J. Numer. Math., Vol. 0, No. 0, pp. 1–20 (2003)
c VSP 2003
Prepared using jnm.sty [Version: 02.02.2002 v1.2]
Regularity estimates for elliptic boundary value problems
with smooth data on polygonal domains
C. BACUTA, J. H. BRAMBLE †, and J. Xu Received 1 January, 2003
Received in revised form 1 March, 2003
Communicated by R. Lazarov
Abstract — We consider the model Dirichlet problem for Poisson’s equation on a plane polygonal
convex domain Ω with data f in a space smoother than L2 . The regularity and the critical case of the
problem depend on the measure of the maximum angle of the domain. Interpolation theory and multilevel theory are used to obtain estimates for the critical case. As a consequence, sharp error estimates
for the corresponding discrete problem are proved. Some classical shift estimates are also proved using the powerful tools of interpolation theory and mutilevel approximation theory. The results can be
extended to a large class of elliptic boundary value problems.
Keywords: interpolation spaces, finite element method, multilevel decomposition, shift theorems,
subspace interpolation
1. INTRODUCTION
Regularity estimates of the solutions of elliptic boundary value problems in terms
of Sobolev norms of fractional order are known as shift theorems or shift estimates.
The shift estimates for the Laplace operator with Dirichlet boundary conditions with
non-smooth data on polygonal domains are well known (see, e.g, [21], [23],[27],
[2]). The classical regularity estimate for the case when Ω is a convex polygonal
domain in 2 , with boundary ∂ Ω, is as follows: If u is the variational solution of
∆u f in
u 0 on
Ω
∂ Ω
(1.1)
H 0 Ω L2 Ω (1.2)
then, u H 2 Ω and
u
H2
C f
H0
for all f
Let ω be the radian measure of the largest corner of Ω ( ω
π ), and let
Dept. of Mathematics, The Pennsylvania State University University Park, PA 16802, USA
† Dept. of Mathematics, Texas A & M University, College Station, TX 77843, USA
The work of the second author was supported under NSF Grant No. DMS-9973328. The work of
the first and the third authors was supported under NSF Grant No.DMS-0209497.
2
C. Bacuta, J. H. Bramble, J. Xu
s0 min 1 π ω 1 . If u is the variational solution of (1.1), then for 0 s s0 , it
is known (see e.g. [16]) that
s C s f
H s Ω (1.3)
s
1
2
Here, H Ω is the interpolation space between H Ω and L Ω s distance from
L2 Ω . In this paper we will prove a stronger version of (1.3) and approach the
problem in the critical value situation s s0 . We prove that, for s0 1, the following
u
estimate holds.
H2
Hs
for all f
for all f Bs10 Ω (1.4)
Ω c f Bs0 Ω 1
where B2∞ s0 Ω and Bs10 Ω are standard interpolation spaces defined in Section 3.
It is worth noting here that the space Bs10 Ω contains all the Hilbert spaces H s with
s s0 . The estimate (1.4) leads to new finite element convergence estimates. The
u
2 s0
B∞
method presented in this paper can be extended to other boundary conditions such
as Neumann or mixed Neumann-Dirichlet conditions.
The technique involved in proving shift results is the real method of interpolation of Lions and Peetre ([3], [24] and [25]), combined with multilevel approximation theory. The cases q 2, q 1 and q ∞, where q is the second index
of interpolation, are of special importance for our problem. The following type of
interpolation problem is essential for our approach. If X and Y are Sobolev spaces
of integer order and XK is a subspace of codimension one of X , then how can one
characterize the interpolation spaces between XK and Y for q 2 and q ∞? The
problem was studied in [21], [20] and [1] for q 2 and particular spaces X and XK .
This paper gives an interpolation result of this type for the case q ∞ .
The remaining part of the paper is structured as follows. The interpolation results presented in Section 2 give a formulas for the norms on the intermediate subspaces XK Y s 2 and XK Y s ∞ when XK is of codimension one. In Section 3 the
main result concerning the codimension-one subspace interpolation problem is presented. Shift theorems for the Poisson equation on polygonal domains are considered in Section 4. In the last section, a straightforward application of the interpolation results is shown to lead to some new estimates for finite element approximations.
2. INTERPOLATION RESULTS
In this section we give some definitions and results concerning interpolation spaces
via real method of interpolation of Lions and Peetre (see [3], [4]), [24]).
2.1. Interpolation between Banach spaces
Let X Y be Banach spaces satisfying for some positive constant c,
X
is a dense
subset of Y and
u Y c u X for all u X (2.1)
New regularity estimates
3
where u X and u Y are the norms on X and Y respectively.
The interpolation spaces X Y s q , 1 q ∞ and s 0 1 are defined using the
K function, where for u Y and t 0,
K t u :
inf
u0 X
Then, for q ∞, the space X Y ∞
0
and X Y u 2 t 2 u u 2 1 2 0 X 0 Y
consists of all u Y such that
sq
t s K t u q dt ∞ t
consists of all u Y such that
sup t 2s K t u 2 ∞ t 0
The norm on X Y s q is defined by
∞
2
t s K t u q dt u X Y s q :
s∞
and the norm on X Y 0
t
is defined by
u X Y s ∞ : sup t s K t u t 0
for all u Y , the interval 0 ∞ in the
Remark 2.1. Since K t u t u Y
above definitions can be replaced by any subinterval A ∞ . The new norms obs∞
tained are equivalent with the original norms.
2.2. Interpolation between Hilbert spaces
An extended Hilbert interpolation theory can be found in [24]. For completeness
and consistence of notation we present in this section the interpolation results used
in this paper.
Let X Y be separable Hilbert spaces with inner products ! ! X and ! ! Y , respectively, and satisfying (2.1). Let D S denote the subset of X consisting of all
elements u such that
v " u v X v X
(2.2)
is continuous in the topology induced by Y .
For any u in D S the anti-linear form (2.2) can be uniquely extended to a continuous
anti-linear form on Y . Then by Riesz representation theorem, there exists an element
Su in Y such that
u v Su v
X
Y
for all v X (2.3)
In this way S is a well defined operator with domain D S in Y . The next result
illustrates the properties of S (see [24]).
4
C. Bacuta, J. H. Bramble, J. Xu
Proposition 2.1. The domain D S of the operator S is dense in X and consequently D S is dense in Y . The operator S : D S $# Y " Y is a bijective, selfadjoint and positive definite operator. The inverse operator S 1 : Y " D S %# Y is
a bounded symmetric positive definite operator and
S 1 z u z u
X
Y
for all z y u X (2.4)
The next lemma provides the relation between K t u and the connecting operator S. A proof of the lemma is given in [2].
Lemma 2.1. For all u Y and t
0,
K t u 2 t 2 I t 2 S 1 1 u u Y For the special case q 2 (X, Y Hilbert spaces), due to the spectral or multilevel
representation of the norm on X Y s 2 , the definition of the norm is slightly changed
as follows:
2
u
2
s 2 : cs
XY
where
∞
cs : 0
∞
0
t
t 1 2s
dt
t2 1
2s 1 K t u 2 dt 1 2
With the new definition for the norm of X Y representation case in Section 3) to define
2 sin π s&
π
it is natural (see the multilevel
s2
X Y 0 2 : X and X Y 1 2 : Y Remark 2.2. Lemma 2.1 yields other expressions for the norms on X Y X Y s ∞ . Namely,
2
u
and
2
X Y s 2 cs
2
u
∞
0
t
2s 1
I t 2 S 1 1 u u dt Y
t 2 1 s I t 2 S 1 ' 1 u u Y s ∞ sup
t 0
XY
and
s2
(2.5)
(2.6)
2.3. Interpolation between subspaces of a Hilbert space
span ϕ be a one-dimensional subspace of X and let X) be the orthogLet (
in X in the ! ! X inner product. We are interested in deonal complement of (
termining the interpolation spaces of X) and Y , where on X) we consider again
the ! ! X inner product. To apply the interpolation results from the previous section
New regularity estimates
5
we need to check that the density part of the condition (2.1) is satisfied for the pair
X) Y .
For ϕ *( , define the linear functional Λϕ : X " C, by
u ϕ u X X
Λϕ u : The following result is an extension of Kellogg’s lemma [21]. The proof can be
found in [1].
Lemma 2.2. Let (
be a closed subspace of X and let X) be the orthogonal
complement of (
in X in the ! ! X inner product. The space X) is dense in Y if
and only if the following condition is satisfied:
Λϕ is not bounded in the topology of Y
for all ϕ +(, ϕ - 0
(2.7)
For the remaining part of this section we assume that Λϕ is not bounded in the
topology of Y, so the condition (2.1) is satisfied for the pair X). Y . We denote X)
by Xϕ . It follows from the previous section that the operator Sϕ : D Sϕ /# Y " Y
defined by
u v Sϕ u v
for all v Xϕ (2.8)
X
Y
has the same properties as S. Consequently, the norms on the intermediate spaces
Xϕ Y s 2 and Xϕ Y s ∞ are given by:
2
u
2
s 2 cs
Xϕ Y
∞
0
t
2s 1
I t 2 Sϕ 1 1 u u dt Y
(2.9)
and
2
u
t 2 1 s I t 2 Sϕ 1 ' 1 u u Y s ∞ : sup
t 0
Xϕ Y
(2.10)
Our aim in this section is to determine sufficient conditions for ϕ such that the norm
Xϕ Y s q (for q 2 and q ∞) can be compared with more familiar intermediate
norms which are independent of ϕ . First, we note that the operators Sϕ and S are
related by the following identity:
Sϕ 1
I Qϕ S 1 (2.11)
where Qϕ : X "0( is the orthogonal projection onto ( span ϕ . The proof of
(2.11) follows easily from the definitions of the operators involved.
Next, (2.11) leads to a new formula for the norms on Xϕ Y s ∞ and X Y s 2 .
Theorem 2.1. For any u Y we have,
2
u
2
u X Y c2s
Xϕ Y s 2 s2
∞
0
t
2s 3
u ϕ 2
Y t
dt ϕ ϕ
Xt
(2.12)
6
C. Bacuta, J. H. Bramble, J. Xu
and
2
u
t 2
s ∞ sup
t 0
2s
Xϕ Y
u u
4
t
u ϕ Y t
ϕ ϕ
2s
Yt
2
(2.13)
Xt
In particular for u Xϕ we have
2
u
u 2 X Y s 2 c2s
Xϕ Y s 2
∞
0
u ϕ 2
Xt
t 2s 1 dt
ϕ ϕ Xt
and
2
u
t 2
s ∞ sup
t 0
2s
Xϕ Y
where
and
u v
Xt
u v
Yt
u u
t
Yt
2s
u ϕ 2
Xt
ϕ ϕ
Xt
(2.14)
(2.15)
:
I t 2S 1 1 u v
X
for all u v X (2.16)
:
I t 2S 1 1u v
Y
for all u v Y (2.17)
Proof. The first two formulas follows immediately from Remark 2.2 and the
following identity
I t 2 S 1 ' 1 u u
ϕ
Y
u ϕ 2
Y t
u u Y t t 2 ϕ ϕ Xt
The proof of the identity is based on (2.11). A detailed proof of it can be found in
[2]. Using (2.4) and a simple manipulation of the operator S we get
t 2 u ϕ u ϕ X u ϕ X t Yt
which, for for u Xϕ leads to (2.14) and (2.15).
1
Theorem 2.1 can be easily extended to the case when K is of finite dimension(see [1]). Such an extension would be needed, for example, to treat the case of
mixed Neumann-Dirichlet conditions or the case of the biharmonic problem.
3. MULTILEVEL REPRESENTATION OF INTERPOLATION SPACES
Let Ω be a domain in
2
with boundary ∂ Ω. Assume that
M1 # M2 #2'# Mk
#3
New regularity estimates
7
is a sequence of finite dimensional subspaces of H 1 Ω whose union is dense in
H 1 Ω , and assume that an equivalent norm on H 1 Ω is given by
u
1
∑ λk Q k Q k 1 u
k4 1
∞
:
1 2
2
where Qk denotes the L2 Ω orthogonal projection onto Mk ,
(3.1)
! !
, Q0 0
L2 Ω and λk 4k 1 .
The sequence Mk can be taken for example the standard sequence of piecewise linear functions associated with a sequence of nested meshes. Proofs for the
multilevel representation of the norm on H 1 , for specific choices of the spaces Mk
can be found in [1], [10], [28] and [30] .
3.1. Scales of multilevel norms
On H 1 Ω we consider the norm given by (3.1) and define H 1 Ω to be the dual of
H 1 Ω . The elements of L2 Ω can be viewed as continuous linear functionals on
H 1 Ω and we have the natural continuous and dense embeddings
H 1 Ω 5# L2 Ω 6# H 1
Ω The projection Qk , k 1 2 , can be extended to be defined on H 1
Ω by
Q u v 2 u v+7 u H 1 Ω v M k
k
L
where ! ! on the right hand side represents the duality between H 1 Ω and H 1 Ω .
One can easily check that the induced inner product on H ε Ω is given by
∞
u v ε : ∑ λ ε q u q v
for all u v H ε Ω ε 1 1 k
k
k
k4 1
where qk Qk Qk 1 .
Then the pair H 1 Ω L2 Ω satisfies the condition (2.1) and the operator S
associated with the pair is given by
Su ∞
∑ λk qk u
4
for all u D S k 1
Thus, for X
H 1 Ω and Y L2 Ω , we have
u v
∞
λ
k
q u q v u v Y Y t ∑
k
2 k
λ
t
k4 1 k (3.2)
8
C. Bacuta, J. H. Bramble, J. Xu
and
∞
u v
λ2
k
q u q v u v X Xt ∑
k
2 k
λ
t
k4 1 k For any s 0 1 , q 1 or q ∞, let
H 2 s Ω : 8 H 3 Ω H 1 Ω 9 1 s 2
H s Ω : 8 H 1 Ω L2 Ω 9
1 s2
Bsq Ω :: H 1 Ω L2 Ω 9
1 sq
and
Bq2 s Ω : 8 H 3 Ω H 1 Ω 9 1 s q By using (2.5) and (2.6), one can easily check that
u
and
∞
λks qk u 2 k∑
4 1
2
Hs Ω
(3.3)
∞
λk 2s
sup
t
qk u 2 (3.4)
∑
2
t 0
k 4 1 λk t
One can verify using the above formula that B0∞ Ω 5 L2 Ω and B1∞ Ω H 1 Ω .
u
2
Bs∞ Ω
In addition we have
sup
t 0
t 2s
λk t 2
ss 1 s 1 s λk 1 s which leads to
u
1
s s 1 s 2
Bs∞ Ω
s
u
for all u H s Ω 2
Hs Ω
(3.5)
a well known embedding property.
Remark 3.1. If we assume that the sequence of subspaces Mk is chosen so
that the following approximation and inverse inequalities hold with a constant c
independent
of j.
(Ap)
(Inv)
u Q ju c 2 j u
u
c 2j u
H1
H1
for all u H 1 Ω ,
for all u M j ,
New regularity estimates
9
then, by a well known approximation theory result, an equivalent norm on H 1 L2
is given by
sj 2 q j u j; 1 lq 0 s 1 1 q ∞ In particular, an equivalent norm on Bs1 : : H 1 L2
∞
∑ 2s j
4
q ju
1 sq
is given by
1 s1
0 s 1
j 1
One can verify now that we have the following norm-inequality:
u
Bs1
cε 1 2 u H s ε for all u H s
ε
(3.6)
with c independent of ε .
3.2. Subspace interpolation results
Next, we focus our attention on a specific case of subspace interpolation associated
with the pair X Y , where X H 1 Ω and Y L2 Ω . For a fixed s0 in the interval
0 1 , let ϑ 1 s and ϕ H 1 Ω . By the Riesz representation theorem there
0
0
exists a function ϕ H 1 Ω such that
∞
v ϕ v ϕ ∑ λ q ϕ v97 v H 1 Ω 1
k k
k4 1
Since qi q j
0 for i - j, we deduce that in fact
qk ϕ
λk 1 qk ϕ
(3.7)
Next, we assume that the function ϕ satisfies the following condition:
(C) There exist two positive constants c1 c2 such that
c1 λks0
qk ϕ
2
c2 λks0 k 1 2&
Note that the above condition is equivalent to
c1 λk
ϑ0
qk ϕ
2
1
c2 λk ϑ0 k 1 2&
Lemma 3.1. Let ϕ H 1 Ω satisfy (C) and let ϕ be the corresponding H 1 representation. Then the following conditions are also satisfied.
(C.0) Hϕ1 is dense in H 1 L2
1 s
for s s0 . (Here, Hϕ1 is the kernel of ϕ . )
10
C. Bacuta, J. H. Bramble, J. Xu
(C.1) There exist two positive constants c1 c2 such that
c1 t 2ϑ0
λ
ϕ ϕ X t ∑ λ k t 2 qk ϕ 2 c2t k4 1 k ∞
2
2ϑ0
f or t 1
Proof. The constants involved in this proof might change at different occurrences. To prove (C.0), by Lemma 2.2, it is enough to show that the functional
u ϕ u ϕ u H 1 Ω 1
u"
(3.8)
is not continuous in the topology induced by H s Ω , s s0 . Let un be the sequence in H 1 Ω defined by
n
∑ λk s
un : 4
0
qk ϕ k 1
Then,
n
un ϕ un ϕ ∑ λ s0 q ϕ 2 " ∞ as n " ∞ 1
k
k
k4 1
On the other hand
un
2
Hs
n
∑ λks k4 1
2s0
qk ϕ
2
is uniformly bounded. Therefore, the functional defined in (3.8) is not continuous
and (C.0) is proved.
To estimate ϕ ϕ X t we observe that qk ϕ 2 λk 1 qk ϕ 21 and that
∞
λk1 ϑ0
t2
∑λ
4
k 1
k
t
2ϑ0
4 t ∑ 4k t 2 1 k4 1
∞
k
2 1 ϑ0
Using the standard convergence criteria for sums via integrals, the last sum can be
estimated below and above by constants which are independent of t.
1
Next theorem is the main subspace interpolation result of the paper.
Theorem 3.1. Let ϕ
H 1 Ω satisfy (C). Then
Hϕ1 L2 1 s 2 8 H 1 L2 1 s 2 : H s Ω for s s0 and
Bs10 : : H 1 L2
#8 Hϕ1 L2 1 s0 ∞ 1 s0 1
(3.9)
(3.10)
New regularity estimates
11
Proof. Recall that L2 Ω < Y and H 1 Ω = X . In order to prove (3.9) it is
enough (by the density property (C.0), (2.14) and Remark 2.1) to prove that
u ϕ 2
Xt
dt t 2 1 s > 1> I :
ϕ ϕ Xt
1
for u H s Ω denote ũk : λks 2 qk u
u Hs ∞
Here ! for all u Xϕ 2
Hs
c u
(3.11)
and ũ : 3 ũk . Then we have
ũ
l2
! is simply the L2 Ω inner product. Then, we have
u ϕ Xt
∞
I t 2 S 1 ' 1 u ϕ
For u Xϕ we have u ϕ X
λ
∑ k 2 qk u qk ϕ X X
k 4 1 λk t
0. Then
∞
∑ qk u ϕ X 0
4
k 1
Consequently,
u ϕ ∞
t 2 ∑ 1 qk u qk ϕ Xt 2
k 4 1 λk t
Thus, using the Cauchy-Schwarz inequality and (C) we obtain the estimate
?
?
u ϕ ∞
c2 t 2 ∑
λks0
2
t2
k 1 λk
Xt
4
qk u
(3.12)
Now we are prepared to estimate the integral I. The constant c, to be used next, may
have different values at different places in which it appears. Let s1 s0 s. Then,
by (C.1) and the estimate (3.12), we have
I
c
∞
1
t 3 2 2s0 4
∞
∑
4
k 1
λk1 s0 2
λk t 2
qk u
2
dt
λm λn s 0 2 @
qm u qn u
dt
∑
2
2
1
m n 4 1 λm t λn t ∞
A
∞
t 3 2s1
c ∑ λm λn s0 2 qm u qn u
λm t 2 λn t 2 dt 1
m n4 1
∞
c
t 3
2s1
∞
12
C. Bacuta, J. H. Bramble, J. Xu
Next, we use the formula
∞
a
t 3 2ϑ
dt
t2 b t2
1 a1 ϑ b1 cϑ2
a b
ϑ
0 ϑ 2 ϑ - 1 a b 0 (3.13)
The integral can be calculated by elementary calculus methods. If a right side of the above identity is replaced by 1c 2ϑ a ϑ . Thus,
b, then the
ϑ
∞
1
λm
t 3 2s1
dt
t 2 λn t 2 ∞
0
λm
t 3 2s1
dt
t 2 λn t 2 λn1 λm λn
λm1 cs 12
s1
s1
Combining the above inequalities, we get
I
1 s1 λn1 s 1 2 λ m
λ
λ
∑ m n
λm λn
m n4 1
∞
c
Let
lmn 1
λm λn s 1 2 λm s1
λms 2 qm u λns
λn1 λm λn
s1
s1
2
qn u
Then, the above estimate becomes
∞
I
c ∑ lmn ũm ũn m n4 1
An elementary calculation gives
2 m lmn 2 m n 1 s1 2 CB m n B s1 m n 1 2D
2 m n 2 m n
n 1 s1
and by elementary estimates we obtain
I
c ũ 2l2 c u
2
Hs
which proves (3.9).
Next, we prove (3.10). Let u Bs10 . Then by (2.13) and Remark 2.1) we have
that
2
u
1E
Xϕ Y
Note that
s0 ∞
c sup t 2s0 u u Y t sup t 4 t 1
t 1
sup t 2s0 u u
t 1
t 2s0 u u Y t sup
t 0
Yt
u
2ϑ0
u ϕ Y t
ϕ ϕ
2
Xt
2
s
B∞0
cu
2
s
B10
New regularity estimates
13
and, with the help of (C.1) we have
sup t 4 u ϕ Y t
ϕ ϕ
2ϑ0
t 1
2
c sup t 4 u ϕ Y t
t 1
Xt
2
(3.14)
For u Y , we have
u ϕ ∞
∞
λ
k ∑ λ t 2 qk u qk ϕ 5 ∑ λ t 2 qk u qk ϕ k4 1 k k4 1 k 1
Yt
and by using condition (C), we obtain
?
?
c2 ∑ λ t 2 qk u k4 1 k u ϕ The function
λks0
∞
2
Yt
u
s
B10
. Therefore,
?
λks0
" t u ϕ Y t t ∑
qk u
2
k 4 1 λk t
is
an
increasing function of t. As t " ∞, the limit of the above function is exactly
t
2
?
(3.15)
2
u
1E
Xϕ Y
s0 ∞
2
∞
2
2
c u X Y 1E
s0 1
1
and the proof is complete.
4. A SHIFT THEOREM FOR THE LAPLACE OPERATOR ON CONVEX
POLYGONAL DOMAINS.
In this section we will prove a stronger version of the estimate (1.3) and (1.4).
4.1. Shift estimates
Let Ω be a convex polygonal domain in R2 , with boundary ∂ Ω and no right angles.
Let V k Ω : H k Ω GF H01 Ω , k 2 and k 3. It is well known that for f L2 Ω the variational problem has a unique solution u V 2 Ω and (1.2) holds. If Ω is an
acute triangular domain then one can prove that for f H 1 Ω the solution of (1.1)
belongs to V 3 Ω and (1.3) holds for s 1. Using (1.2) and interpolation we obtain
u
H2
s c s u V 3 V 2 1E
s2
C s f
Hs
for all f
H s Ω 0 s 1
Thus, without restricting the generality of the problem, we can assume that there
exist one corner of measure ω with π 2 ω π . In fact, by a partition of unity
type argument and using the regularity results for domains with smooth boundary,
14
C. Bacuta, J. H. Bramble, J. Xu
we can reduce the problem to the case when Ω is a domain with only one corner
of measure ω with π 2 ω π . We will call this the “ω -corner” and we will
assume that the vertex of the ω -corner coincides with the origin of polar system
of coordinates. Let α π ω and s0 α 1. Given f H 1 Ω , we consider the
Dirichlet problem (1.1). The variational formulation of (1.1) is : Find u H01 Ω such that
Ω
∇u ! ∇v dx Ω
for all v H01 Ω f v dx
(4.1)
Let ζ IH Ω be a cut-off function, which depends only on the distance r to the
ω -corner, is identically equal to one in a neighborhood of the ω -corner and is identically equal to zero close to the part of ∂ Ω which does not contain the sides of
the ω -corner. Let ψ ϕ uR , ϕ r ϑ $ ζ r α sin αϑ and uR H01 Ω be the
variatonal solution of (1.1) with f ∆ϕ . One can check without difficulty that
ψ H 1 Ω . Let Hψ1 be defined as the kernel of ψ as linear functional on H 1 .
Then, (see e.g., Theorem 9.8 in [16]) for f Hψ1 the (variational) solution of (1.1)
Belongs to V 3 Ω and
u
c f
H3 Ω
H1 Ω
for all u Hψ1 Ω (4.2)
Thus, by interpolation, we have
u
for all f
J 1E
V3 Ω V2 Ω
sq
c f H 1 Ω ψ L2 Ω 1E s q (4.3)
K H 1 Ω ψ L2 Ω 9 1 s q . Then, we have the following theorem.
Theorem 4.1. Let Ω be a convex polygonal domain in R2 with no right angles
and with ω the measure of the largest angle and let s0 min 1 π ω 1 . If u is
the variational solution of (1.1) then, for 0 s s0 there exist positive constant c s
and C s such that
u
s c s u V 3 V 2 1E
H2
and for 0 s0
u
2 s0
B∞
Hs
for all f
H s Ω (4.4)
1,
c s0 u V 3 V 2
1E
Ω
s2
C s f
s0 ∞
C s0 f
s
B10 Ω
for all f
Bs10 Ω (4.5)
Proof. Use (4.3) and apply Theorem 3.1 with ϕ ψ . The proof that (C) is
satisfied is given later. The lower part of (4.4) or (4.5) follows by comparing the K
1
functions associated with the two intermediate spaces.
MM
M
New regularity estimates
M
M
Γ3
M
L
M
M
M
M
M
Γ2
M
M
L NM M
L NM
L
L
L
L
L
L
L M
L
M
M
L
M L
M L MM
M L M L
M
NM
L NM
L NM
M
L M
L
M L
M L MM
ML ML
M
N
N
L
15
M
M
M
M
M
M
M
M
M
M
M
M
Γ1
O
Figure 1. The ω -corner of Ω
4.2. Proving condition (C)
Let Ω be a polygonal convex domain with the only one vertex O of measure ω with
π 2 ω π and the remaining vertices denoted by S1 S2 Sn . Let
Ω
n
4
τi i 1
where, for i 1& n, τi is a triangular domain with vertices Si O Si 1 and O
is taken to be the origin of a Cartesian system of coordinates in the plane. For i 1 & n 1 let Γi denote the segment O Si . We assume, without loss of generality,
that S1 lies on the positive semi-axis (see Figure 1, the case n 2).
Let O 1 : τ1 & τn be the initial triangulation of Ω. We define multilevel
triangulations recursively. For k 1, the triangulation O k is obtained from O k 1 by
the
splitting each triangle in O k 1 into four triangles by connecting the midpoints of
edges. The space Mk is defined to be the space of all functions which are piecewise
linear with respect to O k , vanish on ∂ Ω and are continuous on Ω. Let Qk denote the
L2 Ω orthogonal projection onto Mk .
We verify that the function ψ satisfies the condition (C). To begin with, we will
prove that the function ϕ satisfies (C). First we will prove that there exist a function
wh Mk which is supported in a ball of radius H 1 2k 1 2h centered at origin
and wh is orthogonal on the space Mk 1 . Let ϕ1 ' ϕ7 be the nodal functions in
Mk 1 corresponding to the nodal points in O k 1 marked by P N P in the figure. Next,
we consider the eight nodal functions ϕ1 ϕ8 corresponding to the nodes marked
by PRQSP ). We define wh to be a linear combination wh of ϕ1 ' ϕ8 , with coefficients
independent of h such that
w ϕ 5 0 j 1& 7 and w ϕ T - 0
j
h
h
16
C. Bacuta, J. H. Bramble, J. Xu
Hence, wh is orthogonal on the space Mk
Then,
1
and consequently qk wh wh .
?
? ?
?
ϕ q v
ϕ qk wh ϕ wh k
U
sup
v
wh
wh
v L2 Ω q ϕ v
k qk ϕ sup
v
v L2 Ω Note further that
wh
ch
with c independent of h and by making the change of variable x hx̂ in the integral
which defines the inner product ϕ wh we get
?
ϕ wh ?
U ch2 α for another constant c. Combining the above estimates we conclude that the lower
part of the condition (C) holds.
For the upper part of (C) we first note that qk ϕ 0 λk1 2 qk ϕ 1 . To estimate
qk ϕ 1 we let η ηh to be a cut off function which depends only on r and satisfies
ηh r 1 for r
h 2 ηh r % 0 for r U h
? ?
?
?
ηhP r c h ηhP P r c h2
for all h 2 r h for some positive constant c. Then,
qk ϕ
1
sup
v H1 Ω
q ϕ v
k v
sup
v H1 Ω
1
q ηϕ v
k v
1
sup
v H1 Ω
q 1 η ϕ v
k
M1 M2 v
1
Using polar coordinates we have
ω
ηϕ q v
k
0
h
0
r
η q v dr sin αϑ d ϑ k
α 1
Next, we integrate by parts with respect to the r variable (write r1 α as the derivative
of r2 α 2 α ). Using the Cauchy-Schwarz inequality and the estimate for η P we
get
?
?
ηϕ qk v c h α 1 qk v h α 2 qk v r The L2 and H 1 - stability of the L2 projection give
qk v
ch v 1 and qk v 1 c v 1 with c independent of h (or k). Thus,
?
ηϕ qk v ?
ch2 α
v
1
and M1
ch2 α New regularity estimates
17
To estimate M2 , we observe first that 1 ηh ϕ H 2 Ω . Let Πh : H 2 Ω $" Mk
be the interpolant associated with O h Tk . By applying standard approximation
properties and (1.2) we obtain
M2
h qk 1 ηh ϕ h I Qk 1 1 η ϕ
ch I Πh 1 ηh ϕ ch3 1 ηh ϕ H 2 Ω ch3 ∆ 1 ηh ϕ L2 Ω qk 1 ηh ϕ
1
Using a simple computation in polar coordinates, and the estimates for the derivative
of ηh , we get
∆ ηh ϕ L2 Ω ch 1 α Combining the above inequalities, we have that
M2
ch2 α Hence, the upper part of the condition (C) holds and consequently, (C) holds for the
function ϕ . Since the function uR belongs to H 1 Ω , we have
qk uR
2
cλk 1 k 1 2&
Therefore, the function ψ satisfies condition (C).
5. APPLICATIONS TO FINITE ELEMENT CONVERGENCE ESTIMATES
Let Ω be a convex polygonal domain in 2 , with boundary ∂ Ω and no right angles.
Let ω be the measure of the largest corner and let s0 min 1 π ω 1 and let
u H01 Ω be the variational solution of (4.1) with f L2 Ω . We let Vh to be
a finite dimensional approximation subspace of H01 Ω and consider the discrete
problem:
Find uh Vh such that
Ω
∇uh ! ∇v dx for all v Vh (5.1)
for all u V 2 Ω 5 H 2 Ω VF H01 Ω (5.2)
Ω
f v dx
Further, let us assume that
u uh
H1 Ω
ch u
u uh
H1 Ω
H2 Ω
and
ch2 u H 3 Ω for all u V 3 Ω 5 H 3 Ω VF H01 Ω (5.3)
By interpolation with p 2 and 0 s 1, from (5.3) and (5.2), we obtain that
u uh H 1 Ω ch1 s u V 3 ΩJ V 2 Ω (5.4)
1E s 2 18
C. Bacuta, J. H. Bramble, J. Xu
for all u W V 3 Ω V 2 Ω 9 1 s 2 , where c is a constant independent of h. Interpolating
with p ∞ and s 1 s0 we have
u uh
1
ch
s0
H1 Ω
u
J 1E
V3 Ω V2 Ω
s0 ∞
(5.5)
for all u I V 3 Ω V 2 Ω 9 1 s ∞ where again c is a constant independent of h.
0
We thus have the following result.
Theorem 5.1. Let u uh be the variational solutions of problem (4.1) and (5.1),
respectively. Then, there exists a constant c independent of h such that
u uh
H1
ch1
u uh
Furthermore, for s0
that
u uh
H1
s
f
Hs
ch1
s0
for all f
f
s
B10
H s 0 s s0 (5.6)
Bs10 (5.7)
for all f
s 1 there exists a constant c independent of h and s such
1 2 1 c s s0 ' h
H1 Ω
1
e 2X 1E s0 Y ,
u uh H 1 Ω ch1
s0
f
for all f
H s Ω (5.8)
for all f
H s0 Ω
(5.9)
Hs Ω
and for h
s0
log 1 h 1 2 f H s0 Ω
Proof. The inequalities (5.6) and (5.7) follows from (5.4) and (5.5) respectively,
as a direct consequence of Theorem 4.1. The estimate (5.8) follows from (5.7) and
(3.6) with ε s s0 . The inequality (5.9) is obtained from (5.8) as follows. Let
f H s0 Ω and write f f Qh f Qh f . Next, we denote by uh H01 Ω the
solution of
∇uh ! ∇v dx Qh f v dx for all v H01 Ω Since u Ω
uh
Ω
Qh f instead of f we have
h
u u H 1 Ω c f Qh f H E 1 Ω is the solution of (4.1) with f
and from standard approximation properties we get
f Qh f HE 1 Ω ch1
s0
f
H s0 Ω
Combining the two inequalities we have
h
u u
1
ch
H1 Ω
s0
f
H s0 Ω
(5.10)
New regularity estimates
19
Next, using the estimate (5.8), a standard inverse inequality and the stability of
the L2 projection in H s0 Ω , we get
uh uh
1 2 h1 s0 Qh f HE 1 s Ω
c s s0 1 2 s0 s 1 s0 h Qh f H s0 Ω
s s0 ch
1 2 s0 s 1 s0 h s s0 ' f H s0 Ω ch
H1 Ω
where c is a constant independent of of h and s. The minimum of the function
s " s s0 1 2 hs0 s on the interval s0 1 is 2e log 1 h 1 2 and is attained for
s s0 2 log 1 h 1 , with h e 2X 1E 1 s0 Y . Thus,
h
u uh H 1 Ω ch1 s0 log 1 h 1 2 f
H s0 Ω
(5.11)
Finally, (5.9) follows from (5.10) and (5.11).
1
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