A Weak Galerkin Finite Element Scheme for solving the stationary Stokes Equations

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by  Ruishu Wang

A WEAK GALERKIN FINITE ELEMENT SCHEME FOR SOLVING
THE STATIONARY STOKES EQUATIONS
RUISHU WANG∗, XIAOSHEN WANG†, QILONG ZHAI ‡, AND RAN ZHANG§
6 Abstract. AweakGalerkin(WG)finiteelementmethodforsolvingthestationaryStokesequa-
1 tionsintwo-orthree-dimensionalspacesbyusingdiscontinuouspiecewisepolynomialsisdeveloped
0 and analyzed. The variational form we considered is based on two gradient operators which is dif-
2 ferentfromthe usualgradient-divergence operators. TheWGmethod ishighlyflexiblebyallowing
theuseofdiscontinuousfunctionsonarbitrarypolygonsorpolyhedrawithcertainshaperegularity.
n
Optimal-order errorestimates areestablished for the corresponding WG finite element solutions in
a various norms.Numerical results are presented to illustrate the theoretical analysis of the new WG
J
finiteelementschemeforStokes problems.

1
2
Key words. weakGalerkinfiniteelementmethods,weakgradient,Stokesequations,polytopal

meshes.
]
A
N AMS subject classifications. Primary, 65N30, 65N15, 65N12, 74N20; Secondary, 35B45,
35J50,35J35
.
h
t
a
1. Introduction. TheaimofthispaperistopresentanovelweakGalerkinfinite
m
element method for solving the stationary Stokes equations. Let Ω be a polygonal
[ or polyhedral domain in Rd,d = 2,3. As a model for the flow of an incompressible

1 viscous fluid confined in Ω, we consider the following equations
v
0 (1.1) −µ∆u+∇p=f, in Ω,
8
(1.2) ∇·u=0, in Ω,
6
5 (1.3) u=g, on ∂Ω,
0
. for unknown velocity function u and pressure function p (we require that p has zero
1
0 averagein order to guaranteethe uniqueness of the pressure). Boldsymbols are used
6 to denote vector- or tensor-valued functions or spaces of such functions. Here f is a
1 bodysourceterm,µ>0isthekinematicviscosityandg isaboundaryconditionthat
:
v satisfies the compatibility condition
i
X
g·n ds=0,
r
a Z∂Ω
where n is the unit outward normal vector on the domain boundary ∂Ω.
This problem mainly arises from approximations of low-Reynolds-number flows.
The finite element methods for Stokes and Navier−Stokes problems enforce the
divergence-free property in finite element spaces, which satisfy the inf-sup (LBB)
∗DepartmentofMathematics, JilinUniversity,Changchun, China([email protected]).
†Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204,
UnitedStates([email protected]).
‡DepartmentofMathematics, JilinUniversity,Changchun, China([email protected]).
§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.
1
condition, in order for them to be numerically stable [2, 1, 10, 11, 8]. The Stokes
problem has been studied with various different new numerical methods: [4, 12, 13,
22, 23].
Throughout this paper, we would follow the standard definitions for Lebesgue
and Sobolev spaces: L2(Ω), H1(Ω), [L2(Ω)]d,
[H1(Ω)]d ={v∈[H1(Ω)]d :v=0 on ∂Ω}
0
and
L2(Ω):={q ∈L2(Ω): qdx=0}
0
ZΩ
are the natural spaces for the weak form of the Stokes problem [10, 7]. Denote (·,·)
for inner products in the corresponding spaces.
Next we assume that µ=1 and g=0. Then one of the variational formulations
for the Stokes problem (1.1)-(1.3) is to find u∈[H1(Ω)]d and p∈L2(Ω) such that
0 0
(1.4) (∇u,∇v)−(∇·v,p)=(f,v),
(1.5) (∇·u,q)=0,
for all v ∈ [H1(Ω)]d and q ∈ L2(Ω). Here ∇u denotes the velocity gradient tensor
0 0
(∇u) = ∂ u . It is well known that under our assumptions on the domain and the
ij j i
data, problem (1.4)-(1.5) has a unique solution (u;p)∈[H1(Ω)]d×L2(Ω).
0 0
For any p∈L2(Ω), define a functional ∇p such that
0
h∇p,vi=−(∇·v,p), ∀v∈[H1(Ω)]d.
0
It is easy to know that the weak form (1.4)-(1.5) is also equivalent to the following
variational problem: find (u;p)∈[H1(Ω)]d×L2(Ω) such that
0 0
(1.6) (∇u,∇v)+h∇p,vi=(f,v),
(1.7) h∇q,ui=0,
for all v ∈ [H1(Ω)]d and q ∈ L2(Ω). The unique solvability of (1.6)-(1.7) follows
0 0
directly from that of the (1.4)-(1.5).
TheWGmethodreferstoageneralfiniteelementtechniqueforpartialdifferential
equations where differential operators are approximated as distributions for general-
izedfunctions. Thismethod wasfirstproposedin[20,21,15]forsecondorderelliptic
problem, then extended to other partial differential equations [14, 16, 18, 17, 25, 26].
Weak functions and weak derivatives can be approximatedby polynomials with vari-
ous degrees. The WG method uses weak functions and their weak derivatives which
are defined as distributions. The most prominent features of it are:
• Theusualderivativesarereplacedbydistributionsordiscreteapproximations
of distributions.
• The approximating functions are discontinuous. The flexibility of discontin-
uous functions gives WG methods many advantages, such as high order of
accuracy, high parallelizability, localizability, and easy handling of compli-
cated geometries.
2
The abovefeatures motivate the use ofWG methods for the Stokes equations. It
caneasilyhandlemesheswithhangingnodes,elementsofgeneralshapeswithcertain
shaperegularityandideallysuitedforhp-adaptivity. In[19], Wanget. al. considered
WG methods for the Stokes equations (1.4)-(1.5). Similarly, in [17], they presented
WG methods for the Brinkman equations, which is a model with a high-contrast
parameter dependent combination of the Darcy and Stokes models. The numerical
method of [17] is based on the traditional gradient-divergence variational form for
the Brinkman equations. In [24], we presented a new WG scheme based on the
gradient-gradientvariationalform. It is shown that this scheme is suit for the mixed
formulation of Darcy which would present a better approximation for this case. In
fact, forcomplex porousmediawith interfaceconditions,people oftenuse Brinkman-
Stokesinterfacemodeltodescribethisproblem,whichisanongoingworkforusnow.
In order to present a more efficient WG scheme, we prefer to utilize this gradient-
gradient weak form to approximate the model. In order to unify the weak form of
this interface problem, we need the numerical analysis results of this form for Stokes
problem. However, to the best of our knowledge, the numerical analysis of methods
basedonthevariationalform(1.6)-(1.7)hasneverbeendonebefore. Thereforeinthis
paper, we propose a WG method based on the weak form (1.6)-(1.7) of the primary
problem. In addition, if we choose high order polynomials to approximate the model
and use Schur complement to reduce the interior DOF of the velocity and pressure
by the boundary DOF, the total DOF of this new method is less than the scheme of
[19].
Therestofthispaperisorganizedasfollows. InSection2weshallintroducesome
preliminariesandnotationsforSobolevspaces. Section3isdevotedtothe definitions
ofweakfunctionsandweakderivatives. TheWGfiniteelementschemesforvariational
form of the Stokes equation (1.6)-(1.7) are presented in Section 4. This section also
contains some local L2 projection operators and then derives some approximation
propertieswhichareusefulinaconvergenceanalysis. InSection5,wederiveanerror
equationforthe WGfinite elementapproximation. Optimal-ordererrorestimates for
the WG finite element approximations are derived in Section 6 in an H1-equivalent
normforthe velocity,andL2 normforboth the velocity andthe pressure. InSection
7, we present some numerical results which confirm the theory developed in earlier
sections. Finally, we present some technical estimates in the appendix for quantities
related to the local L2 projections into various finite element spaces.
2. Preliminaries and Notations. Let K ⊂ Ω be an open bounded domain
withLipschitz continuousboundaryinRd,d=2,3. We shalluse standarddefinitions
of the Sobolev spaces Hs(K) and inner products (·,·) , their norms k·k , and
s,K s,K
seminorms |·| , for any s ≥ 0. For instance, for any integer s ≥ 0, the seminorm
s,K
|·| is defined as
s,K
1
2
|v| = |∂αv|2dK ,
s,K
 
|αX|=sZK
 
with notations
d
α=(α1,α2,··· ,αd), |α|=α1+α2+···+αd, ∂α = ∂xαjj.
j=1
Y
3
The Sobolev norm k·k is defined as
m,K
1
m 2
kvk = |v|2 .
m,K  j,K
j=0
X
 
The space H0(K) is same as L2(K), whose norm and inner product are denoted
by k·k and (·,·) , respectively. If K = Ω, we would drop the subscript K in the
K K
notations of the L2 norm and the L2 inner product.
3. Weak Differential Operators. In this section we will define weak func-
tions for both the vector-valued function and the scalar-valued function, also we will
introduce the weak gradients and the corresponding discrete forms.
3.1. Weakgradientforweakvector-valued function. LetT beapolygonal
or polyhedral domain with boundary ∂T. A weak vector-valued function on the
domainT isdefined asv={v ,v } suchthat v ∈[L2(T)]d andv ∈[L2(∂T)]d. Let
0 b 0 b
V(T)={v={v ,v };v ∈[L2(T)]d,v ∈[L2(∂T)]d},
0 b 0 b
where v is not necessarily the trace of v .
b 0
Definition 3.1. ([19]) For any v ∈ V(T), the weak gradient of v, denoted by
∇ v, is defined as a linear functional in the dual space of [H1(T)]d×d whose action
w
on each τ ∈[H1(T)]d×d is given by
(∇ v,τ) =−(v ,∇·τ) +hv ,τ ·ni , ∀τ ∈[H1(T)]d×d,
w T 0 T b ∂T
where n is the outer unit normal vector to ∂T, (v ,∇·τ) is the L2 inner product of
0 T
v and ∇·τ, and hv ,τ ·ni is the inner product of τ ·n and v in [L2(∂T)]d.
0 b ∂T b
Consider the inclusion map i :[H1(T)]d →V(T) defined below
V
i (φ)={φ| ,φ| }, φ∈[H1(T)]d.
V T ∂T
By this map the Sobolev space [H1(T)]d can be embedded into the space V(T).
Withthehelpofmapi ,theSobolevspace[H1(T)]d canbeconsideredasasubspace
V
of V(T) by identifying each φ∈[H1(T)]d with i (φ).
V
Let P (T) be the set of polynomials on T with degree no more than r.
r
Definition 3.2. ([19]) The discrete weak gradient operator ∇ is defined as
w,r,T
follows: for each v∈V(T), ∇ v∈[P (T)]d×d is the unique element such that
w,r,T r
(3.1) (∇ v,τ) =−(v ,∇·τ) +hv ,τ ·ni , ∀τ ∈[P (T)]d×d.
w,r,T T 0 T b ∂T r
3.2. Weak gradient for weak scalar-valued function. We define a weak
scalar-valued function on the domain T as q = {q ,q } such that q ∈ L2(T) and
0 b 0
q ∈L2(∂T). Let
b
W(T)={q ={q ,q };q ∈L2(T),q ∈L2(∂T)},
0 b 0 b
4
where q is not necessarily the trace of q .
b 0
Definition 3.3. ([20]) For any q ∈ W(T), the weak gradient of q, denote by
∇ q, is defined as a linear functional in the dual space of [H1(T)]2 whose action on
w
each w∈[H1(T)]2 is given by
e
(∇ q,w) =−(q ,∇·w) +hq ,w·ni , ∀w ∈[H1(T)]2,
w T 0 T b ∂T
where n is the outer unit normal vector to ∂T, (q ,∇·w) is the L2 inner product
e 0 T
of q and ∇·w, and hq ,w·ni is the inner product of w·n and q in L2(∂T).
0 b ∂T b
Consider the inclusion map i :H1(T)→W(T) defined as follows
W
i (φ)={φ| ,φ| }, φ∈H1(T).
W T ∂T
By which the Sobolev space H1(T) is embedded into the space W(T). With the help
of map i , the Sobolev space H1(T) can be considered as a subspace of W(T) by
W
identifying each φ∈H1(T) with i (φ).
W
Definition 3.4. ([20]) The discrete weak gradient operator ∇ is defined as
w,r,T
follows: for each q ∈W(T), ∇ q ∈[P (T)]d is the unique element such that
w,r,T r
e
(3.2) (∇w,r,Tq,w)T =−(eq0,∇·w)T +hqb,w·ni∂T, ∀w ∈[Pr(T)]d.
e
4. A Weak Galerkin Finite Element Scheme. Let T be a partition of the
h
domain Ω into polygons in 2D or polyhedral in 3D. Assume that T is shape regular
h
in the sense as defined in [18]. 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. Denote by h the
h h T
diameter of T ∈T and h=max h the meshsize for the partition T .
h T∈Th T h
For any interger k ≥1, we define weak Galerkin finite element spaces as follows:
for velocity variable, let
V ={v={v ,v };{v ,v }| ∈[P (T)]d×[P (e)]d,e⊂∂T,v =0 on ∂Ω}.
h 0 b 0 b T k k b
It should be noticed that v is single valued on each edge e ⊂ E . For pressure
b h
variable, we define
W = q ={q ,q }; q dT =0,{q ,q }| ∈P (T)×P (e),e⊂∂T .
h 0 b 0 0 b T k−1 k
( TX∈ThZT )
Also q is single valued on each edge e⊂E .
b h
The discrete weak gradients ∇ and ∇ on the spaces V and W can be
w,k−1 w,k h h
computed by the equations (3.1) and (3.2) on each element T respectively, that is,
e
(∇ v)| =∇ (v| ), ∀v∈V ,
w,k−1 T w,k−1,T T h
(∇ q)| =∇ (q| ), ∀q ∈W .
w,k T w,k,T T h
For the sake of simplicity, we shall drop the subscripts k −1 and k of ∇ and
e e w,k−1
∇ in the rest of the paper.
w,k
5
e
We use the L2 inner product to denote the sum of inner products on each of the
elements as follows:
(∇ v,∇ w)= (∇ v,∇ w) ,
w w w w T
TX∈Th
(∇ q,v)= (∇ q,v) .
w w T
TX∈Th
e e
Lemma 4.1. ([19]) For any v∈V and p∈W the following equations hold true
h h
(4.1) (∇ v,τ) =(∇v ,τ) −hv −v ,τ ·ni , ∀τ ∈[P (T)]d×d,
w T 0 T 0 b ∂T k−1
(4.2) (∇ p,w) =(∇p ,w) −hp −p ,w·ni , ∀w ∈[P (T)]d.
w T 0 T 0 b ∂T k
e
For eachelementT ∈T , denote by Q the L2 projectionoperatorfrom[L2(T)]d
h 0
onto [P (T)]d. For each edge or face e ∈ E , denote by Q the L2 projection from
k h b
[L2(e)]d onto[P (e)]d.WeshallcombineQ withQ asaprojectionontoV ,suchthat
k 0 b h
on each element T ∈T
h
Q u={Q u,Q u}.
h 0 b
On eachelement T ∈T , denote by Q the L2 projectiononto[P (T)]d×d. Denote
h h k−1
by Q the L2 projection operator from L2(T) onto P (T). For each edge or face
0 k−1
e∈E , denote by Q the L2 projection from L2(e) onto P (e). We shall combine Q
h b k 0
witheQ as a projection onto space W , such that on each element T ∈T
b h h
e e
Q q ={Q q,Q q}.
e h 0 b
e e e
Thenweshallpresentausefulpropertywhichindicatesthediscreteweakgradient
operators are good approximation to the gradient operators in the classical sense.
Lemma 4.2. ([19]) The following equations hold true.
(4.3) ∇ Q v=Q ∇v, ∀v∈[H1(Ω)]d,
w h h
(4.4) ∇ Q p=Q ∇p, ∀p∈H1(Ω).
w h 0
e e
Now we introduce four bilinear forms as follows:
(4.5) s(w,v)= h−1hw −w ,v −v i ,
T 0 b 0 b ∂T
TX∈Th
(4.6) a(w,v)=(∇ w,∇ v)+s(w,v),
w w
(4.7) b(w,q)=(w ,∇ q),
0 w
(4.8) c(ρ,q)= h hρ −ρ ,q −q i .
T 0 b 0 b ∂T
e
TX∈Th
Using these bilinear forms we define the following two norms. For any v ∈ V0 and
h
q ∈W ,
h
(4.9) |||v|||2 =a(v,v)=(∇ v,∇ v)+ h−1hv −v ,v −v i ,
w w T 0 b 0 b ∂T
TX∈Th
6
and
(4.10) |||q|||2 =kq k2+|||q|||2,
0 0 ∗
where |||q|||2 =c(q,q) is a seminorm.
∗
It is easy to verify that |||·||| and |||·||| are norms in V and W , respectively,
0 h h
Weak Galerkin Algorithm 1. A numerical approximation for (1.1)-(1.3)
can be obtained by seeking u ={u ,u }∈V and p ={p ,p }∈W such that
h 0 b h h 0 b h
(4.11) a(u ,v)+b(v,p )=(f,v ),
h h 0
(4.12) b(u ,q)−c(p ,q)=0,
h h
for all v={v ,v }∈V and q ∈W .
0 b h h
Next we shallshow that the weak Galerkinfinite element algorithm(4.11)-(4.12)
has only one solution. Since the system is linear, it suffices to show that if f =0, the
only solution is u ={0,0};p ={0,0}.
h h
Lemma 4.3. TheWGfiniteelementscheme(4.11)-(4.12)hasauniquesolution.
Proof. Let f = 0, we shall show that the solution of (4.11)-(4.12) is trivial. To
this end, taking v=u and q =p and subtracting (4.12) from (4.11) we arrive at
h h
a(u ,u )+c(p ,p )=0.
h h h h
By the definition of a(·,·) and c(·,·), we know ∇ u = 0 on each T ∈ T , u = u ,
w h h 0 b
and p =p on each ∂T. Thus u and q are continuous.
0 b 0 0
By (4.1) and the fact that u =u on ∂T we have, for any τ ∈[P (T)]d×d,
b 0 k−1
0=(∇ u ,τ)
w h T
=(∇u ,τ) −hu −u ,τ ·ni
0 T 0 b ∂T
=(∇u ,τ) ,
0 T
which implies ∇u = 0 on each T ∈T and thus u is a constant. Since u = u on
0 h 0 0 b
each ∂T and u = 0 on ∂Ω, we arrive at u = {0,0} in Ω. It follows from (4.11),
b h
u ={0,0}, and f =0 that for any v∈V ,
h h
0=b(v,p )
h
=(v ,∇ p )
0 w h
= (v ,∇p ) − hv ·n,p −p i
0 0 T 0 0 b ∂T
e
TX∈Th TX∈Th
= (v ,∇p ) .
0 0 T
TX∈Th
Hencewehave∇p =0oneachT ∈T . Thusp isaconstantinΩ. Fromp ∈L2(Ω),
0 h 0 0 0
we would obtain p =0 in Ω. Since p =p on each ∂T, p =0.
0 b 0 b
This completes the proof of the lemma.
7
5. Error Equation. In this section, we shall derive the error equations for the
WG finite element solutionwe getfrom(4.11)-(4.12). This errorequationis essential
for the following analysis.
Now we define two bilinear forms
(5.1) l (w,v)= hv −v ,(∇w−Q ∇w)·ni ,
1 0 b h ∂T
TX∈Th
(5.2) l (w,q)= hq −q ,(w−Q w)·ni ,
2 0 b 0 ∂T
TX∈Th
for all w ∈[H1(Ω)]d,v∈V and q ∈W .
h h
Let (u;p) be the exact solution of (1.1)-(1.3), and (u ;p ) ∈ V ×W be the
h h h h
solution of (4.11)-(4.12).
Define
e =Q u−u , ε =Q p−p .
h h h h h h
We shall derive the error equations that e ∈V and ε ∈W satisfy.
h h e h h
Lemma 5.1. Let u ∈ V and p ∈ W be the solution of the numerical scheme
h h h h
(4.11)-(4.12), and (u;p) be the exact solution of (1.1)-(1.3). Then, for any v ∈ V
h
and q ∈W we have
h
(5.3) a(e ,v)+b(v,ε )=s(Q u,v)+l (u,v),
h h h 1
(5.4) b(e ,q)−c(ε ,q)=l (u,q)−c(Q p,q).
h h 2 h
Proof. First, from (4.1) and the property (4.3) weeobtain
(∇ Q u,∇ v) =(Q ∇u,∇ v)
w h w T h w T
=(∇v ,Q ∇u) −hv −v ,(Q ∇u)·ni
0 h T 0 b h ∂T
=(∇v ,∇u) −hv −v ,(Q ∇u)·ni .
0 T 0 b h ∂T
Summing over all elements T ∈T , we have
h
(5.5) (∇ Q u,∇ v)=(∇u,∇v )− hv −v ,(Q ∇u)·ni .
w h w 0 0 b h ∂T
TX∈Th
From the commutative property (4.4) we arrive at
(5.6) b(v,Q p)=(v ,∇ Q p)=(v ,∇p).
h 0 w h 0
It follows from (4.2) that
e e e
(Q u,∇ q) =(∇q ,Q u) −hq −q ,Q u·ni
0 w T 0 0 T 0 b 0 ∂T
=(∇q ,u) −hq −q ,Q u·ni .
0 T 0 b 0 ∂T
e
Summing over all T ∈T yields
h
b(Q u,q)=(Q u,∇ q)
h 0 w
(5.7)
=(u,∇q )− hq −q ,Q u·ni .
0e 0 b 0 ∂T
TX∈Th
8
Next, using v in v={v ,v }∈V to test (1.1), we have
0 0 b h
(−∆u,v )+(v ,∇p)=(f,v ).
0 0 0
Integrating by parts, we obtain
(5.8) (∇u,∇v )+(v ,∇p)=(f,v )+ h∇u·n,v −v i ,
0 0 0 0 b ∂T
TX∈Th
wherewehaveusedthefactthat hv ,∇u·ni =0. Usingq inq ={q ,q }∈
T∈Th b ∂T 0 0 b
W to test (1.2), we arrive at
h
P
(∇·u,q )=0.
0
Using the fact that hu·n,q i =0 one has
T∈Th b ∂T
P
0=(∇·u,q )
0
=−(u,∇q )+ hu·n,q i
0 0 ∂T
(5.9)
TX∈Th
=−(u,∇q )+ hu·n,q −q i .
0 0 b ∂T
TX∈Th
Finally combining the equations (5.5) and (5.6) with (5.8) yields
a(Q u,v)+b(v,Q p)=(∇ Q u,∇ v)+s(Q u,v)+(v ,∇ Q p)
h h w h w h 0 w h
=(f,v )+ hv −v ,(∇u−Q ∇u)·ni
(5.10) e 0 0 b h e e ∂T
TX∈Th
+s(Q u,v).
h
Substituting it into (4.11), then we would have
a(e ,v)+b(v,ε )=s(Q u,v)+l (u,v).
h h h 1
Combining the equations (5.7) with (5.9) we arrive at
b(Q u,q)−c(Q p,q)=(∇q ,u)− hq −q ,Q u·ni −c(Q p,q)
h h 0 0 b 0 ∂T h
TX∈Th
e e
= hu·n,q −q i − hq −q ,Q u·ni
0 b ∂T 0 b 0 ∂T
(5.11) TX∈Th TX∈Th
−c(Q p,q)
h
= hq −q ,(u−Q u)·ni −c(Q p,q).
e0 b 0 ∂T h
TX∈Th
e
Substituting (5.11) into (4.12) yields the following error equation
b(e ,q)−c(ε ,q)=l (u,q)−c(Q p,q),
h h 2 h
for all q ∈Wh, which completes the proof of (5.4). e
9
6. ErrorEstimates. Inthissectionweshallpresenttheerrorestimatesbetween
the exact solution of (1.1)-(1.3) and the numerical solution of WG finite element
method (4.11)-(4.12). The two norms |||·||| and |||·||| are essentially H1 norm and
0
L2 norm on V and W respectively. In this section we always assume T is shape
h h h
regular ([18]).
Theorem 6.1. Let (u,p) be the exact solution of (1.1)-(1.3), (u ,p ) be the
h h
numerical solution of (4.11)-(4.12), then the following error estimates hold true
(6.1) |||e |||+|||ε ||| ≤Chk(kuk +kpk ),
h h ∗ k+1 k
(6.2) kε k≤Chk(kuk +kpk ),
0 k+1 k
and consequently, one has
(6.3) |||e |||+|||ε ||| ≤Chk(kuk +kpk ).
h h 0 k+1 k
Proof. Letting v=e in (5.3) and q =ε in (5.4), we would obtain
h h
|||e |||2+|||ε |||2 =s(Q u,e )+l (u,e )
h h ∗ h h 1 h
−l (u,ε )+c(Q p,ε ).
2 h h h
Then from (8.5)-(8.8) we arrive at
e
|||e |||2+|||ε |||2 ≤Chk(kuk +kpk )(|||e |||+|||ε ||| ),
h h ∗ k+1 k h h ∗
from which we would have
|||e |||+|||ε ||| ≤Chk(kuk +kpk ).
h h ∗ k+1 k
For any given ρ∈W ⊂L2(Ω), it follows from [19, 6, 7, 5, 10, 11] that there is a
h 0
v∈[H1(Ω)]d such that
0
(∇·v,ρ)
(6.4) ≥Ckρk,
e kvk
1
where C is a positive constant which ies dependent only on Ω. Let v=Q v∈V , we
h h
claim that the following inequality hoelds true
(6.5) |||v|||≤Ckvk , e
1
where C is a constant.
e
From (4.3), we have
k∇ vk2 = k∇ (Q v)k2 = kQ (∇v)k2 ≤k∇vk2.
w T w h T h T
TX∈Th TX∈Th TX∈Th
It follows from the definition of Q , (8.1e), and (8.4) that e e
b
h−1kv −v k2 = h−1kQ v−Q vk2
T 0 b ∂T T 0 b ∂T
TX∈Th TX∈Th
= h−1kQ (eQ v)−eQ vk2
T b 0 b ∂T
TX∈Th
≤ h−1kQ v−evk2 e
T 0 ∂T
TX∈Th
≤Ck∇vk2, e e
10
e