A Weak Galerkin Finite Element Scheme for the Biharmonic Equations by Using Polynomials of Reduced Order

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by  Ran Zhang

A WEAK GALERKIN FINITE ELEMENT SCHEME FOR THE
BIHARMONIC EQUATIONS BY USING POLYNOMIALS OF
REDUCED ORDER
RAN ZHANG ∗ AND QILONG ZHAI†
6
1 Abstract. AnewweakGalerkin(WG)finiteelementmethodforsolvingthebiharmonicequa-
0 tion in two or three dimensional spaces by using polynomials of reduced order is introduced and
2 analyzed. TheWGmethodisontheuseofweakfunctionsandtheirweakderivativesdefinedasdis-
tributions. Weak functions andweak derivatives can beapproximated bypolynomials withvarious
n
degrees. Different combination of polynomial spaces leads to different WG finite element methods,
a
whichmakesWGmethodshighlyflexibleandefficientinpracticalcomputation. Thispaperexplores
J
thepossibilityofoptimalcombinationofpolynomialspacesthatminimizethenumberofunknowns
5 in the numerical scheme, yet without compromising the accuracy of the numerical approximation.
2 Errorestimates of optimal order areestablishedfor thecorresponding WG approximations inboth
a discrete H2 norm and the standard L2 norm. In addition, the paper also presents some numeri-

] cal experiments to demonstrate the power of the WG method. The numerical results show a great
A
promiseoftherobustness,reliability,flexibilityandaccuracyoftheWGmethod.
N
. Key words. weak Galerkin finite element methods, weak Laplacian, biharmonic equation,
h
polyhedralmeshes.
t
a
m
AMS subject classifications. Primary, 65N30, 65N15, 65N12, 74N20; Secondary, 35B45,
[ 35J50,35J35

1
v 1. Introduction. This paper will concernwith approximatingthe solutionu of
4
the biharmonic equation
3
6
(1.1) ∆2u=f, in Ω,
6
0
. with clamped boundary conditions
1
0
(1.2) u=g, on ∂Ω,
6
∂u
1
(1.3) =φ, on ∂Ω,
: ∂n
v
i where∆istheLaplacianoperator,Ωisaboundedpolygonalorpolyhedraldomainin
X
Rd for d=2,3 and n denotes the outward unit normal vector along ∂Ω. We assume
r
a that f,g,φ are given, sufficiently smooth functions.
This problem mainly arises in fluid dynamics where the stream functions u of
incompressibleflowsaresoughtandelasticitytheory,inwhichthe deflectionofathin
plate of the clamped plate bending problem is sought [27, 36, 39].
Duetothesignificanceofthebiharmonicproblem,alargenumberofmethodsfor
discretizing(1.1)-(1.3)havebeenproposed. Thesemethodsinclude dealingwiththe
biharmonic operator directly, such as discretizing (1.1)-(1.3) on a uniform grid using
∗Department of Mathematics, Jilin University, Changchun, China ([email protected]).
The research of Zhang was supported in part by China Natural National Science Founda-
tion(11271157, 11371171, 11471141), and by the Program for New Century Excellent Talents in
UniversityofMinistryofEducationofChina.
†DepartmentofMathematics, JilinUniversity,Changchun, China([email protected]).
1
a 13-point or 25-point direct approximation of the fourth order differential operator
[9, 25]; mixed methods, that is, splitting the biharmonic equation into two coupled
Poisson equations [1, 5, 18, 4, 13, 16, 19, 20, 21, 28, 26, 6, 7]. Also there are some
other approaches to the biharmonic problems, like the conformal mapping methods
[12, 37], integralequations [30], orthogonalspline collocationmethod [8] and the fast
multipole methods [24], etc.
Among these methods, finite element methods are one of the most widely used
technique, which is based on variationalformulations of the equations considered. In
fact, the biharmonic equation is also one of the most important applicable problems
of the finite element methods, cf. [17, 23, 2, 44, 14, 15]. The Galerkin methods,
discretizingthe correspondingvariationalformof(1.1)is givenbyseekingu∈H2(Ω)
satisfying
∂u
u| =g, | =φ
∂Ω ∂n ∂Ω
such that
(1.4) (∆u,∆v)=(f,v), ∀v ∈H2(Ω),
0
where H2(Ω) is the subspace of H2(Ω) consisting of functions with vanishing value
0
and normal derivative on ∂Ω.
Standardfinite element methods for solving(1.1) - (1.3) basedonthe variational
form (1.4) with conforming finite element require rather sophisticated finite elements
such as the 21-degrees-of-freedom of Argyris (see [3]) or nonconforming elements of
Hermite type. Since the complexity in the construction for the finite element with
high continuous elements, H2 conforming element are seldom used in practice for
the biharmonic problem. To avoidusing of C1-elements, besides the mixed methods,
an alternative approach, nonconforming and discontinuous Galerkin finite element
methods have been developed for solving the biharmonic equation over the last sev-
eral decades. Morley element [29] is a well known nonconforming element for the
biharmonic equation for its simplicity. A C0 interior penalty method was developed
in [10, 22]. In [31], a hp-version interior penalty discontinuous Galerkin method was
presented for the biharmonic equation.
Recently a new class of finite element methods, called weak Galerkin(WG) finite
element methods were developed for the biharmonic equation for its highly flexible
and robust properties. The WG method refers to a numerical scheme for partial dif-
ferentialequationsinwhichdifferentialoperatorsareapproximatedby weakforms as
distributions over a set of generalized functions. This thought was first proposed in
[41]foramodelsecondorderelliptic problem,andthismethodwasfurtherdeveloped
in [42, 32, 43]. In [34], a weak Galerkin method for the biharmonic equation was
derived by using discontinuous functions of piecewise polynomials on general parti-
tions of polygons or polyhedra of arbitrary shape. After that, in order to reduce the
number of unknowns, a C0 WG method [35] was proposed and analyzed. However,
due to the continuity limitation, the C0 WG scheme only works for the traditional
finite partitions, while not arbitrary polygonal or polyhedral girds as allowed in [34].
In order to realize the aim that reducing the unknown numbers and suit for
general partitions of polygons or polyhedra of arbitrary shape at the same time, in
this paper we construct a reduction WG scheme based on the use of a discrete weak
2
Laplacianplus a new stabilizationthat is also parameter free. The goalof this paper
istospecifyallthedetailsforthereductionWGmethodforthebiharmonicequations
andpresentthe numericalanalysisby presentingamathematicalconvergencetheory.
An outline of the paper is as follows. In the remainder of the introduction we
shall introduce some preliminaries and notations for Sobolev spaces. In Section 2 is
devoted to the definitions of weak functions and weak derivatives. The WG finite
element schemes for the biharmonic equation (1.1)-(1.3) are presented in Section 3.
In Section 4, we establishan optimal order error estimates for the WG finite element
approximation in an H2 equivalent discrete norm. In Section 5, we shall drive an
error estimate for the WG finite element method in the standard L2 norm. Section
6 contains the numerical results of the WG method. The theoretical results are
illustratedbythese numericalexamples. Finally, wepresentsometechnicalestimates
for quantities related to the local L2 projections into various finite element spaces
and some approximation properties which are useful in the convergence analysis in
Appendix A.
Now let us define some notations. Let D be any open bounded domain with
Lipschitz continuous boundaryin Rd,d=2,3. We use the standarddefinition for the
SobloevspaceHs(D)andtheir associatedinner products(·,·) , normsk·k ,and
s,D s,D
seminorms |·| for any s≥0.
s,D
ThespaceH0(D)coincideswithL2(D),forwhichthenormandtheinnerproduct
aredenotedbyk·k and(·,·) ,respectively. WhenD =Ω,weshalldropthesubscript
D D
D in the norm and in the inner product notation.
The space H(div;D) is defined as the set of vector-valuedfunctions on D which,
together with their divergence, are square integrable; i.e.,
H(div;D)={v:v∈[L2(D)]d,∇·v∈L2(D)}.
The norm in H(div;D) is defined by
kvkH(div;D) =(kvk2D+k∇·vk2D)12.
2. Weak Laplacain and Discrete Weak Laplacian. For the biharmonic
equation (1.1), the underlying differential operator is the Laplacian ∆. Thus, we
shall first introduce a weak version for the Laplacian operator defined on a class of
discontinuous functions as distributions [34].
Let K be any polygonalor polyhedral domain with boundary ∂K. A weak func-
tion on the region K refers to a function v = {v ,v ,v } such that v ∈ L2(K),
0 b g 0
v ∈L2(∂K), and v ·n∈L2(∂K), where n is the outward unit normal vector along
b g
∂K. Denote by W(K) the space of all weak functions on K, that is,
(2.1) W(K)={v={v ,v ,v }:v ∈L2(K),v ,v ·n∈L2(∂K)}.
0 b g 0 b g
Recallthat,foranyv ∈W(K),theweakLaplacianofv ={v ,v ,v }isdefinedas
0 b g
a linearfunctional∆ v inthe dualspace ofH2(K)whoseactiononeachϕ∈H2(K)
w
is given by
(2.2) (∆ v,ϕ) =(v ,∆ϕ) −hv ,∇ϕ·ni +hv ·n,ϕi ,
w K 0 K b ∂K g ∂K
3
where(·,·) standsfortheL2-innerproductinL2(K)andh·,·i istheinnerproduct
K ∂K
in L2(∂K).
TheSobolevspaceH2(K)canbeembeddedintothespaceW(K)byaninclusion
map iW :H2(K)→W(K) defined as follows
iW(φ)={φ|K,φ|∂K,(∇φ·n)n|∂K}, φ∈H2(K).
With the help of the inclusion map iW, the Sobolev space H2(K) canbe viewed as a
subspace of W(K) by identifying each φ∈H2(K) with iW(φ).
Analogously,aweakfunctionv ={v ,v ,v }∈W(K)issaidtobeinH2(K)ifit
0 b g
can be identified with a function φ∈H2(K) through the above inclusion map. Here
the first components v can be seen as the value of v in the interior and the second
0
component v represents the value of v on ∂K. Denote ∇v·n by v , then the third
b n
component v represents (∇v·n)n| =v n. Obviously, v ·n=∇v·n. Note that
g ∂K n g
if v 6∈ H2(K), then v and v may not necessarily be related to the trace of v and
b g 0
(∇v ·n)n on ∂K, respectively.
0
For v ∈H2(K), from integration by parts we have
(∆ v,ϕ) =(v,∆ϕ) −hv,∇ϕ·ni +h∇v·n,ϕi
w K K ∂K ∂K
=(v ,∆ϕ) −hv ,∇ϕ·ni +hv ·n,ϕi .
0 K b ∂K g ∂K
Thus the weak Laplacian is identical with the strong Laplacian, i.e.,
∆wiW(v)=∆v
for smooth functions in H2(K).
For numerical implementation purpose, we define a discrete version of the weak
Laplacain operator by approximating ∆ in polynomial subspaces of the dual of
w
H2(K). To this end, for any non-negative integer r ≥ 0, let P (K) be the set of
r
polynomials on K with degree no more than r.
Definition 2.1. ([34]) A discrete weak Laplacian operator, denoted by ∆ ,
w,r,K
is defined as the unique polynomial ∆ v ∈P (K) satisfying
w,r,K r
(2.3) (∆ v,ϕ) =(v ,∆ϕ) −hv ,∇ϕ·ni +hv ·n,ϕi , ∀ϕ∈P (K).
w,r,K K 0 K b ∂K n ∂K r
From the integration by parts, we have
(v ,∆ϕ) =(∆v ,ϕ) +hv ,∇ϕ·ni −h∇v ·n,ϕi .
0 K 0 K 0 ∂K 0 ∂K
Substituting the above identity into (2.3) yields
(2.4) (∆ v,ϕ) −(∆v ,ϕ) =hv −v ,∇ϕ·ni −h(∇v −v )·n,ϕi ,
w,r,K K 0 K 0 b ∂K 0 g ∂K
for all ϕ∈P (K).
r
3. Weak Galerkin Finite Element Scheme. Let T be a partition of the
h
domain Ω into polygons in 2D or polyhedra in 3D. Assume that T is shape regular
h
in the sense as defined in [42]. Denote by E the set of all edges or flat faces in T ,
h h
and let E0 =E \∂Ω be the set of all interior edges or flat faces.
h h
4
Since v represents ∇v ·n, then v is naturally dependent on n. To ensure a
n n
single valued function v on e∈E , we introduce a set of normal directions on E as
n h h
follows
(3.1) N ={n :n is unit and normal to e, e∈E }.
h e e h
For any given integer k ≥ 2,T ∈ T , denote by W (T) the discrete weak function
h k
space given by
(3.2) Wk(T)={{v0,vb,vnne}:v0 ∈Pk(T),vb,vn ∈Pk−1(e),e⊂∂T}.
By patching W (T) over all the elements T ∈ T through a common value on the
k h
interface E0, we arrive at a weak finite element space V given by
h h
V ={{v ,v ,v n }:{v ,v ,v n } ∈W (T), ∀T ∈T }.
h 0 b n e 0 b n e T k h
Denote by V0 the subspace of V constituting d(cid:12)iscrete weak functions with vanishing
h h (cid:12)
traces; i.e.,
V0 ={{v ,v ,v n }:{v ,v ,v n }∈V ,v | =0,v | =0, e∈∂T ∩∂Ω}.
h 0 b n e 0 b n e h b e n e
Denote by Λ the trace of V on ∂Ω from the component v . It is obvious that Λ
h h b h
consists of piecewise polynomials of degree k−1. Similarly, denote by Υ the trace
h
of V from the component of v as piecewise polynomials of degree k−1. Denote by
h n
∆w,k−2 thediscreteweakLaplacianoperatoronthefiniteelementspaceVh computed
by using (2.3) on each element T for k≥2, that is,
(3.3) (∆w,k−2v)|T =∆w,k−2,T(v|T) ∀v ∈Vh.
Forsimplicity,weshalldropthesubscriptk−2inthenotation∆w,k−2 forthediscrete
weak Laplacian operator. We also introduce the following notation
(∆ v,∆ w) = (∆ v,∆ w) .
w w h w w T
TX∈Th
For eachelement T ∈T , denote by Q the L2 projectiononto P (T), k ≥2. For
h 0 k
each edge/face e ⊂ ∂T, denote by Qb the L2 projection onto Pk−1(e). Now for any
u ∈ H2(Ω), we shall combine these two projections together to define a projection
into the finite element space V such that on the element T
h
Q u={Q u,Q u,(Q (∇u·n ))n }.
h 0 b b e e
Theorem 3.1. Let Qh be the local L2 projection onto Pk−2. Then the following
commutative diagram holds true on each element T ∈T :
h
(3.4) ∆ Q u=Q ∆u, ∀u∈H2(T).
w h h
Proof. For any φ ∈ Pk−2(T), from the definition of the discrete weak Laplacian
and the L2 projection
(∆ Q u,φ) =(Q u,∆φ) −hQ u,∇φ·ni +hQ (∇u·n )n ·n,φi
w h T 0 T b ∂T b e e ∂T
=(u,∆φ) −hu,∇φ·ni +h∇u·n,φi
T ∂T ∂T
=(∆u,φ) =(Q ∆u,φ),
T h
5
which implies (3.4).
Thecommutativeproperty(3.4)indicatesthatthediscreteweakLaplacianofthe
L2 projectionofuisagoodapproximationoftheLaplacianofuintheclassicalsense.
This is a good property of the discrete weak Laplacian in application to algorithm
and analysis.
For any u ={u ,u ,u n } and v ={v ,v ,v n } in V , we introduce a bilinear
h 0 b n e 0 b n e h
form as follows
s(u ,v)= h−1h∇u ·n −u ,∇v ·n −v i
h T 0 e n 0 e n ∂T
TX∈Th
+ h−3hQ u −u ,Q v −v i .
T b 0 b b 0 b ∂T
TX∈Th
Weak Galerkin Algorithm 1. Find u = {u ,u ,u n } ∈ V satisfying
h 0 b n e h
u =Q g and u =Q φ on ∂Ω and the following equation:
b b n b
(3.5) (∆ u ,∆ v) +s(u ,v)=(f,v ), ∀v ={v ,v ,v n }∈V0.
w h w h h 0 0 b n e h
Lemma 3.2. For any v ∈V0, let |||v||| be given by
h
(3.6) |||v|||2 =(∆ v,∆ v) +s(v,v).
w w h
Then, |||·||| defines a norm in the linear space V0.
h
Proof. Forsimplicity,weshallonlyprovethepositivitypropertyfor|||·|||. Assume
that |||v|||=0 for some v ∈V0. It followsthat ∆ v =0 on eachelementT, Q v =v
h w b 0 b
and ∇v ·n = v on each edge ∂T. We claim that ∆v = 0 holds true locally on
0 e n 0
each element T. To this end, for any ϕ ∈ Pk−2(T) we use ∆wv = 0 and the identify
(2.4) to obtain
(3.7) 0=(∆ v,ϕ)
w T
=(∆v ,ϕ) +hQ v −v ,∇ϕ·ni +hv n ·n−∇v ·n,ϕi
0 T b 0 b ∂T n e 0 ∂T
=(∆v ,ϕ) ,
0 T
where we have used the fact that Q v −v =0 and
b 0 b
v n ·n−∇v ·n=±(v −∇v ·n )=0
n e 0 n 0 e
inthelastequality. Theidentity(3.7)impliesthat∆v =0holdstruelocallyoneach
0
element T.
Next, we claim that ∇v =0 also holds true locally on each element T. For this
0
purpose, for any φ∈P (T), we utilize the Gauss formula to obtain
k
(3.8) (∇v ,∇φ) =−(∆v ,φ) +h∇v ·n,φi =h∇v ·n,φi .
0 T 0 T 0 ∂T 0 ∂T
By letting φ=v on each element T and summing over all T we obtain
0
(3.9) (∇v ,∇v ) = h∇v ·n,v i .
0 0 T 0 0 ∂T
TX∈Th TX∈Th
6
For two elements T ,T ∈ T , which share e ∈ E \ ∂Ω as a common edge,
1 2 h h
denote v1,v2 the values of v in the interior of T ,T , respectively. It follows from
0 0 1 2
Qbv01 =Qbv02 =vb on edge e and the fact ∇v0·ne =vn ∈Pk−1(e) that
h∇v1·n ,v1i +h∇v2·n ,v2i
0 T1 0 e 0 T2 0 e
=±hv ,v1−v2i =±hv ,Q v1−Q v2i =0,
n 0 0 e n b 0 b 0 e
where n ,n denote the outward unit normal vectors on e according to elements
T1 T2
T ,T , respectively. This, together with ∇v ·n = v = 0 on the boundary edge
1 2 0 n
e∈E ∩∂Ω implies
h
h∇v ·n,v i =0.
0 0 ∂T
TX∈Th
It follows from equation (3.9) that k∇v k =0 on each element T. Thus, v =const
0 T 0
locally on each element and is then continuous across each interior edge e as
v | =Q v =v .
0 e b 0 b
The boundary condition of v =0 then implies that v ≡0 on Ω, which completes the
b
proof.
Lemma 3.3. The weak Galerkin finite element scheme (3.5) has a unique solu-
tion.
Proof. Assume u(1) and u(2) are two solutions of the WG finite element scheme
h h
(3.5). It is obvious that the difference ρ =u(1)−u(2) is a finite element function in
h h h
V0 satisfying
h
(3.10) (∆ ρ ,∆ v) +s(ρ ,v)=0, ∀v ∈V0.
w h w h h h
By letting v =ρ in above equation (3.10) we obtain the following indentity
h
(∆ ρ ,∆ ρ ) +s(ρ ,ρ )=0.
w h w h h h h
Itfollows fromLemma 3.2that ρ ≡0, whichshowsthatu(1) =u(2). This completes
h h h
the proof.
4. AnErrorEstimate. Thegoalofthissectionistoestablishanerrorestimate
for the WG-FEM solution u arising from (3.5).
h
First of all, let us derive an error equation for the WG finite element solution
obtained from (3.5). This error equation is critical in convergence analysis.
Lemma 4.1. Let u and u ∈ V be the solution of (1.1)-(1.3) and (3.5), respec-
h h
tively. Denote by
e =Q u−u
h h h
the error function between the L2 projection of u and its weak Galerkin finite element
solution. Then the error function e satisfies the following equation
h
(4.1) (∆ e ,∆ v) +s(e ,v)=ℓ (v)
ω h ω h h u
7
for all v ∈V0. Here
h
(4.2) ℓ (v)= h∆u−Q ∆u,∇v ·n−v n ·ni
u h 0 n e ∂T
TX∈Th
− h∇(∆u−Q ∆u)·n,v −v i +s(Q u,v).
h 0 b ∂T h
TX∈Th
Proof. Using (2.4) with ϕ=∆ Q u=Q ∆u we obtain
ω h h
(∆ Q u,∆ v)
ω h ω T
=(∆v ,Q ∆u) +hv −v ,∇(Q ∆u)·ni −h(∇v −v n )·n,Q ∆ui
0 h T 0 b h ∂T 0 n e h ∂T
=(∆u,∆v ) +hv −v ,∇(Q ∆u)·ni −h(∇v −v n )·n,Q ∆ui ,
0 T 0 b h ∂T 0 n e h ∂T
which implies that
(4.3) (∆u,∆v ) =(∆ Q u,∆ v) −hv −v ,∇(Q ∆u)·ni
0 T ω h ω T 0 b h ∂T
+h(∇v −v n )·n,Q ∆ui .
0 n e h ∂T
Next, it follows from the integration by parts that
(∆u,∆v ) =(∆2u,v ) +h∆u,∇v ·ni −h∇(∆u)·n,v i .
0 T 0 T 0 ∂T 0 ∂T
By summing over all T and then using the identity (∆2u,v )=(f,v ) we arrive at
0 0
(∆u,∆v ) =(f,v )+ h∆u,∇v ·n−v n ·ni
0 T 0 0 n e ∂T
TX∈Th TX∈Th
− h∇(∆u)·n,v −v i ,
0 b ∂T
TX∈Th
where we have used the fact that v and v vanish on the boundary of the domain.
n b
Combining the above equation with (4.3) yields
(∆ Q u,∆ v) =(f,v )+ h∆u−Q ∆u,(∇v −v n )·ni
ω h ω h 0 h 0 n e ∂T
TX∈Th
− h∇(∆u−Q ∆u)·n,v −v i .
h 0 b ∂T
TX∈Th
Adding s(Q u,v) to both sides of the above equation gives
h
(4.4) (∆ Q u,∆ v) +s(Q u,v)
ω h ω h h
=(f,v )+ h∆u−Q ∆u,(∇v −v n )·ni
0 h 0 n e ∂T
TX∈Th
− h∇(∆u−Q ∆u)·n,v −v i +s(Q u,v).
h 0 b ∂T h
TX∈Th
Subtracting (3.5) from (4.4) leads to the following error equation
(∆ e ,∆ v) +s(e ,v)= h∆u−Q ∆u,(∇v −v n )·ni
ω h ω h h h 0 n e ∂T
TX∈Th
− h∇(∆u−Q ∆u)·n,v −v i +s(Q u,v)
h 0 b ∂T h
TX∈Th
8
for all v ∈V0. This completes the derivation of (4.1).
h
The following Theorem presents an optimal order error estimate for the error
function e in the trip-bar norm. We believe this tripe-bar norm provides a discrete
h
analogue of the usual H2-norm.
Theorem 4.2. Let u ∈V be the weak Galerkin finite element solution arising
h h
from(3.5)withfiniteelementfunctionsoforderk ≥2. Assumethattheexactsolution
of (1.1)-(1.3) is sufficiently regular such that u ∈ Hk+2(Ω). Then, there exists a
constant C such that
(4.5) |||u −Q u|||≤Chk−1 kuk .
h h k+2
The above estimate is of optimal order in terms of the meshsize h, but not in the
regularity assumption on the exact solution of the biharmonic equation.
Proof. By letting v =e in the error equation (4.1), we have
h
(4.6) |||e |||2 =ℓ(e ),
h h
where
(4.7) ℓ(e )= h∆u−Q ∆u,(∇e −e n )·ni
h h 0 n e ∂T
TX∈Th
− h∇(∆u−Q ∆u)·n,e −e i
h 0 b ∂T
TX∈Th
+ h−1h∇Q u·n −Q (∇u·n ),∇e ·n −e i
T 0 e b e 0 e n ∂T
TX∈Th
+ h−3hQ Q u−Q u,Q e −e i .
T b 0 b b 0 b ∂T
TX∈Th
The rest of the proof shall estimate each of the terms on the right-hand side of
(4.7). For the first term, we use the Cauchy-Schwarz inequality and the estimates
(A.5) and (A.6) in Lemma A.4 (see Appendix A) with m=k to obtain
(4.8) h∆u−Q ∆u,(∇e −e n )·ni
h 0 n e ∂T
(cid:12) (cid:12)
(cid:12)TX∈Th (cid:12)
(cid:12) 1 (cid:12) 1
(cid:12) 2 (cid:12) 2
≤(cid:12) h k∆u−Q ∆uk2 h−(cid:12)1k∇e ·n −e k2
T h ∂T T 0 e n ∂T
TX∈Th ! TX∈Th !
≤Chk−1kuk |||e |||.
k+1 h
9
For the second term, using Lemma A.4, A.6 and A.9 we obtain
(4.9) h∇(∆u−Q ∆u)·n,e −e i
h 0 b ∂T
(cid:12) (cid:12)
(cid:12)TX∈Th (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
≤(cid:12) h∇(∆u−Q ∆u)·n,Q e −e i(cid:12)
h b 0 b ∂T
(cid:12) (cid:12)
(cid:12)TX∈Th (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12)+ h∇(∆u−Q ∆u)·n,e −Q e i(cid:12)
h 0 b 0 ∂T
(cid:12) (cid:12)
(cid:12)TX∈Th (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
= (cid:12) h∇(∆u−Q ∆u)·n,Q e −e i (cid:12)
h b 0 b ∂T
(cid:12) (cid:12)
(cid:12)TX∈Th (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12)+ h(∇(∆u)−Q (∇(∆u)))·n,e −(cid:12)Q e i
b 0 b 0 ∂T
(cid:12) (cid:12)
(cid:12)TX∈Th (cid:12)
(cid:12) 1 (cid:12) 1
(cid:12) 2 (cid:12) 2
≤ (cid:12) h3k∇(∆u−Q ∆u)k2 · h−3kQ(cid:12)e −e k2
T h ∂T T b 0 b ∂T
! !
TX∈Th TX∈Th
1 1
2 2
+ k∇(∆u)−Q (∇(∆u))k2 · ke −Q e k2
b ∂T 0 b 0 ∂T
TX∈Th ! TX∈Th !
≤Chk−1 kuk |||e |||,
k+2 h
where the Hk+2-normofu is usedbecause the estimate in Lemma A.9 is notoptimal
in terms of the mesh parameter h.
The third and fourth terms can be estimated by using the Cauchy-Schwarz in-
equality and the estimates (A.7) and (A.8) in Lemma A.4 as follows
(4.10) h−1h∇Q u·n −Q (∇u·n ),∇e ·n −e i ≤Chk−1kuk |||e |||
T 0 e b e 0 e n ∂T k+1 h
(cid:12) (cid:12)
(cid:12)TX∈Th (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
and (cid:12) (cid:12)
(4.11) h−3hQ Q u−Q u,Q e −e i ≤Chk−1kuk |||e |||.
T b 0 b b 0 b ∂T k+1 h
(cid:12) (cid:12)
(cid:12)TX∈Th (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Substituting(cid:12)(4.8)-(4.11) into (4.6) gives (cid:12)
|||e |||2 ≤Chk−1 kuk |||e |||,
h k+2 h
which implies (4.5) and hence completes the proof.
5. Error Estimates in L2. In this section, we shall establish some error esti-
mates for all three components of the error function e in the standard L2 norm.
h
First of all, let us derive an error estimate for the first component of the error
function e by applying the usualduality argumentin the finite element analysis. To
h
10