# A Posteriori Error Estimates with Computable Upper Bound for the Nonconforming RotatedQ

Authors: Jie Liu; Tian Xia; Wei Jiang
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

This paper
discusses the nonconforming rotatedQ
1 finite element computable upper bound a posteriori error estimate of the boundary value problem established by M. Ainsworth and obtains efficient computable upper bound a posteriori error indicators for the eigenvalue problem associated with the boundary value problem. We extend the a posteriori error estimate to the Steklov eigenvalue problem and also derive efficient computable upper bound a posteriori error indicators. Finally, through numerical experiments, we verify the validity of the a posteriori error estimate of the boundary value problem; meanwhile, the numerical results show that the a posteriori error indicators of the eigenvalue problem and the Steklov eigenvalue problem are effective.


---

## Body

## 1. Introduction

A posteriori error estimates and adaptive algorithms are the mainstream directions in the study of finite element methods; however, a posteriori error estimates are the theoretical basis of adaptive finite element method. Under these reasons, it is very meaningful to study the a posteriori error estimates. Particularly, it is well known that the residual type a posteriori error estimates usually contain a general constantC, which often affects the validity of the error estimates. Then, it is significant that exploring a computable upper bound a posteriori error estimate does not include constant C.The residual type a posteriori error estimate of finite element was first proposed by Babushka and Rheinboldt [1] in 1978 and has been studied and applied to many problems. For example, in 2005, Ainsworth [2] gave the a posteriori error estimate of residual type which can provide a computable upper bound for elliptic boundary value problem. In 2007, based on what Ainsworth researched in [2], Carstensen et al. [3] established a framework of a posteriori error estimates of residual type of a class of nonconforming finite element, which includes the nonconforming C
-
R element, the nonconforming rotated Q
1 element, and Han element, and so forth. In 2010, using the a posteriori error estimates of nonconforming finite element established by Carstensen, Yang [4] founded the a posteriori error indicators for elliptic differential operator eigenvalue problem. Recently, Han and Yang [5] gave a class of a posteriori error estimates of spectral element methods for 2nd-order elliptic eigenvalue problems.The finite element method is an important approach to solve the Steklov eigenvalue problem (see [6–10]). A posteriori error estimates of finite element for the Steklov eigenvalue problem has attracted attention from mathematical community in recent years. In 2008, Armentano and Padra [11] proposed and analyzed the a posteriori error estimate of the linear finite element approximation for the Steklov eigenvalue problem, and their residual type error estimate can be obtained by the local computation of approximate eigenpairs. In 2011, Ma et al. [12] studied a posteriori error estimate of the nonconforming E
Q
1
rot
⁡ element for Steklov eigenvalue problem. For the Steklov eigenvalue problems, Yang and Bi [13] have lately obtained the local a priori/a posteriori error estimates of conforming finite elements approximation and Zhang et al. [14] gave certain results of spectral method.The nonconforming rotatedQ
1 element was proposed by Rannacher and Turek [15]. Based on the existing research results, we discuss further a computable upper bound a posteriori error estimate of the boundary value problem established by Ainsworth and discover that this error estimate does not include a general constant C. So, we use the a posteriori error estimate to establish a computable upper bound a posteriori error indicators for the eigenvalue problem associated with the boundary value problem. In addition, we extend the error estimate to the Steklov eigenvalue problem, and obtain an efficient computable upper bound a posteriori error indicators. Finally, we verify that the computable upper bound a posteriori error estimate of the boundary value problem is effective (see Table 1). Through calculating the validity of the computable upper bound a posteriori error indicators on L-shaped domain, we can ascertain that the indicators of the eigenvalue problem and the Steklov eigenvalue problem are effective (see Tables 2 and 3).Table 1
The numerical results of Example1.
n
1
×
n
2
32
×
32
64
×
64
128
×
128
256
×
256
∥
w
1
-
w
1
,
h
∥
h
0.0062
0.0031
0.0015
7.703
e
-
004
∥
w
2
-
w
2
,
h
∥
h
0.0224
0.0112
0.0056
0.0028
∥
w
3
-
w
3
,
h
∥
h
8.215
e
-
004
4.110
e
-
004
2.055
e
-
004
1.028
e
-
004
η
com
/
∥
w
1
-
w
1
,
h
∥
h
1.0038
1.0030
1.0028
1.0027
η
com
/
∥
w
2
-
w
2
,
h
∥
h
1.0007
1.0004
1.0003
1.0003
η
com
/
∥
w
3
-
w
3
,
h
∥
h
1.0106
1.0071
1.0060
1.0056Table 2
The numerical results of the eigenvaluesλ
1, λ
2, and λ
3.
n
1
×
n
2
32
×
32
64
×
64
128
×
128
256
×
256
512
×
512
λ
1
,
h
38.47534
38.52266
38.54373
38.55268
38.55638
λ
2
,
h
60.83608
60.80069
60.79191
60.78973
60.78919
λ
3
,
h
78.95635
78.95680
78.95683
78.95683
78.95683
|
λ
1
,
h
-
λ
1
|
0.08354
0.03622
0.01515
0.00620
0.00250
|
λ
2
,
h
-
λ
2
|
0.04708
0.01169
0.00291
0.00073
0.00019
|
λ
3
,
h
-
λ
3
|
0.00050
0.00003
0.00000
0.00000
0.00000
η
com
2
/
|
λ
1
,
h
-
λ
1
|
3.55176
02.63591
2.13593
1.84850
1.68088
η
com
2
/
|
λ
2
,
h
-
λ
2
|
8.08495
8.12050
8.14971
8.11889
7.89114
η
com
2
/
|
λ
3
,
h
-
λ
3
|
1442.526
5026.540
6625.932
2260.119
578.1650Table 3
The numerical results of the Steklov eigenvaluesλ
2, λ
3, and λ
4.
n
1
×
n
2
8
×
8
16
×
16
32
×
32
64
×
64
128
×
128
λ
2
,
h
0.88320
0.88930
0.89190
0.89295
0.89337
λ
3
,
h
1.68213
1.68662
1.68805
1.68846
1.68856
λ
4
,
h
3.17490
3.20336
3.21370
3.21675
3.21757
|
λ
2
,
h
-
λ
2
|
0.01047
0.00436
0.00177
0.00071
0.00030
|
λ
3
,
h
-
λ
3
|
0.00647
0.00198
0.00055
0.00015
0.00003
|
λ
4
,
h
-
λ
4
|
0.04298
0.01453
0.00418
0.00114
0.00031
η
com
2
/
|
λ
2
,
h
-
λ
2
|
1.57977
1.47585
1.41700
1.36649
1.29035
η
com
2
/
|
λ
3
,
h
-
λ
3
|
1.56752
1.29357
1.17339
1.10827
1.04199
η
com
2
/
|
λ
4
,
h
-
λ
4
|
1.81759
1.35064
1.16560
1.06969
0.97406


## 2. Model Problem and Preliminaries

### 2.1. Model Problem

Consider the following eigenvalue problem:(1)
-
d
i
v
(
α
g
r
a
d
u
)
=
λ
u
,
in
Ω
n
·
α
g
r
a
d
u
=
0
,
on
⁡
Γ
N
u
=
0
,
on
⁡
Γ
D
,
where Ω
⊂
R
2 is a planar polygonal domain with boundary Γ
:
=
∂
Ω, the disjoint sets Γ
D and Γ
N form a partition of the boundary of Ω, and α
∈
L
∞
(
Ω
) is assumed to be nonnegative. For simplicity, we assume that α is piecewise constant on the finite element mesh.Then (1) can be written in a weak form: to seek (
λ
,
u
)
∈
ℝ
×
H
E
1
(
Ω
) with ∥
u
∥
0
=
1 such that
(2)
a
(
u
,
v
)
=
λ
b
(
u
,
v
)
,
∀
v
∈
H
E
1
(
Ω
)
,
where H
E
1
(
Ω
)
=
{
v
∈
H
1
(
Ω
)
:
v
=
0
,
on
⁡
Γ
D
}, and a
(
u
,
v
)
=
∫
Ω
‍
α
∇
u
∇
v
d
x
d
y, b
(
u
,
v
)
=
∫
Ω
‍
u
v
d
x
d
y.Let𝒫
h be a partition with mesh diameters h of the domain Ω consisting of disjoint convex quadrilateral elements, and the nonempty intersection of any two distinct elements is either a single common node or a common edge. In addition, the nonempty intersection of an element with the exterior boundary is a portion of either Γ
D or Γ
N. The family of partitions is assumed to be locally quasi-uniform in the sense that the ratio of the diameters of any adjacent elements is bounded above and below uniformly over the whole family of partitions. Define the generalized energy norm |
∥
v
∥
| by
(3)
|
∥
v
∥
|
2
=
(
α
g
r
a
d
h
v
,
g
r
a
d
h
v
)
,
where the operator g
r
a
d
h satisfies the condition (
g
r
a
d
h
v
)
|
κ
=
g
r
a
d
(
v
|
κ
)
,
∀
κ
∈
𝒫
h and the notation (
·
,
·
)
ϖ is used to denote the L
2-inner product over a domain ϖ. The subscript ϖ is omitted when it is a physical domain ϖ.The nonconforming rotatedQ
1 finite element space (see [15]) is defined by
(4)
X
h
=
{
∫
γ
‍
v
∈
L
2
(
Ω
)
:
v
|
κ
∈
span
⁡
(
1
,
x
,
y
,
x
2
-
y
2
)
,
∀
κ
∈
𝒫
h
,
∫
γ
‍
[
v
]
d
s
=
0
,
∀
γ
∈
∂
𝒫
h
∖
∂
Ω
}
,
where [
v
] denotes the jump across an interface γ and ∂
𝒫
h the set of element edges. The subspace X
h
,
E of X
h is defined by
(5)
X
h
,
E
=
{
v
∈
X
h
:
∫
γ
‍
v
d
s
=
0
,
if
γ
⊂
Γ
D
}
.The nonconforming rotatedQ
1 element approximation of (2) is the following: find (
λ
h
,
u
h
)
∈
ℝ
×
X
h
,
E such that
(6)
a
h
(
u
h
,
v
)
=
λ
h
b
(
u
h
,
v
)
,
∀
v
∈
X
h
,
E
,
where a
h
(
u
h
,
v
)
=
∑
κ
∈
𝒫
h
‍
∫
κ
‍
α
κ
∇
u
h
∇
v
d
x
d
y. Define ∥
·
∥
h
=
∑
κ
∈
𝒫
h
‍
|
∥
·
∥
|
κ
2. Evidently, ∥
·
∥
h is the norm on X
h
,
E.


### 2.2. A Posteriori Error Estimate of Boundary Value Problem

In this subsection we present the computable upper bound a posteriori error estimate of the boundary value problem established by Ainsworth in [2, 16]. It is the key to establishing a computable upper bound a posteriori error indicator for the eigenvalue problem (1).Consider the boundary value problem of findingw such that(7)
-
d
i
v
(
α
g
r
a
d
w
)
=
f
,
in
Ω
n
·
α
g
r
a
d
w
=
g
,
on
⁡
Γ
N
w
=
0
,
on
⁡
Γ
D
,
where f
∈
L
2
(
Ω
), g
∈
L
2
(
Γ
N
).The variational form of (7) consists of seeking w
∈
H
E
1
(
Ω
) such that
(8)
(
a
g
r
a
d
w
,
g
r
a
d
v
)
=
(
f
,
v
)
+
∫
Γ
N
‍
g
v
d
s
,
∀
v
∈
H
E
1
(
Ω
)
.The nonconforming rotatedQ
1 finite element approximation of (8) is the following: find w
h
∈
X
h
,
E such that
(9)
(
α
g
r
a
d
h
w
h
,
g
r
a
d
h
v
)
=
(
f
,
v
)
+
∫
Γ
N
‍
g
v
d
s
,
∀
v
∈
X
h
,
E
.To establish a computable upper bound of nonconforming finite element a posteriori estimate for the errore
=
w
-
w
h in the sense of energy norm (3), we use the following Helmholtz decomposition (see [17]) to divide the error e into the conforming part and the nonconforming part.Lemma 1.
Let(10)
ℋ
=
{
w
∈
H
1
(
Ω
)
:
∫
Ω
‍
w
d
x
d
y
=
0
,
∂
w
∂
τ
=
0
,
o
n
Γ
N
}
,
where ∂
w
/
∂
τ denotes the tangential derivative in direction τ.Then the errore can be decomposed as the form
(11)
α
g
r
a
d
h
e
=
α
g
r
a
d
ε
+
c
u
r
l
ξ
,
where ε
∈
H
E
1
(
Ω
) satisfies
(12)
(
α
g
r
a
d
ε
,
g
r
a
d
v
)
=
(
α
g
r
a
d
h
e
,
g
r
a
d
v
)
,
∀
v
∈
H
E
1
(
Ω
)
and ξ
∈
ℋ satisfies
(13)
(
α
-
1
c
u
r
l
ξ
,
c
u
r
l
w
)
=
(
α
g
r
a
d
h
e
,
g
r
a
d
w
)
,
∀
w
∈
ℋ
,
where c
u
r
l denotes the operator c
u
r
l
w
=
(
-
∂
y
w
,
∂
x
w
). Moreover, it is valid that
(14)
|
∥
e
∥
|
2
=
|
∥
ε
∥
|
2
+
(
α
-
1
c
u
r
l
ξ
,
c
u
r
l
ξ
)
.
Lemma 1 shows that the error e can be decomposed to the conforming part |
∥
ε
∥
|
2 and the nonconforming part (
α
-
1
c
u
r
l
ξ
,
c
u
r
l
ξ
).The following Theorem2 gives the error estimate of the conforming part.Theorem 2.
Letr
∈
L
2
(
κ
) and J
ν
∈
L
2
(
∂
κ
) denote the interior residual and the interelement flux jump, respectively. Then
(15)
|
∥
ε
∥
|
2
≤
∑
κ
∈
𝒫
h
{
∥
σ
κ
+
1
2
c
u
r
l
χ
κ
∥
α
-
1
,
κ
+
Δ
κ
}
2
,
where Δ
κ
=
C
𝒫
h
κ
∥
r
-
π
κ
r
∥
α
-
1
,
κ
+
(
1
/
2
)
C
t
∑
γ
⊂
∂
κ
‍
h
κ
1
/
2
∥
J
ν
-
π
γ
J
ν
∥
α
-
1
,
γ, π
γ
r
∈
L
2
(
κ
), ∥
f
∥
α
-
1
,
κ
=
∥
α
1
/
2
f
∥
κ
,
∀
κ
∈
𝒫
h, while Δ
κ is a quantity of higher order or even negligible compared with ∥
σ
κ
+
(
1
/
2
)
c
u
r
l
χ
κ
∥
α
-
1
,
κ. Both the vector-valued function σ
κ and the scalar-valued function χ
κ contain the interior residual (see [2])
(16)
r
|
κ
=
f
+
d
i
v
h
(
α
g
r
a
d
h
u
h
)
.Moreover, there exists a positive constantc, independent of mesh-size, such that for each element there holds
(17)
c
∥
σ
κ
+
1
2
c
u
r
l
χ
κ
∥
α
-
1
,
κ
≤
|
∥
ε
∥
|
κ
~
+
Δ
κ
,
where κ
~ is a block including the element κ and its adjacent elements.Lemma3 plays a key role for obtaining the error estimate of the nonconforming part.Lemma 3.
Letξ
∈
ℋ, ℋ be defined by (10); then
(18)
(
α
-
1
c
u
r
l
ξ
,
c
u
r
l
ξ
)
=
min
⁡
w
*
∈
H
E
1
(
Ω
)
|
∥
w
*
-
w
h
∥
|
.Evidently, (18) gives an upper bound of the nonconforming part. It is important to note that the right hand side of (18) is the minimum value and the interpolation postprocessing function w
* appears in the right hand side of (18). Reference [18] has emphasized that an appropriate selection of w
* is the key to obtaining an effective computable upper bound a posteriori error estimate. And this requires that the function w
* is of a simple form and computable and makes the error of the nonconforming part effective.Considering these factors, [2, 16] made such selection: w
* is taken to be a piecewise (pullback) biquadratic function on each element κ. The interpolation nodes of the function are the element vertices x
n, edge midpoints m
γ, and element centers x
-
κ. The interpolation conditions are given by
(19)
w
*
(
x
n
)
=
{
0
,
if
x
n
∈
Γ
D
1
|
𝒫
n
|
∑
κ
∈
𝒫
n
‍
w
h
(
x
n
)
|
κ
,
otherwise
,
w
*
(
m
γ
)
=
{
0
,
if
m
γ
∈
Γ
D
h
γ
-
1
∫
γ
‍
w
h
d
s
,
otherwise
,
w
*
(
x
-
κ
)
=
w
h
(
x
-
κ
)
,
where 𝒫
n
⊂
𝒫
h denotes the set of elements which share common vertex x
n, |
𝒫
n
|
=
card
{
𝒫
n
}.It is obvious that the functionw
* defined above satisfies w
*
∈
H
E
1
(
Ω
) and can be used to obtain an upper bound for the nonconforming part of the a posteriori error estimates.Theorem4 gives the reliability and validity of the nonconforming part.Theorem 4.
Letw
*
∈
H
E
1
(
Ω
) be constructed as described above; then
(20)
∥
c
u
r
l
ξ
∥
α
-
1
,
Ω
≤
|
∥
w
h
-
w
*
∥
|
.
Moreover, there exists a positive constant C, independent of any mesh-size, such that
(21)
|
∥
w
h
-
w
*
∥
|
κ
≤
C
∥
c
u
r
l
ξ
∥
α
-
1
,
κ
~
.Combining (14), (15), and (20), we have the following overall a posteriori error estimate:
(22)
|
∥
e
∥
|
2
≤
∑
κ
∈
𝒫
h
‍
(
∥
σ
κ
+
1
2
c
u
r
l
χ
κ
∥
α
-
1
,
κ
+
Δ
κ
)
2
+
|
∥
w
h
-
w
*
∥
|
2
.Note thatΔ
κ is a quantity of higher order compared with ∥
σ
κ
+
(
1
/
2
)
c
u
r
l
χ
κ
∥
α
-
1
,
κ, or even negligible. Let w
h be the approximate solution of (8); we define a computable upper bound a posteriori error indicator by
(23)
η
com
2
=
η
c
2
+
η
nc
2
in which η
c
2
=
∑
κ
∈
𝒫
h
‍
∥
σ
κ
+
(
1
/
2
)
c
u
r
l
χ
κ
∥
α
-
1
,
κ
2 denotes the a posteriori error indicator of conforming part and η
nc
2
=
|
∥
w
h
-
w
*
∥
|
2 the a posteriori error indicator of nonconforming part. Hence, we can use η
com as the error estimate indicator of w
h.Obviously, the error indicatorη
com does not include a general constant C and is an effective error indicator (see Table 1). So, we are very interested in the error indicator η
com and decide to apply the indicator η
com to eigenvalue problem (1).


## 2.1. Model Problem

Consider the following eigenvalue problem:(1)
-
d
i
v
(
α
g
r
a
d
u
)
=
λ
u
,
in
Ω
n
·
α
g
r
a
d
u
=
0
,
on
⁡
Γ
N
u
=
0
,
on
⁡
Γ
D
,
where Ω
⊂
R
2 is a planar polygonal domain with boundary Γ
:
=
∂
Ω, the disjoint sets Γ
D and Γ
N form a partition of the boundary of Ω, and α
∈
L
∞
(
Ω
) is assumed to be nonnegative. For simplicity, we assume that α is piecewise constant on the finite element mesh.Then (1) can be written in a weak form: to seek (
λ
,
u
)
∈
ℝ
×
H
E
1
(
Ω
) with ∥
u
∥
0
=
1 such that
(2)
a
(
u
,
v
)
=
λ
b
(
u
,
v
)
,
∀
v
∈
H
E
1
(
Ω
)
,
where H
E
1
(
Ω
)
=
{
v
∈
H
1
(
Ω
)
:
v
=
0
,
on
⁡
Γ
D
}, and a
(
u
,
v
)
=
∫
Ω
‍
α
∇
u
∇
v
d
x
d
y, b
(
u
,
v
)
=
∫
Ω
‍
u
v
d
x
d
y.Let𝒫
h be a partition with mesh diameters h of the domain Ω consisting of disjoint convex quadrilateral elements, and the nonempty intersection of any two distinct elements is either a single common node or a common edge. In addition, the nonempty intersection of an element with the exterior boundary is a portion of either Γ
D or Γ
N. The family of partitions is assumed to be locally quasi-uniform in the sense that the ratio of the diameters of any adjacent elements is bounded above and below uniformly over the whole family of partitions. Define the generalized energy norm |
∥
v
∥
| by
(3)
|
∥
v
∥
|
2
=
(
α
g
r
a
d
h
v
,
g
r
a
d
h
v
)
,
where the operator g
r
a
d
h satisfies the condition (
g
r
a
d
h
v
)
|
κ
=
g
r
a
d
(
v
|
κ
)
,
∀
κ
∈
𝒫
h and the notation (
·
,
·
)
ϖ is used to denote the L
2-inner product over a domain ϖ. The subscript ϖ is omitted when it is a physical domain ϖ.The nonconforming rotatedQ
1 finite element space (see [15]) is defined by
(4)
X
h
=
{
∫
γ
‍
v
∈
L
2
(
Ω
)
:
v
|
κ
∈
span
⁡
(
1
,
x
,
y
,
x
2
-
y
2
)
,
∀
κ
∈
𝒫
h
,
∫
γ
‍
[
v
]
d
s
=
0
,
∀
γ
∈
∂
𝒫
h
∖
∂
Ω
}
,
where [
v
] denotes the jump across an interface γ and ∂
𝒫
h the set of element edges. The subspace X
h
,
E of X
h is defined by
(5)
X
h
,
E
=
{
v
∈
X
h
:
∫
γ
‍
v
d
s
=
0
,
if
γ
⊂
Γ
D
}
.The nonconforming rotatedQ
1 element approximation of (2) is the following: find (
λ
h
,
u
h
)
∈
ℝ
×
X
h
,
E such that
(6)
a
h
(
u
h
,
v
)
=
λ
h
b
(
u
h
,
v
)
,
∀
v
∈
X
h
,
E
,
where a
h
(
u
h
,
v
)
=
∑
κ
∈
𝒫
h
‍
∫
κ
‍
α
κ
∇
u
h
∇
v
d
x
d
y. Define ∥
·
∥
h
=
∑
κ
∈
𝒫
h
‍
|
∥
·
∥
|
κ
2. Evidently, ∥
·
∥
h is the norm on X
h
,
E.


## 2.2. A Posteriori Error Estimate of Boundary Value Problem

In this subsection we present the computable upper bound a posteriori error estimate of the boundary value problem established by Ainsworth in [2, 16]. It is the key to establishing a computable upper bound a posteriori error indicator for the eigenvalue problem (1).Consider the boundary value problem of findingw such that(7)
-
d
i
v
(
α
g
r
a
d
w
)
=
f
,
in
Ω
n
·
α
g
r
a
d
w
=
g
,
on
⁡
Γ
N
w
=
0
,
on
⁡
Γ
D
,
where f
∈
L
2
(
Ω
), g
∈
L
2
(
Γ
N
).The variational form of (7) consists of seeking w
∈
H
E
1
(
Ω
) such that
(8)
(
a
g
r
a
d
w
,
g
r
a
d
v
)
=
(
f
,
v
)
+
∫
Γ
N
‍
g
v
d
s
,
∀
v
∈
H
E
1
(
Ω
)
.The nonconforming rotatedQ
1 finite element approximation of (8) is the following: find w
h
∈
X
h
,
E such that
(9)
(
α
g
r
a
d
h
w
h
,
g
r
a
d
h
v
)
=
(
f
,
v
)
+
∫
Γ
N
‍
g
v
d
s
,
∀
v
∈
X
h
,
E
.To establish a computable upper bound of nonconforming finite element a posteriori estimate for the errore
=
w
-
w
h in the sense of energy norm (3), we use the following Helmholtz decomposition (see [17]) to divide the error e into the conforming part and the nonconforming part.Lemma 1.
Let(10)
ℋ
=
{
w
∈
H
1
(
Ω
)
:
∫
Ω
‍
w
d
x
d
y
=
0
,
∂
w
∂
τ
=
0
,
o
n
Γ
N
}
,
where ∂
w
/
∂
τ denotes the tangential derivative in direction τ.Then the errore can be decomposed as the form
(11)
α
g
r
a
d
h
e
=
α
g
r
a
d
ε
+
c
u
r
l
ξ
,
where ε
∈
H
E
1
(
Ω
) satisfies
(12)
(
α
g
r
a
d
ε
,
g
r
a
d
v
)
=
(
α
g
r
a
d
h
e
,
g
r
a
d
v
)
,
∀
v
∈
H
E
1
(
Ω
)
and ξ
∈
ℋ satisfies
(13)
(
α
-
1
c
u
r
l
ξ
,
c
u
r
l
w
)
=
(
α
g
r
a
d
h
e
,
g
r
a
d
w
)
,
∀
w
∈
ℋ
,
where c
u
r
l denotes the operator c
u
r
l
w
=
(
-
∂
y
w
,
∂
x
w
). Moreover, it is valid that
(14)
|
∥
e
∥
|
2
=
|
∥
ε
∥
|
2
+
(
α
-
1
c
u
r
l
ξ
,
c
u
r
l
ξ
)
.
Lemma 1 shows that the error e can be decomposed to the conforming part |
∥
ε
∥
|
2 and the nonconforming part (
α
-
1
c
u
r
l
ξ
,
c
u
r
l
ξ
).The following Theorem2 gives the error estimate of the conforming part.Theorem 2.
Letr
∈
L
2
(
κ
) and J
ν
∈
L
2
(
∂
κ
) denote the interior residual and the interelement flux jump, respectively. Then
(15)
|
∥
ε
∥
|
2
≤
∑
κ
∈
𝒫
h
{
∥
σ
κ
+
1
2
c
u
r
l
χ
κ
∥
α
-
1
,
κ
+
Δ
κ
}
2
,
where Δ
κ
=
C
𝒫
h
κ
∥
r
-
π
κ
r
∥
α
-
1
,
κ
+
(
1
/
2
)
C
t
∑
γ
⊂
∂
κ
‍
h
κ
1
/
2
∥
J
ν
-
π
γ
J
ν
∥
α
-
1
,
γ, π
γ
r
∈
L
2
(
κ
), ∥
f
∥
α
-
1
,
κ
=
∥
α
1
/
2
f
∥
κ
,
∀
κ
∈
𝒫
h, while Δ
κ is a quantity of higher order or even negligible compared with ∥
σ
κ
+
(
1
/
2
)
c
u
r
l
χ
κ
∥
α
-
1
,
κ. Both the vector-valued function σ
κ and the scalar-valued function χ
κ contain the interior residual (see [2])
(16)
r
|
κ
=
f
+
d
i
v
h
(
α
g
r
a
d
h
u
h
)
.Moreover, there exists a positive constantc, independent of mesh-size, such that for each element there holds
(17)
c
∥
σ
κ
+
1
2
c
u
r
l
χ
κ
∥
α
-
1
,
κ
≤
|
∥
ε
∥
|
κ
~
+
Δ
κ
,
where κ
~ is a block including the element κ and its adjacent elements.Lemma3 plays a key role for obtaining the error estimate of the nonconforming part.Lemma 3.
Letξ
∈
ℋ, ℋ be defined by (10); then
(18)
(
α
-
1
c
u
r
l
ξ
,
c
u
r
l
ξ
)
=
min
⁡
w
*
∈
H
E
1
(
Ω
)
|
∥
w
*
-
w
h
∥
|
.Evidently, (18) gives an upper bound of the nonconforming part. It is important to note that the right hand side of (18) is the minimum value and the interpolation postprocessing function w
* appears in the right hand side of (18). Reference [18] has emphasized that an appropriate selection of w
* is the key to obtaining an effective computable upper bound a posteriori error estimate. And this requires that the function w
* is of a simple form and computable and makes the error of the nonconforming part effective.Considering these factors, [2, 16] made such selection: w
* is taken to be a piecewise (pullback) biquadratic function on each element κ. The interpolation nodes of the function are the element vertices x
n, edge midpoints m
γ, and element centers x
-
κ. The interpolation conditions are given by
(19)
w
*
(
x
n
)
=
{
0
,
if
x
n
∈
Γ
D
1
|
𝒫
n
|
∑
κ
∈
𝒫
n
‍
w
h
(
x
n
)
|
κ
,
otherwise
,
w
*
(
m
γ
)
=
{
0
,
if
m
γ
∈
Γ
D
h
γ
-
1
∫
γ
‍
w
h
d
s
,
otherwise
,
w
*
(
x
-
κ
)
=
w
h
(
x
-
κ
)
,
where 𝒫
n
⊂
𝒫
h denotes the set of elements which share common vertex x
n, |
𝒫
n
|
=
card
{
𝒫
n
}.It is obvious that the functionw
* defined above satisfies w
*
∈
H
E
1
(
Ω
) and can be used to obtain an upper bound for the nonconforming part of the a posteriori error estimates.Theorem4 gives the reliability and validity of the nonconforming part.Theorem 4.
Letw
*
∈
H
E
1
(
Ω
) be constructed as described above; then
(20)
∥
c
u
r
l
ξ
∥
α
-
1
,
Ω
≤
|
∥
w
h
-
w
*
∥
|
.
Moreover, there exists a positive constant C, independent of any mesh-size, such that
(21)
|
∥
w
h
-
w
*
∥
|
κ
≤
C
∥
c
u
r
l
ξ
∥
α
-
1
,
κ
~
.Combining (14), (15), and (20), we have the following overall a posteriori error estimate:
(22)
|
∥
e
∥
|
2
≤
∑
κ
∈
𝒫
h
‍
(
∥
σ
κ
+
1
2
c
u
r
l
χ
κ
∥
α
-
1
,
κ
+
Δ
κ
)
2
+
|
∥
w
h
-
w
*
∥
|
2
.Note thatΔ
κ is a quantity of higher order compared with ∥
σ
κ
+
(
1
/
2
)
c
u
r
l
χ
κ
∥
α
-
1
,
κ, or even negligible. Let w
h be the approximate solution of (8); we define a computable upper bound a posteriori error indicator by
(23)
η
com
2
=
η
c
2
+
η
nc
2
in which η
c
2
=
∑
κ
∈
𝒫
h
‍
∥
σ
κ
+
(
1
/
2
)
c
u
r
l
χ
κ
∥
α
-
1
,
κ
2 denotes the a posteriori error indicator of conforming part and η
nc
2
=
|
∥
w
h
-
w
*
∥
|
2 the a posteriori error indicator of nonconforming part. Hence, we can use η
com as the error estimate indicator of w
h.Obviously, the error indicatorη
com does not include a general constant C and is an effective error indicator (see Table 1). So, we are very interested in the error indicator η
com and decide to apply the indicator η
com to eigenvalue problem (1).


## 3. A Posteriori Error Estimate of the Eigenvalue Problem

In this section, we apply the error indicatorη
com to the eigenvalue problem (1) and obtain a computable upper bound a posteriori error indicator η
com
2 with f
=
λ
h
u
h in (16), where (
λ
h
,
u
h
) is the kth eigenpair of (6).In order to establish the error indicatorη
com
2, we need the following results, cited from [4, 19, 20], respectively, as our Lemmas 5, 6, and 7.Lemma 5.
Let(
λ
h
,
u
h
) be the kth eigenpair of (6) with ∥
u
h
∥
b
=
1, let λ be the kth eigenvalue of (2), and let M
(
λ
)
⊂
H
1
+
r
(
Ω
) be the eigenspace corresponding to λ. Then λ
h
→
λ, and there exists u
∈
M
(
λ
) with ∥
u
∥
b
=
1, such that
(24)
|
λ
h
-
λ
|
≤
C
λ
2
h
2
r
,
∥
u
h
-
u
∥
b
≤
C
h
2
r
,
∥
u
h
-
u
∥
h
≤
C
λ
h
r
,
where ω is the largest inner angle of Ω with the edges parallel with axis. If ω
>
π then r
<
π
/
ω and sufficiently close to π
/
ω, and ω
<
π; then r
=
1.LetΩ
⊂
R
2 be the bounded domain. Define operators T
:
L
2
(
Ω
)
→
H
E
1
(
Ω
) satisfies
(25)
a
(
T
f
,
v
)
=
b
(
f
,
v
)
,
∀
f
∈
L
2
(
Ω
)
,
∀
v
∈
H
E
1
(
Ω
)
,
and T
h
:
L
2
(
Ω
)
→
X
h
,
E satisfies
(26)
a
h
(
T
h
f
,
v
)
=
b
(
f
,
v
)
,
∀
f
∈
L
2
(
Ω
)
,
∀
v
∈
X
h
,
E
.It is easy to know that (2) and (6) have the equivalent operator forms λ
T
u
=
u and λ
h
T
h
u
h
=
u
h, respectively. Meanwhile, we have the following estimates.Lemma 6.
Under the assumption of Lemma5. Moreover, if ∥
T
-
T
h
∥
b
→
0
(
h
→
0
) and there exists a positive constant C independent of mesh-size h and q
1
<
q
2, such that ∀
f
∈
L
2
(
Ω
), ∥
T
f
-
T
h
f
∥
h
≤
C
h
q
1
∥
f
∥
b, and ∥
T
f
-
T
h
f
∥
b
≤
C
h
q
2
∥
f
∥
b, then there exists u
∈
M
(
λ
) with ∥
u
∥
b
=
1, such that ∀
v
∈
H
E
1
(
Ω
)
∩
X
h
,
E
(27)
∥
u
h
-
u
∥
h
,
D
=
λ
h
∥
T
u
h
-
T
h
u
h
∥
h
,
D
+
R
1
,
(28)
λ
h
-
λ
=
λ
h
2
∥
T
u
h
-
T
h
u
h
∥
h
2
+
2
λ
h
a
h
(
T
u
h
-
T
h
u
h
,
u
h
-
v
)
+
R
2
,
where D
⊂
Ω, R
1, R
2 are infinitesimals of higher order.Lemma 7.
Let(
λ
,
u
) and (
λ
h
,
u
h
) be the solutions of problems (2) and (6), respectively. Then ∀
v
∈
X
h
,
E
(29)
λ
-
λ
h
=
∥
u
-
u
h
∥
h
2
-
λ
h
∥
v
-
u
h
∥
b
2
-
2
λ
h
b
(
u
-
v
,
u
h
)
+
2
a
h
(
u
-
v
,
u
h
)
.Under the above preparations, we can obtain the following error estimates for the eigenvalue and eigenfunction of problem (1).Theorem 8.
Let(
λ
h
,
u
h
) be the kth nonconforming rotated Q
1 element eigenpair of (6) with ∥
u
h
∥
b
=
1, and let λ be the kth eigenvalue of (2). Moreover, let Ω be a concave domain and let the eigenfunction u be singular. Then we have
(30)
∥
u
h
-
u
∥
h
≤
η
c
o
m
+
R
1
,
(31)
|
λ
-
λ
h
|
=
∥
u
h
-
u
∥
h
2
+
o
(
h
2
r
)
;
Thus,
(32)
|
λ
-
λ
h
|
≤
η
c
o
m
2
+
o
(
h
2
r
)
+
R
2
,
where R
2
=
R
1
2
+
2
R
1
η
c
o
m is infinitesimal of higher order compared with η
c
o
m
2.Proof.
Takingf
=
λ
h
u
h, g
=
0 in (8), then w
=
λ
h
T
u
h and w
h
=
λ
h
T
h
u
h
=
u
h. By (22), we have
(33)
∥
λ
h
T
u
h
-
λ
h
T
h
u
h
∥
h
≤
η
com
.
Combining (33) and (27) (taking D
=
Ω), we get
(34)
∥
u
h
-
u
∥
h
=
λ
h
∥
T
u
h
-
T
h
u
h
∥
h
+
R
1
≤
η
com
+
R
1
,
which shows that (30) holds.
In order to prove (31) and (32), we define interpolation operator I
h
:
H
E
1
(
Ω
)
→
X
h
,
E by
(35)
∫
γ
‍
I
h
u
d
s
=
∫
γ
‍
u
d
s
,
∀
γ
∈
∂
κ
,
∀
κ
∈
𝒫
h
,
∀
u
∈
H
E
1
(
Ω
)
.
Let v
=
I
h
u in Lemma 7; the fourth term on the right-hand side of (29) vanishes (see [21, 22]).
Considering the third term of (29), from the interpolation error estimate, we have
(36)
|
-
2
λ
h
b
(
u
-
I
h
u
,
u
h
)
|
≤
C
h
1
+
r
∥
u
∥
1
+
r
,
and that, according to Lemma 5, we know that the second and the third terms are infinitesimals of higher order comparing with the first term. Hence, the error λ
-
λ
h completely hinges on the first term on the right-hand side of (29); that is, (31) holds. Combining (30) and (31), we obtain (32).Remark 9.
For the nonconforming rotatedQ
1 finite element, only when π
<
ω
<
2
π would the estimation |
-
2
λ
h
b
(
u
-
I
h
u
,
u
h
)
|
≤
C
h
1
+
r
∥
u
∥
1
+
r be valid (see [19]). Therefore, it is necessary to assume that Ω be a concave domain.
From (32) and (30), we can obtain the computable upper bound a posteriori error indicators η
com
2 and η
com for the eigenvalue λ
h and the associated eigenfunction u
h, respectively.


## 4. Extension and Application

In this section, we extend the error indicatorη
com to the Steklov eigenvalue problem and also obtain an effective error indicator η
com
2 with f
=
λ
h
u
h and α
=
1 in (16), where λ
h and u
h are the approximations of (37).The Steklov eigenvalue problem reads as follows:(37)
-
Δ
u
+
u
=
0
,
in
Ω
,
n
·
g
r
a
d
u
=
λ
u
,
on
⁡
∂
Ω
,
where Ω
⊂
R
2 is a bounded convex polygonal domain.We have Steklov eigenvalue problem in its variational formulation: find(
λ
,
u
)
∈
ℝ
×
H
1
(
Ω
) with ∥
u
∥
b
=
1, so that
(38)
a
(
u
,
v
)
=
λ
b
(
u
,
v
)
,
∀
v
∈
H
1
(
Ω
)
,
where a
(
u
,
v
)
=
∫
Ω
‍
∇
u
∇
v
+
u
v
d
x, b
(
u
,
v
)
=
∫
∂
Ω
‍
u
v
d
s, ∥
u
∥
b
=
b
(
u
,
v
). Clearly a
(
·
,
·
) is a symmetric, continuous, and H
1
(
Ω
)-elliptic bilinear form defined on H
1
(
Ω
)
×
H
1
(
Ω
).The nonconforming finite element approximation of (38) is the following: find (
λ
h
,
u
h
)
∈
ℝ
×
X
h with ∥
u
h
∥
b
=
1, such that
(39)
a
h
(
u
h
,
v
)
=
λ
h
b
(
u
h
,
v
)
,
∀
v
∈
X
h
,
where a
h
(
u
h
,
v
)
=
∑
κ
∈
𝒫
h
‍
∫
κ
‍
∇
u
h
∇
v
+
u
h
v
d
x. Define ∥
v
∥
h
=
a
h
(
v
,
v
). Evidently, ∥
·
∥
h is the norm on X
h and a
h
(
·
,
·
) is uniformly X
h-elliptic. In fact, a
h
(
v
,
v
)
=
∥
v
∥
h
2
,
∀
v
∈
X
h.To define two useful operators, we need the source problem (40) associated with (38) and the discrete problem (41).Findφ
∈
H
1
(
Ω
), satisfies
(40)
a
(
φ
,
v
)
=
b
(
ρ
,
v
)
,
∀
v
∈
H
1
(
Ω
)
,
and find φ
h
∈
X
h, such that
(41)
a
h
(
φ
h
,
v
)
=
b
(
ρ
,
v
)
,
∀
v
∈
X
h
.Using the source problem (40), we define the operators A and T:
(42)
A
:
L
2
(
∂
Ω
)
⟶
H
3
/
2
(
Ω
)
⊂
H
1
(
Ω
)
,
a
(
A
ρ
,
v
)
=
b
(
ρ
,
v
)
,
∀
v
∈
H
1
(
Ω
)
,
T
ρ
=
(
A
ρ
)
′
,
T
:
L
2
(
∂
Ω
)
⟶
H
1
(
∂
Ω
)
,
where the symbol “′” denotes the restriction to ∂
Ω. Bramble and Osborn [8] proved that (38) has the operator form T
u
=
(
1
/
λ
)
u.Sincea
h
(
·
,
·
) is uniformly elliptic with respect to h, the problem (41) has unique solution. We then define the operators:
(43)
A
h
:
L
2
(
∂
Ω
)
⟶
X
h
,
a
h
(
A
h
ρ
,
v
)
=
b
(
ρ
,
v
)
,
∀
v
∈
X
h
,
T
h
ρ
=
(
A
h
ρ
)
′
,
T
h
:
L
2
(
∂
Ω
)
⟶
∂
X
h
⊂
L
2
(
∂
Ω
)
.
From [10], (39) has the operator form T
h
u
h
′
=
(
1
/
λ
h
)
u
h
′. T and T
h are self-adjoint, completely continuous operators and ∥
T
h
-
T
∥
b
→
0
,
(
h
→
0
).For the Steklov eigenvalue problem, we need the following error estimates (see [10]) and expansion (see [21]) which will be used in our subsequent analysis.Lemma 10.
Let(
λ
h
,
u
h
) be the kth nonconforming rotated Q
1 element eigenpair of (39) with ∥
u
h
∥
b
=
1, and let λ be the kth eigenvalue of (38). Then λ
h
→
λ, and there exists u
∈
M
(
λ
) with ∥
u
∥
b
=
1, such that
(44)
|
λ
h
-
λ
|
≤
C
h
2
,
(45)
∥
u
h
-
u
∥
b
+
∥
A
h
u
-
A
u
∥
b
≤
C
h
3
/
2
,
(46)
∥
u
h
-
u
∥
h
+
λ
∥
A
h
u
-
A
u
∥
h
≤
C
h
,
where M
(
λ
) is the space spanned by eigenvector corresponding λ.Lemma 11.
Let(
λ
,
u
)
∈
ℝ
×
H
1
(
Ω
) be an eigenpair of (37) and let (
λ
h
,
u
h
)
∈
ℝ
×
X
h be an eigenpair of (39). Then ∀
v
∈
X
h
(47)
λ
-
λ
h
=
∥
u
-
u
h
∥
h
2
-
λ
h
∥
v
-
u
h
∥
b
2
+
λ
h
(
∥
v
∥
b
2
-
∥
u
∥
b
2
)
+
2
a
h
(
u
-
v
,
u
h
)
.According to the above consequences, we have the following theorem which can be proved with the approach in [4].Theorem 12.
Under the assumption of Lemma10. Then there exists u
∈
M
(
λ
) with ∥
u
∥
b
=
1, such that
(48)
∥
u
h
-
u
∥
h
=
λ
h
∥
A
h
u
h
-
A
u
h
∥
h
+
R
,
where |
R
|
≤
C
h
3
/
2.Proof.
From the definitions ofA and A
h, we obtain
(49)
u
h
-
u
=
λ
h
A
h
u
h
-
λ
A
u
=
λ
h
A
h
u
h
-
λ
h
A
u
h
+
λ
h
A
u
h
-
λ
h
A
u
+
λ
h
A
u
-
λ
A
u
=
λ
h
(
A
h
-
A
)
u
h
+
λ
h
A
(
u
h
-
u
)
+
(
λ
h
-
λ
)
A
u
.
Let R
=
∥
u
h
-
u
∥
h
-
λ
h
∥
A
u
u
h
-
A
u
h
∥
h, combining the triangle inequality, (49), (44), and (45), we deduce
(50)
|
R
|
=
|
∥
u
h
-
u
∥
h
-
λ
h
∥
A
u
u
h
-
A
u
h
∥
h
|
≤
∥
u
h
-
u
-
λ
h
(
A
h
-
A
)
u
h
∥
h
≤
C
(
|
λ
h
-
λ
|
+
λ
h
∥
u
h
-
u
∥
b
)
≤
C
h
3
/
2
.
That is, (48) is obtained.Based on (48), we have the following computable upper bound a posteriori error estimate of the eigenfunction of (39).Theorem 13.
Under the assumption of Lemma10. Then exists u
∈
M
(
λ
) with ∥
u
∥
b
=
1, such that
(51)
∥
u
h
-
u
∥
h
≤
η
c
o
m
+
o
(
h
3
/
2
)
+
o
(
∥
u
h
-
w
∥
-
1
)
.Proof.
Consider the auxiliary problem(52)
-
Δ
w
=
-
u
h
,
in
Ω
,
n
·
g
r
a
d
w
=
λ
h
u
h
,
on
⁡
∂
Ω
;
under the condition of ∫
Ω
‍
w
d
x
=
0, the auxiliary problem exists a unique solution only up to additive constant. Let w be the exact solution and let w
h be the approximate solution of (52) and (
λ
h
,
u
h
) be a rotated Q
1 element eigenpair of (39) obviously, w
h
=
u
h. Taking f
=
-
u
h, α
=
1, and g
=
λ
h
u
h in (7), from the a posteriori error estimate (22) and the definition of η
com, we have
(53)
∥
w
h
-
w
∥
h
≤
η
com
.
For the source problem (40) and (41), taking ρ
=
λ
h
u
h, then φ
=
λ
h
A
u
h, φ
h
=
λ
h
A
h
u
h
=
u
h. We deduce
(54)
a
(
φ
-
w
,
v
)
=
a
(
φ
-
u
h
+
u
h
-
w
,
v
)
=
a
(
φ
,
v
)
-
∫
Ω
‍
∇
u
h
∇
v
-
∫
Ω
‍
u
h
v
+
∫
Ω
‍
∇
(
u
h
-
w
)
∇
v
+
(
u
h
-
w
,
v
)
=
a
(
φ
,
v
)
-
∫
Ω
‍
u
h
v
+
∫
Ω
‍
Δ
w
v
-
∫
∂
Ω
‍
(
n
·
g
r
a
d
w
)
v
+
(
u
h
-
w
,
v
)
=
a
(
φ
,
v
)
-
b
(
ρ
,
v
)
+
(
u
h
-
w
,
v
)
=
(
u
h
-
w
,
v
)
.
LetH
-
1
(
Ω
) denote the dual space of H
1
(
Ω
) with norm
(55)
∥
f
∥
-
1
=
sup
⁡
v
∈
H
1
(
Ω
)
(
f
,
v
)
∥
v
∥
1
,
∀
f
∈
H
-
1
(
Ω
)
.
Setting v
=
φ
-
w, it follows that
(56)
∥
φ
-
w
∥
1
2
≤
sup
⁡
v
∈
H
1
(
Ω
)
(
u
h
-
w
,
v
)
∥
v
∥
1
∥
v
∥
1
=
∥
u
h
-
w
∥
-
1
∥
φ
-
w
∥
1
;
thus,
(57)
∥
φ
-
w
∥
1
≤
∥
u
h
-
w
∥
-
1
.
Further,
(58)
∥
φ
-
u
h
∥
h
=
∥
φ
-
w
+
w
-
u
h
∥
h
=
∥
u
h
-
w
∥
h
+
o
(
∥
u
h
-
w
∥
-
1
)
.
From (53) and (58), we find
(59)
∥
λ
h
A
u
u
-
λ
h
A
h
u
h
∥
h
=
∥
φ
-
φ
h
∥
h
=
∥
φ
-
u
h
∥
h
=
∥
w
h
-
w
∥
h
+
o
(
∥
u
h
-
w
∥
-
1
)
≤
η
com
+
o
(
∥
u
h
-
w
∥
-
1
)
.
Substituting (59) into (48), we obtain (51).In Theorem13, o
(
h
3
/
2
) and o
(
∥
u
h
-
w
∥
-
1
) are generally infinitesimals of higher order comparing with η
com. Therefore, we can use η
com as a computable upper bound a posteriori error indicator for the eigenfunction u
h of (39).The next corollary gives a relation between the error in the eigenvalue and eigenfunction approximations.Corollary 14.
Under the assumption of Lemma10, we have
(60)
λ
-
λ
h
≤
η
c
o
m
2
+
2
∫
Ω
‍
(
u
-
I
h
u
)
u
h
+
O
(
h
5
/
2
)
+
R
,
where R is infinitesimal of higher order comparing with η
c
o
m
2.Proof.
We define interpolation operatorI
h
:
H
1
(
Ω
)
→
X
h, such that
(61)
∫
γ
‍
I
h
u
d
s
=
∫
γ
‍
u
d
s
,
∀
γ
⊂
∂
κ
,
∀
κ
∈
𝒫
h
,
∀
u
∈
H
1
(
Ω
)
.
Taking v
=
I
h
u in Lemma 11, for the fourth term and the third term on the right-hand side of (47), we have (see [10], pp: 2397–2398)
(62)
a
h
(
u
-
I
h
u
,
u
h
)
=
∫
Ω
‍
(
u
-
I
h
u
)
u
h
d
x
,
|
∥
I
h
u
∥
b
2
-
∥
u
∥
b
2
|
≤
C
h
5
/
2
.
For the second term on the right-hand side of (47), by (45), we get
(63)
|
-
λ
h
∥
I
h
u
-
u
h
∥
b
2
|
≤
C
h
3
.
Thus,
(64)
λ
-
λ
h
=
∥
u
-
u
h
∥
h
2
+
2
∫
Ω
‍
(
u
-
I
h
u
)
u
h
+
O
(
h
5
/
2
)
.
Combining (51) and (64), we obtain (60).Remark 15.
Under certain conditions, we can prove that2
∫
Ω
‍
(
u
-
I
h
u
)
u
h is a infinitesimal of higher order than ∥
u
-
u
h
∥
h
2. From Lemma 11 of Yang et al. [10], we have |
∫
Ω
‍
(
u
-
I
h
u
)
u
h
|
≤
C
h
2 and ∥
u
-
u
h
∥
h
≤
C
h provided that u
∈
H
2
(
Ω
). Similarly, if u
∈
H
1
+
r
(
0
<
r
<
1
), Ω is a concave domain and the eigenfunction u
h is singular. Then, |
∫
Ω
‍
(
u
-
I
h
u
)
u
h
| may reach O
(
h
1
+
r
) convergence rate and ∥
u
-
u
h
∥
h may reach O
(
h
r
). So, the error λ
-
λ
h hinges on ∥
u
-
u
h
∥
h
2. By (60) and (64), we can obtain a computable upper a posteriori error indicator η
com
2 for the eigenvalue λ
h. The numerical results (see Table 3) show that this hypothesis is appropriate.


## 5. Numerical Examples

This section will report some computational results for the computable upper bound a posteriori error indicatorsη
com and η
com
2. For the sake of simplicity, we take α
=
1, and the partition 𝒫
h is uniform square meshes in problems (1) and (7).We now verify that the error indicatorη
com is effective for the boundary value problem (7) by the following three different types of test functions. The corresponding boundary conditions are shown in Figures 1, 2, and 3. The numerical results are listed in Table 1.Figure 1Figure 2Figure 3Example 1.
Consider the equation-
Δ
w
=
f on the square domain Ω
=
[
0,1
]
2 and the L-shaped domain Ω
=
[
0,1
]
2
∖
[
0.5,1
]
2, respectively.
Test function 1:w
1
=
x
(
1
-
x
)
y
(
1
-
y
), corresponding f
=
2
x
(
1
-
x
)
+
2
y
(
1
-
y
).
Test function 2:w
2
=
sin
(
(
π
/
2
)
x
)
(
e
y
-
1
-
e
y
), corresponding f
=
sin
(
(
π
/
2
)
x
)
[
-
(
π
2
/
4
)
(
e
y
-
1
-
e
y
)
+
e
y
].
Test function 3:w
3
=
(
x
-
0.5
)
(
y
-
0.5
)
x
(
x
-
1
)
y
(
y
-
1
), corresponding f
=
(
3
/
2
)
(
2
x
-
1
)
(
2
y
-
1
)
(
-
x
2
+
x
-
y
2
+
y
), where w
i
,
h
(
i
=
1,2
,
3
) denotes the nonconforming rotate Q
1 element approximations. From Table 1 we find out the ratio η
com
/
∥
w
i
-
w
i
,
h
∥
h converges that to 1 rapidly, when the number n
1
×
n
2 of the elements increases gradually. Namely, the a posteriori error indicator η
com is effective (see Figure 4).
Next we will compute the validity of the error indicatorη
com
2 of the eigenvalue problem (1). The numerical results are listed in Table 2.Figure 4
The error for test functions 1, 2, and 3.Example 2.
Consider the eigenvalue problem-
Δ
u
=
λ
u on the L-shaped domain Ω
=
[
0,1
]
2
∖
[
0.5,1
]
2 (see Figure 3). Here we take λ
1
≈
4
*
9.63972, λ
2
≈
4
*
15.19725, and λ
3
≈
4
*
19.73921, respectively. λ
i
,
h denotes the ith approximate eigenvalue.In Table2, we can see that the indicators η
com
2 for the first eigenvalue and second eigenvalue are effective and reliable, respectively. But the indicator η
com
2 for the third eigenvalue is distortion, obviously, for which reason is that the eigenfunction is smooth corresponding to the eigenvalue λ
3. So, in Theorem 8, the assumptions, in which Ω is a concave domain and the eigenfunction is singular, are necessary.According to the explanation in Remark15, we compute the validity of the error indicator η
com
2 of the Steklov eigenvalue problem on the L-shaped domain Ω
=
[
0,1
]
2
∖
[
0.5,1
]
2 (see Figure 3).Example 3.
Consider the problems (37), where λ
2, λ
3, and λ
4 were taken at the midpoint value of the interval [0.893602779, 0.893736398], [1.688598128, 1.688606742], and [3.21786760, 3.217900202], respectively. Taking into account the possibility that M
(
λ
1
)
⊂
H
2
(
Ω
), λ
1 can be omitted.We can see that the ratioη
com
2
/
|
λ
i
,
h
-
λ
i
|
(
i
=
2,3
,
4
) converges to 1 from Table 3, which proves that our supposition is rational in Remark 15.



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*Source: 891278-2014-01-28.xml*