Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 863125, 19 pages
doi:10.1155/2012/863125
Research Article
Constrained C0 Finite Element Methods for
Biharmonic Problem
Rong An and Xuehai Huang
College of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China
Correspondence should be addressed to Xuehai Huang, xuehaihuang@wzu.edu.cn
Received 12 September 2012; Revised 29 November 2012; Accepted 29 November 2012
Academic Editor: Allan Peterson
Copyright q 2012 R. An and X. Huang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
This paper presents some constrained C0 finite element approximation methods for the biharmonic
problem, which include the C0 symmetric interior penalty method, the C0 nonsymmetric interior
penalty method, and the C0 nonsymmetric superpenalty method. In the finite element spaces, the
C1 continuity across the interelement boundaries is obtained weakly by the constrained condition.
For the C0 symmetric interior penalty method, the optimal error estimates in the broken H 2 norm
and in the L2 norm are derived. However, for the C0 nonsymmetric interior penalty method,
the error estimate in the broken H 2 norm is optimal and the error estimate in the L2 norm is
suboptimal because of the lack of adjoint consistency. To obtain the optimal L2 error estimate, the
C0 nonsymmetric superpenalty method is introduced and the optimal L2 error estimate is derived.
1. Introduction
The discontinuous Galerkin methods DGMs have become a popular method to deal with
the partial differential equations, especially for nonlinear hyperbolic problem, which exists
the discontinuous solution even when the data is well smooth, and the convection-dominated
diffusion problem, and the advection-diffusion problem. For the second-order elliptic
problem, according to the different numerical fluxes, there exist different discontinuous
Galerkin methods, such as the interior penalty method IP, the nonsymmetric interior
penalty method NIPG, and local discontinuous Galerkin method LDG. A unified analysis
of discontinuous Galerkin methods for the second-order elliptic problem is studied by Arnold
et al. in 1.
The DGM for the fourth-order elliptic problem can be traced back to 1970s. Baker in
2 used the IP method to study the biharmonic problem and obtained the optimal error
estimates. Moreover, for IP method, the C0 and C1 continuity can be achieved weakly by
the interior penalty. Recently, using IP method and NIPG method, Süli and Mozolevski in
3–5 studied the hp-version DGM for the biharmonic problem, where the error estimates
2
Abstract and Applied Analysis
are optimal with respect to the mesh size h and are suboptimal with respect to the degree
of the piecewise polynomial approximation p. However, we observe that the bilinear forms
and the norms corresponding to the IP method in 3–5 are much complicated. A method to
simplify the bilinear forms and the norms is using C0 interior penalty method. C0 interior
penalty method for the biharmonic problem was introduced by Babuška and Zlámal in 6,
where they used the nonconforming element and considered the inconsistent formulation
and obtained the suboptimal error estimate. Motivated by the Engel and his collaborators’
work 7, Brenner and Sung in 8 studied the C0 interior penalty method for fourth-order
problem on polygonal domains. They used the C0 finite element solution to approximate C1
solution by a postprocessing procedure, and the C1 continuity can be achieved weakly by the
penalty on the jump of the normal derivatives on the interelement boundaries.
In this paper, thanks to Rivière et al.’s idea in 9, we will study some constrained
C0 finite element approximation methods for the biharmonic problem. The C1 continuity
can be weakly achieved by a constrained condition that integrating the jump of the normal
derivatives over the inter-element boundaries vanish. Under this constrained condition, we
discuss three C0 finite element methods which include the C0 symmetric interior penalty
method based on the symmetric bilinear form, the C0 nonsymmetric interior penalty method,
and C0 nonsymmetric superpenalty method based on the nonsymmetric bilinear forms. First,
we study the C0 symmetric interior penalty method and obtain the optimal error estimates
in the broken H 2 norm and in L2 norm. However, for the C0 nonsymmetric interior penalty
method, the L2 norm is suboptimal because of the lack of adjoint consistency. Finally, in order
to improve the order of the L2 error estimate, we give the C0 nonsymmetric superpenalty
method and show the optimal L2 error estimates.
2. C0 Finite Element Approximation
Let Ω ⊂ R2 be a bounded and convex domain with boundary ∂Ω. Consider the following
biharmonic problem:
Δ2 u f,
u
∂u
0,
∂n
in Ω,
2.1
on ∂Ω,
where n denotes the unit external normal vector to ∂Ω. We assume that f is sufficiently
smooth such that the problem 2.1 admits a unique solution u ∈ H 4 Ω ∩ H02 Ω.
Let Th be a family of nondegenerate triangular partition of Ω into triangles. The
corresponding ordered triangles are denoted by K1 , K2 , . . . , KN . Let hi diam Ki , i 1, . . . , N, and h max{h1 , h2 , . . . , hN }. The nondegenerate requirement is that there exists
ρ > 0 such that Ki contains a ball of radius ρhi in its interior. Conventionally, the boundary of
Ki is denoted by ∂Ki . We denote
eij ∂Ki ∩ ∂Kj ,
ei ∂Ki ∩ ∂Ω,
heij diam eij ,
hei diamei .
2.2
Assume that the partition Th is quasiuniform; that is, there exists a positive constant ν
such that
h νhi ,
i 1, . . . , N.
2.3
Abstract and Applied Analysis
3
Let EI and EB be the set of interior edges and boundary edges of Th , respectively. Let E EI ∪ EB . Denote by vi the restriction of v to Ki . Let e eij ∈ EI with i > j. Then we denote the
jump v and the average {v} of v on e by
v|e vi −vj ,
e
e
1 i v vj .
e
e
2
{v}|e 2.4
If e ei ∈ EB , we denote v and {v} of v on e by
v|e {v}|e vi .
2.5
e
Define V by
V v ∈ H01 Ω, v|K ∈ H s K, ∀K ∈ Th ,
s3
2.6
with broken H s norm
|v|s 1/2
v2s,K
2.7
∀v ∈ V,
,
K∈Th
where || · ||s,K is the standard Sobolev norm in H s K. Define the broken H 2 norm by
vh 1/2
|v|22,K
,
2.8
∀v ∈ V,
K∈Th
where | · |2,K is the seminorm in H 2 K.
For every K ∈ Th and any v ∈ V , we apply the integration by parts formula to obtain
Δ2 u vdx −
∇Δu · ∇vdx ∂Δu
vds
K
∂K ∂n
∂Δu
∂v
vds.
ΔuΔvdx −
Δu ds ∂n
K
∂K
∂K ∂n
K
2.9
Summing all K ∈ Th , we have
K∈Th
ΔuΔvdx −
K
e∈EI
∂Δu ∂v
∂v
Δu
ds {v}
{Δu}
∂n
∂n
∂n
e
e∈E e
∂Δu
∂v
Δu ds fvdx.
vds −
∂n
∂n
∂Ω
Ω
I
2.10
4
Abstract and Applied Analysis
Since u ∈ H 4 Ω ∩ H02 Ω and v ∈ V , then Δu ∂Δu/∂n v 0 on e ∈ EI . Thus, the
previous identity can be simplified as follows:
K∈Th
ΔuΔvdx −
K
∂v
∂v
Δu ds fvdx.
ds −
{Δu}
∂n
∂n
e
∂Ω
Ω
e∈EI
2.11
Now, we introduce the following two bilinear forms:
aS u, v K∈Th
ΔuΔvdx −
K
e∈E
{Δu}
e
∂v
∂u
ds −
ds
{Δv}
∂n
∂n
e∈E e
β
∂u ∂v
ds,
h
∂n ∂n
e∈E e e
∂v
∂u
aNS u, v ds ds
ΔuΔvdx −
{Δu}
{Δv}
∂n
∂n
K
K∈T
e∈E e
e∈E e
2.12
h
1 ∂u ∂v ds.
h
∂n ∂n
e∈E e e
It is clear that aS ·, · is a symmetric bilinear form and aNS ·, · is a nonsymmetric bilinear
form. In terms of 2.11 and
∂u
∂u
ds Δv ds 0,
{Δv}
∂n
∂n
e
∂Ω
e∈EI
2.13
the solution u to problem 2.1 satisfies the following variational problems:
aS u, v f, v , ∀v ∈ V,
aNS u, v f, v , ∀v ∈ V.
SP
NSP
Let Pr K denote the space of the polynomials on K of degree at most r. Define the
following constrained C0 finite element space:
Vh ∂vh
ds 0, ∀e ∈ E ,
vh ∈ H01 Ω, vh |K ∈ Pr K, r 4, ∀K ∈ Th ,
e ∂n
2.14
from which we note that the C1 continuity of vh ∈ Vh can be weakly achieved by the
constrained condition e ∂vh /∂nds 0 for all e ∈ E. Next, we define the degrees of freedom
for this finite element space. To this end, for any K ∈ Th , denote by pi i 1, 2, 3 the three
Abstract and Applied Analysis
5
vertices of K. Recall that the degrees of freedom of Lagrange element on K are vp, for all
p ∈ C with cf. 10
⎫
3
3
⎬
r
−
1
1
C : p ,1 , 1 j 3 .
λj pj ;
λj 1, λj ∈ 0, , . . . ,
⎭
⎩
r
r
j1
j1
⎧
⎨
2.15
Then we modify the degrees of freedom of Lagrange element to suit the constraint of normal
derivatives over the edges in Vh . Specifically speaking, the degrees of freedom of Vh are given
by
v p ,
∀p ∈ C with C : C \
e
∂v
ds
∂n
r−3
1
pi , i 1, 2, 3 ,
p1 p2 p3 r
r
2.16
for each edge e of K.
Based on the symmetric bilinear form aS ·, ·, the C0 symmetric interior penalty finite
element approximation of 2.1 is
find uSh ∈ Vh such that,
aS uSh , vh f, vh , ∀vh ∈ Vh .
2.17
Based on the nonsymmetric bilinear form aNS ·, ·, the C0 nonsymmetric interior penalty finite
element approximation of 2.1 is
find uNS
h ∈ Vh such that,
f, vh , ∀vh ∈ Vh .
aNS uNS
,
v
h
h
2.18
Moreover, the following orthogonal equations hold:
aS uSh − u, vh 0,
∀vh ∈ Vh ,
aNS uNS
h − u, vh 0,
2.19
∀vh ∈ Vh .
2.20
In order to introduce a global interpolation operator, we first define φhi ∈ Pr Ki for
φ ∈ H Ki and Ki ∈ Th according to the degrees of freedom of Vh by
s
φhi p φ p ,
∂φhi
e
∂n
ds e
∂φ
ds,
∂n
if p ∈ C \ ∂Ω,
φhi p 0,
∂φhi
if edge e ⊂ ∂Ki \ ∂Ω,
e
∂n
if p ∈ C ∩ ∂Ω,
ds 0,
if edge e ⊂ ∂Ki ∩ ∂Ω.
2.21
6
Abstract and Applied Analysis
Due to standard scaling argument and Sobolev embedding theorem cf. 10, we have that
for every Ki ∈ Th , e ∈ ∂Ki , φ ∈ H s Ki with s 2, i 1, . . . , N,
φ − φhi φ − φhi q,Ki
chμ−q φs,Ki ,
0 q μ,
φs,Ki ,
μ−1/2−m m,e
che
∂ φ − φi h
∂n
m 0, 1, 2,
2.22
φs,Ki ,
μ−3/2 che
0,e
where μ min{s, r 1} and c > 0 is independent of h. We also suppose that the following
inverse inequalities hold:
i
φh 0,e
i
ch−1/2
φh e
0,Ki
i
φh ,
1,e
i
ch−1/2
φh e
1,Ki
,
i
φh 1,Ki
ch−1 φhi 0,Ki
,
2.23
where c > 0 is independent of h. Then for every φ ∈ V , we define the global interpolation
operator Ph : V → Vh by Ph φ|Ki φhi . Moreover, from 2.22 there holds
φ − Ph φ chμ−2 φ ,
h
s
2.24
where c > 0 is independent of h.
The following lemma is useful to establish the existence and uniqueness of the finite
element approximation solution.
Lemma 2.1. There exists some constant κ0 > 0 independent of h such that
κ0 vh 2h
Δvh 20,K
K∈Th
1
∂vh 2
∂n ,
h
e
0,e
e∈E
∀vh ∈ Vh .
2.25
Proof. Introduce a H02 conforming finite element space Zh thanks to Guzmán and Neilan 11.
The advantage of Zh is that the degrees of freedom depend only on the values of functions
and their first derivatives. Denote by Lh the interpolation operator from Vh to Zh . Then there
holds
K∈Th
|vh −
Lh vh |22,K
1
∂vh 2
,
C1
h ∂n 0,e
e∈E e
∀vh ∈ Vh ,
2.26
Abstract and Applied Analysis
7
where C1 > 0 is independent of h. Thus, we have
vh 2h |vh − Lh vh |22,K K∈Th
|Lh vh |22,K C1
K∈Th
1
∂vh 2
C2
ΔLh vh 20,K
h
∂n
e
0,e
e∈E
K∈T
h
1
∂vh 2
C2
C1
ΔLh vh − vh 20,K C2
Δvh 20,K
h
∂n
e
0,e
e∈E
K∈T
K∈T
h
h
1
∂vh 2
C2
C1
|Lh vh − vh |22,K C2
Δvh 20,K
h
∂n
e
0,e
e∈E
K∈T
K∈T
h
h
1
∂vh 2
C2
C1 1 C2 Δvh 20,K ,
h
∂n
e
0,e
e∈E
K∈T
h
2.27
which completes the proof of 2.25 with κ0 1/ max{C1 1 C2 , C2 }.
3. C0 Symmetric Interior Penalty Method
In this section, we will show the optimal error estimates in the broken H 2 norm and in the L2
norm between the solution u to problem 2.1 and the solution uSh to the problem 2.17. First,
concerning the symmetric aS ·, ·, we have the following coercive property in Vh .
Lemma 3.1. For sufficiently large β, there exists some constant κ1 > 0 such that
aS vh , vh κ1 vh 2h ,
3.1
∀vh ∈ Vh .
Proof. According to the definition of aS ·, ·, we have
aS vh , vh β ∂vh 2
∂vh
ds |Δvh |2 dx − 2
{Δvh }
∂n ds.
∂n
h
K
e∈E e
e∈E e e
K∈Th
3.2
Using the Hölder’s inequality and the Young’s inequality, we have
e∈E
1/2 2 1/2
∂vh
2
−1 ∂vh ds he {Δvh }0,e
he {Δvh }
∂n
∂n 0,e
e
e∈E
e∈E
c
1/2 Δvh 20,K
K∈Th
ε
K∈Th
Δvh 20,K
e∈E
cε
−1
e∈E
1/2
∂vh 2
∂n h−1
e 0,e
∂vh 2
,
∂n h−1
e 0,e
3.3
8
Abstract and Applied Analysis
where c > 0 is independent of h and ε > 0 is a sufficiently small constant. Thus
aS vh , vh Δvh 20,K − 2ε
K∈Th
Δvh 20,K − cε−1
K∈Th
∂vh 2
h−1
e ∂n
0,e
e∈E
3.4
β
∂vh 2
ds.
h ∂n 0,e
e∈E e
Taking ε 1/4, then, for sufficiently large β such that β > 4c, using 2.25 we have
−1 ∂vh 2
1 2
he Δvh 20,K β − 4c
∂n κ1 vh h ,
2 K∈T
0,e
e∈E
aS vh , vh 3.5
h
with κ1 min{1/2, β − 4c}κ0 .
A direct result of Lemma 3.1 is that the discretized problem 2.17 admits a unique
solution uSh ∈ Vh for sufficiently large β.
Lemma 3.2. For all φ ∈ V , there holds
aS φ − Ph φ, vh chμ−2 vh h φs ,
3.6
∀vh ∈ Vh ,
where μ min{s, r 1} and c > 0 is independent of h.
Proof. For all φ ∈ V and vh ∈ Vh , we have
∂vh
aS φ − Ph φ, vh Δ φ − Ph φ Δvh dx −
Δ φ − Ph φ
ds
∂n
K
K∈T
e∈E e
h
−
e∈E
β ∂ φ − Ph φ ∂vh ∂ φ − Ph φ
ds ds
{Δvh }
∂n
h
∂n
∂n
e
e∈E e e
φ − Ph φh vh h Δ φ − Ph φ 2
0,e
e∈E
e∈E
he {Δvh }20,e
1/2 1/2
∂vh 2
∂n e∈E
0,e
1/2 ⎛
2 ⎞1/2
∂ φ − Ph φ ⎠
−1
⎝ he ∂n
e∈E
0,e
⎛
⎞1/2 1/2
2
∂vh 2
∂
φ
−
P
φ
h
⎠
−2 ⎝
c
he ,
∂n ∂n
0,e
e∈E
e∈E
0,e
3.7
Abstract and Applied Analysis
9
where c > 0 is independent of h. Note that
constant Ce depending on e there holds
e
∂vh /∂nds 0 for e ∈ E. Thus, for some
2 1/2
1/2 ∂vh 2
∂vh 2
∂vh
∂v
∂v
h
h
e
e
ds
− C ds − C ds
.
∂n ds ∂n
∂n
∂n
e
e ∂n
e
e
3.8
That is
2
∂vh 2
ds ∂vh − Ce ds.
∂n ∂n
e
e
3.9
∂vh
e 1/2
−
C
∂n
ch |vh |2,K ,
0,e
3.10
From 9, we have
where c > 0 is independent of h. Thus
∂vh 2
∂n 0,e
e∈E
1/2
ch1/2 vh h .
3.11
Substituting 3.11 into 3.7 and using 2.22-2.23 give
aS φ − Ph φ, vh
1/2
2
1/2
Δ φ − Ph φ φ − Ph φh vh h ch
vh h
0,e
e∈E
⎞1/2
⎛
2
∂
φ
−
P
φ
h
⎠
c⎝ h−1
vh h
e ∂n
e∈E
0,e
⎛
1/2 ⎝
ch
e∈E
⎞1/2
∂ φ − Ph φ 2
⎠
vh h chμ−2 φs vh h .
∂n
h−2
e 0,e
3.12
Theorem 3.3. Suppose that u ∈ V and uSh ∈ Vh are the solutions to problems SP and 2.17,
respectively; then the following optimal broken H 2 error estimate holds:
u − uSh chμ−2 |u|s ,
h
where μ min{s, r 1} and c > 0 is independent of h.
3.13
10
Abstract and Applied Analysis
Proof. According to Lemma 3.1, we have
2
κ1 Ph u − uSh aS Ph u − uSh , Ph u − uSh aS Ph u − u, Ph u − uSh aS u − uSh , Ph u − uSh
h
aS Ph u − u, Ph u − uSh
chμ−2 |u|s Ph u − uSh ,
h
3.14
where we use the orthogonal equation 2.19 and Lemma 3.2. The previous estimate implies
Ph u − uSh chμ−2 |u|s .
h
3.15
Finally, the triangular inequality and 2.24 yield
u − uSh u − Ph uh Ph u − uSh chμ−2 |u|s .
h
h
3.16
Next, we will show the optimal L2 error estimate in terms of the duality technique.
Suppose g ∈ L2 Ω and consider the following biharmonic problem:
Δ2 w g,
w
∂w
0,
∂n
in Ω,
3.17
on ∂Ω.
Suppose that problem 3.17 admits a unique solution w ∈ H02 Ω ∩ H 4 Ω such that
w4 cg ,
3.18
where || · ||4 denotes the H 4 norm in Ω and || · || denotes the L2 norm in Ω and c > 0 is
independent of h.
Denote by Πh the C1 continuous interpolate operator from V to H02 Ω ∩ Vh , and Πh
satisfies the approximation property 2.22. Then for the solution w ∈ H 4 Ω ∩ H02 Ω to
problem 3.17, there hold
w − Πh w2 ch2 w4 ch2 g ,
3/2 g , ∀e ∈ E,
w − Πh w2,e ch3/2
e w4 che
where c > 0 is independent of h.
3.19
Abstract and Applied Analysis
11
Theorem 3.4. Suppose that u ∈ V and uSh ∈ Vh are the solutions to problems SP and 2.17,
respectively; then the following optimal L2 error estimate holds:
u − uSh chμ |u|s ,
3.20
where μ min{s, r 1} and c > 0 is independent of h.
Proof. Setting g u − uSh in 3.17, multiplying 3.17 by u − uSh , and integrating over Ω, we
have
2 S
Δ2 w u − uSh dx aS w, u − uSh aS w − Πh w, u − uSh
u − uh Ω
aS w − Πh w, Ph u −
uSh
3.21
aS w − Πh w, u − Ph u,
where we use the orthogonal equation 2.19. We estimate two terms in the right-hand side
of 3.21 as follows:
aS w − Πh w, Ph u − uSh
∂ Ph u − uSh
S
ds
Δw − Πh wΔ Ph u − uh dx −
{Δw − Πh w}
∂n
K
e∈E e
K∈Th
∂w − Π w β ∂w − Πh w ∂ Ph u − uSh
h
S
Δ P h u − uh
ds ds
−
∂n
h
∂n
∂n
e∈E e
e∈E e e
∂ Ph u − uSh
S
ds
Δw − Πh wΔ Ph u − uh dx −
{Δw − Πh w}
∂n
K
K∈T
e∈E e
h
w − Πh w2 Ph u − uSh h
e∈E
1/2 w −
Πh w22,e
ch2 Ph u − uSh u − uSh chμ |u|s u − uSh ,
2
Ph u − uSh e∈E
1/2
1,e
h
3.22
where we use the estimate 3.15. In terms of the inequalities 2.22-2.23, we have
aS w − Πh w, u − Ph u
∂u − Ph u
ds
Δw − Πh wΔu − Ph udx −
{Δw − Πh w}
∂n
K
K∈T
e∈E e
h
−
e∈E
β ∂w − Πh w ∂u − Ph u ∂w − Πh w
ds ds
{Δu − Ph u}
∂n
h
∂n
∂n
e
e∈E e e
12
Abstract and Applied Analysis
∂u − Ph u
ds
Δw − Πh wΔu − Ph udx −
{Δw − Πh w}
∂n
K
K∈T
e∈E e
h
w − Πh w22,e
w − Ph w2 u − Ph uh 1/2 1/2
∂u − Ph u 2
∂n
e∈E
chμ |u|s u − uSh .
e∈E
0,e
3.23
Substituting the estimates 3.22-3.23 into 3.21 yields
u − uSh chμ |u|s .
3.24
4. C0 Nonsymmetric Interior Penalty Method
In this section, we will show the error estimates in the broken H 2 norm and in L2 norm
to the problem 2.18. The
between the solution u to problem 2.1 and the solution uNS
h
2
2
optimal broken H error estimate is derived. However, the L error estimate is suboptimal
because of the lack of adjoint consistency. According to Lemma 2.1, we have
aNS vh , vh κ0 vh 2h ,
∀vh ∈ Vh .
4.1
Moreover, for the nonsymmetric bilinear form aNS ·, ·, proceeding as in the proof of
Lemma 3.2, we have the following lemma.
Lemma 4.1. For all φ ∈ V , there holds
aNS φ − Ph φ, vh chμ−2 vh h φs ,
∀vh ∈ Vh ,
4.2
where μ min{s, r 1} and c > 0 is independent of h.
∈ Vh are the solutions to problems NSP and 2.18,
Theorem 4.2. Suppose that u ∈ V and uNS
h
respectively; then there holds
μ−2
|u|s ,
u − uNS
h ch
h
where μ min{s, r 1} and c > 0 is independent of h.
4.3
Abstract and Applied Analysis
13
Proof. According to 2.20, 4.1, and Lemma 4.1, we have
2
NS
NS
P
a
u
−
u
,
P
u
−
u
κ0 Ph u − uNS
NS
h
h
h
h
h
h
NS
aNS u − uNS
aNS Ph u − u, Ph u − uNS
h
h , P h u − uh
4.4
μ−2
NS ch
aNS Ph u − u, Ph u − uNS
u
−
u
|u|
P
h
s
h
h ,
h
where c > 0 is independent of h. That is
μ−2
|u|s .
Ph u − uNS
h ch
h
4.5
Using the triangular inequality yields
NS μ−2
−
P
u
u
−
u
u
|u|s .
u − uNS
P
h
h
h
h
h ch
h
h
4.6
Theorem 4.3. Suppose that u ∈ V and uNS
∈ Vh are the solutions to problems NSP and 2.18,
h
respectively; then there holds
μ−1
|u|s ,
u − uNS
h ch
4.7
where μ min{s, r 1} and c > 0 is independent of h.
Proof. Setting g u − uNS
in 3.17, multiplying 3.17 by u − uNS
, and integrating over Ω, we
h
h
have
2 Δ2 w u − uNS
dx aNS w, u − uNS
u − uNS
h h
h
Ω
aNS w − Πh w, u − uNS
h
⎡ ⎤
∂ u − uNS
h
⎥
⎢
2
{ΔΠh w}⎣
⎦ds
∂n
e∈E e
∂Π w h
NS
ds
Δ u − uh
−2
∂n
e∈E e
aNS w − Πh w, u − uNS
h
⎡ ⎤
∂ u − uNS
h
⎥
⎢
2
{ΔΠh w}⎣
⎦ds
∂n
e∈E e
14
Abstract and Applied Analysis
K∈Th
K
dx
Δw − Πh wΔ u − uNS
h
⎡ ⎡ ⎤
⎤
∂ u − uNS
∂ u − uNS
h
h
⎥
⎥
⎢
⎢
3
{ΔΠh w − w}⎣
{Δw}⎣
⎦ds 2
⎦ds
∂n
∂n
e
e
e∈E
e∈E
I1 I2 I3 ,
4.8
where we use Πh w ∈ H02 Ω. We estimate I1 as follows:
I1 K∈Th
K
NS dx
−
u
Δw − Πh wΔ u − uNS
u
h
h w − Πh w2
h
chμ |u|s u − uNS
h .
4.9
We estimate I2 as follows:
⎡ ⎤
∂ u − uNS
h
⎥
⎢
I2 3
{ΔΠh w − w}⎣
⎦ds
∂n
e∈E e
⎡ ⎤
NS
u
−
u
∂
P
h
h
∂u − Ph u
⎥
⎢
3
ds 3
{ΔΠh w − w}
{ΔΠh w − w}⎣
⎦ds
∂n
∂n
e∈E e
e∈E e
1/2 1/2
∂u − Ph u 2
2
c
w − Πh w2,e
∂n
0,e
e∈E
e∈E
w − Πh w22,e
e∈E
ch3/2
1/2
⎛ ⎡ ⎞1/2
⎤
2
NS
∂
P
u
−
u
h
h
⎜
⎢
⎥ ⎟
⎝ ⎣
⎦ ⎠
∂n
e∈E 0,e
NS μ
NS hμ−3/2 |u|s ch1/2 Ph u − uNS
h u − uh ch |u|s u − uh .
h
We estimate I3 as follows:
⎡ ⎡ ⎤
⎤
NS
∂ u − uNS
∂
u
−
u
h
h
⎥
⎥
⎢
⎢
I3 2
{Δw}⎣
{Δw − Ce }⎣
⎦ds 2
⎦ds
∂n
∂n
e∈E e
e∈E e
⎡ ⎤
NS
u
−
u
∂
P
h
h
∂u − Ph u
⎥
⎢
2
ds 2
{Δw − Ce }
{Δw − Ce }⎣
⎦ds
∂n
∂n
e∈E e
e∈E e
4.10
Abstract and Applied Analysis
c
15
1/2 Δw −
Ce 20,e
e∈E
Δw − Ce 20,e
1/2
e∈E
c
u −
e∈E
1/2
Ph u21,e
⎛ ⎡ ⎞1/2
⎤
2
∂ Ph u − uNS
h
⎜ ⎢
⎥ ⎟
⎝ ⎣
⎦ ⎠
∂n
e∈E 0,e
1/2
Δw − Ce 20,e
hμ−3/2 |u|s ch1/2 Ph u − uNS
h h
e∈E
ch
μ−3/2
Δw − Ce 20,e
1/2
|u|s ,
e∈E
4.11
where Ce is some positive constant. Substituting the following estimate
2
Δw − Ce 20,e c he w23,e chw24 chu − uNS
h e∈E
4.12
e∈E
into 4.11 gives
I3 chμ−1 |u|s u − uNS
h .
4.13
Finally, substituting 4.9–4.13 into 4.8, we obtain
μ−1
|u|s .
u − uNS
h ch
4.14
5. C0 Superpenalty Nonsymmetric Method
In order to obtain the optimal L2 error estimate for the nonsymmetric method, in this section
we will consider the C0 superpenalty nonsymmetric method. First, we introduce a new
nonsymmetric bilinear form:
aSNS u, v K∈Th
ΔuΔvdx −
K
1
3
e∈E he
e
e∈E
∂u
∂n
∂v
.
∂n
∂v
∂u
ds ds
{Δu}
{Δv}
∂n
∂n
e
e∈E e
5.1
16
Abstract and Applied Analysis
The broken H 2 norm is modified to
|vh |h |vh |22,Ki
K∈Th
1/2
1
∂vh 2
,
3 ∂n 0,e
e∈E he
∀vh ∈ Vh .
5.2
From Lemma 2.1, it is easy to show that there exists some constant κ2 > 0 such that
aSNS vh , vh κ2 |vh |2h ,
∀vh ∈ Vh .
5.3
In fact, we have
1
∂vh 2
3 ∂n 0,e
K∈Th
e∈E he
2
1
∂vh 2
∂vh 1
1 1 2
Δvh 0,K ∂n 2 e∈E h3e ∂n 0,e 2 K∈T
h
e
0,e
e∈E
aSNS vh , vh Δvh 20,K 5.4
h
2
∂vh 1
1 κ0
|vh |22,K κ2 |vh |2h
2 e∈E h3e
∂n 0,e 2 K∈T
h
for κ2 min{1/2, κ0 /2}. Since the solution u to problem 2.1 belongs to H 4 Ω ∩ H02 Ω,
then it satisfies
aSNS u, v f, v ,
∀v ∈ V.
SNSP
In this case, the the C0 superpenalty nonsymmetric finite element approximation of 2.1 is
∈ Vh such that,
find uSNS
h
aSNS uSNS
h , vh f, vh ,
∀vh ∈ Vh .
5.5
Then, we have the following orthogonal equation:
aSNS uSNS
− u, vh 0,
h
∀vh ∈ Vh .
5.6
Let Πh be the C1 continuous interpolated operator defined in Section 3. Observe that
∂u − Πh u/∂n 0 on every e ∈ E, and proceeding as in the proof of Lemmas 3.2 and 4.1,
we have the following.
Lemma 5.1. For all φ ∈ V , there holds
aNS φ − Πh φ, vh chμ−2 |vh |h φs ,
where μ min{s, r 1} and c > 0 is independent of h.
∀vh ∈ Vh ,
5.7
Abstract and Applied Analysis
17
Using Lemma 5.1, the following optimal broken H 2 error estimate holds.
Theorem 5.2. Suppose that u ∈ V and uSNS
∈ Vh are the solutions to problems SNSP and 5.5,
h
respectively; then there holds
μ−2
|u|s ,
u − uSNS
h ch
h
5.8
where μ min{s, r 1} and c > 0 is independent of h.
The main result in this section is the following optimal L2 error estimate.
Theorem 5.3. Suppose that u ∈ V and uSNS
∈ Vh are the solutions to problems SNSP and 5.5,
h
respectively; then there holds
μ
u − uSNS
h ch |u|s ,
5.9
where μ min{s, r 1} and c > 0 is independent of h.
Proof. Let g u − uSNS
. Multiplying 3.17 by u − uSNS
and integrating over Ω, we have
h
h
2 SNS −
u
Δ2 w u − uSNS
dx aSNS w, u − uSNS
u
h h
h
Ω
aSNS w − Πh w, u − uSNS
h
⎡ ⎤
∂ u − uSNS
h
⎥
⎢
2
{ΔΠh w}⎣
⎦ds
∂n
e∈E e
∂Π w h
SNS
Δ u − uh
−2
ds
∂n
e∈E e
aSNS w − Πh w, u − uSNS
h
K∈Th
K
⎡ ⎤
∂ u − uSNS
h
⎥
⎢
2
{ΔΠh w}⎣
⎦ds
∂n
e∈E e
dx
Δw − Πh wΔ u − uSNS
h
⎡ ⎡ ⎤
⎤
SNS
SNS
∂
u
−
u
∂
u
−
u
h
h
⎥
⎥
⎢
⎢
3
{ΔΠh w − w}⎣
{Δw}⎣
⎦ds 2
⎦ds
∂n
∂n
e∈E e
e∈E e
I4 I 5 I 6 ,
5.10
18
Abstract and Applied Analysis
where we use Πh w ∈ C1 Ω ∩ H02 Ω. Proceeding as in the proof of Theorem 4.2, we can
estimate I4 and I5 as follows:
I4 K∈Th
K
dx chμ |u|s u − uSNS
Δw − Πh wΔ u − uSNS
h
h ,
⎡ ⎤
∂ u − uSNS
h
⎥
⎢
μ
SNS I5 3
{ΔΠh w − w}⎣
⎦ds ch |u|s u − uh .
∂n
e∈E e
5.11
5.12
The different estimate compared to Theorem 4.3 is the estimate of I6 . Under the new norm
||| · |||h , we have
⎡ ⎡ ⎤
⎤
SNS
∂ u − uSNS
∂
u
−
u
h
h
⎥
⎥
⎢
⎢
I6 2
{Δw}⎣
{Δw − c}⎣
⎦ds 2
⎦ds
∂n
∂n
e∈E e
e∈E e
c
1/2
h3e Δw − C20,e
e∈E
⎡ ⎞1/2
⎤
2
SNS
∂
u
−
u
h
⎜
1 ⎢
⎥ ⎟
⎝
⎦ ⎠
3 ⎣
∂n
h
e
e∈E
⎛
5.13
0,e
SNS μ
SNS ch2 u − uSNS
h · u − uh ch |u|s u − uh .
h
Substituting 5.11–5.13 into 5.10 yields the following optimal L2 error estimate:
μ
u − uSNS
h ch |u|s .
5.14
Acknowledgments
The authors would like to thank the anonymous reviewers for their careful reviews and
comments to improve this paper. This material is based upon work funded by the National
Natural Science Foundation of China under Grants no. 10901122, no. 11001205, and no.
11126226 and by Zhejiang Provincial Natural Science Foundation of China under Grants no.
LY12A01015 and no. Y6110240.
References
1 D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, “Unified analysis of discontinuous Galerkin
methods for elliptic problems,” SIAM Journal on Numerical Analysis, vol. 39, no. 5, pp. 1749–1779, 2002.
2 G. A. Baker, “Finite element methods for elliptic equations using nonconforming elements,”
Mathematics of Computation, vol. 31, no. 137, pp. 45–59, 1977.
3 I. Mozolevski and E. Süli, “A priori error analysis for the hp-version of the discontinuous Galerkin
finite element method for the biharmonic equation,” Computational Methods in Applied Mathematics,
vol. 3, no. 4, pp. 596–607, 2003.
Abstract and Applied Analysis
19
4 I. Mozolevski, E. Süli, and P. R. Bösing, “hp-version a priori error analysis of interior penalty
discontinuous Galerkin finite element approximations to the biharmonic equation,” Journal of
Scientific Computing, vol. 30, no. 3, pp. 465–491, 2007.
5 E. Süli and I. Mozolevski, “hp—version interior penalty DGFEMs for the biharmonic equation,”
Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 13–16, pp. 1851–1863, 2007.
6 I. Babuška and M. Zlámal, “Nonconforming elements in the finite element method with penalty,”
SIAM Journal on Numerical Analysis, vol. 10, pp. 863–875, 1973.
7 G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei, and R. L. Taylor, “Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and
continuum mechanics with applications to thin beams and plates, and strain gradient elasticity,”
Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 34, pp. 3669–3750, 2002.
8 S. C. Brenner and L.-Y. Sung, “C0 interior penalty methods for fourth order elliptic boundary value
problems on polygonal domains,” Journal of Scientific Computing, vol. 22-23, pp. 83–118, 2005.
9 B. Rivière, M. F. Wheeler, and V. Girault, “Improved energy estimates for interior penalty, constrained
and discontinuous Galerkin methods for elliptic problems. I,” Computational Geosciences, vol. 3, no. 34, pp. 337–360, 1999.
10 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing, Amsterdam,
The Netherlands, 1978, Studies in Mathematics and Its Applications.
11 J. Guzmán and M. Neilan, “Conforming and divergence free stokes elements on general triangularmeshes,” Mathematics of Computation. In press.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014