator EkT G is therefore
given by
k
EkT G = Rkm (Idk − Ik−1
Pkk−1 )Rkm
Two-Grid Algorithm
In the two grid algorithm the coarse grid residual equation is
solved exactly. The error propagation operator EkT G is therefore
given by
k
EkT G = Rkm (Idk − Ik−1
Pkk−1 )Rkm
V -Cycle
k
k
V
EkV = Rkm (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1
Pkk−1 )Rkm
W -Cycle
k
k
W 2 k−1
EkW = Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
) Pk )Rkm
F -Cycle
k
k
V
F
EkF = Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
Ek−1
)Pkk−1 )Rkm
Two-Grid Algorithm
In the two grid algorithm the coarse grid residual equation is
solved exactly. The error propagation operator EkT G is therefore
given by
k
EkT G = Rkm (Idk − Ik−1
Pkk−1 )Rkm
It is clear that the analysis of the two-grid algorithm depends on
the property of the operator Rkm (smoothing property) and the
k P k−1 (approximation propproperty of the operator Idk − Ik−1
k
erty).
Smoothing and Approximation
Mesh-Dependent Norms
Mesh-Dependent Norms
Mesh-dependent inner product
(v, w)k = hBk v, wi
∀ v, w ∈ Vk
Mesh-Dependent Norms
Mesh-dependent inner product
(v, w)k = hBk v, wi
∀ v, w ∈ Vk
The operator Bk−1 Ak is SPD with respect to (·, ·)k .
((Bk−1 Ak )v, w)k = hAk v, wi
= ak (v, w)
= ak (w, v)
= ((Bk−1 Ak )w, v)k
((Bk−1 Ak )v, v)k = ak (v, v) > 0
∀ v, w ∈ Vk
if v 6= 0
Mesh-Dependent Norms
Mesh-dependent inner product
(v, w)k = hBk v, wi
∀ v, w ∈ Vk
The operator Bk−1 Ak is SPD with respect to (·, ·)k .
For t ∈ R,
|||v|||t,k =
q
((Bk−1 Ak )t v, v)k
∀ v ∈ Vk
Mesh-Dependent Norms
Mesh-dependent inner product
(v, w)k = hBk v, wi
∀ v, w ∈ Vk
The operator Bk−1 Ak is SPD with respect to (·, ·)k .
For t ∈ R,
|||v|||t,k =
q
((Bk−1 Ak )t v, v)k
∀ v ∈ Vk
In particular
|||v|||20,k = (v, v)k = hBk v, vi
∀ v ∈ Vk
|||v|||21,k = ((Bk−1 Ak )v, v)k = ak (v, v)
∀ v ∈ Vk
Mesh-Dependent Norms
Lemma
(Generalized Cauchy-Schwarz Inequality)
ak (v, w) ≤ |||v|||1+t,k |||w|||1−t,k
∀ v, w ∈ Vk
Mesh-Dependent Norms
Lemma
(Generalized Cauchy-Schwarz Inequality)
ak (v, w) ≤ |||v|||1+t,k |||w|||1−t,k
∀ v, w ∈ Vk
Proof.
ak (v, w) = hAk v, wi
= hBk (Bk−1 Ak )v, wi
= (Bk−1 Ak )v, w k
= (Bk−1 Ak )(1+t)/2 v, (Bk−1 Ak )(1−t)/2 w k
1
≤ (Bk−1 Ak )(1+t)/2 v, (Bk−1 Ak )(1+t)/2 v k2
× (Bk−1 Ak )(1−t)/2 w, (Bk−1 Ak )(1−t)/2 w
1
1
≤ (Bk−1 Ak )(1+t) v, v k2 (Bk−1 Ak )(1−t) w, w k2
= |||v|||1+t,k |||w|||1−t,k
1
2
k
Spectral Radius of Bk−1 Ak
Spectral Radius of Bk−1 Ak
((Bk−1 Ak )v, v)k
v∈Vk
(v, v)k
hAk v, vi
= max
v∈Vk hBk v, vi
ρ(Bk−1 Ak ) = max
(Rayleigh Quotient)
((·, ·)k = hBk v, vi)
Spectral Radius of Bk−1 Ak
((Bk−1 Ak )v, v)k
v∈Vk
(v, v)k
hAk v, vi
= max
v∈Vk hBk v, vi
ρ(Bk−1 Ak ) = max
(Rayleigh Quotient)
((·, ·)k = hBk v, vi)
Standard Smoother
hBk v, vi ≈ kvk2L2 (Ω)
∀ v ∈ Vk
hAk v, vi = ak (v, v) ≈ |v|H 2 (Ω;Tk )
∀ v ∈ Vk
ρ(Bk−1 Ak ) ≈ h−4
k
Spectral Radius of Bk−1 Ak
((Bk−1 Ak )v, v)k
v∈Vk
(v, v)k
hAk v, vi
= max
v∈Vk hBk v, vi
ρ(Bk−1 Ak ) = max
(Rayleigh Quotient)
((·, ·)k = hBk v, vi)
Nonstandard Smoother
hBk v, vi ≈ |v|2H 1 (Ω)
∀ v ∈ Vk
hAk v, vi = ak (v, v) ≈ |v|2H 2 (Ω;Tk )
∀ v ∈ Vk
ρ(Bk−1 Ak ) ≈ h−2
k
Smoothing Property
There exists a constant C > 0 (independent of k) such that
(s−t)/2
|||Rkm v|||s,k ≤ Cρk
m(t−s)/2 |||v|||t,k
where ρk = ρ(Bk−1 Ak ).
∀ v ∈ Vk , 0 ≤ t ≤ s ≤ 2
Smoothing Property
There exists a constant C > 0 (independent of k) such that
(s−t)/2
|||Rkm v|||s,k ≤ Cρk
m(t−s)/2 |||v|||t,k
∀ v ∈ Vk , 0 ≤ t ≤ s ≤ 2
where ρk = ρ(Bk−1 Ak ).
Proof.
Recall
Rkm = (Idk − ωk Bk−1 Ak )m
ρ(ωk Bk−1 Ak ) = ωk ρk ≤ 1
Smoothing Property
There exists a constant C > 0 (independent of k) such that
(s−t)/2
|||Rkm v|||s,k ≤ Cρk
m(t−s)/2 |||v|||t,k
∀ v ∈ Vk , 0 ≤ t ≤ s ≤ 2
where ρk = ρ(Bk−1 Ak ).
Proof.
Recall
Rkm = (Idk − ωk Bk−1 Ak )m
ρ(ωk Bk−1 Ak ) = ωk ρk ≤ 1
Therefore we can take
ωk ≈ ρ−1
k
Smoothing Property
There exists a constant C > 0 (independent of k) such that
(s−t)/2
|||Rkm v|||s,k ≤ Cρk
m(t−s)/2 |||v|||t,k
∀ v ∈ Vk , 0 ≤ t ≤ s ≤ 2
where ρk = ρ(Bk−1 Ak ).
Proof.
Since Bk−1 Ak is SPD with respect to the inner
product (·, ·)k = hBk ·, ·i, there exist, by the Spectral Theorem,
v1 , . . . , vnk (nk = dim Vk ) such that
Bk−1 Ak vj = λj vj
and (vj , v` )k = δj`
where λ1 , . . . , λnk > 0 are the eigenvalues of Bk−1 Ak .
Smoothing Property
There exists a constant C > 0 (independent of k) such that
(s−t)/2
|||Rkm v|||s,k ≤ Cρk
m(t−s)/2 |||v|||t,k
∀ v ∈ Vk , 0 ≤ t ≤ s ≤ 2
where ρk = ρ(Bk−1 Ak ).
Proof.
Since Bk−1 Ak is SPD with respect to the inner
product (·, ·)k = hBk ·, ·i, there exist, by the Spectral Theorem,
v1 , . . . , vnk (nk = dim Vk ) such that
Bk−1 Ak vj = λj vj
and (vj , v` )k = δj`
where λ1 , . . . , λnk > 0 are the eigenvalues of Bk−1 Ak .
(Bk−1 Ak )s vj = λsj vj
for any s
Rk vj = (Idk − ωk Bk−1 Ak )vj = (1 − ωk λj )vj
Smoothing Property
|||Rkm v|||2s,k = (Bk−1 Ak )s Rkm v, Rkm v
k
Smoothing Property
|||Rkm v|||2s,k = (Bk−1 Ak )s Rkm v, Rkm v
k
= (Bk−1 Ak )t (Bk−1 Ak )(s−t) Rkm v, Rkm v
k
Smoothing Property
|||Rkm v|||2s,k = (Bk−1 Ak )s Rkm v, Rkm v
k
= (Bk−1 Ak )t (Bk−1 Ak )(s−t) Rkm v, Rkm v
=
(t−s)
ωk
k
−1
−1
t
(s−t) m
(Bk Ak ) (ωk Bk Ak )
Rk v, Rkm v k
Smoothing Property
|||Rkm v|||2s,k = (Bk−1 Ak )s Rkm v, Rkm v
k
= (Bk−1 Ak )t (Bk−1 Ak )(s−t) Rkm v, Rkm v
=
=
k
−1
−1
t
(s−t) m
(Bk Ak ) (ωk Bk Ak )
Rk v, Rkm v k
nk
X
(t−s)
ωk
λtj (ωk λj )(s−t) (1 − ωk λj )2m c2j
j=1
(t−s)
ωk
v=
nk
X
cj vj
j=1
(Bk−1 Ak )vj = λj vj
(vj , v` )k = δj`
Rkm vj = (1 − ωk λj )m vj
Smoothing Property
|||Rkm v|||2s,k = (Bk−1 Ak )s Rkm v, Rkm v
k
= (Bk−1 Ak )t (Bk−1 Ak )(s−t) Rkm v, Rkm v
=
=
≤
k
−1
−1
t
(s−t) m
(Bk Ak ) (ωk Bk Ak )
Rk v, Rkm v k
nk
X
(t−s)
ωk
λtj (ωk λj )(s−t) (1 − ωk λj )2m c2j
j=1
nk
X
(t−s)
(s−t)
2m
ωk
max (ωk λj )
(1 − ωk λj )
λtj c2j
1≤j≤nk
j=1
(t−s)
ωk
Smoothing Property
|||Rkm v|||2s,k = (Bk−1 Ak )s Rkm v, Rkm v
k
= (Bk−1 Ak )t (Bk−1 Ak )(s−t) Rkm v, Rkm v
=
=
≤
k
−1
−1
t
(s−t) m
(Bk Ak ) (ωk Bk Ak )
Rk v, Rkm v k
nk
X
(t−s)
ωk
λtj (ωk λj )(s−t) (1 − ωk λj )2m c2j
j=1
nk
X
(t−s)
(s−t)
2m
ωk
max (ωk λj )
(1 − ωk λj )
λtj c2j
1≤j≤nk
j=1
(t−s)
ωk
(t−s)
≤ ωk
max x(s−t) (1 − x)2m (Bk−1 Ak )t v, v k
0≤x≤1
ωk λj ≤ 1
since ρ(ωk Bk−1 Ak ) ≤ 1
Smoothing Property
|||Rkm v|||2s,k = (Bk−1 Ak )s Rkm v, Rkm v
k
= (Bk−1 Ak )t (Bk−1 Ak )(s−t) Rkm v, Rkm v
=
=
≤
k
−1
−1
t
(s−t) m
(Bk Ak ) (ωk Bk Ak )
Rk v, Rkm v k
nk
X
(t−s)
ωk
λtj (ωk λj )(s−t) (1 − ωk λj )2m c2j
j=1
nk
X
(t−s)
(s−t)
2m
ωk
max (ωk λj )
(1 − ωk λj )
λtj c2j
1≤j≤nk
j=1
(t−s)
ωk
(t−s)
≤ ωk
≤
max x(s−t) (1 − x)2m (Bk−1 Ak )t v, v k
0≤x≤1
(t−s)
ωk
Cm(t−s) |||v|||2t,k
max x(s−t) (1 − x)2m ≤ Cm(t−s)
0≤x≤1
(calculus)
Smoothing Property
|||Rkm v|||2s,k = (Bk−1 Ak )s Rkm v, Rkm v
k
= (Bk−1 Ak )t (Bk−1 Ak )(s−t) Rkm v, Rkm v
=
=
≤
k
−1
−1
t
(s−t) m
(Bk Ak ) (ωk Bk Ak )
Rk v, Rkm v k
nk
X
(t−s)
ωk
λtj (ωk λj )(s−t) (1 − ωk λj )2m c2j
j=1
nk
X
(t−s)
(s−t)
2m
ωk
max (ωk λj )
(1 − ωk λj )
λtj c2j
1≤j≤nk
j=1
(t−s)
ωk
(t−s)
≤ ωk
≤
≤
max x(s−t) (1 − x)2m (Bk−1 Ak )t v, v k
0≤x≤1
(t−s)
ωk
Cm(t−s) |||v|||2t,k
(s−t)
Cρk m(t−s) |||v|||2t,k
ωk ≈ ρ−1
k
Smoothing Property
In the special case where t = s, we have
|||Rk v|||s,k ≤ |||v|||s,k
∀ v ∈ Vk
Approximation Property
Assuming that Ω is convex, we have
k
|||(Idk − Ik−1
Pkk−1 )v|||0,k ≤ Ch2k |||v|||2,k
∀ v ∈ Vk
where the mesh-dependent norms are defined in terms of the
nonstandard smoother.
Approximation Property
Assuming that Ω is convex, we have
k
|||(Idk − Ik−1
Pkk−1 )v|||0,k ≤ Ch2k |||v|||2,k
∀ v ∈ Vk
where the mesh-dependent norms are defined in terms of the
nonstandard smoother.
Proof.
Recall that
|||v|||20,k = hBk v, vi ≈ |v|2H 1 (Ω)
∀ v ∈ Vk
Approximation Property
Assuming that Ω is convex, we have
k
|||(Idk − Ik−1
Pkk−1 )v|||0,k ≤ Ch2k |||v|||2,k
∀ v ∈ Vk
where the mesh-dependent norms are defined in terms of the
nonstandard smoother.
Proof.
Recall that
|||v|||20,k = hBk v, vi ≈ |v|2H 1 (Ω)
∀ v ∈ Vk
By duality
k
k
|||(Idk − Ik−1
Pkk−1 )v|||0,k ≈ |(Idk − Ik−1
Pkk−1 )v|H 1 (Ω)
=
k P k−1 )v)
φ((Idk − Ik−1
k
kφk
−1
−1
H
(Ω)
φ∈H (Ω)
sup
Approximation Property
Let φ ∈ H −1 (Ω) be arbitrary and define ζ ∈ H02 (Ω), ζk ∈ Vk and
ζk−1 ∈ Vk−1 by
a(ζ, v) = φ(v)
ak (ζk , v) = φ(v)
ak−1 (ζk−1 , v) = φ(v)
∀ v ∈ H02 (Ω)
∀ v ∈ Vk
∀ v ∈ Vk−1
ζ is the solution of the clamped plate problem where the righthand side φ belongs to H −1 (Ω).
ζk is the approximation of ζ obtained by the k-th level C 0 interior penalty method.
ζk−1 is the approximation of ζ obtained by the (k − 1)-st level
C 0 interior penalty method.
Approximation Property
Let φ ∈ H −1 (Ω) be arbitrary and define ζ ∈ H02 (Ω), ζk ∈ Vk and
ζk−1 ∈ Vk−1 by
a(ζ, v) = φ(v)
ak (ζk , v) = φ(v)
ak−1 (ζk−1 , v) = φ(v)
∀ v ∈ H02 (Ω)
∀ v ∈ Vk
∀ v ∈ Vk−1
Error Estimates in H 1 (Ω)
|ζ − ζk |H 1 (Ω) ≤ Ch2k |φ|H −1 (Ω)
|ζ − ζk−1 |H 1 (Ω) ≤ Ch2k−1 |φ|H −1 (Ω) ≤ Ch2k |φ|H −1 (Ω)
Approximation Property
Let φ ∈ H −1 (Ω) be arbitrary and define ζ ∈ H02 (Ω), ζk ∈ Vk and
ζk−1 ∈ Vk−1 by
a(ζ, v) = φ(v)
ak (ζk , v) = φ(v)
ak−1 (ζk−1 , v) = φ(v)
∀ v ∈ H02 (Ω)
∀ v ∈ Vk
∀ v ∈ Vk−1
Error Estimates in H 1 (Ω)
|ζ − ζk |H 1 (Ω) ≤ Ch2k |φ|H −1 (Ω)
|ζ − ζk−1 |H 1 (Ω) ≤ Ch2k−1 |φ|H −1 (Ω) ≤ Ch2k |φ|H −1 (Ω)
Relation between ζk and ζk−1
ζk−1 = Pkk−1 ζk
Approximation Property
k
k
φ((Idk − Ik−1
Pkk−1 )v) = ak (ζk , (Idk − Ik−1
Pkk−1 )v)
ak (ζk , v) = φ(v)
∀ v ∈ Vk
Approximation Property
k
k
φ((Idk − Ik−1
Pkk−1 )v) = ak (ζk , (Idk − Ik−1
Pkk−1 )v)
k
= ak (ζk , v) − ak (ζk , Ik−1
Pkk−1 v)
Approximation Property
k
k
φ((Idk − Ik−1
Pkk−1 )v) = ak (ζk , (Idk − Ik−1
Pkk−1 )v)
k
= ak (ζk , v) − ak (ζk , Ik−1
Pkk−1 v)
= ak (ζk , v) − ak−1 (Pkk−1 ζk , Pkk−1 v)
k
ak−1 (Pkk−1 v, w) = ak (v, Ik−1
w)
∀ v ∈ Vk , w ∈ Vk−1
Approximation Property
k
k
φ((Idk − Ik−1
Pkk−1 )v) = ak (ζk , (Idk − Ik−1
Pkk−1 )v)
k
= ak (ζk , v) − ak (ζk , Ik−1
Pkk−1 v)
= ak (ζk , v) − ak−1 (Pkk−1 ζk , Pkk−1 v)
= ak (ζk , v) − ak−1 (ζk−1 , Pkk−1 v)
ζk−1 = Pkk−1 ζk
Approximation Property
k
k
φ((Idk − Ik−1
Pkk−1 )v) = ak (ζk , (Idk − Ik−1
Pkk−1 )v)
k
= ak (ζk , v) − ak (ζk , Ik−1
Pkk−1 v)
= ak (ζk , v) − ak−1 (Pkk−1 ζk , Pkk−1 v)
= ak (ζk , v) − ak−1 (ζk−1 , Pkk−1 v)
k
= ak (ζk − Ik−1
ζk−1 , v)
k
ak−1 (w, Pkk−1 v) = ak (Ik−1
w, v)
∀ v ∈ Vk , w ∈ Vk−1
Approximation Property
k
k
φ((Idk − Ik−1
Pkk−1 )v) = ak (ζk , (Idk − Ik−1
Pkk−1 )v)
k
= ak (ζk , v) − ak (ζk , Ik−1
Pkk−1 v)
= ak (ζk , v) − ak−1 (Pkk−1 ζk , Pkk−1 v)
= ak (ζk , v) − ak−1 (ζk−1 , Pkk−1 v)
k
= ak (ζk − Ik−1
ζk−1 , v)
k
≤ |||ζk − Ik−1
ζk−1 |||0,k |||v|||2,k
ak (w, v) ≤ |||w|||1−t,k |||v|||1+t,k
∀ v, w ∈ Vk
Approximation Property
k
k
φ((Idk − Ik−1
Pkk−1 )v) = ak (ζk , (Idk − Ik−1
Pkk−1 )v)
k
= ak (ζk , v) − ak (ζk , Ik−1
Pkk−1 v)
= ak (ζk , v) − ak−1 (Pkk−1 ζk , Pkk−1 v)
= ak (ζk , v) − ak−1 (ζk−1 , Pkk−1 v)
k
= ak (ζk − Ik−1
ζk−1 , v)
k
≤ |||ζk − Ik−1
ζk−1 |||0,k |||v|||2,k
≤ C|ζk − ζk−1 |H 1 (Ω) |||v|||2,k
k
Ik−1
= natural injection
||| · |||0,k ≈ | · |H 1 (Ω)
on Vk
Approximation Property
k
k
φ((Idk − Ik−1
Pkk−1 )v) = ak (ζk , (Idk − Ik−1
Pkk−1 )v)
k
= ak (ζk , v) − ak (ζk , Ik−1
Pkk−1 v)
= ak (ζk , v) − ak−1 (Pkk−1 ζk , Pkk−1 v)
= ak (ζk , v) − ak−1 (ζk−1 , Pkk−1 v)
k
= ak (ζk − Ik−1
ζk−1 , v)
k
≤ |||ζk − Ik−1
ζk−1 |||0,k |||v|||2,k
≤ C|ζk − ζk−1 |H 1 (Ω) |||v|||2,k
≤ C |ζk − ζ|H 1 (Ω) + |ζ − ζk−1 |H 1 (Ω) |||v|||2,k
Approximation Property
k
k
φ((Idk − Ik−1
Pkk−1 )v) = ak (ζk , (Idk − Ik−1
Pkk−1 )v)
k
= ak (ζk , v) − ak (ζk , Ik−1
Pkk−1 v)
= ak (ζk , v) − ak−1 (Pkk−1 ζk , Pkk−1 v)
= ak (ζk , v) − ak−1 (ζk−1 , Pkk−1 v)
k
= ak (ζk − Ik−1
ζk−1 , v)
k
≤ |||ζk − Ik−1
ζk−1 |||0,k |||v|||2,k
≤ C|ζk − ζk−1 |H 1 (Ω) |||v|||2,k
≤ C |ζk − ζ|H 1 (Ω) + |ζ − ζk−1 |H 1 (Ω) |||v|||2,k
≤ Ch2k kφkH −1 (Ω) |||v|||2,k
|ζ − ζk |H 1 (Ω) ≤ Ch2k kφkH −1 (Ω)
|ζ − ζk−1 |H 1 (Ω) ≤ Ch2k kφkH −1 (Ω)
Approximation Property
k
k
φ((Idk − Ik−1
Pkk−1 )v) = ak (ζk , (Idk − Ik−1
Pkk−1 )v)
k
= ak (ζk , v) − ak (ζk , Ik−1
Pkk−1 v)
= ak (ζk , v) − ak−1 (Pkk−1 ζk , Pkk−1 v)
= ak (ζk , v) − ak−1 (ζk−1 , Pkk−1 v)
k
= ak (ζk − Ik−1
ζk−1 , v)
k
≤ |||ζk − Ik−1
ζk−1 |||0,k |||v|||2,k
≤ C|ζk − ζk−1 |H 1 (Ω) |||v|||2,k
≤ C |ζk − ζ|H 1 (Ω) + |ζ − ζk−1 |H 1 (Ω) |||v|||2,k
≤ Ch2k kφkH −1 (Ω) |||v|||2,k
|||(Idk −
k
Ik−1
Pkk−1 )v|||0,k
k P k−1 )v)
φ((Idk − Ik−1
k
≈ sup
kφk
−1
H (Ω)
φ∈H −1 (Ω)
≤ Ch2k |||v|||2,k
Approximation Property
Assuming that Ω is convex, we have
k
|||(Idk − Ik−1
Pkk−1 )v||| 1 ,k ≤ Ch2k |||v||| 3 ,k
2
2
∀ v ∈ Vk
where the mesh-dependent norms are defined in terms of the
standard smoother.
Approximation Property
Assuming that Ω is convex, we have
k
|||(Idk − Ik−1
Pkk−1 )v||| 1 ,k ≤ Ch2k |||v||| 3 ,k
2
2
∀ v ∈ Vk
where the mesh-dependent norms are defined in terms of the
standard smoother.
This approximation property has a similar derivation. The difference is that in this case
||| · ||| 1 ,k ≈ | · |H 1 (Ω)
2
on Vk
because
|||v|||0,k ≈ kvkL2 (Ω)
∀ v ∈ Vk
|||v|||1,k ≈ kvkH 2 (Ω;Tk )
∀ v ∈ Vk
Approximation Property
Assuming that Ω is convex, we have
k
|||(Idk − Ik−1
Pkk−1 )v||| 1 ,k ≤ Ch2k |||v||| 3 ,k
2
2
∀ v ∈ Vk
where the mesh-dependent norms are defined in terms of the
standard smoother.
This approximation property has a similar derivation. The difference is that in this case
||| · ||| 1 ,k ≈ | · |H 1 (Ω)
2
on Vk
Therefore the generalized Cauchy-Schwarz inequality dictates
that
k
|||(Idk − Ik−1
Pkk−1 )v||| 1 ,k ≤ Ch2k |||v||| 3 ,k
2
2
∀ v ∈ Vk
Convergence of W - Cycle
Convergence of the Two-Grid Algorithm
Convergence of the Two-Grid Algorithm
Standard Smoother
(ρk = ρ(Bk−1 Ak ) ≈ h−4
k )
k
|||EkT G v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k
k
EkT G = Rkm (Idk − Ik−1
Pkk−1 )Rkm
Convergence of the Two-Grid Algorithm
Standard Smoother
(ρk = ρ(Bk−1 Ak ) ≈ h−4
k )
k
|||EkT G v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k
1/4
k
≤ C(ρk m−1/4 )|||(Idk − Ik−1
Pkk−1 )Rkm v||| 1 ,k
2
Smoothing Property
(s−t)/2
|||Rkm v|||s,k ≤ Cρk
m(t−s)/2 |||v|||t,k
1/4
|||Rkm v|||1,k ≤ Cρk m−1/4 |||v||| 1 ,k
2
Convergence of the Two-Grid Algorithm
Standard Smoother
(ρk = ρ(Bk−1 Ak ) ≈ h−4
k )
k
|||EkT G v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k
1/4
k
≤ C(ρk m−1/4 )|||(Idk − Ik−1
Pkk−1 )Rkm v||| 1 ,k
2
≤
1/4
C(ρk m−1/4 )h2k |||Rkm v||| 3 ,k
2
Approximation Property
k
|||(Idk − Ik−1
Pkk−1 )v||| 1 ,k ≤ Ch2k |||v||| 3 ,k
2
2
Convergence of the Two-Grid Algorithm
(ρk = ρ(Bk−1 Ak ) ≈ h−4
k )
Standard Smoother
k
|||EkT G v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k
1/4
k
≤ C(ρk m−1/4 )|||(Idk − Ik−1
Pkk−1 )Rkm v||| 1 ,k
2
≤
1/4
C(ρk m−1/4 )h2k |||Rkm v||| 3 ,k
2
1/4
1/4
≤ C(ρk m−1/4 )h2k (ρk m−1/4 )|||v|||1,k
Smoothing Property
(s−t)/2
|||Rkm v|||s,k ≤ Cρk
1/4
m(t−s)/2 |||v|||t,k
|||Rkm v||| 3 ,k ≤ Cρk m−1/4 |||v|||1,k
2
Convergence of the Two-Grid Algorithm
Standard Smoother
(ρk = ρ(Bk−1 Ak ) ≈ h−4
k )
k
|||EkT G v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k
1/4
k
≤ C(ρk m−1/4 )|||(Idk − Ik−1
Pkk−1 )Rkm v||| 1 ,k
2
≤
1/4
C(ρk m−1/4 )h2k |||Rkm v||| 3 ,k
2
1/4
1/4
≤ C(ρk m−1/4 )h2k (ρk m−1/4 )|||v|||1,k
≤ Cm−1/2 |||v|||1,k
ρk ≈ h−4
k
Convergence of the Two-Grid Algorithm
Standard Smoother
(ρk = ρ(Bk−1 Ak ) ≈ h−4
k )
k
|||EkT G v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k
1/4
k
≤ C(ρk m−1/4 )|||(Idk − Ik−1
Pkk−1 )Rkm v||| 1 ,k
2
≤
1/4
C(ρk m−1/4 )h2k |||Rkm v||| 3 ,k
2
1/4
1/4
≤ C(ρk m−1/4 )h2k (ρk m−1/4 )|||v|||1,k
≤ Cm−1/2 |||v|||1,k
The two-grid algorithm is a contraction if m is sufficiently large
√
and the contraction number will decrease at the rate of 1/ m.
Convergence of the Two-Grid Algorithm
Nonstandard Smoother
(ρk = ρ(Bk−1 Ak ) ≈ h−2
k )
k
|||EkT G v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k
k
EkT G = Rkm (Idk − Ik−1
Pkk−1 )Rkm
Convergence of the Two-Grid Algorithm
Nonstandard Smoother
(ρk = ρ(Bk−1 Ak ) ≈ h−2
k )
k
|||EkT G v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k
1/2
k
≤ C(ρk m−1/2 )|||(Idk − Ik−1
Pkk−1 )Rkm v|||0,k
Smoothing Property
(s−t)/2
|||Rkm v|||s,k ≤ Cρk
1/2
m(t−s)/2 |||v|||t,k
|||Rkm v|||1,k ≤ Cρk m−1/2 |||v|||0,k
Convergence of the Two-Grid Algorithm
Nonstandard Smoother
(ρk = ρ(Bk−1 Ak ) ≈ h−2
k )
k
|||EkT G v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k
1/2
k
≤ C(ρk m−1/2 )|||(Idk − Ik−1
Pkk−1 )Rkm v|||0,k
1/2
≤ C(ρk m−1/2 )h2k |||Rkm v|||2,k
Approximation Property
k
|||(Idk − Ik−1
Pkk−1 )v|||0,k ≤ Ch2k |||v|||2,k
Convergence of the Two-Grid Algorithm
Nonstandard Smoother
(ρk = ρ(Bk−1 Ak ) ≈ h−2
k )
k
|||EkT G v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k
1/2
k
≤ C(ρk m−1/2 )|||(Idk − Ik−1
Pkk−1 )Rkm v|||0,k
1/2
≤ C(ρk m−1/2 )h2k |||Rkm v|||2,k
1/2
1/2
≤ C(ρk m−1/2 )h2k (ρk m−1/2 )|||v|||1,k
Smoothing Property
(s−t)/2
|||Rkm v|||s,k ≤ Cρk
1/2
m(t−s)/2 |||v|||t,k
|||Rkm v|||2,k ≤ Cρk m−1/2 |||v|||1,k
Convergence of the Two-Grid Algorithm
Nonstandard Smoother
(ρk = ρ(Bk−1 Ak ) ≈ h−2
k )
k
|||EkT G v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k
1/2
k
≤ C(ρk m−1/2 )|||(Idk − Ik−1
Pkk−1 )Rkm v|||0,k
1/2
≤ C(ρk m−1/2 )h2k |||Rkm v|||2,k
1/2
1/2
≤ C(ρk m−1/2 )h2k (ρk m−1/2 )|||v|||1,k
≤ Cm−1 |||v|||1,k
ρk ≈ h−2
k
Convergence of the Two-Grid Algorithm
Nonstandard Smoother
(ρk = ρ(Bk−1 Ak ) ≈ h−2
k )
k
|||EkT G v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k
1/2
k
≤ C(ρk m−1/2 )|||(Idk − Ik−1
Pkk−1 )Rkm v|||0,k
1/2
≤ C(ρk m−1/2 )h2k |||Rkm v|||2,k
1/2
1/2
≤ C(ρk m−1/2 )h2k (ρk m−1/2 )|||v|||1,k
≤ Cm−1 |||v|||1,k
The two-grid algorithm is a contraction if m is sufficiently large
and the contraction number will decrease at the rate of 1/m.
Convergence of the Two-Grid Algorithm
Convex Polygonal Domain
Standard Smoother
|||EkT G v|||1,k ≤ Cm−1/2 |||v|||1,k
∀ v ∈ Vk
Nonstandard Smoother
|||EkT G v|||1,k ≤ Cm−1 |||v|||1,k
∀ v ∈ Vk
Convergence of the Two-Grid Algorithm
General Polygonal Domain
Standard Smoother
|||EkT G v|||1,k ≤ Cm−α/2 |||v|||1,k
∀ v ∈ Vk
Nonstandard Smoother
|||EkT G v|||1,k ≤ Cm−α |||v|||1,k
∀ v ∈ Vk
where α ∈ ( 21 , 2] is the index of elliptic regularity for clamped
plates.
Convergence of the Two-Grid Algorithm
For the mesh-dependent norms defined in terms of the the nonstandard smoother, the proof of the approximation property in
the case of convex polygonal domains depends on the fact that
|||v|||0,k ≈ |v|H 1 (Ω)
∀ v ∈ Vk
The proof of the approximation property for general polygonal
domains requires the relation
|||v|||1−α,k ≈ |v|H 2−α (Ω)
∀ v ∈ Vk
whose proof depends on the existence of a one-to-one enriching operator Eh : Vh −→ H02 (Ω).
Convergence of the Two-Grid Algorithm
General Polygonal Domain
Standard Smoother
|||EkT G v|||1,k ≤ Cm−α/2 |||v|||1,k
∀ v ∈ Vk
Nonstandard Smoother
|||EkT G v|||1,k ≤ Cm−α |||v|||1,k
∀ v ∈ Vk
Comparing these two estimates we see that the effect of 100
smoothing steps by the standard smoother is (roughly) equivalent to the effect of 10 smoothing steps by the nonstandard
smoother.
Convergence of the Two-Grid Algorithm
General Polygonal Domain
Standard Smoother
|||EkT G v|||1,k ≤ Cm−α/2 |||v|||1,k
∀ v ∈ Vk
Nonstandard Smoother
|||EkT G v|||1,k ≤ Cm−α |||v|||1,k
∀ v ∈ Vk
This difference in performance is due to the difference in the
spectral radius of the preconditioned operators.

h−4
standard smoother
k
−1
ρ(Bk Ak ) ≈
h−2
nonstandard smoother
k
Convergence of the W -Cycle Algorithm
Convergence of the W -Cycle Algorithm
k
Estimate for the Operator Ik−1
: Vk−1 −→ Vk
k
|||Ik−1
v|||1,k ≤ CCF |||v|||1,k−1
∀ v ∈ Vk−1
where the positive constant CCF is independent of k.
Convergence of the W -Cycle Algorithm
k
Estimate for the Operator Ik−1
: Vk−1 −→ Vk
k
|||Ik−1
v|||1,k ≤ CCF |||v|||1,k−1
∀ v ∈ Vk−1
where the positive constant CCF is independent of k.
k
k
k
|||Ik−1
v|||21,k = ak (Ik−1
v, Ik−1
v)
Convergence of the W -Cycle Algorithm
k
Estimate for the Operator Ik−1
: Vk−1 −→ Vk
k
|||Ik−1
v|||1,k ≤ CCF |||v|||1,k−1
∀ v ∈ Vk−1
where the positive constant CCF is independent of k.
k
k
k
|||Ik−1
v|||21,k = ak (Ik−1
v, Ik−1
v)
X
X σ
k
≈
|Ik−1
v|2H 2 (T ) +
kI k vk2
|e| k−1 L2 (e)
T ∈Tk
e∈Ek
Convergence of the W -Cycle Algorithm
k
Estimate for the Operator Ik−1
: Vk−1 −→ Vk
k
|||Ik−1
v|||1,k ≤ CCF |||v|||1,k−1
∀ v ∈ Vk−1
where the positive constant CCF is independent of k.
k
k
k
|||Ik−1
v|||21,k = ak (Ik−1
v, Ik−1
v)
X
X σ
k
≈
|Ik−1
v|2H 2 (T ) +
kI k vk2
|e| k−1 L2 (e)
T ∈Tk
e∈Ek
X
X σ
2
=
kvk2L2 (e)
|v|H 2 (T ) +
|e|
T ∈Tk
e∈Ek
Convergence of the W -Cycle Algorithm
k
Estimate for the Operator Ik−1
: Vk−1 −→ Vk
k
|||Ik−1
v|||1,k ≤ CCF |||v|||1,k−1
∀ v ∈ Vk−1
where the positive constant CCF is independent of k.
k
k
k
|||Ik−1
v|||21,k = ak (Ik−1
v, Ik−1
v)
X
X σ
k
≈
|Ik−1
v|2H 2 (T ) +
kI k vk2
|e| k−1 L2 (e)
T ∈Tk
e∈Ek
X
X σ
2
=
kvk2L2 (e)
|v|H 2 (T ) +
|e|
T ∈Tk
e∈Ek
X
X σ
2
=
|v|H 2 (T ) +
kvk2L2 (e)
|e|
T ∈Tk−1
e∈Ek
Convergence of the W -Cycle Algorithm
k
Estimate for the Operator Ik−1
: Vk−1 −→ Vk
k
|||Ik−1
v|||1,k ≤ CCF |||v|||1,k−1
∀ v ∈ Vk−1
where the positive constant CCF is independent of k.
k
k
k
|||Ik−1
v|||21,k = ak (Ik−1
v, Ik−1
v)
X
X σ
k
≈
|Ik−1
v|2H 2 (T ) +
kI k vk2
|e| k−1 L2 (e)
T ∈Tk
e∈Ek
X
X σ
2
=
kvk2L2 (e)
|v|H 2 (T ) +
|e|
T ∈Tk
e∈Ek
X
X σ
2
=
|v|H 2 (T ) +
kvk2L2 (e)
|e|
T ∈Tk−1
e∈Ek
X
X σ
2
kvk2L2 (e)
=
|v|H 2 (T ) + 2
|e|
T ∈Tk−1
e∈Ek−1
Convergence of the W -Cycle Algorithm
k
Estimate for the Operator Ik−1
: Vk−1 −→ Vk
k
|||Ik−1
v|||1,k ≤ CCF |||v|||1,k−1
∀ v ∈ Vk−1
where the positive constant CCF is independent of k.
k
k
k
|||Ik−1
v|||21,k = ak (Ik−1
v, Ik−1
v)
X
X σ
k
≈
|Ik−1
v|2H 2 (T ) +
kI k vk2
|e| k−1 L2 (e)
T ∈Tk
e∈Ek
X
X σ
2
=
kvk2L2 (e)
|v|H 2 (T ) +
|e|
T ∈Tk
e∈Ek
X
X σ
2
=
|v|H 2 (T ) +
kvk2L2 (e)
|e|
T ∈Tk−1
e∈Ek
X
X σ
2
kvk2L2 (e)
=
|v|H 2 (T ) + 2
|e|
T ∈Tk−1
≈
|||v|||21,k−1
e∈Ek−1
Convergence of the W -Cycle Algorithm
k
Estimate for the Operator Ik−1
: Vk−1 −→ Vk
k
|||Ik−1
v|||1,k ≤ CCF |||v|||1,k−1
∀ v ∈ Vk−1
where the positive constant CCF is independent of k.
Estimate for the Operator Pkk−1 : Vk −→ Vk−1
|||Pkk−1 v|||1,k−1 ≤ CCF |||v|||1,k
∀ v ∈ Vk
Convergence of the W -Cycle Algorithm
Two-Grid Estimates
|||EkT G v|||1,k ≤ CTG m−α/γ |||v|||1,k
∀ v ∈ Vk
where γ = 1 for the nonstandard smoother and γ = 2 for the
standard smoother.
Convergence of the W -Cycle Algorithm
Two-Grid Estimates
|||EkT G v|||1,k ≤ CTG m−α/γ |||v|||1,k
∀ v ∈ Vk
where γ = 1 for the nonstandard smoother and γ = 2 for the
standard smoother.
Theorem Given any C∗ > CTG , there exists a positive integer
m∗ independent of k such that
|||EkW v|||1,k ≤ C∗ m−α/γ |||v|||1,k
provided m ≥ m∗ .
∀ v ∈ Vk
Convergence of the W -Cycle Algorithm
Two-Grid Estimates
|||EkT G v|||1,k ≤ CTG m−α/γ |||v|||1,k
∀ v ∈ Vk
where γ = 1 for the nonstandard smoother and γ = 2 for the
standard smoother.
Theorem Given any C∗ > CTG , there exists a positive integer
m∗ independent of k such that
|||EkW v|||1,k ≤ C∗ m−α/γ |||v|||1,k
∀ v ∈ Vk
provided m ≥ m∗ .
The W -cycle algorithm is a contraction if the number of smoothing steps is sufficiently large (but independent of k). The contraction numbers are bounded away from 1 on all levels, i.e., the
W -cycle algorithm is uniformly convergent.
Convergence of the W -Cycle Algorithm
Two-Grid Estimates
|||EkT G v|||1,k ≤ CTG m−α/γ |||v|||1,k
∀ v ∈ Vk
where γ = 1 for the nonstandard smoother and γ = 2 for the
standard smoother.
Theorem Given any C∗ > CTG , there exists a positive integer
m∗ independent of k such that
|||EkW v|||1,k ≤ C∗ m−α/γ |||v|||1,k
∀ v ∈ Vk
provided m ≥ m∗ .
As in the case of the two-grid algorithm, the contraction numbers decrease at the rate of m−α/γ , where γ = 1 for the
nonstandard smoother and γ = 2 for the standard smoother.
The algorithm based on the nonstandard smoother is therefore
more efficient.
Convergence of the W -Cycle Algorithm
Two-Grid Estimates
|||EkT G v|||1,k ≤ CTG m−α/γ |||v|||1,k
∀ v ∈ Vk
where γ = 1 for the nonstandard smoother and γ = 2 for the
standard smoother.
Theorem Given any C∗ > CTG , there exists a positive integer
m∗ independent of k such that
|||EkW v|||1,k ≤ C∗ m−α/γ |||v|||1,k
provided m ≥ m∗ .
Proof.
(mathematical induction)
The case k = 0 is obvious since EkW = 0.
∀ v ∈ Vk
Convergence of the W -Cycle Algorithm
k
k
W 2 k−1
|||EkW v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
) Pk )Rkm v|||1,k
Recursive Relation for W -cycle
k
k
W 2 k−1
EkW = Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
) Pk )Rkm
Convergence of the W -Cycle Algorithm
k
k
W 2 k−1
|||EkW v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
) Pk )Rkm v|||1,k
k
k
W 2 k−1 m
(Ek−1
) Pk Rk
≤ |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k + |||Rkm Ik−1
Convergence of the W -Cycle Algorithm
k
k
W 2 k−1
|||EkW v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
) Pk )Rkm v|||1,k
k
k
W 2 k−1 m
(Ek−1
) Pk Rk
≤ |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k + |||Rkm Ik−1
k
W 2 k−1 m
= |||EkT G v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
Two-Grid Error Propagation Operator
k
EkT G = Rkm (Idk − Ik−1
Pkk−1 )Rkm
Convergence of the W -Cycle Algorithm
k
k
W 2 k−1
|||EkW v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
) Pk )Rkm v|||1,k
k
k
W 2 k−1 m
(Ek−1
) Pk Rk
≤ |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k + |||Rkm Ik−1
k
W 2 k−1 m
= |||EkT G v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
k
W 2 k−1 m
≤ CTG m−α/γ |||v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
Two-Grid Estimate
|||EkT G v|||1,k ≤ CTG m−α/γ |||v|||1,k
∀ v ∈ Vk
Convergence of the W -Cycle Algorithm
k
k
W 2 k−1
|||EkW v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
) Pk )Rkm v|||1,k
k
k
W 2 k−1 m
(Ek−1
) Pk Rk
≤ |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k + |||Rkm Ik−1
k
W 2 k−1 m
= |||EkT G v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
k
W 2 k−1 m
≤ CTG m−α/γ |||v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
W 2 k−1 m
≤ CTG m−α/γ |||v|||1,k + CCF |||(Ek−1
) Pk Rk v|||1,k−1
|||Rk v|||1,k ≤ |||v|||1,k
k
|||Ik−1
v|||1,k ≤ CCF |||v|||1,k−1
∀ v ∈ Vk
∀ v ∈ Vk−1
Convergence of the W -Cycle Algorithm
k
k
W 2 k−1
|||EkW v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
) Pk )Rkm v|||1,k
k
k
W 2 k−1 m
(Ek−1
) Pk Rk
≤ |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k + |||Rkm Ik−1
k
W 2 k−1 m
= |||EkT G v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
k
W 2 k−1 m
≤ CTG m−α/γ |||v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
W 2 k−1 m
≤ CTG m−α/γ |||v|||1,k + CCF |||(Ek−1
) Pk Rk v|||1,k−1
≤ CTG m−α/γ |||v|||1,k + CCF C∗2 m−2α/γ |||Pkk−1 Rkm v|||1,k−1
Induction Hypothesis
W
|||Ek−1
v|||1,k ≤ C∗ m−α/γ |||v|||1,k−1
∀ v ∈ Vk−1
Convergence of the W -Cycle Algorithm
k
k
W 2 k−1
|||EkW v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
) Pk )Rkm v|||1,k
k
k
W 2 k−1 m
(Ek−1
) Pk Rk
≤ |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k + |||Rkm Ik−1
k
W 2 k−1 m
= |||EkT G v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
k
W 2 k−1 m
≤ CTG m−α/γ |||v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
W 2 k−1 m
≤ CTG m−α/γ |||v|||1,k + CCF |||(Ek−1
) Pk Rk v|||1,k−1
≤ CTG m−α/γ |||v|||1,k + CCF C∗2 m−2α/γ |||Pkk−1 Rkm v|||1,k−1
2
≤ CTG m−α/γ |||v|||1,k + CCF
C∗2 m−2α/γ |||v|||1,k
|||Pkk−1 v|||1,k−1 ≤ CCF |||v|||1,k
∀ v ∈ Vk
|||Rk v|||1,k ≤ |||v|||1,k
∀ v ∈ Vk
Convergence of the W -Cycle Algorithm
k
k
W 2 k−1
|||EkW v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
) Pk )Rkm v|||1,k
k
k
W 2 k−1 m
(Ek−1
) Pk Rk
≤ |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k + |||Rkm Ik−1
k
W 2 k−1 m
= |||EkT G v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
k
W 2 k−1 m
≤ CTG m−α/γ |||v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
W 2 k−1 m
≤ CTG m−α/γ |||v|||1,k + CCF |||(Ek−1
) Pk Rk v|||1,k−1
≤ CTG m−α/γ |||v|||1,k + CCF C∗2 m−2α/γ |||Pkk−1 Rkm v|||1,k−1
2
≤ CTG m−α/γ |||v|||1,k + CCF
C∗2 m−2α/γ |||v|||1,k
2
≤ CTG + CCF
C∗2 m−α/γ m−α/γ |||v|||1,k
Convergence of the W -Cycle Algorithm
k
k
W 2 k−1
|||EkW v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
) Pk )Rkm v|||1,k
k
k
W 2 k−1 m
(Ek−1
) Pk Rk
≤ |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k + |||Rkm Ik−1
k
W 2 k−1 m
= |||EkT G v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
k
W 2 k−1 m
≤ CTG m−α/γ |||v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
W 2 k−1 m
≤ CTG m−α/γ |||v|||1,k + CCF |||(Ek−1
) Pk Rk v|||1,k−1
≤ CTG m−α/γ |||v|||1,k + CCF C∗2 m−2α/γ |||Pkk−1 Rkm v|||1,k−1
2
≤ CTG m−α/γ |||v|||1,k + CCF
C∗2 m−2α/γ |||v|||1,k
2
≤ CTG + CCF
C∗2 m−α/γ m−α/γ |||v|||1,k
≤ CTG + (C∗ − CTG ) m−α/γ |||v|||1,k
If m ≥ m∗ and m∗ is chosen so that
−α/γ
2
CCF
C∗2 m∗
≤ C∗ − CTG
Convergence of the W -Cycle Algorithm
k
k
W 2 k−1
|||EkW v|||1,k = |||Rkm (Idk − Ik−1
Pkk−1 + Ik−1
(Ek−1
) Pk )Rkm v|||1,k
k
k
W 2 k−1 m
(Ek−1
) Pk Rk
≤ |||Rkm (Idk − Ik−1
Pkk−1 )Rkm v|||1,k + |||Rkm Ik−1
k
W 2 k−1 m
= |||EkT G v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
k
W 2 k−1 m
≤ CTG m−α/γ |||v|||1,k + |||Rkm Ik−1
(Ek−1
) Pk Rk v|||1,k
W 2 k−1 m
≤ CTG m−α/γ |||v|||1,k + CCF |||(Ek−1
) Pk Rk v|||1,k−1
≤ CTG m−α/γ |||v|||1,k + CCF C∗2 m−2α/γ |||Pkk−1 Rkm v|||1,k−1
2
≤ CTG m−α/γ |||v|||1,k + CCF
C∗2 m−2α/γ |||v|||1,k
2
≤ CTG + CCF
C∗2 m−α/γ m−α/γ |||v|||1,k
≤ CTG + (C∗ − CTG ) m−α/γ |||v|||1,k
≤ C∗ m−α/γ |||v|||1,k
Summary
The convergence analysis of the W -cycle algorithm for C 0 interior penalty methods is based on the convergence analysis of
the two-grid algorithm and a perturbation argument.
Summary
The convergence analysis of the W -cycle algorithm for C 0 interior penalty methods is based on the convergence analysis of
the two-grid algorithm and a perturbation argument.
Ingredients for the Convergence Analysis
I
calculus
I
spectral theorem
I
error estimates in lower order norms
elliptic regularity
enriching operator
interpolation of operators
References
• Bank and Dupont, An optimal order process for solving fi-
nite element equations
Math. Comp., 1981
• B., Convergence of nonconforming multigrid methods with-
out full elliptic regularity
Math. Comp., 1999
Convergence of V - Cycle
Conforming Finite Element Methods
Conforming Finite Element Methods
Second Order Model Problem Find u ∈ H01 (Ω) such that
Z
Z
∇u · ∇v dx =
f v dx
∀ v ∈ H01 (Ω)
Ω
Ω
Conforming Finite Element Methods
Second Order Model Problem Find u ∈ H01 (Ω) such that
Z
Z
∇u · ∇v dx =
f v dx
∀ v ∈ H01 (Ω)
Ω
Ω
Discrete Problem Find uh ∈ Vh such that
Z
Z
∇uh · ∇v dx =
f v dx
∀ v ∈ Vh
Ω
Ω
where Vh ⊂ H01 (Ω) is the P1 finite element space associated
with a triangulation Th .
Conforming Finite Element Methods
Second Order Model Problem Find u ∈ H01 (Ω) such that
Z
Z
∇u · ∇v dx =
f v dx
∀ v ∈ H01 (Ω)
Ω
Ω
Discrete Problem Find uh ∈ Vh such that
Z
Z
∇uh · ∇v dx =
f v dx
∀ v ∈ Vh
Ω
Ω
where Vh ⊂ H01 (Ω) is the P1 finite element space associated
with a triangulation Th .
Multigrid Setting There is a sequence of triangulations generated by uniform refinements and a corresponding sequence
of nested conforming finite element spaces
V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ H01 (Ω)
Conforming Finite Element Methods
With Full Elliptic Regularity
(Ω convex)
kukH 2 (Ω) ≤ CΩ kf kL2 (Ω)
1983
Braess-Hackbusch
|Ek v|H 1 (Ω) ≤
C
|v| 1
C + m H (Ω)
∀ v ∈ Vk
for m, k ≥ 1, where C is independent of m and k. In particular,
the V -cycle is a contraction with only one smoothing step.
Conforming Finite Element Methods
Without Full Elliptic Regularity
(Ω nonconvex)
kukH 1+α (Ω) ≤ CΩ kf kL2 (Ω)
for
1
2
<α<1
Conforming Finite Element Methods
Without Full Elliptic Regularity
(Ω nonconvex)
kukH 1+α (Ω) ≤ CΩ kf kL2 (Ω)
for
1
2
1987
<α<1
Bramble-Pasciak
1988
Decker-Mandel-Parter
1
|v|H 1 (Ω)
|Ek v|H 1 (Ω) ≤ 1 −
Ck (1−α)/α
∀ v ∈ Vk , k ≥ 1
Conforming Finite Element Methods
Without Full Elliptic Regularity
(Ω nonconvex)
kukH 1+α (Ω) ≤ CΩ kf kL2 (Ω)
for
1
2
1987
<α<1
Bramble-Pasciak
1988
Decker-Mandel-Parter
1
|v|H 1 (Ω)
|Ek v|H 1 (Ω) ≤ 1 −
Ck (1−α)/α
1991
Bramble-Pasciak-Wang-Xu
1 |Ek v|H 1 (Ω) ≤ 1 −
|v|H 1 (Ω)
Ck
∀ v ∈ Vk , k ≥ 1
(no regularity assumption)
∀ v ∈ Vk , k ≥ 1
Conforming Finite Element Methods
1992
Zhang, Xu
1993
Bramble-Pasciak
There exists δ ∈ (0, 1) such that
|Ek v|H 1 (Ω) ≤ δ|v|H 1 (Ω)
∀ v ∈ Vk
for m, k ≥ 1. In particular, the V -cycle is a contraction with one
smoothing step.
Conforming Finite Element Methods
1992
Zhang, Xu
1993
Bramble-Pasciak
There exists δ ∈ (0, 1) such that
|Ek v|H 1 (Ω) ≤ δ|v|H 1 (Ω)
∀ v ∈ Vk
for m, k ≥ 1. In particular, the V -cycle is a contraction with one
smoothing step.
Braess-Hackbusch
|Ek v|H 1 (Ω) ≤
C
|v| 1
C + m H (Ω)
∀ v ∈ Vk
for m, k ≥ 1, where C is independent of m and k. In particular,
the V -cycle is a contraction with only one smoothing step.
Conforming Finite Element Methods
2002
B.
|Ek v|H 1 (Ω) ≤
C
|v| 1
C + mα H (Ω)
for m, k ≥ 1, where C is independent of m and k.
Conforming Finite Element Methods
2002
B.
|Ek v|H 1 (Ω) ≤
C
|v| 1
C + mα H (Ω)
for m, k ≥ 1, where C is independent of m and k.
This is a complete generalization of the Braess-Hackbusch result:
C
|v| 1
∀ v ∈ Vk
|Ek v|H 1 (Ω) ≤
C + m H (Ω)
for m, k ≥ 1, where C is independent of m and k.
Multiplicative Theory
Recursive Relation for V -Cycle
k
k
Ek = Rkm (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1 Pkk−1 )Rkm
E0 = 0
Multiplicative Theory
Recursive Relation for V -Cycle
k
k
Ek = Rkm (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1 Pkk−1 )Rkm
E0 = 0
Notation
I
I
(j ≤ `)
Ij` : Vj −→ V` is the natural injection
P`j : V` −→ Vj is the transpose of Ij` with respect to the
variational form, i.e.,
a(P`j v, w) = a(v, Ij` w)
I
Ijj = Idj = Pjj
∀ v ∈ V` , w ∈ Vj
Multiplicative Theory
Properties of Ij` and P`j
For j ≤ i ≤ `
I
Ij` = Ii` ◦ Iji
I
P`j = Pij ◦ P`i
I
Iji = P`i ◦ Ij`
I
Pij = P`j ◦ Ii`
I
(Ij` P`j )2 = Ij` P`j
I
(Id` − Ij` P`j )2 = (Id` − Ij` P`j )
(in particular Idj = Ijj = P`j ◦ Ij` )
Multiplicative Theory
k−1
k P k−1 ) + I k E
A Recursive Relation for (Idk − Ik−1
k−1 k−1 Pk
k
Multiplicative Theory
k−1
k P k−1 ) + I k E
A Recursive Relation for (Idk − Ik−1
k−1 k−1 Pk
k
k
k
(Idk − Ik−1
Pkk−1 ) + Ik−1
Ek−1 Pkk−1
k
= (Idk − Ik−1
Pkk−1 )
k−1
k−1 k−2
k−1
k−2
k
m
m
+ Ik−1
Rk−1
(Idk−1 − Ik−2
Pk−1 + Ik−2
Ek−2 Pk−1
)Rk−1
Pk
Multiplicative Theory
k−1
k P k−1 ) + I k E
A Recursive Relation for (Idk − Ik−1
k−1 k−1 Pk
k
k
k
(Idk − Ik−1
Pkk−1 ) + Ik−1
Ek−1 Pkk−1
k
= (Idk − Ik−1
Pkk−1 )
k−1
k−1 k−2
k−1
k−2
k
m
m
+ Ik−1
Rk−1
(Idk−1 − Ik−2
Pk−1 + Ik−2
Ek−2 Pk−1
)Rk−1
Pk
k
k
m
= (Idk − Ik−1
Pkk−1 ) + Ik−1
Rk−1
Pkk−1
k
k
· (Idk − Ik−2
Pkk−2 ) + Ik−2
Ek−2 Pkk−2 )
k
k
m
· (Idk − Ik−1
Pkk−1 ) + Ik−1
Rk−1
Pkk−1
Multiplicative Theory
k−1
k P k−1 ) + I k E
A Recursive Relation for (Idk − Ik−1
k−1 k−1 Pk
k
k
k
(Idk − Ik−1
Pkk−1 ) + Ik−1
Ek−1 Pkk−1
k
= (Idk − Ik−1
Pkk−1 )
k−1
k−1 k−2
k−1
k−2
k
m
m
+ Ik−1
Rk−1
(Idk−1 − Ik−2
Pk−1 + Ik−2
Ek−2 Pk−1
)Rk−1
Pk
k
k
m
= (Idk − Ik−1
Pkk−1 ) + Ik−1
Rk−1
Pkk−1
k
k
· (Idk − Ik−2
Pkk−2 ) + Ik−2
Ek−2 Pkk−2 )
k
k
m
· (Idk − Ik−1
Pkk−1 ) + Ik−1
Rk−1
Pkk−1
k
k
m
= (Idk − Ik−1
Pkk−1 ) + Ik−1
Rk−1
Pkk−1
k
k
m
· (Idk − Ik−2
Pkk−2 ) + Ik−2
Rk−2
Pkk−2
k
k
· (Idk − Ik−3
Pkk−3 ) + Ik−3
Ek−3 Pkk−3
k
k
m
· (Idk − Ik−2
Pkk−2 ) + Ik−2
Rk−2
Pkk−2
k
k
m
· (Idk − Ik−1
Pkk−1 ) + Ik−1
Rk−1
Pkk−1
Multiplicative Theory
k
k
Ek = Rkm (Idk − Ik−1
Pkk−1 ) + Ik−1
Ek−1 Pkk−1 Rkm
k
k
m
Pkk−1 ) + Ik−1
Rk−1
Pkk−1 · · ·
= Rkm (Idk − Ik−1
· (Idk − I1k Pk1 ) + I1k R1m Pk1 (Idk − I0k Pk0 ) (Idk − I1k Pk1 ) + I1k R1m
k
k
m
· · · (Idk − Ik−1
Pkk−1 ) + Ik−1
Rk−1
Pkk−1 Rkm
Multiplicative Theory
k
k
Ek = Rkm (Idk − Ik−1
Pkk−1 ) + Ik−1
Ek−1 Pkk−1 Rkm
k
k
m
Pkk−1 ) + Ik−1
Rk−1
Pkk−1 · · ·
= Rkm (Idk − Ik−1
· (Idk − I1k Pk1 ) + I1k R1m Pk1 (Idk − I0k Pk0 ) (Idk − I1k Pk1 ) + I1k R1m
k
k
m
· · · (Idk − Ik−1
Pkk−1 ) + Ik−1
Rk−1
Pkk−1 Rkm
Notation
For 1 ≤ j ≤ k
j
k
m
Tj def
= Ij (Idj − Rj )Pk
Multiplicative Theory
k
k
Ek = Rkm (Idk − Ik−1
Pkk−1 ) + Ik−1
Ek−1 Pkk−1 Rkm
k
k
m
Pkk−1 ) + Ik−1
Rk−1
Pkk−1 · · ·
= Rkm (Idk − Ik−1
· (Idk − I1k Pk1 ) + I1k R1m Pk1 (Idk − I0k Pk0 ) (Idk − I1k Pk1 ) + I1k R1m
k
k
m
· · · (Idk − Ik−1
Pkk−1 ) + Ik−1
Rk−1
Pkk−1 Rkm
Notation
For 1 ≤ j ≤ k
j
k
m
Tj def
= Ij (Idj − Rj )Pk
In particular
Tk = Ikk (Idk − Rkm )Pkk = Idk − Rkm =⇒ Rkm = Idk − Tk
Multiplicative Theory
k
k
Ek = Rkm (Idk − Ik−1
Pkk−1 ) + Ik−1
Ek−1 Pkk−1 Rkm
k
k
m
Pkk−1 ) + Ik−1
Rk−1
Pkk−1 · · ·
= Rkm (Idk − Ik−1
· (Idk − I1k Pk1 ) + I1k R1m Pk1 (Idk − I0k Pk0 ) (Idk − I1k Pk1 ) + I1k R1m
k
k
m
· · · (Idk − Ik−1
Pkk−1 ) + Ik−1
Rk−1
Pkk−1 Rkm
Notation
For 1 ≤ j ≤ k
j
k
m
Tj def
= Ij (Idj − Rj )Pk
In particular
Tk = Ikk (Idk − Rkm )Pkk = Idk − Rkm =⇒ Rkm = Idk − Tk
(Idk − Ijk Pkj ) + Ijk Rjm Pkj = Idk − Ijk (Idj − Rjm )Pkj = Idk − Tj
Multiplicative Theory
k
k
Ek = Rkm (Idk − Ik−1
Pkk−1 ) + Ik−1
Ek−1 Pkk−1 Rkm
k
k
m
Pkk−1 ) + Ik−1
Rk−1
Pkk−1 · · ·
= Rkm (Idk − Ik−1
· (Idk − I1k Pk1 ) + I1k R1m Pk1 (Idk − I0k Pk0 ) (Idk − I1k Pk1 ) + I1k R1m
k
k
m
· · · (Idk − Ik−1
Pkk−1 ) + Ik−1
Rk−1
Pkk−1 Rkm
Notation
For 1 ≤ j ≤ k
j
k
m
Tj def
= Ij (Idj − Rj )Pk
In particular
Tk = Ikk (Idk − Rkm )Pkk = Idk − Rkm =⇒ Rkm = Idk − Tk
(Idk − Ijk Pkj ) + Ijk Rjm Pkj = Idk − Ijk (Idj − Rjm )Pkj = Idk − Tj
Multiplicative Theory
Multiplicative Expression for Ek
Ek = (Idk − Tk )(Idk − Tk−1 ) . . . (Idk − T1 )
· (Idk − T0 )(Idk − T1 ) . . . (Idk − Tk−1 )(Idk − Tk )
Multiplicative Theory
Multiplicative Expression for Ek
Ek = (Idk − Tk )(Idk − Tk−1 ) . . . (Idk − T1 )
· (Idk − T0 )(Idk − T1 ) . . . (Idk − Tk−1 )(Idk − Tk )
Strengthened Cauchy-Schwarz Inequality
For 0 ≤ j ≤ k, vj ∈ Vj and vk ∈ Vk ,
Z
∇vj · ∇vk dx ≤ C2−(k−j)/2 |vj |H 1 (Ω) h−1
k kvk kL2 (Ω)
Ω
Multiplicative Theory
Multiplicative Expression for Ek
Ek = (Idk − Tk )(Idk − Tk−1 ) . . . (Idk − T1 )
· (Idk − T0 )(Idk − T1 ) . . . (Idk − Tk−1 )(Idk − Tk )
Strengthened Cauchy-Schwarz Inequality
For 0 ≤ j ≤ k, vj ∈ Vj and vk ∈ Vk ,
Z
∇vj · ∇vk dx ≤ C2−(k−j)/2 |vj |H 1 (Ω) h−1
k kvk kL2 (Ω)
Ω
Standard Cauchy-Schwarz Inequality
Z
∇vj · ∇vk dx ≤ |vj |H 1 (Ω) |vk |H 1 (Ω)
Ω
which implies
Z
Ω
∇vj · ∇vk dx ≤ C|vj |H 1 (Ω) h−1
k kvk kL2 (Ω)
Multiplicative Theory
Theorem
There exists δ ∈ (0, 1) such that
|Ek v|H 1 (Ω) ≤ δ|v|H 1 (Ω)
for m, k ≥ 1.
∀ v ∈ Vk
Multiplicative Theory
Theorem
There exists δ ∈ (0, 1) such that
|Ek v|H 1 (Ω) ≤ δ|v|H 1 (Ω)
∀ v ∈ Vk
for m, k ≥ 1.
Details can be found in the book by Bramble and the survey
article by Bramble and Zhang. Refinements of the multiplicative
theory can be found in the following papers.
References
• Xu and Zikatanov, The method of alternating projections
and the method of subspace corrections in Hilbert space
J. Amer. Math. Soc., 2002
• B., An additive analysis of multiplicative Schwarz methods
Numer. Math., 2013
Multiplicative Theory
Theorem
There exists δ ∈ (0, 1) such that
|Ek v|H 1 (Ω) ≤ δ|v|H 1 (Ω)
∀ v ∈ Vk
for m, k ≥ 1.
The multiplicative theory cannot be applied to nonconforming
finite element methods since many of the algebraic relations
are no longer valid. For example,
(Idk − I`k Pk` )2 6= (Idk − I`k Pk` )
Additive Theory
Recursive Relation
k
k
Ek = Rkm (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1 Pkk−1 )Rkm
E0 = 0
Additive Theory
Recursive Relation
k
k
Ek = Rkm (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1 Pkk−1 )Rkm
E0 = 0
Ek
k
= Rkm (Idk − Ik−1
Pkk−1 )Rkm
k−1 k−2
k−1
k−2
k
m
m
Rk−1
(Idk−1 − Ik−2
Pk−1 + Ik−2
Ek−2 Pk−1
)Rk−1
Pkk−1 Rkm
+ Rkm Ik−1
k
= Rkm (Idk − Ik−1
Pkk−1 )Rkm
k−1 k−2
k
m
m
+ Rkm Ik−1
Rk−1
(Idk−1 − Ik−2
Pk−1 )Rk−1
Pkk−1 Rkm
k−1 m
k−2 k−3
k−2 m
k
m
m
+ Rkm Ik−1
Rk−1
Ik−2
Rk−2 (Idk−2 − Ik−3
Pk−2 )Rk−2
Pk−1
Rk−1 Pkk−1
+ ···
Additive Theory
Recursive Relation
k
k
Ek = Rkm (Idk − Ik−1
Pkk−1 + Ik−1
Ek−1 Pkk−1 )Rkm
E0 = 0
Ek
k
= Rkm (Idk − Ik−1
Pkk−1 )Rkm
k−1 k−2
k−1
k−2
k
m
m
Rk−1
(Idk−1 − Ik−2
Pk−1 + Ik−2
Ek−2 Pk−1
)Rk−1
Pkk−1 Rkm
+ Rkm Ik−1
k
= Rkm (Idk − Ik−1
Pkk−1 )Rkm
k−1 k−2
k
m
m
+ Rkm Ik−1
Rk−1
(Idk−1 − Ik−2
Pk−1 )Rk−1
Pkk−1 Rkm
k−1 m
k−2 k−3
k−2 m
k
m
m
+ Rkm Ik−1
Rk−1
Ik−2
Rk−2 (Idk−2 − Ik−3
Pk−2 )Rk−2
Pk−1
Rk−1 Pkk−1
=
k
X
j=1
+ ···
j
k
m
Rkm Ik−1
· · · Rj+1
Ijj+1 Rjm (Idj − Ij−1 Pjj−1 )Rjm Pj+1
Rjm · · · Pkk−1 R
Additive Theory
Ideas
The operator Rjm (Idj −Ij−1 Pjj−1 )Rjm has already been analyzed in the two-grid analysis.
Additive Theory
Ideas
The operator Rjm (Idj −Ij−1 Pjj−1 )Rjm has already been analyzed in the two-grid analysis.
The key is to analyze (for 0 ≤ j ≤ k) the multi-level operator
j+1
m k
m
Tk,j def
= Rk Ik−1 · · · Rj+1 Ij : Vj −→ Vk
and its transpose with respect to the variational forms
j
k−1 m
m
Tj,k def
= Pj+1 Rj · · · Pk Rk : Vk −→ Vj
Additive Theory
Ideas
The operator Rjm (Idj −Ij−1 Pjj−1 )Rjm has already been analyzed in the two-grid analysis.
The key is to analyze (for 0 ≤ j ≤ k) the multi-level operator
j+1
m k
m
Tk,j def
= Rk Ik−1 · · · Rj+1 Ij : Vj −→ Vk
and its transpose with respect to the variational forms
j
k−1 m
m
Tj,k def
= Pj+1 Rj · · · Pk Rk : Vk −→ Vj
We will need a strengthened Cauchy-Schwarz inequality
with smoothing and estimates that compare the meshdependent norms on consecutive levels.
Additive Theory
Ideas
The operator Rjm (Idj −Ij−1 Pjj−1 )Rjm has already been analyzed in the two-grid analysis.
The key is to analyze (for 0 ≤ j ≤ k) the multi-level operator
j+1
m k
m
Tk,j def
= Rk Ik−1 · · · Rj+1 Ij : Vj −→ Vk
and its transpose with respect to the variational forms
j
k−1 m
m
Tj,k def
= Pj+1 Rj · · · Pk Rk : Vk −→ Vj
We will need a strengthened Cauchy-Schwarz inequality
with smoothing and estimates that compare the meshdependent norms on consecutive levels.
We need to circumvent the fact that for nonconforming
methods in general (Idk − I`k Pk` )2 6= (Idk − I`k Pk` )
Additive Theory
Strengthened Cauchy-Schwarz Inequality with Smoothing
Let 0 ≤ j, ` ≤ k, vj ∈ Vj and v` ∈ V` .
ak (Tk,j Rjm vj , Tk,` R`m v` ) ≤
C |`−j|
δ
|||vj |||1−α,j |||v` |||1−α,`
mα
where C is a positive constant, 0 < δ < 1 and α ∈ ( 12 , 1] is the
index of elliptic regularity, provided the number of smoothing
steps m is sufficiently large.
Additive Theory
Strengthened Cauchy-Schwarz Inequality with Smoothing
Let 0 ≤ j, ` ≤ k, vj ∈ Vj and v` ∈ V` .
ak (Tk,j Rjm vj , Tk,` R`m v` ) ≤
C |`−j|
δ
|||vj |||1−α,j |||v` |||1−α,`
mα
where C is a positive constant, 0 < δ < 1 and α ∈ ( 12 , 1] is the
index of elliptic regularity, provided the number of smoothing
steps m is sufficiently large.
A Nonconforming Estimate
k
||| Idk−1 − Pkk−1 Ik−1
v|||1−α,k−1 ≤ Chαk |||v|||1,k−1
∀ v ∈ Vk−1
(This will allow us to handle (Idk − I`k Pk` )2 6= (Idk − I`k Pk` ).)
Additive Theory
Two-Level Estimates
(0 < θ < 1)
k
2
|||Ik−1
v|||21,k ≤ (1 + θ2 )|||v|||21,k−1 + Cθ−2 h2α
k |||v|||1+α,k−1
∀ v ∈ Vk−1
k
2
|||Ik−1
v|||21−α,k ≤ (1 + θ2 )|||v|||21−α,k−1 + Cθ−2 h2α
k |||v|||1,k−1
∀ v ∈ Vk−1
2
|||Pkk−1 v|||21−α,k−1 ≤ (1 + θ2 )|||v|||21−α,k + Cθ−2 h2α
k |||v|||1,k
∀ v ∈ Vk
Important aspect: the constant C is independent of k and θ.
Additive Theory
Two-Level Estimates
(0 < θ < 1)
k
2
|||Ik−1
v|||21,k ≤ (1 + θ2 )|||v|||21,k−1 + Cθ−2 h2α
k |||v|||1+α,k−1
∀ v ∈ Vk−1
k
2
|||Ik−1
v|||21−α,k ≤ (1 + θ2 )|||v|||21−α,k−1 + Cθ−2 h2α
k |||v|||1,k−1
∀ v ∈ Vk−1
2
|||Pkk−1 v|||21−α,k−1 ≤ (1 + θ2 )|||v|||21−α,k + Cθ−2 h2α
k |||v|||1,k
∀ v ∈ Vk
Important aspect: the constant C is independent of k and θ.
θ is a parameter that calibrates the meaning of high/low
frequency.
The freedom to choose different θ on different levels allows
us to build multi-level estimates from these two-level estimates.
Additive Theory
Theorem There exists a positive constant C independent of k
and m, such that
C
|Ek v|H 1 (Ω) ≤ α |v|H 1 (Ω)
∀ v ∈ Vk
m
provided that the number of smoothing steps m is larger than
a number m∗ which is independent of k. In particular the V cycle algorithm is a contraction with contraction number uniformly bounded away from 1 if m is sufficiently large.
Additive Theory
Theorem There exists a positive constant C independent of k
and m, such that
C
|Ek v|H 1 (Ω) ≤ α |v|H 1 (Ω)
∀ v ∈ Vk
m
provided that the number of smoothing steps m is larger than
a number m∗ which is independent of k. In particular the V cycle algorithm is a contraction with contraction number uniformly bounded away from 1 if m is sufficiently large.
This result holds for both conforming and nonconforming finite
element methods.
Additive Theory
Theorem There exists a positive constant C independent of k
and m, such that
C
|Ek v|H 1 (Ω) ≤ α |v|H 1 (Ω)
∀ v ∈ Vk
m
provided that the number of smoothing steps m is larger than
a number m∗ which is independent of k. In particular the V cycle algorithm is a contraction with contraction number uniformly bounded away from 1 if m is sufficiently large.
This result holds for both conforming and nonconforming finite
element methods.
In the conforming case we can combine this with the result from
the multiplicative theory to show that
C
|v| 1
∀ v ∈ Vk
|Ek v|H 1 (Ω) ≤
C + mα H (Ω)
Additive Theory
The additive theory has been successfully applied to discontinuous Galerkin methods for second order problems and C 0
interior penalty methods for fourth order problems.
Additive Theory
The additive theory has been successfully applied to discontinuous Galerkin methods for second order problems and C 0
interior penalty methods for fourth order problems.
References
• B., Convergence of the multigrid V-cycle algorithm for sec-
ond order boundary value problems without full elliptic regularity
Math. Comp., 2002
• B., Convergence of nonconforming V -cycle and F -cycle
multigrid algorithms for second order elliptic boundary
value problems
Math. Comp., 2004
Additive Theory
• B. and Zhao, Convergence of multigrid algorithms for inte-
rior penalty methods
Appl. Numer. Anal. Comput. Math., 2005
• B. and Sung, Multigrid algorithms for C 0 interior penalty
methods
SIAM J. Numer. Anal., 2006
• B., Cui, Gudi and Sung, Multigrid algorithms for symmetric
discontinuous Galerkin methods on graded meshes
Numer. Math., 2011
Numerical Results (Clamped Plates)
k=0
Boundary conditions
k=1
k=2
u = ∂u/∂n = 0
σ=5
Vk (⊂ H01 (Ω)) = Q2 rectangular finite element space
nk = dim(Vk ) = (2k+1 − 1)2
κ(Ak ) ≈ h−4
k
Numerical Results (Clamped Plates)
k=0
n0
n1
n2
n3
n4
n5
n6
n7
=1
=9
= 49
= 225
= 961
= 3969
= 16129
= 65025
k=1
k=2
κ0
κ1
κ2
κ3
κ4
κ5
κ6
κ7
= 1.0 × 100
= 4.0 × 101
= 1.6 × 103
= 2.9 × 104
= 4.8 × 105
= 7.8 × 106
= 1.3 × 108
= 2.0 × 109
Contraction Numbers in the Energy Norm
m
4
5
6
7
8
1
0.08
0.04
0.02
0.011
0.006
2
0.27
0.22
0.18
0.15
0.13
0.11
0.09
3
0.43
0.32
0.29
0.26
0.23
0.21
0.19
4
0.56
0.35
0.34
0.31
0.28
0.25
0.23
5
0.64
0.42
0.37
0.34
0.31
0.29
0.27
6
0.70
0.43
0.39
0.35
0.33
0.30
0.27
7
0.75
0.44
0.39
0.36
0.34
0.31
0.29
k
9
10
0.0032 0.0017
V -Cycle Algorithm With the Nonstandard Preconditioner
Contraction Numbers in the Energy Norm
m
k
1
2
3
4
5
6
7
8
9
10
1
0.53 0.28 0.15 0.08 0.04 0.02 0.01
0.006
0.003
0.002
2
0.72 0.49 0.24 0.27 0.22 0.18 0.15
0.13
0.11
0.09
3
0.71 0.51 0.40 0.34 0.30 0.26 0.24
0.22
0.19
0.17
4
0.80 0.51 0.41 0.37 0.34 0.31 0.28
0.26
0.24
0.22
5
0.76 0.53 0.42 0.38 0.34 0.31 0.29
0.26
0.24
0.23
6
0.82 0.53 0.42 0.38 0.34 0.32 0.29
0.26
0.25
0.22
7
0.83 0.53 0.42 0.38 0.34 0.32 0.29
0.27
0.25
0.23
W -Cycle Algorithm With the Nonstandard Preconditioner
Contraction Numbers in the Energy Norm
m
k
2
3
4
5
6
7
8
9
10
1
0.28 0.15 0.08 0.04 0.02 0.01
0.0060
0.0032
0.0017
2
0.50 0.35 0.27 0.22 0.18 0.15
0.13
0.11
0.09
3
0.52 0.40 0.34 0.30 0.27 0.24
0.22
0.19
0.18
4
0.53 0.42 0.37 0.34 0.31 0.28
0.26
0.24
0.22
5
0.53 0.43 0.37 0.34 0.31 0.29
0.27
0.25
0.23
6
0.53 0.44 0.38 0.34 0.32 0.29
0.27
0.25
0.23
7
0.54 0.46 0.38 0.35 0.32 0.29
0.27
0.25
0.23
F -Cycle Algorithm With the Nonstandard Preconditioner
Contraction Numbers in the Energy Norm
m
75
76
77
78
79
80
81
82
83
1
0.06
0.06
0.06
0.06
0.06
0.06
0.05
0.05
0.05
2
0.47
0.47
0.46
0.46
0.46
0.46
0.46
0.45
0.45
3
0.64
0.47
0.42
0.64
0.42
0.40
0.41
0.36
0.63
4
0.60
0.60
0.58
0.57
0.54
0.52
0.50
0.51
0.49
5
0.71
0.69
0.66
0.64
0.63
0.61
0.57
0.56
0.52
6
0.76
0.74
0.72
0.70
0.68
0.65
0.62
0.60
0.56
7
0.80
0.78
0.76
0.73
0.71
0.68
0.65
0.61
0.56
k
V -Cycle Algorithm With the Standard Preconditioner
Other Multigrid Algorithms
Variable V -Cycle
This is the V -cycle algorithm where the number of smoothing
steps can vary from level to level.
Variable V -Cycle
This is the V -cycle algorithm where the number of smoothing
steps can vary from level to level.
Suppose we want to solve the finite element equation on level k.
Then mj , the number of smoothing steps for level j, is chosen
according to the rule
β1 mj ≤ mj−1 ≤ β2 mj
for 0 ≤ j ≤ k, where 1 < β1 ≤ β2 .
Variable V -Cycle
This is the V -cycle algorithm where the number of smoothing
steps can vary from level to level.
Suppose we want to solve the finite element equation on level k.
Then mj , the number of smoothing steps for level j, is chosen
according to the rule
β1 mj ≤ mj−1 ≤ β2 mj
for 0 ≤ j ≤ k, where 1 < β1 ≤ β2 .
The variable V -cycle algorithm is mostly used as an optimal
preconditioner, and its analysis is based on the same ingredients that appear in the analysis of the W -cycle algorithm.
Full Multigrid
Full Multigrid
The k th level multigrid algorithm solves the equation
Ak z = ψ
with an (arbitrary) initial guess z0 .
Full Multigrid
The k th level multigrid algorithm solves the equation
Ak z = ψ
with an (arbitrary) initial guess z0 .
When we are solving finite element equations, the finite element
solutions on different levels are related, because they are all
approximations of the solution of the continuous problem.
Full Multigrid
The k th level multigrid algorithm solves the equation
Ak z = ψ
with an (arbitrary) initial guess z0 .
When we are solving finite element equations, the finite element
solutions on different levels are related, because they are all
approximations of the solution of the continuous problem.
Therefore the initial guess for the k th level multigrid algorithm
should come from the solution on the (k − 1)st level.
Full Multigrid
Finite Element Equation
Ak uk = φk
Z
f v dx
hφk , vi =
∀ v ∈ Vk
Ω
Full Multigrid Algorithm
For k = 0, û0 = A−1
0 φ0
For k ≥ 1,
k
ûk−1
uk0 = Ik−1
uk` = M G(k, φk , uk`−1 , m)
ûk =
ukr
for
1≤`≤r
Full Multigrid
Suppose the multigrid algorithm is uniformly convergent. For a
sufficiently large r, the full multigrid algorithm, which is a nested
iteration of the k th level multigrid algorithms, will produce an
approximate solution of the continuous problem that is accurate
to the same order as the exact solution of the finite element
equation. Moreover the computational cost of the full multigrid
algorithm remains proportional to the number of unknowns.