0
C Interior Penalty Methods
Isoparametric Version
Current Trends in Computational Methods for PDEs
CIMPA-NPDE-NBHM Research School
Bangalore, July 2013
Tuesday, July 16, 2013
Outline
�
Bending of Kirchhoff Plates
�
Numerical Difficulties
�
Isoparametric C0 Interior Penalty Methods
�
Convergence Analysis
�
Numerical Results
�
Concluding Remarks
Tuesday, July 16, 2013
Bending of Kirchhoff Plates
Ω (⊂ R2 ) = smooth configuration domain of the middle surface
of a thin plate
∂Ω is the union of disjoint closed sets where clamped, simply
supported or free boundary conditions are posed.
(|ΓS | + |ΓC | > 0)
ΓF
PSfrag replacements
ΓC
ΓS
u = vertical displacement of the middle surface
Tuesday, July 16, 2013
Bending of Kirchhoff Plates
Weak Formulation
Find u ∈ V such that
�
a(u, v) =
f v dx
Ω
a(w, v) =
�
Ω
�
�
(∆w)(∆v) − (1 − τ )[w, v] dx
[w, v] = wx1 x1 vx2 x2 + wx2 x2 vx1 x1 − 2wx1 x2 vx1 x2
τ = Poisson ratio
(0 < τ < 1/2)
f = load density/flexural rigidity
(f ∈ L2 (Ω))
V = {v ∈ H 2 (Ω) : v = 0 on ΓC ∪ ΓS and ∂v/∂n = 0 on ΓC }.
Tuesday, July 16, 2013
Bending of Kirchhoff Plates
a(w, v) =
�
Ω
Bounded
�
�
(∆w)(∆v) − (1 − τ )[w, v] dx
|a(w, v)| ≤ C|w|H 2 (Ω) |v|H 2 (Ω)
∀ v, w ∈ H 2 (Ω)
Coercive
a(v, v) =
�
Ω
� �
� 2
��
2
2
2
τ ∆v) + (1 − τ ) vx1 x1 + vx2 x2 + 2vx1 x2 dx
≥ (1 − τ )|v|2H 2 (Ω)
≥ c�v�2H 2 (Ω)
∀v ∈ V
The plate bending problem is well-posed.
Tuesday, July 16, 2013
Bending of Kirchhoff Plates
Integration by Parts
�
{(∆w)(∆v) − (1 − τ )[w, v]} dx
D
�
� �
�
= (∆2 w)v dx −
M∂D (w)(∇v · n) + N∂D (w)v ds
D
w ∈ H 4 (D),
∂D
v ∈ H 2 (D)
M∂D (w) = −∆w + (1 − τ )(D2 w)t · t
�
�
�
d� 2
(D w)t · n
N∂D (w) = ∇(∆w) · n + (1 − τ )
ds
D2 w = Hessian matrix of the second order derivatives of w
Tuesday, July 16, 2013
Bending of Kirchhoff Plates
Integration by Parts
�
{(∆w)(∆v) − (1 − τ )[w, v]} dx
D
�
� �
�
= (∆2 w)v dx −
M∂D (w)(∇v · n) + N∂D (w)v ds
D
∂D
In particular
�
�
f v dx =
{(∆u)(∆v) − (1 − τ )[u, v]} dx
Ω
�Ω
� �
�
2
= (∆ u)v dx −
M∂D (u)(∇v · n) + N∂D (u)v ds
Ω
∂Ω
for all v ∈ H 2 (Ω) such that
v = 0 on ΓC ∪ ΓS and ∂v/∂n = 0 on ΓC
Tuesday, July 16, 2013
Bending of Kirchhoff Plates
Strong Form
Find u ∈ H 4 (Ω) such that
∆2 u = f
Tuesday, July 16, 2013
in Ω
u = ∂u/∂n = 0
on ΓC
u = M∂Ω (u) = 0
on ΓS
M∂Ω u = N∂Ω (u) = 0
on ΓF
Bending of Kirchhoff Plates
Strong Form
Find u ∈ H 4 (Ω) such that
∆2 u = f
Remark
in Ω
u = ∂u/∂n = 0
on ΓC
u = M∂Ω (u) = 0
on ΓS
M∂Ω u = N∂Ω (u) = 0
on ΓF
The bending moment
MC (w) = −∆w + (1 − τ )(D2 w)t · t
can be defined on any curve C intrinsically (i.e., independent of
the orientation of C).
Tuesday, July 16, 2013
Numerical Difficulties
Since plate bending problems are fourth order problems, the
standard finite element approach requires C1 finite element
methods, which are difficult to construct for non-polygonal domains.
Tuesday, July 16, 2013
Numerical Difficulties
A natural approach is to first approximate Ω by a polygonal domain Ω̃, and then solve the corresponding plate bending problem on Ω̃ by a C1 finite element method to obtain an approximate solution of the original problem.
Tuesday, July 16, 2013
Numerical Difficulties
A natural approach is to first approximate Ω by a polygonal domain Ω̃, and then solve the corresponding plate bending problem on Ω̃ by a C1 finite element method to obtain an approximate solution of the original problem.
This approach works for the clamped plate, but fails for the simply supported plate due to the plate paradox.
Tuesday, July 16, 2013
Numerical Difficulties
Plate Paradox
Let a circular disc Ω be approximated
by inscribed regular polygons Ωn with n
sides. The solutions of the problem of
the simply supported plate on the polygons do not converge to the solution
of the problem of the simply supported
plate on the disc as n ↑ ∞.
Tuesday, July 16, 2013
Numerical Difficulties
Plate Paradox
Let a circular disc Ω be approximated
by inscribed regular polygons Ωn with n
sides. The solutions of the problem of
the simply supported plate on the polygons do not converge to the solution
of the problem of the simply supported
plate on the disc as n ↑ ∞.
In fact, if un is the solution of the simply supported plate on the
regular n-gon, then un converges to the solution of the simply
supported plate on the disc with τ = 1.
Tuesday, July 16, 2013
Numerical Difficulties
Plate Paradox
Boundary Value Problem for un
∆ 2 un = f
un = 0
on ∂Ωn
M∂Ωn (un ) = 0
on ∂Ωn
M∂Ωn (un ) = −∆un + (1 − τ )(D2 un )t · t
Tuesday, July 16, 2013
in Ωn
Numerical Difficulties
Plate Paradox
Boundary Value Problem for un
∆ 2 un = f
in Ωn
un = 0
on ∂Ωn
M∂Ωn (un ) = 0
on ∂Ωn
M∂Ωn (un ) = −∆un + (1 − τ )(D2 un )t · t
However, on a straight boundary edge,
2u
d
n
2
=0
(D un )t · t =
2
ds
Therefore the second boundary condition becomes
∆un = 0
Tuesday, July 16, 2013
Numerical Difficulties
Plate Paradox
Boundary Value Problem for un
∆ 2 un = f
in Ωn
un = 0
on ∂Ωn
M∂Ωn (un ) = 0
on ∂Ωn
M∂Ωn (un ) = −∆un + (1 − τ )(D2 un )t · t
Hence the limit u of un satisfies
∆2 u = f
in Ω
u=0
on ∂Ω
∆u = 0
on ∂Ω
which is the boundary value problem for the simply supported
plate with (the nonphysical) Poisson ratio τ = 1.
Tuesday, July 16, 2013
Numerical Difficulties
References
• Babuška, The theory of small changes in the domain of
existence in the theory of partial differential equations and
its applications
(1963)
• Maz’ya and Nazarov, Paradoxes of limit passage in solu-
tions of boundary value problems involving the approximation of smooth domains by polygonal domains
(1986)
• Babuška and Pitkäranta, The plate paradox for hard and
soft simple support
Tuesday, July 16, 2013
(1990)
Numerical Difficulties
If we want to have higher order convergence, then we must
approximate Ω to higher order, which means that we need to
approximate Ω by curvilinear polygons. But it is very difficult to
construct C1 finite element methods on curvilinear polygons.
Tuesday, July 16, 2013
Numerical Difficulties
Summary
C1 finite element methods are automatically stable and consistent for fourth order problems. But they are not appropriate for
problems on curved domains. We need to develop alternative
stable and consistent finite element methods for fourth order
problems on curvilinear polygons.
Reference
• Scott, A survey of displacement methods for the plate
bending problem
Tuesday, July 16, 2013
(1977)
Numerical Difficulties
Summary
C1 finite element methods are automatically stable and consistent for fourth order problems. But they are not appropriate for
problems on curved domains. We need to develop alternative
stable and consistent finite element methods for fourth order
problems on curvilinear polygons.
Isoparametric finite element methods, which are originally designed for second order boundary value problems on curved
domains, can be applied to fourth order problems with optimal
orders of convergence, provided we combine them with the approach of interior penalty methods.
Tuesday, July 16, 2013
Numerical Difficulties
Summary
C1 finite element methods are automatically stable and consistent for fourth order problems. But they are not appropriate for
problems on curved domains. We need to develop alternative
stable and consistent finite element methods for fourth order
problems on curvilinear polygons.
Isoparametric finite element methods, which are originally designed for second order boundary value problems on curved
domains, can be applied to fourth order problems with optimal
orders of convergence, provided we combine them with the approach of interior penalty methods.
Isoparametric mixed finite element methods for clamped plates
were investigated by Bhattacharyya and Nataraj (2002).
Tuesday, July 16, 2013
0
Isoparametric C Interior Penalty Methods
Ωh = curvilinear polygonal approximation of Ω
Th = isoparametric triangulation of Ωh
Each triangle in Th has at most one curved edge, and only those
that have more than one vertex on ∂Ω have a curved edge.
∂Ω
Each triangle in Th is the image of the reference triangle (with vertices (0,
0), (1, 0)
PSfrag replacements
and (0, 1)) under a polynomial
map of degree ≤ k.
Tuesday, July 16, 2013
0
Isoparametric C Interior Penalty Methods
Ωh = curvilinear polygonal approximation of Ω
Th = isoparametric triangulation of Ωh
Each triangle in Th has at most one curved edge, and only those
that have more than one vertex on ∂Ω have a curved edge.
Vh is the curvilinear Pk Lagrange finite element space
whose members vanish on
ΓS,h ∪ ΓC,h .
PSfrag replacements
All the nodes associated with
Vh belong to Ω̄ and the nodes
on ∂Ωh also belong to ∂Ω.
Tuesday, July 16, 2013
∂Ω
0
Isoparametric C Interior Penalty Methods
References for Isoparametric Finite Elements
• Ciarlet and Raviart, Interpolation theory over curved ele-
ments, with applications to finite element methods
(1972)
• Lenoir, Optimal isoparametric finite elements and error es-
timates for domains involving curved boundaries
(1986)
• Bernardi, Optimal finite-element interpolation on curved
domains
(1989)
Tuesday, July 16, 2013
0
Isoparametric C Interior Penalty Methods
Notation
�
�
�
�
�
�
�
�
�
�
�
Tuesday, July 16, 2013
Eh is the set of the edges of the triangles in Th .
EhI is the subset of Eh consisting of edges interior to Ωh .
EhC is the subset of Eh consisting of edges on ΓC,h .
EhS is the subset of Eh consisting of edges on ΓS,h .
EhF is the subset of Eh consisting of edges on ΓF,h .
EhB = EhC ∪ EhS ∪ EhF
|e| is the length of the edge e.
hT = diameter of T and h = maxT∈Th hT .
σ ≥ 1 is a penalty parameter.
{{M(v)}} is the average of the bending moment of v across
an edge.
[ ∂v/∂n]] is the jump of the normal derivative of v across
an edge.
0
Isoparametric C Interior Penalty Methods
Let e ∈ Eh be
� shared by the triangles
T± , v± = v�T± , ne be the unit normal
of e pointing from T− to T+ , and te be
a unit tangent along the edge e.
T+
I
te
T_
ne
e
The average {{M(v)}} and jump [ ∂v/∂n]] on the edge e are
defined by
� 2
�
�
1�
{{M(v)}} =
− ∆(v− + v+ ) + (1 − τ ) D (v− + v+ ) te · te
2
[ ∂v/∂n]] = (∇v+ − ∇v− ) · ne
Tuesday, July 16, 2013
0
Isoparametric C Interior Penalty Methods
Let e ∈ EhB , ne be the outward pointing
unit normal, and te be a unit tangent.
te
e
ne
The average {{M(v)}} and jump [ ∂v/∂n]] on the edge e are
defined by
{{M(v)}} = M(v) = −∆v + (1 − τ )(D2 v)te · te
[ ∂v/∂n]] = −∇v · ne
Tuesday, July 16, 2013
0
Isoparametric C Interior Penalty Methods
Discrete Problem
Find uh ∈ Vh such that
�
ah (uh , v) =
fh v dx
∀ v ∈ Vh
Ωh
where fh is the interpolant of f and
�
�� �
ah (w, v) =
(∆w)(∆v) − (1 − τ )[w, v] dx
T
T∈Th
−
�
�
�
e∈EhI ∪EhC
+
e
{{M(w)}}[[∂v/∂n]] + {{M(v)}}[[∂w/∂n]] ds
�
e∈EhI ∪EhC
Tuesday, July 16, 2013
�
σ
|e|
�
e
[ ∂w/∂n]][ ∂v/∂n]] ds,
Convergence Analysis
We assume f ∈ Wpk−1 (Ω), where p > 2 if k = 2 and p = 2 when
k ≥ 3.
(k ≥ 2 is the degree of the polynomials in the finite element
space.)
This implies in particular that fh is well-defined.
Then u ∈ Wpk+3 (Ω) by elliptic regularity and we can extend u
to R2 (still denoted by u) so that
�u�Wpk+3 (R2 ) ≤ CΩ �u�Wpk+3 (Ω)
f = ∆2 u is defined on R2 and belongs to Wpk−1 (R2 ).
Tuesday, July 16, 2013
Convergence Analysis
Energy Norm (mesh-dependent)
�
��
2
2
2
�v�h =
τ �∆v�L2 (T) + (1 − τ )|v|H 2 (T)
T∈Th
+
�
e∈EhI ∪EhC
Tuesday, July 16, 2013
�
|e|
�{{M(v)}}�2L2 (e) +
σ
I
e∈Eh ∪EhC
σ
�[[∂v/∂n]]�2L2 (e) .
|e|
Convergence Analysis
Energy Norm (mesh-dependent)
�
��
2
2
2
�v�h =
τ �∆v�L2 (T) + (1 − τ )|v|H 2 (T)
T∈Th
+
�
e∈EhI ∪EhC
�
|e|
�{{M(v)}}�2L2 (e) +
σ
I
e∈Eh ∪EhC
σ
�[[∂v/∂n]]�2L2 (e) .
|e|
Bounded
ah (w, v) ≤ 2�w�h �v�h
H 3 (Ω
Tuesday, July 16, 2013
∀ v, w ∈ H 3 (Ωh , Th )
�
3 (T)
�
,
T
)
=
{v
∈
L
(Ω
)
:
v
=
v
∈
H
T
h h
2
h
T
∀ T ∈ Th }
Convergence Analysis
Energy Norm (mesh-dependent)
�
��
2
2
2
�v�h =
τ �∆v�L2 (T) + (1 − τ )|v|H 2 (T)
T∈Th
+
�
e∈EhI ∪EhC
�
|e|
�{{M(v)}}�2L2 (e) +
σ
I
e∈Eh ∪EhC
σ
�[[∂v/∂n]]�2L2 (e) .
|e|
Bounded
ah (w, v) ≤ 2�w�h �v�h
Coercive
1
ah (v, v) ≥ �v�2h
2
(for σ sufficiently large)
Tuesday, July 16, 2013
∀ v, w ∈ H 3 (Ωh , Th )
∀ v ∈ Vh
Convergence Analysis
Abstract Error Estimate
For any v ∈ Vh ,
�u − uh �h ≤ �u − v�h + �v − uh �h
ah (v − uh , w)
≤ �u − v�h + C sup
�w�h
w∈Vh
ah (v − u, w) + ah (u − uh , w)
= �u − v�h + C sup
�w�h
w∈Vh
�
ah (u − uh , w) �
≤ C �u − v�h + sup
�w�h
w∈Vh
Tuesday, July 16, 2013
Convergence Analysis
Abstract Error Estimate
For any v ∈ Vh ,
�u − uh �h ≤ �u − v�h + �v − uh �h
ah (v − uh , w)
≤ �u − v�h + C sup
�w�h
w∈Vh
ah (v − u, w) + ah (u − uh , w)
= �u − v�h + C sup
�w�h
w∈Vh
�
ah (u − uh , w) �
≤ C �u − v�h + sup
�w�h
w∈Vh
�
ah (u − uh , w) �
�u − uh �h ≤ C inf �u − v�h + sup
v∈Vh
�w�h
w∈Vh
Tuesday, July 16, 2013
Convergence Analysis
Abstract Error Estimate
For any v ∈ Vh ,
�u − uh �h ≤ �u − v�h + �v − uh �h
ah (v − uh , w)
≤ �u − v�h + C sup
�w�h
w∈Vh
ah (v − u, w) + ah (u − uh , w)
= �u − v�h + C sup
�w�h
w∈Vh
�
ah (u − uh , w) �
≤ C �u − v�h + sup
�w�h
w∈Vh
�
ah (u − uh , w) �
�u − uh �h ≤ C inf �u − v�h + sup
v∈Vh
�w�h
w∈Vh
Tuesday, July 16, 2013
Convergence Analysis
Abstract Error Estimate
For any v ∈ Vh ,
�u − uh �h ≤ �u − v�h + �v − uh �h
ah (v − uh , w)
≤ �u − v�h + C sup
�w�h
w∈Vh
ah (v − u, w) + ah (u − uh , w)
= �u − v�h + C sup
�w�h
w∈Vh
�
ah (u − uh , w) �
≤ C �u − v�h + sup
�w�h
w∈Vh
�
ah (u − uh , w) �
�u − uh �h ≤ C inf �u − v�h + sup
v∈Vh
�w�h
w∈Vh
Tuesday, July 16, 2013
Convergence Analysis
Approximation Property
inf �u − v�h ≤ �u − Πh u�h ≤ Chk−1 �u�Wpk+1 (R2 )
v∈Vh
Πh u ∈ Vh equals u at all the nodes of the Pk finite element space.
�v�2h =
��
T∈Th
+
τ �∆v�2L2 (T) + (1 − τ )|v|2H 2 (T)
�
e∈EhI ∪EhC
Tuesday, July 16, 2013
�
�
|e|
�{{M(v)}}�2L2 (e) +
σ
I
e∈Eh ∪EhC
σ
�[[∂v/∂n]]�2L2 (e) .
|e|
Convergence Analysis
Approximation Property
inf �u − v�h ≤ �u − Πh u�h ≤ Chk−1 �u�Wpk+1 (R2 )
v∈Vh
Πh u ∈ Vh equals u at all the nodes of the Pk finite element space.
It remains only to bound the consistency error
ah (u − uh , w)
sup
�w�h
w∈Vh
Tuesday, July 16, 2013
Convergence Analysis
ah (u, w) =
��
T
T∈Th
−
� � �
e∈EhI ∪EhC
+
{(∆u)(∆w) − (1 − τ )[u, w]} dx
e
�
{{M(u)}}[[∂w/∂n]] + {{M(w)}}[[∂u/∂n]] ds
�
e∈EhI ∪EhC
σ
|e|
�
e
[ ∂u/∂n]][ ∂w/∂n]] ds
�
{(∆u)(∆w) − (1 − τ )[u, w]} dx
D
�
� �
�
2
= (∆ u)w dx −
M∂D (u)(∇w · n) + N∂D (u)w ds
D
Tuesday, July 16, 2013
∂D
Convergence Analysis
ah (u, w) =
��
T
T∈Th
−
� � �
e∈EhI ∪EhC
+
{(∆u)(∆w) − (1 − τ )[u, w]} dx
e
{{M(u)}}[[∂w/∂n]] + {{M(w)}}[[∂u/∂n]] ds
�
e∈EhI ∪EhC
�
�
�
σ
|e|
�
e
[ ∂u/∂n]][ ∂w/∂n]] ds
�
�
∂w
=
(∆ u)w dx −
M(u) ds
∂n
T
e
S
F
T∈Th
e∈Eh ∪Eh
�
� ��
∂u
σ
∂u ∂w �
+
M(w) ds +
ds
∂n
|e| e ∂n ∂n
e
C
e∈Eh
�
�
−
N∂Ωh (u)w ds
2
e∈EhF
Tuesday, July 16, 2013
e
Convergence Analysis
�
� �
∂w
ah (u, w) =
fw dx −
M(u) ds
∂n
Ωh
e
S
F
e∈Eh ∪Eh
�
� ��
∂u
σ
∂u ∂w �
+
M(w) ds +
ds
∂n
|e| e ∂n ∂n
e
C
e∈Eh
��
−
N∂Ωh (u)w ds
e∈EhF
Tuesday, July 16, 2013
e
Convergence Analysis
�
� �
∂w
ah (u, w) =
fw dx −
M(u) ds
∂n
Ωh
e
S
F
e∈Eh ∪Eh
�
� ��
∂u
σ
∂u ∂w �
+
M(w) ds +
ds
∂n
|e| e ∂n ∂n
e
C
e∈Eh
��
−
N∂Ωh (u)w ds
e∈EhF
ah (uh , w) =
�
Ωh
Tuesday, July 16, 2013
fh w dx
e
Convergence Analysis
�
� �
∂w
ah (u − uh , w) =
(f − fh )w dx −
M(u) ds
∂n
e
Ωh
S
F
e∈Eh ∪Eh
�
� ��
∂u
σ
∂u ∂w �
+
M(w) ds +
ds
∂n
|e| e ∂n ∂n
e
C
e∈Eh
��
−
N∂Ωh (u)w ds
e∈EhF
= I + II + III + IV
Tuesday, July 16, 2013
e
Convergence Analysis
�
� �
∂w
ah (u − uh , w) =
(f − fh )w dx −
M(u) ds
∂n
e
Ωh
S
F
e∈Eh ∪Eh
�
� ��
∂u
σ
∂u ∂w �
+
M(w) ds +
ds
∂n
|e| e ∂n ∂n
e
C
e∈Eh
��
−
N∂Ωh (u)w ds
e∈EhF
e
= I + II + III + IV
Goal
so that
Tuesday, July 16, 2013
|I| + |II| + |III| + |IV| ≤ Chk−1 �w�h
ah (u − uh , w)
sup
≤ Chk−1
�w�h
w∈Vh
Convergence Analysis
��
�
�
�
|I| = �
(f − fh )w dx� ≤ �f − fh �L2 (Ωh ) �w�L2 (Ωh ) ≤ Chk−1 �w�h
Ωh
�
f ∈ Wpk−1 (Ωh )
�
Poincaré-Friedrichs inequality
(p ≥ 2)
�w�L2 (Ωh ) ≤ C�w�h
Tuesday, July 16, 2013
∀ w ∈ Vh
Convergence Analysis
�
�
�
� �
�
∂w �
�
k+ 12
|II| = �
M(u) ds� ≤ Ch �w�h
� S F e
∂n �
e∈Eh ∪Eh
�
The arc on ∂Ω between the two endpoints of a boundary
edge is approximated by the edge to the order of hk+1 in
the C0 norm and to the order of hk in the C1 seminorm.
�
3 (R2 )
M∂Ω (u) = 0 on ΓS ∪ ΓF and u ∈ W∞
�
�
Tuesday, July 16, 2013
�
�
|M(u)| = � − ∆u + (1 − τ )(∇2 u)te · te � ≤ Chk
��
−1 �∂w/∂n�2
|e|
S
F
e∈Eh ∪Eh
L2 (e)
�1
2
≤ C�w�h
Convergence Analysis
�
�
�
�
���
��
σ
∂u
∂w
∂u
�
�
k+ 12
|III| = �
M(w) ds +
ds � ≤ Ch �w�h
� C e
�
∂n
|e| e ∂n ∂n
e∈Eh
�
The arc on ∂Ω between the two endpoints of a boundary
edge is approximated by the edge to the order of hk+1 in
the C0 norm and to the order of hk in the C1 seminorm.
�
2 (R2 )
∇u = 0 on ΓC and u ∈ W∞
�
|∂u/∂n| ≤ |∇u| ≤ Chk+1
�
Tuesday, July 16, 2013
��
e∈EhC
�
|e|�M(w)�2L2 (e) + |e|−1 �∂w/∂n�2L2 (e)
�� 21
≤ C�w�h
Convergence Analysis
�
�
�
�
��
�
�
|IV| = �
N∂Ωh (u)w ds� ≤ Chk−1 �w�h
� F e
�
e∈Eh
�
The arc on ∂Ω between the two endpoints of a boundary
edge is approximated by the edge to the order of hk+1 in
the C0 norm and to the order of hk in the C1 seminorm.
�
4 (R2 )
N∂Ω u = 0 on ΓF and u ∈ W∞
�
�
Tuesday, July 16, 2013
�
��
�
�
�
�
�
d
2
|N∂Ωh (u)| = � ∇(∆u) · ne + (1 − τ ) ds (D u)te · ne � ≤ Chk−1
�w�L∞ (Ωh ) ≤ �w�h
Convergence Analysis
Theorem
�
�
ah (u − uh , w)
�u − uh �h ≤ C inf �u − v�h + sup
v∈Vh
�w�h
w∈Vh
≤ Chk−1
This result also holds for nonhomogeneous boundary conditions.
Tuesday, July 16, 2013
Convergence Analysis
Theorem
�
�
ah (u − uh , w)
�u − uh �h ≤ C inf �u − v�h + sup
v∈Vh
�w�h
w∈Vh
≤ Chk−1
By a duality argument, we have

Ch2
�u − uh �L2 (Ωh ) ≤
Chk+1
provided ΓF = ∅
Tuesday, July 16, 2013
k=2
and
f ∈ H 2 (Ω)
k≥3
and
f ∈ H k+1 (Ω)
Numerical Results
Example 1
A general plate with τ = 0.3
ΓF : x14 + x24 = 1
1 2
1
2
ΓC : (x1 + ) + x2 =
2
16
1 2
1
2
ΓS : (x1 − ) + x2 =
2
16
Exact solution
�2 � 1
� 2
�1
1 2
1
2
2
2 x1 +x2
− (x1 + ) − x2
− (x1 − ) − x2 e
u=
16
2
16
2
(solved by an isoparametric quadratic C0 interior penalty
method)
Tuesday, July 16, 2013
Numerical Results
h
0.283
0.202
0.141
0.094
0.064
0.043
0.029
0.019
�u − uh �L2
1.25E+00
7.15E-01
4.65E-01
2.00E-01
1.27E-01
7.33E-02
4.03E-02
2.40E-02
rate
1.66
1.21
2.08
1.18
1.36
1.51
1.31
k=2
�D2h (u − uh )�L2
1.27E+02
9.42E+01
7.38E+01
4.94E+01
3.26E+01
2.44E+01
1.64E+01
1.12E+01
σ = 50
�u − uh �h ≤ Chk−1
Tuesday, July 16, 2013
rate
0.89
0.68
0.99
1.09
0.71
1.01
0.95
Numerical Results
Example 2
A clamped plate with τ = 0.3
ΓC : x14 + x24 = 1
Exact solution
u = (1 − x14 − x24 )2
(solved by isoparametric C0 interior penalty methods)
Tuesday, July 16, 2013
Numerical Results
h
0.141
0.094
0.064
0.043
0.029
0.019
�u − uh �L2
2.80E-01
1.43E-01
7.08E-02
3.35E-02
1.65E-02
7.24E-03
rate
1.65
1.84
1.88
1.79
1.96
k=2
�u − uh �h ≤ Chk−1
Tuesday, July 16, 2013
�D2h (u − uh )�L2
1.08E+01
6.89E+00
5.58E+00
3.68E+00
2.57E+00
1.64E+00
rate
1.10
0.55
1.05
0.91
1.06
σ = 50
�u − uh �L2 (Ωh ) ≤

Ch2
Chk+1
k=2
k≥3
Numerical Results
h
0.141
0.094
0.064
0.043
0.029
0.019
�u − uh �L2
4.10E-03
1.10E-03
2.81E-04
5.38E-05
4.16E-05
5.70E-06
rate
3.24
3.56
4.15
0.65
4.70
k=3
�u − uh �h ≤ Chk−1
Tuesday, July 16, 2013
�D2h (u − uh )�L2
1.80E+00
7.03E-01
3.80E-01
1.58E-01
7.52E-02
3.23E-02
rate
2.32
1.60
2.22
1.88
2.00
σ = 50
�u − uh �L2 (Ωh ) ≤

Ch2
Chk+1
k=2
k≥3
Concluding Remarks
�
Tuesday, July 16, 2013
C0 interior penalty methods are simpler than C1 finite element methods and they can be naturally implemented on
isoparametric meshes to solve plate bending problems on
smooth domains with optimal order of convergence.
Concluding Remarks
�
C0 interior penalty methods are simpler than C1 finite element methods and they can be naturally implemented on
isoparametric meshes to solve plate bending problems on
smooth domains with optimal order of convergence.
�
This is due to the fact that isoparametric finite element
methods are designed for C0 Lagrange finite element
spaces, which are exactly the spaces for C0 interior penalty
methods.
Tuesday, July 16, 2013
Concluding Remarks
�
C0 interior penalty methods are simpler than C1 finite element methods and they can be naturally implemented on
isoparametric meshes to solve plate bending problems on
smooth domains with optimal order of convergence.
�
This is due to the fact that isoparametric finite element
methods are designed for C0 Lagrange finite element
spaces, which are exactly the spaces for C0 interior penalty
methods.
�
Unlike other ad hoc methods for plate bending problems,
isoparametric C0 interior penalty methods come in a natural hierarchy.
Tuesday, July 16, 2013
Reference
• Brenner, Neilan and S., Isoparametric C0 interior penalty
methods for plate bending problems on smooth domains
Calcolo, 2013
Tuesday, July 16, 2013