ON THE INTERPOLATION ERROR ESTIMATES FOR Q1
QUADRILATERAL FINITE ELEMENTS ∗
SHIPENG MAO† , SERGE NICAISE
‡ , AND
ZHONG-CI SHI§
Abstract. In this paper, we study the relation between the error estimate of the bilinear
interpolation on a general quadrilateral and the geometric characters of the quadrilateral.
Some explicit bounds of the interpolation error are obtained based on some sharp estimates
of the integral |J|1p−1 for 1 ≤ p ≤ ∞ on the reference element, where J is the Jacobian of
the non-affine mapping. This allows us to introduce weak geometric conditions leading to
interpolation error estimates in W 1,p norm, conditions that can be regarded as a generalization of the RDP (regular decomposition property) condition introduced in [2]. We avoid
the use of the reference family elements, that allows us to extend the results to a larger
class of elements and to introduce the GRDP condition in a more unified way. As far as we
know, the GRDP condition presented in this paper is weaker than any other mesh conditions
proposed in the literatures for any p with 1 ≤ p ≤ ∞.
Key words. Error estimates, quadrilateral elements, isoparametric finite elements,
maximal angle condition
AMS subject classification. 65N30, 65N15
1. Introduction. Quadrilateral finite elements, particularly low order quadrilateral
elements, are widely used in engineering computations due to their flexibility and simplicity.
However, numerical accuracy can not be achieved over arbitrary quadrilateral mesh and
certain geometric conditions are indispensable to guarantee the optimal convergence error
estimates. It is known that Q1 quadrilateral finite element is the mostly used quadrilateral
element, in order to obtain the optimal interpolation error of it, many mesh conditions have
been introduced in the references, let us give a review of them.
Denoting by Q the Lagrange interpolation operator and using the standard notation for
Sobolev spaces (cf. [11]), the first interpolation error estimate for the operator Q goes back
to Ciarlet and Raviart in [14], where the regular quadrilateral is supposed to satisfy
(1.1)
hK /h̄K ≤ µ1
and
(1.2)
| cos θK | ≤ µ2 ≤ 1
∗ The research is supported by the National Basic Research Program of China under the Grant
2005CB321701.
† Institute of Computational Mathematics, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100080,
People’s Republic of China, e-mail:[email protected]
‡ Universite de Valenciennes et du Hainaut Cambresis, LAMAV, ISTV, F-59313 - Valenciennes Cedex 9,
France, e-mail: [email protected], http://www.univvalenciennes. fr/macs/nicaise
§ Institute of Computational Mathematics, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100080,
People’s Republic of China, e-mail:[email protected]
1
for all angle θK of the quadrilateral K, here hK is the diameter of K and h̄K is the length
of the shortest side of K. Under the above so-called “nondegenerate” condition, Ciarlet and
Raviart proved the following interpolation error estimate
(1.3)
|u − Qu|1,K ≤ ChK |u|2,K .
On the other hand, error estimates for degenerate elements have attracted much attention since the works by Babus̆ka and Aziz [9] and by Jamet [19], interested readers are
referred to the works [5-7, 12, 13, 16-18, 21-24, 31], the book [8] by Apel, the ICM report [15]
by Durán and references therein. For triangular elements, the constant C in the estimate
(1.3) depends only on the maximal angle of the element. For quadrilaterals, the situation
may be different since the maximal angle condition is not necessary due to the work [20].
In such a case, the term “degenerate” is used in the two following situations: one is referred
to elements which are close to a triangle, while the other one is referred to narrow elements
or anisotropic elements, interested readers are referred to Jamet [20] for the first case and
to Z̆enisek and Vanmaele [29, 30], Apel [5] for the other one.
In the first case, Jamet [20] considered a quadrilateral that can degenerate into a regular
triangle and proved the error estimate (1.3) under the condition that there exists a constant
σ such that
hK /ρK ≤ σ,
where ρK denotes the diameter of the maximum circle contained in quadrilateral K.
In view of this result, one may believe that the maximal angle condition is not necessary
for the optimal interpolation error of the Q1 Lagrange interpolation operator. Recently,
Acosta and Durán [2] made great contribution to this aspect and obtained the optimal
interpolation error of Q1 interpolation under the RDP condition (cf. the definition in the
next section), which needs that, if we divide the quadrilateral into two triangles by one
diagonal, the ratio of the length of the other diagonal with the first one is bounded and both
the divided triangles verify the maximal angle condition. The above RDP condition is so
weak that almost all the degenerate quadrilateral conditions proposed before fall into this
scope. Furthermore, the authors of [2] assert that this condition is necessary and state it as an
open problem in the conclusion of their paper. For related papers concerning this assertion,
we refer to [25, 26, 32]. More recently, the same conclusion is extended to the case 1 ≤ p < 3
by Acosta and Monzón in [3]. Meanwhile, the authors of [3] introduce the double angle
condition (DAC) and show that (1.2) is a sufficient condition for the optimal interpolation
error in W 1,p norm with p ≥ 3. Though the DAC condition is much stronger than the
RDP condition, so far, it is the weakest mesh condition for the optimal interpolation error
in W 1,p norm with p ≥ 3. One of the key techniques employed in [2] and [3] is to introduce
an appropriate affine change of variables which reduces the problem to a reference family of
elements.
In this paper, we revisit the optimal error estimates of Q1 isoparametric Lagrange
interpolation for degenerate quadrilaterals. Our motivation comes from the observation
that, if we divide the quadrilateral into two triangles by the longest diagonal, when the
two triangles have the comparable area, we should impose the maximal angle condition for
both triangles, otherwise, we may only need to impose the maximal angle condition for
the big triangle T1 , and because the error on the small triangle T3 contributes little to the
2
interpolation error on the global quadrilateral, its maximal angle may become very large as
|T3 |
|T1 | approaches zero. Based on this observation, we introduce a generalized RDP condition
which involves the ratio between the area of the two divided triangles in the mesh condition.
Interpolation error of Q1 Lagrange interpolation in W 1,p norm with 1 ≤ p ≤ ∞ are proved
in the same spirit. Note that for p ≥ 3, the proposed GRDP condition is much weaker than
the DAC (double angle condition) condition proposed in [3]. The technique developed in
this paper is a combination of the ones in [2] and [20].
The rest of the paper is organized as follow. In section 2, we present our motivation
for the geometric condition by revisiting a simple example considered in [2], based on some
observations we propose our GRDP condition for the optimal H 1 interpolation error. In
section 3, following the techniques developed in [2] and [20], we prove the optimal interpolation error in the H 1 norm for the Q1 quadrilateral finite element. In section 4, we develop
a generic approach that will be applied in the next section to deduce interpolation error
estimates for all p ≥ 1.
2. The generalized regular decomposition property. In this section, we will introduce a mesh condition that is sufficient for (1.3). This can be regarded as a generalization
of the regular decomposition property presented by Acosta and Durán in [2].
We will adopt the following notation. Let K be a general quadrilateral with its vertices
M1 , M2 , M3 , M4 enumerated in anticlockwise order. In order to define the isoparametric
b = [0, 1]2 denotes the reference element, then there exists a bijection
elements on K, if K
b −→ K which is defined as
mapping FK : K
(2.1)
c) =
M = FK (M
4
X
c = (ξ, η) ∈ K,
b
Mi φbi (ξ, η), ∀ M
i=1
ci , i.e., φbi (M
ci ) = δ j , i =
where φbi is the bilinear basis function associated with the vertex M
i
1, 2, 3, 4 and j = 1, 2, 3, 4.
c) =
Let the basis functions on general quadrilateral K be defined as φi (M ) = φbi (M
b
φi (FK (M )) for any point M ∈ K. Then the Q1 isoparametric interpolation operator is
defined as by
bu(M
c), for any point M ∈ K
Qu(M ) = Qb
b is the bilinear Lagrange interpolation operator on K.
b
and Q
There are several mesh conditions in the literature that lead to (1.3). Among them, the
RDP condition proposed by Acosta and Durán in [2] is the weakest one. It is defined as
follows.
Definition 2.1. A quadrilateral or a triangle verifies the maximal angle condition with
constant ψ < π, or shortly MAC(ψ), if the interior angles of K are less than or equal to ψ.
Definition 2.2. Let K be a convex quadrilateral. We say that K satisfies the regular
decomposition property with constants N ∈ R+ and 0 < ψ < π, or shortly RDP(N, ψ), if we
can divide K into two triangles along one of its diagonals, always called d1 , the other one
being denoted by d2 in such a way that |d2 |/|d1 | ≤ N and both triangles satisfy MAC(ψ).
In order to motivate our mesh condition introduced below, we first analyze the following
examples. Let K = K(a, b, ã, b̃) be a convex quadrilateral with vertices M1 = (0, 0), M2 =
3
(a, 0), M3 = (ã, b̃), M4 = (0, b). Consider the case K(1, a, a, a) (cf. the left side of Figure 1)
and take u = x2 . Straightforward computations show that
°
°
° ∂(u − Qu) °2
°
°
≥ Ca ln(a−1 ) and |u|22,K ≤ Ca.
°
°
∂y
0,K
Then the constant on the right hand of (1.3) can not be bounded when a approaches zero.
This is just the counterexample given in [2].
If one consider the case K(1, a, as , a) with s > 1 (the right side of Figure 1), we have
°
°
° ∂(u − Qu) °2
°
°
≤ Ca2s−1 ln(a−1 ), |u|22,K ≥ Ca.
°
°
∂y
0,K
However, in this case the error constant
k ∂(u−Qu)
k20,K
∂y
|u|22,K
≤ Ca2s−2 ln(a−1 )
can be bounded with a constant independent of a. Both cases do not satisfy the RDP
condition since the maximal angle condition of 4M2 M3 M4 is violated if we divide the
quadrilateral by the diagonal M2 M4 .
What is the difference between this two examples? One reasonable interpretation is
S
2 M3 M4
that the ratio S4M
for K(1, a, as , a) is much smaller than the one for K(1, a, a, a).
4M1 M2 M4
This suggests to relax the maximal angle condition on 4M2 M3 M4 because the error on
4M2 M3 M4 contributes less compared to that on 4M2 M1 M4 .
6
a
6
M3
b
b
M4
(a, a)
HH
HH
b
b
b
b
HH
K(1, a, a, a)
H
H1
b
b
b M2
K(1, a, as , a), s > 1
M1
Figure 1.
Based on these considerations, we introduce the following geometric condition, which
can be regarded as a generalized RDP condition and will be proved to be sufficient for the
optimal interpolation error estimate for the Q1 Lagrange interpolation in the next section.
Definition 2.3. Let K be a convex quadrilateral (illustrated by Figure 2). We say
that K satisfies the generalized regular decomposition property with constant N ∈ R+ and
0 < ψ < π, or shortly GRDP(N, ψ), if we can divide K into two triangles along one of its
diagonals, always called d1 , in such a way that the big triangle satisfies MAC(ψ) and that
(2.2)
hK
|d1 | sin α
µ
|T3 | |T1 |
ln
|T1 | |T3 |
¶ 12
≤ N,
where the big triangle will always be called T1 , the other one is denoted by T3 , hK denotes
the diameter of the quadrilateral K and α is the maximal angle of T3 .
4
M4
* T3
`
`M3 ©©
H`
Hαb©©
HH
·
b
·O
b
H
b
·
dH
1H
b
· T1
H
bH
b
·
b
H
M1
M2
Figure 2. A general convex quadrilateral K
|T3 |
3|
= |a
Remark 2.4. Noticing that the term |T
|a1 | , where a3 = d2 ∩ T3 and a1 = d2 ∩ T1
1|
denote the two parts of the diagonal d2 divided by the diagonal d1 , thus the condition (2.2)
can be easily checked in practice computations, particularly, if we choose the longest diagonal
³
´1
|a1 | 2
3|
for d1 , the condition (2.2) becomes sin1 α |a
ln
≤ N.
|a1 |
|a3 |
Remark 2.5. It is easy to see that if a quadrilateral K satisfies the RDP condition,
then it also satisfies the GRDP condition. However, the converse is not true, as shown by
the example K(1, a, as , a) with s > 2.
Remark 2.6. The ratio of the area of the little triangle T3 and that of the big one T1
is also considered as an impact factor of the dependence of the interpolation error constant
in the GRDP condition, this is the main difference between it and the RDP condition. In
fact, this consideration is reasonable since if one divides the quadrilateral into two triangles
by the longest diagonal, if the small triangle is much more smaller compared to the big one
the quadrilateral K is almost degenerated into the big triangle, the maximal condition of the
3|
small triangle should be relaxed under the control of |T
|T1 | because the error on T3 contributes
little to the interpolation error on the global quadrilateral. This is just our motivation for
the presentation of the GRDP condition.
Remark 2.7. Let us finish this section by showing that the condition in Definition
2.3 that the big triangle satisfies the maximal angle condition is necessary. Indeed consider
the family of quadrilaterals Qα of vertices M1 = (−1 + cos α, − sin α), M2 = (1, 0), M3 =
(1 − cos α, sin α) and M4 = (−1, 0), with the parameter α ∈ ( π2 , π). Obviously this family
does not satisfy the maximal angle condition from Definition 2.3 since the angles at M2 and
M4 are larger than α. Hence any triangle obtained by subdividing Qα by one diagonal does
not satisfy MAC(ψ) for some ψ < π independent on α. If we consider u(x, y) = x2 , we
directly see that
|u|2,Qα =
√
2|Qα |1/2 .
On the other hand by using the affine transformation that maps Qα into the reference K̂,
we check that
∂
∂Qu 2
(u − Qu)|20,Qα = |
|
∂y
∂y 0,Qα
Z
|Qα |
(2 cos αŷ + 2(1 − cos α)x̂ + (1 − cos α)2 ) dx̂dŷ.
≥
4(sin α)2 K̂
|u − Qu|21,Qα ≥ |
As α goes to π, we see that
Z
Z
2
(2 cos αŷ + 2(1 − cos α)x̂ + (1 − cos α) ) dx̂ →
(−2ŷ + 4x̂ + 4) dx̂dŷ = C 2 > 0.
K̂
K̂
5
Hence there exists β0 > 0 small enough, such that for all α ∈ (π − β0 , π)
Z
C2
(2 cos αŷ + 2(1 − cos α)ŷ + (1 − cos α)2 ) dx̂dŷ ≥
.
2
K̂
This finally shows that for all α ∈ (π − β0 , π), the ratio
|u − Qu|1,Qα
C
≥
|u|2,Qα
4 sin α
and hence goes to infinity as α tends to π.
3. Interpolation error estimate in H 1 for Q1 elements. In this section, we shall
prove the optimal order error estimate for Q1 Lagrange elements satisfying the GRDP
condition by following the idea from [2] and [20]. Let Π be the conforming P1 Lagrange
interpolation operator on the big triangle T1 , i.e., Πu is the linear function which admits
the same values with the function u at the three vertices M1 , M2 and M4 . Then we have
|u − Qu|1,K ≤ |Πu − Qu|1,K + |u − Πu|1,K .
Because Πu − Qu belongs to the isoparametric finite element space and vanishes at M1 , M2
and M4 , it follows that
(Πu − Qu)(x) = (Πu − u)(M3 )φ3 (x),
where φ3 is the basis function corresponding to M3 . Hence we obtain
(3.1)
|u − Qu|1,K ≤ |(Πu − u)(M3 )||φ3 |1,K + |u − Πu|1,K .
The goal of the rest of this section is to estimate the above two terms on the right hand side
of (3.1). We first give an estimate for the term |φ3 |1,K following the idea developed in [20]
and leave the terms |(Πu − u)(M3 )| and |u − Πu|1,K for the end.
R 1
dξdη, where
In order to estimate |φ3 |1,K , we start with a new bound for the term Kb |J|
J is the Jacobian of the mapping FK .
Lemma 3.1. Let K be a general convex quadrilateral with consecutive vertices M1 , M2 , M3
and M4 (cf. Figure 2). Let θ be the angle of the two diagonals M1 M3 (denoted by d2 ) and
M2 M4 (denoted by d1 ) and let O be the point at which they intersect. Let ai = |OMi | with
ai > 0 for i = 1, 2, 4 and a3 ≥ 0. Let α, s be the maximal angle and the shortest edge of the
triangle T3 , respectively. Without loss of generality, we can assume that |M3 M4 | = s. Then
we have
µ
¶
Z
1
4
|T3 |
|T1 |
(3.2)
dξdη <
2 + ln
.
|d1 ||s| sin α |T1 |
|T3 |
b |J|
K
Proof. Let (Ox̃, Oỹ) be two auxiliary axes oriented along the vectors M1 M3 and M2 M4 .
Let J˜ be the Jacobian of the affine mapping (ξ, η) → (x̃, ỹ) and J1 the Jacobian of the affine
˜ 1 with |J1 | = sin θ. It follows from (2.1) that
mapping (x̃, ỹ) → (x, y). Then we have J = JJ
(
x = −(1 − ξ)(1 − η)a1 + ξηa3 ,
y = −ξ(1 − η)a2 + (1 − ξ)ηa4 .
First we assume that a2 ≥ a4 . It is easy to see that
J˜ = a2 a3 ξ + a3 a4 η + a1 a4 (1 − ξ) + a1 a2 (1 − η)
≥ a2 a3 ξ + a1 a2 (1 − η) ≥ 0.
6
Then we can derive that
Z
Z
1
dξdη
b a2 a3 ξ + a1 a2 (1 − η)
b
K
K
¶
Z 1 µ
a3
1
ln 1 +
dη
=
a2 a3 0
a1 (1 − η)
à Z a3 Z ! µ
¶
1
a1
1
a3
=
+
ln 1 +
dt
a3
a2 a3
a1 t
0
a1
!
à Z a3 r
Z 1
a1
1
a3
a3
<
dt +
dt .
a3 a t
a2 a3
a1 t
1
0
a1
µ
¶
1
a3
=
2 − ln
,
a1 a2
a1
√
where we have used the inequalities ln(1 + x) < x, ∀x ∈ [1, ∞) and ln(1 + x) < x, ∀x ∈
[ aa31 , 1].
On the other hand, an application of the Sin theorem in the triangle M3 OM4 yields
1
dξdη ≤
˜
|J|
(3.3)
sin θ =
sin ∠M2 M4 M3 |s|
.
a3
Furthermore, it can be easily proved that sin ∠M2 M4 M3 ≥ 12 sin α. Indeed, if α = ∠M2 M4 M3 ,
the assertion is obvious; otherwise, α = ∠M2 M3 M4 , and then
sin ∠M2 M4 M3 = sin(∠M2 M4 M3 + ∠M4 M2 M3 ) ≤ 2 sin ∠M2 M4 M3
because ∠M4 M2 M3 ≤ ∠M2 M4 M3 . Therefore,
µ
¶
Z
1
2
a3
a3
dξdη <
2 − ln
,
a2 |s| sin α a1
a1
b |J|
K
a3
a1
|T3 |
|T1 |
and a2 ≥ 12 |d1 |, implies (3.2).
In the case a2 < a4 , we just use the inequality J˜ ≥ a3 a4 η + a1 a4 (1 − ξ) and prove the
assertion by the same argument.
Lemma 3.2. Let K be a general convex quadrilateral with the same hypotheses as Lemma
3.1. Then we have
µ
¶¶ 12
µ
¯ ¯
|T1 |
8hK
|T3 |
¯ ¯
(3.4)
2
+
ln
.
≤
¯φ3 ¯
1
|T3 |
1,K
(|d1 ||s| sin α) 2 |T1 |
together with the fact that
=
Proof. By Lemma 2.2 in [20], we have
¯ ¯
¯ ¯
¯φ3 ¯
µZ
1,K
≤ 4hK
b
K
1
dξdη
|J|
¶ 12 ¯ ¯
¯b ¯
¯φ 3 ¯
b
1,∞,K
.
The conclusion follows from Lemma 3.1 and the fact that |φb3 |1,∞,Kb = 1.
Remark 3.3. As mentioned in [2], the error estimate of the term |φ3 |1,K is the most
technical one. It is estimated therein by introducing an appropriate affine change of variables
that reduces the problem to a reference family of elements. Here we did not adopt the
technique developed in [2] because we have not imposed the maximal angle condition on the
small triangle. Meanwhile, the estimate of |φ3 |1,K in [2] (see Lemma 4.6 of [2]) can be easily
2|
recovered under the assumption that |d
|d1 | is bounded.
7
Remark 3.4.
Note that (3.4) gives a sharp estimate of the term |φ3 |1,K up to a
generic constant. In fact, one can just consider the example of the quadrilateral K(1, b, a, b)
under the assumption 0 < a, b ¿ 1. Some immediate calculations yield
¯ ¯
¯ ¯
¯φ3 ¯
1,K
¶1
µ
° ∂φ °
1
1 2
° 3°
≥°
≥ Cp
ln
°
∂y 0,K
a
b(1 − a)
µ
µ
¶¶ 12
hK
|T3 |
|T1 |
≥C
2
+
ln
1
|T3 |
(|d1 ||s| sin α) 2 |T1 |
3|
since |s| = a, sin α > b and |T
|T1 | = a.
The next lemma gives an estimate for |(u − Πu)(M3 )|.
Lemma 3.5. Let K be a general convex quadrilateral, then we have
(3.5)
¯
¯
¯(u − Πu)(M3 )¯ ≤
µ
4|s|
|d1 | sin α
¶ 12 n
o
|u − Πu|1,T3 + hK |u|2,T3 .
Proof. The proof is just the same as that of Lemma 4.2 in [2], which exploited a trace
theorem with a sharp dependence of the constant given by [28], we then omit it here.
Remark 3.6. The result of lemma 3.5 gives a sharp estimate up to a generic constant.
Consider the example of the quadrilateral K(1, b, a, b) under the assumption 0 < a, b ¿ 1
and the function u(x, y) = x2 . We then see that Πu = x and therefore
µ
|(Πu − u)(M3 )| = a(1 − a) ≥ C
|s|
|d1 | sin α
¶ 21 n
o
|u − Πu|1,T3 + hK |u|2,T3
√
since |s| = a, sin α > b and |u − Πu|1,T3 + hK |u|2,T3 ≤ ab.
It remains to bound the term |u − Πu|1,K . This is the goal of the following lemma.
M4``
3
¶γbb` M
b ll
¶
b
l¶
1
bl
bl
T
1
¶
b
l
b
¶
l
b
M1
M2
l2
Figure 3.
Lemma 3.7. Let K be a general convex quadrilateral and Π be the linear Lagrange
interpolation operator defined on T1 , then
(3.6)
|u − Πu|1,K
4
≤
sin γ
µ
¶µ
¶1
2
2|K| 2
1+
hK |u|2,K ,
π
|T1 |
where γ is the maximal angle of T1 .
Proof. Without loss of generality, we can assume that ∠M1 M4 M2 = γ is the maximal
angle of T1 and adopt the notations of Figure 3. Let v1 , v2 be the directions of the edges l1
and l2 , respectively. Then we have
|u − Πu|1,K ≤
´
1 ³
k∇(u − Πu) · v1 k0,K + k∇(u − Πu) · v2 k0,K .
sin γ
8
Consider A = ∇(u − Πu) · v1 = ∂(u−Πu)
and let AK be the mean value of A on K. The
∂v1
well known Poincaré inequality gives
(3.7)
kA − AK k0,K ≤
hK
|A|1,K .
π
Remark that the constant in the above Poincaré inequality can be taken explicitly and
independent of the shape (i.e., depending only on the diameter) for a general convex domain.
However, the original proof of it in [27] contains a mistake, and recently [10] gave a corrected
proof, fortunately, the optimal constant π1 in the Poincaré inequality remains valid.
Now, we bound kAK k0,K following an idea from [2]. By the sharp trace theorem of [28],
R
Cauchy-Schwarz inequality, and the fact that l1 Adv1 = 0, we have
kAK k0,K
¯
1 ¯Z
¯
|K| 2 ¯¯
(A − AK )dv1 ¯¯
= |K| |AK | =
¯
|l1 |
l1
µ
¶ 12 ³
´
2|K|
≤
kA − AK k0,K + hK |A|1,K ,
|T1 |
1
2
which, together with (3.7) gives
k∇(u − Πu) · v1 k0,K ≤
¶µ
¶1
µ
2|K| 2
2
hK |u|2,K .
1+
π
|T1 |
The term k∇(u − Πu) · v2 k0,K is estimated similarly. The proof of the lemma follows.
Collecting all the above lemmas we can obtain the main theorem of this section, which
gives the optimal error estimate in H 1 norm for convex quadrilaterals.
Theorem 3.8. Let K be a convex quadrilateral satisfying GRDP(N, ψ), then we have
(3.8)
|u − Qu|1,K ≤ ChK |u|2,K ,
with C > 0 depending only on N and ψ.
Proof. A combination of (3.4) and (3.5) yields
|(Πu − u)(M3 )||φ3 |1,K
µ
µ
¶¶ 12
16hK
|T3 |
|T1 |
≤
2 + ln
|d1 | sin α |T1 |
|T3 |
o
n
|u − Πu|1,T3 + hK |u|2,T3 ,
together with (3.1) and (3.6) gives
)
(
µ
µ
¶¶ 12
¯
¯
|T
|
|T
|
32h
3
1
K
¯u − Qu¯
2 + ln
+1
≤
1,K
|d1 | sin α |T1 |
|T3 |
(3.9)
Ã
µ
¶1 !
¯ ¯
2 2|K| 2
4
hK ¯u¯2,K .
1+
sin γ
π |T1 |
Since T1 satisfies the maximal angle condition and |T1 | ≥ 21 |K|, (3.8) follows from the
assumption (2.2) and (3.9).
Remark 3.9. In fact, (3.9) gives an explicit error bound for the bilinear interpolation
3|
operator. If the two divided triangles have comparable areas, i.e., |T
|T1 | = O(1), then the results
9
of [2] can be recovered from (3.9) with the RDP condition. Otherwise, if the quadrilateral
3|
is nearly degenerated into the triangle T1 , i.e., |T
|T1 | −→ 0, we can see that the interpolation
error of Q1 element is dominated by that of the P1 Lagrange interpolation operator on T1 .
Obviously, this is a quite reasonable conclusion. This reinforces the fact that the maximal
angle condition imposed on T1 can not be relaxed because it is also a necessary condition for
the optimal interpolation error of the P1 Lagrange interpolation operator (cf. [8, 9]).
Remark 3.10. One may ask whether the GRDP condition is necessary in the sense that,
given a family of elements that does not satisfy the GRDP condition, then the interpolation
error estimate (3.8) can not be uniformly bounded. Indeed we have already shown in Remark
2.7 that the MAC condition on the biggest triangle is necessary. In its full generality we
cannot hope to show that (2.2) is also necessary. From Remark 3.4 and Remark 3.6 we
present a family of quadrilaterals that satisfies the MAC condition but not the condition
(2.2) and for which the interpolation error estimate (3.8) is not uniformly satisfied. In that
sense our condition (2.2) is almost necessary. For that purpose, we take the example from
Remark 3.4 and the function u(x, y) = x2 . We then see that Πu(x, y) = x and therefore
(Πu − u)(M3 ) = a(1 − a),
√
|u|2,K ≤ C b.
By the triangular inequality, we then have
|u − Qu|1,K ≥ |(Πu − u)(M3 )||φ3 |1,K − |u − Πu|1,K .
Hence by Lemma 3.7, we have
|u − Qu|1,K
|(Πu − u)(M3 )||φ3 |1,K
|u − Πu|1,K
≥
−
hK |u|2,K
hK |u|2,K
hK |u|2,K
p
a | ln b|
≥ C1
− C2 ,
b
for some positive constants C1 , C2 independent on a and b. By choosing b = a| ln a|α , with
α ≥ 0, the above right hand side tends to infinity and therefore (3.8) is not uniformly
satisfied. Furthermore we easily check that this family of quadrilaterals satisfies the MAC
condition but not the condition (2.2). In our proof of (3.8), the point where we make an
overestimation is when we use the estimates ku−Πuk1,T3 ≤ ku−Πuk1,K and |u|2,T3 ≤ |u|2,K .
In may cases (for instance in the case a = b2 ), the right-hand sides are much larger than
the corresponding left-hand sides.
4. Interpolation error estimates in W 1,p . Until now, we have studied the interpolation error estimate of the Q1 element in H 1 norm. In this section and the next one, we
will extend the GRDP condition for the error estimate in W 1,p for p ≥ 1. Here we develop
a generic approach that will be applied in the next one for different values of p. Our generic
approach is inspired from the results of the previous section and then closely follows it.
With the notations from the previous section, let us denote by C(K, p) a positive constant such that
(4.1)
|φ3 |1,p,K ≤ hK C(K, p).
Any method that furnishes a computable value of C(K, p) will drive to a sufficient condition
for interpolation error estimates in W 1,p . The next section will describe such methods.
10
With this concept, we can state the
Definition 4.1. Let K be a convex quadrilateral. We say that K satisfies the generalized regular decomposition property with constants N ∈ R+ , 0 < ψ < π and p ∈ [1, ∞),
or shortly GRDP(N, ψ, p), if we can divide K into two triangles along one of its diagonals,
always called d1 , in such a way that the big triangle satisfies MAC(ψ) and that
1
(4.2)
|s|1− p hK C(K, p)
1
(|d1 | sin α) p
≤ N,
where the big triangle will always be called T1 , the other one is called T3 , hK denotes the
diameter of the quadrilateral K, α the maximal angle of T3 and s the smallest edge of T3 .
We first state an estimate for |(u − Πu)(M3 )|, which follows from the proof of Lemma
5.2 of [3].
Lemma 4.2. Let K be a general convex quadrilateral and Π be the linear Lagrange
interpolation operator defined on T1 , then
µ p p−1 ¶ p1 n
o
2 |s|
(4.3)
|(u − Πu)(M3 )| ≤
|u − Πu|1,p,T3 + hK |u|2,p,T3 , p ≥ 1.
|d1 | sin α
Remark 4.3. Since K is convex, it is well known that the Poincaré inequality holds for
general p ≥ 1, i.e., there exists a constant Cp depending only on p such that
(4.4)
kvk0,p,K ≤ Cp h|v|1,p,K
for any v ∈ W 1,p (K) with vanishing average on K.
Now we are in a position to bound the term |u − Πu|1,p,K .
Lemma 4.4. Let K be a general convex quadrilateral and Π be the linear Lagrange
interpolation operator defined on T1 , then
¶1
1 µ
22− p |K| p
|u − Πu|1,p,K ≤ (1 + 2Cp )
hK |u|2,p,K ,
sin γ |T1 |
where γ is the maximal angle of T1 .
Proof. The proof is just a combination of the arguments of Lemma 3.7 in section 3
(using here (4.4)) and of Lemma 5.3 of [3], so we omit it.
Now we come to the main theorem of this section.
Theorem 4.5. Let K be a convex quadrilateral satisfying GRDP(N, ψ, p) with p ∈
[1, ∞), then we have
(4.5)
|u − Qu|1,p,K ≤ ChK |u|2,p,K ,
with C only depending on N, p and ψ.
Proof. A combination of (4.1) and (4.3) yields
o
|s|1− p hK C(K, p) n
|u
−
Πu|
+
h
|u|
1,p,T3
K
2,p,T3 ,
1
(|d1 | sin α) p
1
|(Πu − u)(M3 )||φ3 |1,p,K ≤ Cp
where Cp > 0 depends only on p.
Now invoking Lemma 4.4, we get
|u − Qu|1,p,K ≤ |(Πu − u)(M3 )||φ3 |1,p,K + |u − Πu|1,p,K
≤ κ(K, p)Cp0 hK |u|2,p,K ,
11
where Cp0 > 0 depends only on p and κ(K, p) is defined by
1
κ(K, p) =
|s|1− p hK C(K, p)
1
(|d1 | sin α) p
µ
¶1
1
³
|s|1− p hK C(K, p) ´ 1
|K| p
.
+ 1 + Cp
1
sin γ |T1 |
(|d1 | sin α) p
Hence we will obtain (4.5) if κ(K, p) is bounded uniformly. But due to the maximal angle
condition satisfied by T1 , we have
µ
¶1
|K| p
≤ Kp ,
|T1 |
for some Kp depending only on ψ and p. We therefore conclude by using the assumption
4.2.
5. Interpolation error estimates in W 1,p continued. In this section, we will study
a proper geometric condition for error estimate in W 1,p with p ≥ 1 by estimating explicitly
the constant C(K, p) appearing in (4.1).
5.1. Case 1 ≤ p < 3. In order to estimate |φ3 |1,p,K , we start with a new bound for the
R
term Kb |J|1p−1 dξdη.
Lemma 5.1. Let K be a general convex quadrilateral, then we have
µ
¶p−1
Z
1
4p−1
|T3 |
(5.1)
dξdη <
, for p ∈ [1, 2),
(2 − p)(|d1 ||s| sin α)p−1 |T1 |
b |J|p−1
K
µ
¶p−1
Z
4p−1
|T3 |
1
(5.2)
dξdη
<
, for p ∈ (2, 3).
(p − 2)(3 − p)(|d1 ||s| sin α)p−1 |T1 |
b |J|p−1
K
Proof. We adopt the notations introduced in Lemma 3.1. By direct computations, if
a2 ≥ a4 we can derive
Z
Z
1
1
dξdη ≤
dξdη
p−1
p−1
˜
(a
a
ξ
+
a
a
b
b
2 3
1 2 (1 − η))
K |J|
K
!
¶2−p
Z 1 õ
1
a3
2−p
=
1+ ξ
−ξ
dξ
(2 − p)(a1 a2 )p−1 0
a1
õ
!
¶3−p µ ¶3−p
a3
1
a3
=
1+
−
−1 ,
a1
a1
a3
ap−1
(2 − p)(3 − p)ap−2
2
1
If p ∈ [1, 2), it can be easily proved that (1 + x)3−p < 1 + (3 − p)x + x3−p , ∀x ∈ (0, 1], then
we have
Z
1
1
dξdη <
.
p−1
˜
b
(2 − p)ap−1
ap−1
K |J|
1
2
˜ sin θ and (3.3), then we get
Recalling that |J| = |J|
µ ¶p−1
Z
4p−1
a3
1
dξdη <
,
p−1
p−1
(2 − p)(|d1 ||s| sin α)
a1
b |J|
K
which implies (5.1).
Otherwise, if p ∈ (2, 3), we have
¶2−p µ
¶2−p !
Z 1 õ
Z
1
a1
a1
1
dξdη ≤
1 + (1 − η)
−
(1 − η)
dη
˜ p−1
(2 − p)(a2 a3 )p−1 0
a3
a3
b |J|
K
õ ¶
µ
¶3−p !
3−p
1
a1
a1
=
+1− 1+
.
a3
a3
(p − 2)(3 − p)a3p−2 a2p−1 a1
12
On the other hand, we directly see that
(1 + x)3−p ≥ x3−p , ∀ x > 0,
then we have
Z
b
K
1
1
dξdη <
˜ p−1
|J|
(p − 2)(3 − p)a3p−2 ap−1
a1
2
and
Z
b
K
4p−1
1
a3
dξdη <
,
p−1
|J|
(p − 2)(3 − p)(|d1 ||s| sin α)p−1 a1
which implies (5.2).
The case a2 < a4 is treated similarly. The proof is completed.
Lemma 5.2. Let K be a general convex quadrilateral, then we have
µ
1
(5.3)
C(K, p) ≤
(5.4)
C(K, p) ≤
42− p
1
1
(p − 2) p (|d1 ||s| sin α)1− p
|T3 |
|T1 |
¶1− p1
, for p ∈ [1, 2).
µ
1
42− p
1
1
1
|T3 |
|T1 |
¶ p1
, for p ∈ (2, 3).
(p − 2) p (3 − p) p (|d1 ||s| sin α)1− p
¯
¯
¯
¯
¯ 3 ¯ 2hK ¯ ∂φ3 ¯ 2hK
Proof. Noticing that ¯ ∂φ
∂x ¯ ≤ |J| , ¯ ∂y ¯ ≤ |J| , we have
¯ ¯
¯ ¯
¯φ3 ¯
(5.5)
µZ
1,p,K
≤ 4hK
b
K
1
dξdη
|J|p−1
¶ p1
,
which together with the results of Lemma 5.1 completes the proof.
Corollary 5.3. Condition (4.2) from Definition 4.1 is satisfied if
¶1− p1
|T3 |
≤ N, for p ∈ [1, 2),
1
(2 − p) p |d1 | sin α |T1 |
¶1
µ
hK
|T3 | p
≤ N, for p ∈ (2, 3),
1
1
(p − 2) p (3 − p) p |d1 | sin α |T1 |
hK
(5.6)
(5.7)
µ
Remark 5.4. Comparing the three geometric hypotheses (5.6), (5.7) and (2.2), we can
make the following conclusions: Conditions (5.6) and (5.7) are not valid for p = 2. Hence it
is reasonable to have added a log factor in (2.2). Indeed in the proof of Lemma 5.1, passing
R
1
to the limit p → 2 in the estimates of Kb |J|
˜ p−1 dξdη, we can show that
Z
b
K
1
dξdη = lim
˜
p→2
|J|
Z
b
K
1
dξdη
˜
|J|p−1
õ ¶
µ
¶3−p !
3−p
1
a1
a1
≤ lim
+1− 1+
p→2 (p − 2)(3 − p)ap−2 ap−1 a
a3
a3
1
3
2
µ
¶
1
a1 a1
a1
a1
=
ln
− (1 + ) ln(1 + ) .
a1 a2 a3 a3
a3
a3
³
´
This leads to (2.2) because aa13 ln aa13 − (1 + aa31 ) ln(1 + aa31 ) ≤ 2 + ln aa13 .
13
On the other hand, the hypothesis (5.7) is not valid for the critical point p = 3, so it
is natural to impose a stronger geometric condition for p ≥ 3. This will be discussed in the
next subsection, where we introduce a GRDP condition for p ≥ 3, which is weaker than the
double angle condition (DAC) proposed in [3].
Remark 5.5. When p approaches 2, the estimate (5.3) can be improved in order to
avoid a blowing up. In fact, for p < 2, |φ3 |1,p,K can be bounded directly from (3.2). Indeed
1
taking into account p−1
> 1, applying Hölder’s inequality in the right hand side of (5.5), it
holds
µZ
¶1− p1
¯ ¯
1
¯ ¯
≤ 4hK
dξdη
¯φ3 ¯
1,p,K
b |J|
K
¶1− p1 µ
¶1− p1
µ
1
1+ p
4
hK
|T3 |
|T1 |
2 + ln
≤
.
1
|T3 |
(|d1 ||s| sin α)1− p |T1 |
Therefore, the condition (5.6) could be replaced by
hK
|d1 | sin α
µ
|T3 | |T1 |
ln
|T1 | |T3 |
¶1− p1
≤ N.
This condition is more advantageous than (5.6) for p close to 2, while it is the converse for
p far from 2.
5.2. Case p ≥ 3. In this subsection, we will study a proper geometric condition for
error estimate in W 1,p with p ≥ 3.
As before we only need to estimate |φ3 |1,p,K in order to derive the optimal error estimate
for p ≥ 3.
Lemma 5.6. Let K be a general convex quadrilateral and let Ti = 4Mi−1 Mi Mi+1 , i =
1, 2, 3, 4 with Mi±4 = Mi . Without loss of generality we may assume that |T3 | = min |Ti |,
i=1,3
|T4 | = min |Ti |. Then we have
i=2,4
1
(5.8)
3
1
p
|T1 | |T3 |
2
1− p
2
(5.9)
C(K, p) ≤
1
21+ p M 1− p + 2p
C(K, p) ≤
1
21+ p M p
1
p
1
p
(p − 2) |T1 | |T3 |
1
(5.10)
7
for p ∈ [3, ],
2
,
C(K, p) <
2
1− p
2
3
(p − 3) p 21+ p M 1− p
1
1
2
(p − 2) p |T1 | p |T3 |1− p
n
o
|T3 |
where M = max 1, |T
.
|
4
Proof. We only need to bound
the results in Lemma 3.1, we have
R
1
b |J|p−1 dξdη.
K
7
for p ∈ ( , 4],
2
,
,
for p > 4,
First consider the case p = 3. In view of
¡
¢
˜ 1 = 2 |T1 | + (|T2 | − |T1 |)ξ + (|T4 | − |T1 |)η .
J = JJ
Noticing the fact
|T1 | + |T3 | = |T2 | + |T4 | = |K|,
14
then we can derive that
Z
Z
1
1
1
dξdη
=
¡
¢2 dξdη
2
|J|
4
b
b
K
K |T1 | + (|T2 | − |T1 |)ξ + (|T4 | − |T1 |)η
¶
Z 1µ
1
1
1
=
−
dη
4(|T1 | − |T2 |) 0
|T2 | + (|T4 | − |T1 |)η |T1 | + (|T4 | − |T1 |)η
1
|T1 ||T3 |
ln
=
4(|T1 | − |T2 |)(|T4 | − |T1 |) |T2 ||T4 |
1
|T1 ||T3 |
=
ln
.
4(|T1 ||T3 | − |T2 ||T4 |) |T2 ||T4 |
Now, let us discuss the above result. If |T3 | ≤ |T4 |, then we have
(5.11)
|T1 | ≥ |T2 |, and |T1 ||T3 | ≤ |T2 ||T4 |,
which implies
Z
(5.12)
b
K
1
1
dξdη ≤
,
2
|J|
4|T1 ||T3 |
ln x
where we have used the inequality 1−
1 ≤ 1, ∀x ∈ (0, 1].
x
On the other hand if |T3 | > |T4 |, then
(5.13)
|T2 | > |T1 |, and |T1 ||T3 | > |T2 ||T4 |,
thus we can derive
Z
b
K
(5.14)
µ
¶
1
|T1 ||T3 | − |T2 ||T4 |
1
dξdη
=
ln
1
+
|J|2
4(|T1 ||T3 | − |T2 ||T4 |)
|T2 ||T4 |
1
p
p
≤ p
4( |T1 ||T3 | + |T2 ||T4 |) |T2 ||T4 |
µ
¶1
1
1
|T3 | 2
< p
,
<
4|T1 ||T3 | |T4 |
4 |T1 ||T3 ||T2 ||T4 |
√
where we have used the inequality ln(1 + x) ≤ x, ∀x ∈ (0, ∞).
Then a combination of (5.12), (5.14) and (5.5) gives (5.8) for the case p = 3.
For p ∈ (3, 72 ], we write
Z
Z
1
1
1
dξdη
=
dξdη
b |J|p−1
b |J|p−3 |J|2
K
K
µ
¶p−3 Z
M
1
≤
dξdη,
2
2|T3 |
|J|
b
K
3|
where M = max{1, |T
|T4 | }, since we notice that
min |J| = 2 min{|T1 |, |T2 |, |T3 |, |T4 |} = 2 min{|T3 |, |T4 |} =
b
K
By the estimate (5.12) and (5.14), we deduce that
Z
b
K
1
dξdη ≤
|J|p−1
µ
M
2|T3 |
15
¶p−3
1
M2
.
4|T1 ||T3 |
2|T3 |
.
M
This estimate yields
¯ ¯
¯ ¯
¯φ3 ¯
Now we come to the case
Z
b
K
(5.15)
1
1,p,K
≤
21+ p hK
1
p
|T1 | |T3 |
2
1− p
3
1
M 1− p + 2p .
7
2
< p. An immediate computation gives that
ÃZ µ
¶2−p
1
1
1
1
dξdη
=
dη
|J|p−1
(2 − p)2p−1 (|T2 | − |T1 |)
|T2 | + (|T4 | − |T1 |)η
0
¶2−p !
Z 1µ
1
−
dη
|T1 | + (|T4 | − |T1 |)η
0
=
|T1 |3−p + |T3 |3−p − |T2 |3−p − |T4 |3−p
.
(p − 2)(p − 3)2p−1 (|T1 | − |T2 |)(|T1 | − |T4 |)
If |T3 | > |T4 |, using (5.13) and the fact that |T2 | − |T1 | = |T3 | − |T4 |, and by the Cauchy’s
Mean Value theorem repeatedly, we have
|T1 |3−p + |T3 |3−p − |T2 |3−p − |T4 |3−p
(|T2 ||T4 |)p−3 (|T1 |p−3 + |T3 |p−3 ) − (|T1 ||T3 |)p−3 (|T2 |p−3 + |T4 |p−3 )
|T1 |p−3 |T2 |p−3 |T3 |p−3 |T4 |p−3
(|T2 ||T4 |)p−3 (|T1 |p−3 + |T3 |p−3 − |T2 |p−3 − |T4 |p−3 )
=
|T1 |p−3 |T2 |p−3 |T3 |p−3 |T4 |p−3
((|T2 ||T4 |)p−3 − (|T1 ||T3 |)p−3 )(|T2 |p−3 + |T4 |p−3 )
+
|T1 |p−3 |T2 |p−3 |T3 |p−3 |T4 |p−3
=
(5.16)
=
p−4
(p − 3)(|T2 ||T4 |)p−3 (mp−4
34 − m12 )(|T2 | − |T1 |)
|T1 |p−3 |T2 |p−3 |T3 |p−3 |T4 |p−3
+
p−3
(p − 3)(|T2 ||T4 | − |T1 ||T3 |)mp−4
+ |T4 |p−3 )
1324 (|T2 |
|T1 |p−3 |T2 |p−3 |T3 |p−3 |T4 |p−3
=
(p − 3)(p − 4)(|T2 ||T4 |)p−3 mp−5
1234 (m12 − m34 )(|T1 | − |T2 |)
p−3
p−3
|T1 | |T2 | |T3 |p−3 |T4 |p−3
+
p−3
(p − 3)(|T1 | − |T2 |)(|T1 | − |T4 |)mp−4
+ |T4 |p−3 )
1324 (|T2 |
,
|T1 |p−3 |T2 |p−3 |T3 |p−3 |T4 |p−3
here the constants m12 , m34 , m1324 , m1234 are constants produced by the Mean Value theorem, which satisfy |T1 | ≤ m12 ≤ |T2 |, |T4 | ≤ m34 ≤ |T3 |, |T2 ||T4 | ≤ m1324 ≤ |T1 ||T3 | and
|T34 | ≤ m1234 ≤ |T12 |.
Since, for 27 < p ≤ 4, we have
(p − 4)(|T2 ||T4 |)p−3 mp−5
1234 (m12 − m34 ) ≤ 0
and
p−4
m1324
≤ (|T2 ||T4 |)p−4 ,
by (5.15) and (5.16), there holds
Z
|T4 |p−4 |T2 |2p−7
1
dξdη
≤
(p − 2)2p−2 |T1 |p−3 |T2 |p−3 |T3 |p−3 |T4 |p−3
b |J|p−1
K
(5.17)
1
|T3 |
=
.
(p − 2)2p−2 |T1 ||T3 |p−2 |T4 |
16
When p > 4, further noticing that
p−5
|T4 |p−3 m1234
≤

 |T4 |2p−8 ,
4 < p ≤ 5,
 |T |p−5 |T |p−3 , p > 5
2
4
≤ (|T1 ||T3 |)p−4
and
m12 − m34 ≤ |T2 | − |T4 | ≤ 2(|T1 | − |T4 |),
we can derive
Z
(5.18)
b
K
1
(p − 3)(|T1 ||T3 |)p−4 |T2 |p−3
dξdη
≤
|J|p−1
(p − 2)2p−2 |T1 |p−3 |T2 |p−3 |T3 |p−3 |T4 |p−3
µ
¶p−3
(p − 3)
|T3 |
≤
.
(p − 2)2p−2 |T1 ||T3 |p−2 |T4 |
In the case |T3 | ≤ |T4 |, we proved similarly that (5.17) and (5.18) hold by (5.11)). This
yields (5.9) and (5.10).
Corollary 5.7. With the assumption from Lemma 5.6, Condition (4.2) holds if
3
1
(5.19)
hK M 1− p + 2p
|d1 | sin α
(5.20)
hK M p
|d1 | sin α
1
µ
3
(5.21)
hK M 1− p
|d1 | sin α
µ
|T3 |
|T1 |
|T3 |
|T1 |
µ
¶ p1
≤ N,
¶ p1
|T3 |
|T1 |
7
for p ∈ [3, ],
2
7
≤ N, for p ∈ ( , 4],
2
¶ p1
≤ N, for p > 4,
where hK denotes the diameter of the quadrilateral K and α the maximal angle of T3 .
Remark 5.8. The technique developed in Lemma 5.6 renders the continuity of the
GRDP condition at the turning points p = 3, 27 , 4.
Let us finish our paper by comparing these conditions with the double angle condition
(DAC) introduced in [3], that we here recall
Definition 5.9. Let K be a convex quadrilateral. We say that K satisfies the double
angle condition with constants ψm , ψM , or shortly DAC(ψm , ψM ), if the interior angles ω of
K verify 0 < ψm ≤ ω ≤ ψM < π.
Obviously, the above DAC(ψm , ψM ) is equivalent to (1.2). In [3], the authors proved
that it a sufficient condition for the optimal error estimate in W 1,p with p ≥ 3 and showed
that the restriction on the maximal angle can not be relaxed by some counterexamples. In
fact, the DAC(ψm , ψM ) condition is a quite strong geometric condition and the following
elementary implications hold:
DAC(ψm , ψM ) =⇒ MAC(ψM ) =⇒ RDP(N, ψM ) =⇒ GRDP(N, ψM )
One can see that the above GRDP condition for p ≥ 3 is much weaker than the DAC
condition. In fact, if the quadrilateral K satisfies DAC(ψm , ψM ), it can be proved easily
17
3|
that |T
|T4 | ≤ C(ψm , ψM ) and d1 = O(hK ), hence there exists a constant N = N (ψm , ψM )
such that K satisfies GRDP (N (ψm , ψM ), ψM , p).
3|
If K satisfies |T
|T4 | ≤ C which is satisfied in many cases, the above GRDP condition for
p ≥ 3 is even weaker than the RDP condition. However, there are some examples such that
K satisfies the RDP condition but not the above GRDP condition, e.g., K(1, 1, s, s) with
s −→ 12 (cf. Figure 4), which is employed as counterexample in [3]. In this sense we may
say that our GRDP conditions are as weak as the RDP condition.
3|
If K satisfies |T
|T4 | −→ ∞, the quadrilateral K is almost degenerated into the triangle
T2 . In such a case, the constant of the interpolation error in W 1,p -norm (p > 3) may not
be bounded even if K satisfies the RDP condition. This can be partly interpreted by the
©
ª
3|
fact that max 1, |T
|T4 | appears as a factor in our GRDP condition. Taking K(1, 1, s, s) with
s −→ 21 (see Figure 4) as an example, if one choose d1 to divide K into two triangles, K
|T3 |
does satisfy the RDP condition, but (4.5) does not hold because |T
−→ ∞, which violates
4|
the GRDP condition.
(0, 1)
l
l
l
l
l
l (s, s)
¡e
¡
e
¡d1
e
¡
e
¡
e
¡
e
(0, 0)
(1, 0)
Figure 4. K(1, 1, s, s)
6. Conclusion. Interpolation error estimates of the finite elements play an important
role in the finite element literatures. In this paper, we have introduced generalized RDP (in
short GRDP) conditions and prove some interpolation error estimates in the W 1,p norm with
1 ≤ p < ∞ for the Q1 isoparametric finite elements. As far as we know, our GRDP conditions
presented here are weaker than any other mesh conditions proposed in the literatures for
the same p with 1 ≤ p < ∞.
The results of this paper are only valid for bilinear elements and it seems difficult to extend them to higher order Lagrange quadrilateral elements or other quadrilateral elements,
e.g., mixed elements and nonconforming elements. Some preliminary results for some mixed
elements and nonconforming elements are already obtained and will be investigated in our
future work.
Acknowledgments The authors would like to thank Professor Thomas Apel, Professor
Roland Becker and Professor Ricardo Durán for some useful discussions.
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