# A Type of Multigrid Method Based on the Fixed-Shift Inverse Iteration for the Steklov Eigenvalue Problem

Authors: Feiyan Li; Hai Bi
Journal: Advances in Mathematical Physics
 (2016)

---

## Abstract

For the Steklov eigenvalue problem, we establish a type of multigrid discretizations based on the fixed-shift inverse iteration and study in depth its a priori/a posteriori error estimates. In addition, we also propose an adaptive algorithm on the basis of the a posteriori error estimates. Finally, we present some numerical examples to validate the efficiency of our method.


---

## Body

## 1. Introduction

Due to the wide applications in physical and mechanical field (see, e.g., [1–3]), there has been a lot of research on the numerical methods for Steklov eigenvalue problems; for instance, [4] studied the conforming linear finite element approximation, [5, 6] studied the nonconforming finite elements approximation, [7, 8] discussed a two-grid method of the conforming and nonconforming finite element method based on the inverse iteration, respectively, [9] studied multiscale asymptotic method, [10] studied multilevel method, [11] studied the spectral method, and [12] studied an adaptive algorithm based on the shifted inverse iteration.In this paper we establish a type of multigrid discretizations based on the fixed-shift inverse iteration for the Steklov eigenvalue problem. The multilevel method in [10] made use of the inverse iteration and the extended finite element method. Compared with [10], our method has less computational complexity since we have no correction step in each iteration. On the other hand, compared with [12], we adopt the fixed-shift and thus avoid selecting appropriate shift to ensure the efficiency of shifted inverse iteration; meanwhile, we also do not face the difficulty of solving an almost singular algebraic system in the shifted inverse iteration.We analyze elaborately the a priori and the a posteriori error estimates of the method proposed in this paper. Then, based on the a posteriori error estimates we design an adaptive algorithm of fixed-shift inverse iteration type. Moreover, we also compare the performance of three types of multigrid methods. Numerical results illustrate that our method is also an efficient method for solving the Steklov eigenvalue problem.The rest of this paper is organized as follows. In the subsequent section, some preliminaries needed in this paper are presented. In Section3, a scheme of the inverse iteration with fixed-shift based on multigrid discretizations is established, and the a priori error estimates are also given. The a posteriori error estimates of the inverse iteration with fixed-shift are analyzed in Section 4. Numerical experiments are presented in the final section.In this paper,C with or without subscript denotes a constant independent of mesh size and iterative times.


## 2. Preliminaries

Consider the Steklov eigenvalue problem(1)
-
Δ
u
+
u
=
0
i
n
Ω
,
∂
u
∂
n
=
λ
u
o
n
∂
Ω
,where Ω
⊂
R
2 is a polygonal domain with θ being the largest inner angle of Ω and ∂
u
/
∂
n is the outward normal derivative.We denote the real order Sobolev spaces with norm∥
·
∥
t and ∥
·
∥
t
,
∂
Ω by H
t
(
Ω
) and H
t
(
∂
Ω
), respectively; H
0
(
∂
Ω
)
=
L
2
(
∂
Ω
).The variational form of (1) is given by the following: find λ
∈
R and u
∈
H
1
(
Ω
), u
≠
0, such that(2)
a
u
,
v
=
λ
b
u
,
v
,
∀
v
∈
H
1
Ω
,where (3)
a
u
,
v
=
∫
Ω
∇
u
·
∇
v
+
u
v
d
x
,
b
u
,
v
=
∫
∂
Ω
u
v
d
s
,
u
b
=
b
u
,
u
1
/
2
=
u
0
,
∂
Ω
.As we know,a
(
·
,
·
) is a symmetric, continuous, and H
1
(
Ω
)-elliptic bilinear form on H
1
(
Ω
)
×
H
1
(
Ω
). Thus, we use a
(
·
,
·
) and ·
a
=
a
(
·
,
·
)
=
·
1 as the inner product and norm on H
1
(
Ω
), respectively.LetH
-
1
/
2
(
∂
Ω
) be the dual space of H
1
/
2
(
∂
Ω
) with norm given by (4)
w
-
1
/
2
,
∂
Ω
=
s
u
p
v
∈
H
1
/
2
∂
Ω
w
,
v
v
1
/
2
,
∂
Ω
, where 〈
w
,
v
〉 is the dual product on H
-
1
/
2
(
∂
Ω
)
×
H
1
/
2
(
∂
Ω
). When w
∈
L
2
(
∂
Ω
), 〈
w
,
v
〉
=
b
(
w
,
v
).Let{
π
h
} be a family of regular triangulations of Ω with the mesh diameter h, and let V
h
⊂
H
1
(
Ω
) be a space of piecewise polynomials defined on π
h. For any w
∈
H
1
(
Ω
), the following conclusion holds:(5)
l
i
m
h
→
0
i
n
f
v
∈
V
h
w
-
v
a
=
0
.The conforming finite element approximation of (2) is the following: find λ
h
∈
R and u
h
∈
V
h, u
h
≠
0, such that(6)
a
u
h
,
v
=
λ
h
b
u
h
,
v
,
∀
v
∈
V
h
.Define the operatorsT
:
H
1
(
Ω
)
→
H
1
(
Ω
) and T
h
:
H
1
(
Ω
)
→
V
h
⊂
H
1
(
Ω
) satisfying(7)
a
T
g
,
v
=
b
g
,
v
,
∀
v
∈
H
1
Ω
,
(8)
a
T
h
g
,
v
=
b
g
,
v
,
∀
v
∈
V
h
.Define the Ritz projectionP
h
:
H
1
(
Ω
)
→
V
h by (9)
a
u
-
P
h
u
,
v
=
0
,
∀
v
∈
V
h
.From [13], we know that T
-
T
h
a
→
0
(
h
→
0
); (2) and (6) have the equivalent operator forms T
u
=
μ
u and T
h
u
h
=
μ
h
u
h, respectively, where T
h
=
P
h
T, μ
=
1
/
λ, and μ
h
=
1
/
λ
h.Suppose thatλ and λ
h are the kth eigenvalue of (2) and (6), respectively, and the algebraic multiplicity of λ is equal to q, λ
=
λ
k
=
λ
k
+
1
=
⋯
=
λ
k
+
q
-
1. Let M
(
λ
) be the space spanned by all eigenfunctions corresponding to λ and let M
h
(
λ
) be the direct sum of eigenspaces corresponding to all eigenvalues of (6) that converge to λ. Let M
^
(
λ
)
=
{
v
:
v
∈
M
λ
,
v
a
=
1
}.Denote(10)
σ
h
=
s
u
p
f
∈
H
1
Ω
,
f
a
=
1
i
n
f
v
∈
V
h
T
f
-
v
a
,
ρ
h
=
s
u
p
f
∈
L
2
∂
Ω
,
f
0
,
∂
Ω
=
1
i
n
f
v
∈
V
h
T
f
-
v
a
,
δ
h
λ
=
s
u
p
u
∈
M
^
λ
i
n
f
v
∈
V
h
u
-
v
a
.It is obvious thatδ
h
(
λ
)
≤
λ
σ
(
h
)
≤
C
ρ
(
h
). It follows from Lemma 3.3 in [14] that (11)
σ
h
⟶
0
h
⟶
0
.By using the trace theorem we have(12)
v
0
,
∂
Ω
≤
v
1
/
2
,
∂
Ω
≤
C
1
v
a
,
∀
v
∈
H
1
Ω
.Moreover, ifv
∈
H
1
(
Ω
) and w
∈
H
1
/
2
(
∂
Ω
) we know that 〈
v
,
w
〉
=
b
(
v
,
w
)
≤
v
0
,
∂
Ω
w
0
,
∂
Ω and, consequently,(13)
v
-
1
/
2
,
∂
Ω
≤
v
0
,
∂
Ω
≤
C
1
v
a
,
∀
v
∈
H
1
Ω
.For anyg
∈
H
1
(
Ω
), T
h
g
∈
H
1
(
Ω
). Taking v
=
T
h
g in (8) we deduce (14)
a
T
h
g
,
T
h
g
=
b
g
,
T
h
g
≤
g
-
1
/
2
,
∂
Ω
T
h
g
1
/
2
,
∂
Ω
≤
C
1
g
-
1
/
2
,
∂
Ω
T
h
g
a
, and thus we get(15)
T
h
g
a
≤
C
1
g
-
1
/
2
,
∂
Ω
.The following lemmas are needed in our analysis.Lemma 1.
Let(
λ
,
u
) be an eigenpair of (2); then for any v
∈
H
1
(
Ω
) with v
a
=
1, the Rayleigh quotient R
(
v
)
=
a
(
v
,
v
)
/
v
b
2 satisfies(16)
R
v
-
λ
=
v
-
u
a
2
v
b
2
-
λ
v
-
u
b
2
v
b
2
.Proof.
See page 699 of [13].Lemma 2.
For any nonzerou
,
v
∈
H
1
(
Ω
), (17)
u
u
a
-
v
v
a
a
≤
2
u
-
v
a
u
a
,
u
u
a
-
v
v
a
a
≤
2
u
-
v
a
v
a
.Proof.
See [15].Lemma 3.
Letλ and λ
h be the kth eigenvalue of (2) and (6), respectively. Then for any eigenfunction u
h corresponding to λ
h with u
h
a
=
1, there exist u
∈
M
(
λ
) and h
0
>
0 such that if h
≤
h
0,(18)
u
h
-
u
a
≤
C
2
δ
h
λ
,
(19)
u
h
-
u
0
,
∂
Ω
≤
C
2
ρ
h
δ
h
λ
,
(20)
u
h
-
u
-
1
/
2
,
∂
Ω
≤
C
2
σ
h
δ
h
λ
;for any u
∈
M
^
(
λ
), there exists u
h
∈
M
h
(
λ
) such that if h
≤
h
0,(21)
u
-
u
h
a
≤
C
3
δ
h
λ
,
u
-
u
h
-
1
/
2
,
∂
Ω
≤
C
3
σ
h
δ
h
λ
,where constants C
2 and C
3 are positive and only depend on λ.Proof.
See page 699 of [13] and Lemma 3.7 and (3.29
b) of [14].Ifu
∈
M
(
λ
), v
∈
H
1
(
Ω
), v
a
=
1, and v
-
u
a
≤
4
λ
C
1
-
1, then by Lemma 2 we have (22)
v
-
u
u
a
a
≤
2
v
-
u
a
≤
2
λ
C
1
-
1
,
v
-
u
u
a
0
,
∂
Ω
≤
C
1
v
-
u
u
a
a
≤
1
2
λ
.From (2) we have u
/
u
a
0
,
∂
Ω
=
1
/
λ; then (23)
v
0
,
∂
Ω
≥
u
u
a
0
,
∂
Ω
-
v
-
u
u
a
0
,
∂
Ω
≥
1
2
λ
.Hence, from Lemma1 we get(24)
R
v
-
λ
≤
4
λ
1
+
λ
C
1
2
v
-
u
a
2
.Denote(25)
C
4
=
4
λ
1
+
λ
C
1
2
,and then whenv
a
=
1 and v
-
u
a
≤
4
λ
C
1
-
1, (24) becomes(26)
R
v
-
λ
≤
C
4
v
-
u
a
2
.Since (6) implies λ
h
=
R
(
u
h
), then combining (26) and (18) we deduce that(27)
0
≤
λ
h
-
λ
≤
C
4
u
h
-
u
a
2
≤
C
4
C
2
2
δ
h
2
λ
.


## 3. A Priori Error Estimates of the Inverse Iteration with Fixed-Shift

LetV
h
i
0
∞ be a family of conforming finite element spaces that satisfy V
h
0
=
V
H, V
h
i
⊂
V
h
i
+
1
⊂
H
1
(
Ω
) (i
=
0,1
,
…), and σ
h
i
→
0
(
i
→
∞
). Referring to [16], we establish the following scheme of the inverse iteration with fixed-shift based on multigrid discretizations.Scheme 4 (the inverse iteration with fixed-shift based on multigrid discretizations).
Given the iterative timesl and i
0. Execute the following.Step 1.
Solve (2) on V
H: find (
λ
H
,
u
H
)
∈
R
×
V
H such that u
H
a
=
1 and(28)
a
u
H
,
v
=
λ
H
b
u
H
,
v
,
∀
v
∈
V
H
.Step 2.
Letu
h
0
⇐
u
H, λ
h
0
⇐
λ
H, i
⇐
1.Step 3.
Solve a linear system onV
h
i: find u
′
∈
V
h
i such that(29)
a
u
′
,
v
-
λ
h
i
-
1
b
u
′
,
v
=
b
u
h
i
-
1
,
v
,
∀
v
∈
V
h
i
;
setu
h
i
=
u
′
/
u
′
a.Step 4.
Compute the Rayleigh quotient(30)
λ
h
i
=
a
u
h
i
,
u
h
i
b
u
h
i
,
u
h
i
.Step 5.
Ifi
>
i
0, then λ
h
i
0
⇐
λ
h
i
-
1, i
⇐
i
+
1; turn to Step 6; else, i
⇐
i
+
1, and return to Step 3.Step 6.
Solve a linear system onV
h
i: find u
′
∈
V
h
i such that(31)
a
u
′
,
v
-
λ
h
i
0
b
u
′
,
v
=
b
u
h
i
-
1
,
v
,
∀
v
∈
V
h
i
;
setu
h
i
=
u
′
/
u
′
a.Step 7.
Compute the Rayleigh quotient(32)
λ
h
i
=
a
u
h
i
,
u
h
i
b
u
h
i
,
u
h
i
.Step 8.
Ifi
=
l, then output (
λ
h
l
,
u
h
l
) and stop; else, i
⇐
i
+
1, and return to Step 6.Let(
λ
H
,
u
H
) be the kth eigenpair of (28); then (
λ
h
l
,
u
h
l
) derived from Scheme 4 is the kth eigenpair approximation of (2).In the following analysis, we also denote(
λ
H
,
u
H
)
=
(
λ
k
,
H
,
u
k
,
H
) and (
λ
h
l
,
u
h
l
)
=
(
λ
k
h
l
,
u
k
h
l
).Now, we will analyze the a priori error estimates of Scheme4.Denoted
i
s
t
(
u
,
S
)
=
inf
v
∈
S
⁡
u
-
v
a.Our analysis makes use of the following lemma (see Lemma4.1 in [16]) for the shifted inverse iteration method. Let (
λ
k
,
u
k
) and (
λ
k
,
h
,
u
k
,
h
) denote the kth eigenpair of (2) and (6), respectively, and μ
k
=
1
/
λ
k, μ
k
,
h
=
1
/
λ
k
,
h, M
(
μ
k
)
=
M
(
λ
k
), and M
h
(
μ
k
)
=
M
h
(
λ
k
).Lemma 5.
Let(
μ
0
,
u
0
) be an approximation for (
μ
k
,
u
k
), where μ
0 is not an eigenvalue of T
h, and u
0
∈
V
h with u
0
a
=
1. Suppose that (
C
1
)
d
i
s
t
(
u
0
,
M
h
(
μ
k
)
)
≤
1
/
2;
(
C
2
)
|
μ
0
-
μ
k
|
≤
ρ
/
4 and |
μ
j
,
h
-
μ
j
|
≤
ρ
/
4, for j
=
k
-
1
,
k
,
k
+
q
(
j
≠
0
), where ρ
=
min
μ
j
≠
μ
k
⁡
|
μ
j
-
μ
k
| is the separation constant of the eigenvalue μ
k;
(
C
3
)
u
′
∈
V
h
,
u
k
h
∈
V
h satisfy(33)
μ
0
-
T
h
u
′
=
u
0
,
u
k
h
=
u
′
u
′
a
.Then(34)
d
i
s
t
u
k
h
,
M
h
μ
k
≤
4
ρ
m
a
x
k
≤
j
≤
k
+
q
-
1
μ
0
-
μ
j
,
h
d
i
s
t
u
0
,
M
h
μ
k
.Letδ
0 be a positive constant satisfying the following inequalities:(35)
m
a
x
1
,
C
2
,
C
3
δ
0
≤
m
i
n
1
2
,
4
λ
j
C
1
-
1
,
(36)
4
C
1
C
1
C
4
δ
0
2
+
λ
k
C
1
δ
0
+
λ
k
C
3
δ
0
2
+
C
1
C
5
q
1
/
2
C
4
C
2
2
δ
0
2
≤
1
2
,
(37)
δ
0
λ
k
-
δ
0
λ
k
≤
ρ
4
,
δ
0
≤
λ
k
2
,
(38)
C
4
C
2
2
λ
j
2
δ
0
2
≤
ρ
4
,
j
=
k
-
1
,
k
,
…
,
k
+
q
,
j
≠
0
.Condition 6.
There existsu
¯
∈
M
^
(
λ
k
) for j
=
k
-
1
,
k
,
k
+
q
(
j
≠
0
) such that (39)
u
k
h
l
-
1
-
u
¯
a
≤
δ
0
,
λ
0
-
λ
k
≤
δ
0
,
δ
h
l
λ
j
≤
δ
0
,
σ
h
l
≤
δ
0
, where λ
0 is an approximate eigenvalue of λ
k, u
k
h
l
-
1 is an approximate eigenfunction obtained by Scheme 4, and ρ is the separation constant of the eigenvalue μ
k
=
1
/
λ
k.Let the eigenvectors{
u
j
,
h
l
}
k
k
+
q
-
1 be an orthonormal basis of M
h
l
(
λ
k
) with respect to a
(
·
,
·
), and denote(40)
u
∗
=
∑
j
=
k
k
+
q
-
1
a
u
k
h
l
,
u
j
,
h
l
u
j
,
h
l
.From Lemma 3, we know that there exist eigenvectors {
u
j
0
}
k
k
+
q
-
1
⊂
M
(
λ
k
) making u
j
,
h
l
-
u
j
0 satisfy (18), (19), and (20). Let(41)
u
k
=
∑
j
=
k
k
+
q
-
1
a
u
k
h
l
,
u
j
,
h
l
u
j
0
,and then u
k
∈
M
(
λ
k
) and(42)
u
k
-
u
∗
=
∑
j
=
k
k
+
q
-
1
a
u
k
h
l
,
u
j
,
h
l
u
j
0
-
u
j
,
h
l
.To estimate the error, we split(43)
u
k
h
l
-
u
k
=
u
k
h
l
-
u
∗
-
u
k
-
u
∗
.Now, we will analyze the first termu
k
h
l
-
u
∗.Theorem 7.
Let(
λ
k
h
l
,
u
k
h
l
) be an approximate eigenpair obtained by Scheme 4 with λ
0
=
λ
k
h
i
0. Assume that Lemma 3 and Condition 6 hold; then(44)
u
k
h
l
-
u
∗
a
≤
C
0
4
λ
0
-
λ
k
λ
k
h
l
-
1
-
λ
k
+
u
k
h
l
-
1
-
u
¯
-
1
/
2
,
∂
Ω
+
σ
h
l
δ
h
l
λ
k
,
l
≥
1
,where C
0 is independent of mesh parameters and l.Proof.
We use Lemma5 to complete the proof. First, we will verify that the conditions of Lemma 5 are satisfied.
From Lemma3, we know that, for any given u
¯
∈
M
^
(
λ
k
), there exists u
~
k
,
h
l
∈
M
h
l
(
λ
k
) such that(45)
u
¯
-
u
~
k
,
h
l
a
≤
C
3
δ
h
l
λ
k
,
(46)
u
¯
-
u
~
k
,
h
l
-
1
/
2
,
∂
Ω
≤
C
3
σ
h
l
δ
h
l
λ
k
,where(47)
u
~
k
,
h
l
=
∑
i
=
k
k
+
q
-
1
α
i
u
i
,
h
l
,
u
~
k
,
h
l
a
≤
C
5
.Select μ
0
=
1
/
λ
0 and u
0
=
λ
k
h
l
-
1
T
h
l
u
k
h
l
-
1
/
λ
k
h
l
-
1
T
h
l
u
k
h
l
-
1
a. Then, by (15) and (13) we have (48)
λ
k
h
l
-
1
T
h
l
u
k
h
l
-
1
-
u
~
k
,
h
l
a
=
λ
k
h
l
-
1
T
h
l
u
k
h
l
-
1
-
∑
i
=
k
k
+
q
-
1
λ
i
,
h
l
T
h
l
α
i
u
i
,
h
l
a
≤
C
1
λ
k
h
l
-
1
u
k
h
l
-
1
-
∑
i
=
k
k
+
q
-
1
λ
i
,
h
l
α
i
u
i
,
h
l
-
1
/
2
,
∂
Ω
=
C
1
λ
k
h
l
-
1
u
k
h
l
-
1
-
λ
k
u
k
h
l
-
1
+
λ
k
u
k
h
l
-
1
-
λ
k
u
¯
+
λ
k
u
¯
-
λ
k
u
~
k
,
h
l
+
λ
k
u
~
k
,
h
l
-
∑
i
=
k
k
+
q
-
1
λ
i
,
h
l
α
i
u
i
,
h
l
-
1
/
2
,
∂
Ω
≤
C
1
C
1
λ
k
h
l
-
1
-
λ
k
+
λ
k
u
k
h
l
-
1
-
u
¯
-
1
/
2
,
∂
Ω
+
λ
k
u
¯
-
u
~
k
,
h
l
-
1
/
2
,
∂
Ω
+
∑
i
=
k
k
+
q
-
1
λ
k
-
λ
i
,
h
l
α
i
u
i
,
h
l
-
1
/
2
,
∂
Ω
, noting that u
~
k
,
h
l
a
≥
u
¯
a
-
u
¯
-
u
~
k
,
h
l
a
≥
1
-
C
3
δ
h
l
(
λ
k
)
≥
1
-
C
3
δ
0
≥
1
/
2; then using Lemma 2, (46), the Cauchy-Schwartz inequality, (26), Condition 6, and (36) we obtain(49)
d
i
s
t
u
0
,
M
h
l
λ
k
≤
u
0
-
u
~
k
,
h
l
u
~
k
,
h
l
a
a
≤
2
u
~
k
,
h
l
a
λ
k
h
l
-
1
T
h
l
u
k
h
l
-
1
-
u
~
k
,
h
l
a
≤
4
C
1
C
1
λ
k
h
l
-
1
-
λ
k
+
λ
k
u
k
h
l
-
1
-
u
-
-
1
/
2
,
∂
Ω
+
λ
k
C
3
σ
h
l
δ
h
l
λ
k
+
C
1
C
5
q
1
/
2
C
4
C
2
2
δ
h
l
2
λ
k
≤
4
C
1
C
1
C
4
δ
0
2
+
λ
k
C
1
δ
0
+
λ
k
C
3
δ
0
2
+
C
1
C
5
q
1
/
2
C
4
C
2
2
δ
0
2
≤
1
2
,and then Condition (
C
1
) in Lemma 5 holds.
By using the same arguments in [16], it is clear that the other two conditions in Lemma 5 are valid.
Hence, we see that the conditions of Lemma5 hold.
Then, by the same proof method in [16], we derive that (50)
u
k
h
l
-
u
∗
a
≤
16
ρ
C
1
2
λ
k
2
λ
0
-
λ
k
+
2
λ
k
2
C
4
C
2
2
δ
h
l
2
λ
k
C
1
λ
k
h
l
-
1
-
λ
k
+
λ
k
u
k
h
l
-
1
-
u
¯
-
1
/
2
,
∂
Ω
+
λ
k
C
3
σ
h
l
δ
h
l
λ
k
+
C
1
C
5
q
1
/
2
C
4
C
2
2
δ
h
l
2
λ
k
.Noting that the constants C
1, C
2, C
3, C
4, C
5, and ρ are independent of mesh parameters and l and Condition 6 holds, then based on the above inequality we conclude that there exists a positive constant C
0 that is independent of mesh parameters and l such that (44) holds. And we can have m
i
n
C
0
/
4
,
C
0
C
1
/
4
>
q
1
/
2
C
2. The proof is completed.Next, we will analyze the erroru
k
-
u
∗.Theorem 8.
The erroru
k
-
u
∗ satisfies(51)
u
k
-
u
∗
a
≤
C
0
4
δ
h
l
λ
k
,
(52)
u
k
-
u
∗
-
1
/
2
,
∂
Ω
≤
C
0
C
1
4
σ
h
l
δ
h
l
λ
k
.Proof.
The estimates (51) and (52) can be obtained by the proof arguments in [16].Based on the above two theorems, we now analyze the a priori error estimates of Scheme4.Condition 9.
For any givenβ
0
,
β
0
′
∈
(
0,1
), there exist 0
<
β
0
≤
β
i
<
1 and 0
<
β
0
′
≤
β
i
′
<
1
(
i
=
1,2
,
…
) such that δ
h
i
(
λ
k
)
=
β
i
δ
h
i
-
1
(
λ
k
) and σ
(
h
i
)
=
β
i
′
σ
(
h
i
-
1
), respectively, σ
h
i
→
0
(
i
→
∞
).In the practice Condition9 is not a restrictive condition. For example, let π
h
i be obtained from π
h
i
-
1 via regular refinement (producing 4 congruent elements) such that h
i
=
(
1
/
2
)
h
i
-
1; then, when M
(
λ
k
)
⊂
H
1
+
γ
(
Ω
) and {
T
f
:
f
∈
H
1
(
Ω
)
}
⊂
H
1
+
γ
(
Ω
) we have δ
h
i
(
λ
k
)
≈
(
1
/
2
)
γ
δ
h
i
-
1
(
λ
k
) and σ
(
h
i
)
≈
(
1
/
2
)
γ
σ
(
h
i
-
1
) (see [10]), where γ
=
1 if Ω is convex and 0
<
γ
<
1 if Ω is concave.Theorem 10.
Let(
λ
k
h
l
,
u
k
h
l
) be an approximate eigenpair obtained by Scheme 4. Suppose that Condition 9 holds; then there exist u
k
∈
M
(
λ
k
) and H
0
>
0 such that if H
≤
H
0 it is valid that(53)
u
k
h
l
-
u
k
a
≤
C
0
δ
h
l
λ
k
,
(54)
u
k
h
l
-
u
k
-
1
/
2
,
∂
Ω
≤
C
0
C
1
σ
h
l
δ
h
l
λ
k
,
(55)
λ
k
h
l
-
λ
k
≤
C
4
C
0
2
δ
h
l
2
λ
k
,
l
≥
i
0
.Proof.
We only prove the result (54) since (53) and (55) can be proved analogously by referring to [16].
The proof is completed by using induction, Theorems7 and 8. Note that δ
H
λ
k
≤
λ
k
σ
H
→
0
(
H
→
0
); then there exists a proper small H
0
>
0 such that if H
≤
H
0, Lemma 3 and the following inequalities hold:(56)
C
0
δ
H
λ
k
≤
δ
0
,
C
4
C
0
2
δ
H
2
λ
k
≤
δ
0
,
δ
H
λ
j
≤
δ
0
,
σ
H
≤
δ
0
,
(57)
C
4
2
C
0
4
δ
H
2
λ
k
λ
k
1
β
0
′
1
β
0
+
C
4
C
0
3
C
1
δ
H
2
λ
k
1
β
0
′
1
β
0
≤
1
,where j
=
k
-
1
,
k
,
k
+
q
(
j
≠
0
).
Whenl
=
i
0, it is easy to know that (53)–(55) are valid (see [12, 16]). Suppose that Theorem 10 holds for l
-
1; that is, there exists u
¯
∈
M
(
λ
k
) such that (58)
u
k
h
l
-
1
-
u
¯
a
≤
C
0
δ
h
l
-
1
λ
k
,
u
k
h
l
-
1
-
u
¯
-
1
/
2
,
∂
Ω
≤
C
0
C
1
σ
h
l
-
1
δ
h
l
-
1
λ
k
,
λ
k
h
l
-
1
-
λ
k
≤
C
4
C
0
2
δ
h
l
-
1
2
λ
k
.Then we infer from (56) that the conditions of Theorem 7 hold.
From Theorems7 and 8 we get(59)
u
k
h
l
-
u
k
-
1
/
2
,
∂
Ω
≤
C
0
C
1
2
λ
0
-
λ
k
λ
k
h
l
-
1
-
λ
k
+
u
k
h
l
-
1
-
u
¯
-
1
/
2
,
∂
Ω
+
σ
h
l
δ
h
l
λ
k
,
l
≥
1
.Therefore, for l, from (59) we derive that (60)
u
k
h
l
-
u
k
-
1
/
2
,
∂
Ω
≤
C
0
C
1
2
C
4
2
C
0
4
δ
h
i
0
2
λ
k
δ
h
l
-
1
2
λ
k
+
C
4
C
0
3
C
1
δ
h
i
0
2
λ
k
σ
h
l
-
1
δ
h
l
-
1
λ
k
+
σ
h
l
δ
h
l
λ
k
≤
C
0
C
1
2
C
4
2
C
0
4
δ
h
i
0
2
λ
k
λ
k
1
β
l
′
1
β
l
+
C
4
C
0
3
C
1
δ
h
i
0
2
λ
k
1
β
l
′
1
β
l
+
1
σ
h
l
δ
h
l
λ
k
≤
C
0
C
1
2
C
4
2
C
0
4
δ
H
2
λ
k
λ
k
1
β
0
′
1
β
0
+
C
4
C
0
3
C
1
δ
H
2
λ
k
1
β
0
′
1
β
0
+
1
σ
h
l
δ
h
l
λ
k
, which together with (57) we get (54) immediately.


## 4. A Posteriori Error Estimates of the Inverse Iteration with Fixed-Shift

Based on the work of [4, 12, 17–19], in this section, we will discuss the a posteriori error estimates of Scheme 4 for the Steklov eigenvalue problem.Consider the boundary value problem corresponding to (2): find w
∈
H
1
(
Ω
) such that(61)
a
w
,
v
=
b
f
,
v
,
∀
v
∈
H
1
Ω
,and its finite element approximation states: find w
h
∈
V
h such that(62)
a
w
h
,
v
=
b
f
,
v
,
∀
v
∈
V
h
.For any elementT
∈
π
h with diameter h
T, we denote by E
T the set of edges, and (63)
E
=
⋃
T
∈
π
h
E
T
. We decompose E
=
E
Ω
∪
E
Γ, where E
Ω and E
Γ refer to interior edges and edges on the boundary Γ
=
∂
Ω, respectively. For each l
∈
E
Ω, we choose an arbitrary unit normal vector n
l and denote the two triangles sharing this edge by T
in and T
out, where n
l  points outwards T
in.Forv
h
∈
V
h we set (64)
∂
v
h
∂
n
l
l
=
∇
v
h
∣
T
o
u
t
·
n
l
-
∇
v
h
∣
T
i
n
·
n
l
. Let (65)
λ
^
k
,
h
l
=
1
q
∑
j
=
k
k
+
q
-
1
λ
j
,
h
l
. For each l
∈
E we define the jump residual:(66)
J
l
u
k
h
l
=
1
2
∂
u
k
h
l
∂
n
l
l
l
∈
E
Ω
,
λ
k
h
l
u
k
h
l
-
∂
u
k
h
l
∂
n
l
l
∈
E
Γ
.Now, the local error indicator is defined as(67)
η
T
u
k
h
l
=
h
T
2
u
k
h
l
0
,
T
2
+
∑
l
∈
E
T
l
J
l
u
k
h
l
0
,
l
2
1
/
2
,and then the global error estimator is given by(68)
η
Ω
u
k
h
l
=
∑
T
∈
π
h
η
T
2
u
k
h
l
1
/
2
.Substituting u
∗ for u
k
h
l, we can get the definitions of J
l
(
u
∗
), η
T
(
u
∗
), and η
Ω
(
u
∗
) similarly.Now, we will estimate the errore
=
u
k
-
u
∗.From [4, 12], we give the following two lemmas among which Lemma 11 provides the global upper bound of e, while Lemma 12 provides the local lower bound of e.Lemma 11.
The errore
=
u
k
-
u
∗ satisfies(69)
e
a
≤
C
6
η
Ω
u
∗
+
σ
h
l
δ
h
l
λ
k
.Lemma 12.
The errore
=
u
k
-
u
∗ satisfies the following:
(a) ForT
∈
π
h, if ∂
T
∩
Γ
=
∅, then(70)
η
T
u
∗
≤
C
7
e
1
,
T
∗
,where T
∗ denotes the union of T and the triangles sharing an edge with T.
(b) ForT
∈
π
h, if ∂
T
∩
Γ
≠
∅, then(71)
η
T
u
∗
≤
C
8
e
1
,
T
+
∑
l
∈
E
T
∩
E
Γ
l
1
/
2
λ
k
u
k
-
λ
^
k
,
h
l
u
∗
0
,
l
.Next, we will analyze the erroru
k
h
l
-
u
∗.Theorem 13.
Suppose that the conditions of Theorem10 are satisfied; then(72)
u
k
h
l
-
u
∗
a
≤
C
0
4
σ
h
l
δ
h
l
λ
k
.Proof.
Note thatδ
H
λ
k
≤
λ
k
σ
H
→
0
(
H
→
0
); then there exists a proper small H
0
>
0 such that if H
≤
H
0, the following inequality holds:(73)
C
4
2
C
0
4
δ
H
2
λ
k
λ
k
1
β
0
′
1
β
0
+
C
4
C
0
3
C
1
δ
H
2
λ
k
1
β
0
′
1
β
0
+
C
4
C
0
2
δ
H
2
λ
k
≤
1
.From (44), Theorem 10, and Condition 9, we have (74)
u
k
h
l
-
u
∗
a
≤
C
0
4
C
4
2
C
0
4
δ
h
i
0
2
λ
k
δ
h
l
-
1
2
λ
k
+
C
4
C
0
3
C
1
δ
h
i
0
2
λ
k
σ
h
l
-
1
δ
h
l
-
1
λ
k
+
C
4
C
0
2
δ
h
i
0
2
λ
k
σ
h
l
δ
h
l
λ
k
≤
C
0
4
C
4
2
C
0
4
δ
h
i
0
2
λ
k
λ
k
1
β
l
′
1
β
l
+
C
4
C
0
3
C
1
δ
h
i
0
2
λ
k
1
β
l
′
1
β
l
+
C
4
C
0
2
δ
h
i
0
2
λ
k
σ
h
l
δ
h
l
λ
k
≤
C
0
4
C
4
2
C
0
4
δ
H
2
λ
k
λ
k
1
β
0
′
1
β
0
+
C
4
C
0
3
C
1
δ
H
2
λ
k
1
β
0
′
1
β
0
+
C
4
C
0
2
δ
H
2
λ
k
σ
h
l
δ
h
l
λ
k
, which together with (73) yields (72) immediately.We give the following lemma by referring to [12] (see Lemma 3.4 in [12]).Lemma 14.
Suppose that the conditions of Theorem10 are satisfied; then(75)
η
T
u
∗
-
η
T
u
k
h
l
≤
C
9
δ
h
l
2
λ
k
u
∗
1
,
T
+
u
k
h
l
-
u
∗
1
,
T
,
(76)
η
Ω
u
∗
-
η
Ω
u
k
h
l
≤
C
10
δ
h
l
2
λ
k
u
∗
a
+
u
k
h
l
-
u
∗
a
.In the following discussion, combining Lemmas11, 12, and 14 and Theorem 13, we give the global upper bound and the local lower bound of the error.Theorem 15.
Suppose that the conditions of Theorem10 are satisfied; then there exists u
k
∈
M
(
λ
k
) such that(77)
u
k
-
u
k
h
l
a
≤
C
6
η
Ω
u
k
h
l
+
R
1
,where R
1
=
C
6
C
10
δ
h
l
2
(
λ
k
)
u
∗
a
+
(
C
6
+
C
0
/
4
+
C
6
C
10
(
C
0
/
4
)
)
σ
(
h
l
)
δ
h
l
(
λ
k
).Proof.
Selectu
k
∈
M
(
λ
k
) which is given by (41); then from Lemma 11, Theorem 13, and (76) we get(78)
u
k
-
u
k
h
l
a
≤
u
k
-
u
∗
a
+
u
∗
-
u
k
h
l
a
≤
C
6
η
Ω
u
∗
+
σ
h
l
δ
h
l
λ
k
+
C
0
4
σ
h
l
δ
h
l
λ
k
≤
C
6
η
Ω
u
k
h
l
+
C
10
δ
h
l
2
λ
k
u
∗
a
+
u
k
h
l
-
u
∗
a
+
σ
h
l
δ
h
l
λ
k
+
C
0
4
σ
h
l
δ
h
l
λ
k
≤
C
6
η
Ω
u
k
h
l
+
R
1
.The proof is completed.It is obvious thatR
1 is a higher order term. Hence, we obtain that η
Ω
(
u
k
h
l
) is a global reliable error indicator of u
k
-
u
k
h
l
a.Theorem 16.
Under the conditions of Theorem10, there exists u
k
∈
M
(
λ
k
) such that the following hold:
(a) ForT
∈
π
h
l, if ∂
T
∩
Γ
=
∅, then(79)
η
T
u
k
h
l
≤
C
7
u
k
-
u
k
h
l
1
,
T
∗
+
R
2
,where R
2
=
(
C
7
+
C
9
)
u
k
h
l
-
u
∗
1
,
T
∗
+
C
9
δ
h
l
2
(
λ
k
)
u
∗
1
,
T.
(b) ForT
∈
π
h
l, if ∂
T
∩
Γ
≠
∅, then(80)
η
T
u
k
h
l
≤
C
8
u
k
-
u
k
h
l
1
,
T
+
R
3
,where (81)
R
3
=
C
8
+
C
9
u
k
h
l
-
u
∗
1
,
T
+
C
9
δ
h
l
2
λ
k
u
∗
1
,
T
+
C
8
∑
l
∈
E
T
∩
E
Γ
l
1
/
2
λ
k
u
k
-
λ
^
k
,
h
l
u
∗
0
,
l
.Proof.
We can prove the desired results by using the proof method of Theorem3.4 in [12].According to Remark3.1 in [4] and Remark 3.2 in [12] we know that the term ∑
l
∈
E
T
∩
E
Γ
‍
l
1
/
2
λ
k
u
k
-
λ
^
k
,
h
l
u
∗
0
,
l is a higher order term. From Theorem 13, we know that u
k
h
l
-
u
∗
1
,
T
∗ and u
k
h
l
-
u
∗
1
,
T are also higher order terms. And it is obvious that δ
h
l
2
(
λ
k
)
u
∗
1
,
T is a higher order term. Therefore, from (79) and (80) we know that η
T
(
u
k
h
l
) is an efficient local error indicator of u
k
-
u
k
h
l
1
,
T
∗ and u
k
-
u
k
h
l
1
,
T.In the following theorem, we give the estimate for approximate eigenvalue.Theorem 17.
Suppose that the conditions of Theorem10 are satisfied; then(82)
λ
k
h
l
-
λ
k
=
O
η
Ω
2
u
k
h
l
.Proof.
From Theorem10 and Lemma 1 it is easy to prove that (83)
λ
k
h
l
-
λ
k
=
O
u
k
h
l
-
u
k
a
2
; then combining with Theorem 16 and (77), we can get the desired result (82).


## 5. Numerical Experiments

In this section we first give an adaptive algorithm of the Rayleigh quotient iteration type and establish an adaptive algorithm of fixed-shift inverse iteration type for the Steklov eigenvalue problem.The following Algorithm1 of the Rayleigh quotient iteration type refers to Algorithm 4.3 in [12] or Algorithm 6.1 in [16].Algorithm 1.
Choose parameter0
<
ω
<
1.Step 1.
Pick any initial meshπ
h
0.Step 2.
Solve (2) on π
h
0 for discrete solution (
λ
h
0
,
u
h
0
).Step 3.
Letl
⇐
0
,
λ
0
⇐
λ
h
0.Step 4.
Compute the local indicatorsη
T
(
u
h
l
).Step 5.
Constructπ
^
h
l
⊂
π
h
l byMarking Strategy E and ω.Step 6.
Refineπ
h
l to get a new mesh π
h
l
+
1 by Procedure REFINE.Step 7.
Findu
′
∈
V
h
l
+
1 such that(84)
a
u
′
,
v
-
λ
0
b
u
′
,
v
=
b
u
h
l
,
v
,
∀
v
∈
V
h
l
+
1
;denote u
h
l
+
1
=
u
′
/
u
′
a and compute the Rayleigh quotient(85)
λ
h
l
+
1
=
a
u
h
l
+
1
,
u
h
l
+
1
b
u
h
l
+
1
,
u
h
l
+
1
.Step 8.
Letλ
0
⇐
λ
h
l
+
1,  l
⇐
l
+
1 and go to Step 4.
Marking Strategy E. Give parameter 0
<
ω
<
1.Step 1.
Construct a minimal subsetπ
^
h
l of π
h
l by selecting some elements in π
h
l such that(86)
∑
T
∈
π
^
h
l
η
T
2
u
h
l
≥
ω
η
Ω
2
u
h
l
.Step 2.
Mark all the elements inπ
^
h
l.
η
T
(
u
h
l
) and η
Ω
(
u
h
l
) are defined by (67) and (68) with u
k
h
l and λ
k
h
l replaced by u
h
l and λ
h
l, respectively.
Note that when|
λ
0
-
λ
| is too small, (84) is an almost singular linear equation. Although it has no difficulty in solving (84) numerically (see [12]), one would like to think of selecting a proper integer i
0
≥
0 to establish the following adaptive algorithm.Algorithm 2.
Choose parameter0
<
ω
<
1.
Steps  1–7. Execute Steps  1–7 of Algorithm 1.
Step  8. If l
<
i
0
,
λ
0
⇐
λ
h
l
+
1
,
l
⇐
l
+
1, go to Step 4; else l
⇐
l
+
1, go to Step 4.Marking Strategy E in Algorithm 2 is the same as that in Algorithm 1.Now, we will implement some numerical experiments to validate our theoretical analysis and show the efficiency of Algorithm2 with i
0
=
0. We use MATLAB 2012 together with the package of Chen [20] to solve Examples 1, 2, and 3, and we take ω
=
0.5.For reading conveniently, we use the following notations in our tables:⁡
λ
k
h
l
(
m
): the kth eigenvalue derived from the lth iteration obtained by Algorithm ⁡
m (m
=
1,2).
⁡
λ
k
h
l
m
-
λ
k: the error of λ
k
h
l
(
m
) obtained by Algorithm m (m
=
1,2).
⁡
N
k
,
l
(
m
): the degrees of freedom of the lth iteration for λ
k
h
l
(
m
) (m
=
1,2).
⁡
CPU
k
,
l
(
m
)
(
s
): the CPU time(s) from the program starting to calculate result of the lth iteration appearing by using Algorithm m (m
=
1,2).Example 1.
We use Algorithms1 and 2 to compute the approximations of the 1st and the 2nd eigenvalue of (1) with the triangle linear finite element on Ω
=
[
0,1
]
×
[
0,1
]. The numerical results are listed in Table 1.
Since the exact eigenvalues are unknown, we useλ
1
≈
0.24007908542 and λ
2
≈
1.49230313453 obtained by the spectral element method  (see [21]) as the reference eigenvalues. We show the error curves and the a posteriori estimators obtained by two algorithms for λ
1 and λ
2 in Figure 1. It can be seen from Figure 1 that the error curves are approximately parallel to the line with slope -
1, which indicates that Algorithm 2 achieves the optimal convergence rate of O
(
N
l
-
1
) as well as Algorithm 1.
Observing the numerical results in Table1, we can find that when the degrees of freedom are almost the same, the approximate eigenvalues obtained by Algorithm 2 are nearly as accurate as those obtained by Algorithm 1 and their CPU time are roughly the same.Table 1
The 1st and the 2nd eigenvalues of Example1 obtained by Algorithms 1 and 2 with H
=
2
/
32.
k
l
N
k
,
l
(
1
)
λ
k
h
l
(
1
)
CPU
k
,
l
(
1
)
N
k
,
l
(
2
)
λ
k
h
l
(
2
)
CPU
k
,
l
(
2
)
1
5
5700
0.24008040
0.54
5700
0.24008040
0.57
1
10
27344
0.24007936
1.60
27343
0.24007936
1.66
1
15
117001
0.24007915
7.67
117001
0.24007915
7.82
1
19
408971
0.24007910
29.06
408971
0.24007910
29.25
1
20
509032
0.24007910
39.88
509032
0.24007910
40.07
1
21
764069
0.24007909
55.90
764069
0.24007909
55.96
2
5
4526
1.49245505
0.51
4529
1.49245448
0.54
2
12
37855
1.49231870
2.31
37941
1.49231871
2.33
2
18
223337
1.49230601
15.77
224277
1.49230600
15.89
2
24
1329617
1.49230362
109.68
1334272
1.49230362
109.90
2
25
1805637
1.49230347
151.36
1812726
1.49230347
151.02
2
26
2439573
1.49230337
211.29
2447212
1.49230337
210.12Figure 1
The curves of error and the a posteriori error estimators of two algorithms for the 1st (a) and 2nd (b) eigenvalues on the square domain.
(a)
(b)Example 2.
We use Algorithms1 and 2 to compute the approximations of the 1st and the 3rd eigenvalue of (1) with the triangle linear finite element on Ω
=
(
[
0,1
]
×
[
0
,
1
/
2
]
)
∪
(
[
0
,
1
/
2
]
×
[
1
/
2
,
1
]
). The numerical results are presented in Table 2.
In Figure2 we depict the error curves and the a posteriori estimators obtained by two algorithms for λ
1 and λ
3. Here we use λ
1
≈
0.18296423687 and λ
3
≈
1.68860048358 obtained by the spectral element method (see [21]) as the reference eigenvalues. It can be seen from Figure 2 that the error curves are approximately parallel to the line with slope -
1, which indicates that Algorithm 2 achieves the optimal convergence rate of O
(
N
l
-
1
) as well as Algorithm 1.
It also can be seen from Table2 that when the degrees of freedom are the same, one can use Algorithms 1 and 2 to get the same accurate approximations with nearly the same CPU time.Table 2
The 1st and the 3rd eigenvalues of Example2 obtained by two algorithms with H
=
2
/
32.
k
l
N
k
,
l
(
1
)
λ
k
h
l
(
1
)
CPU
k
,
l
(
1
)
N
k
,
l
(
2
)
λ
k
h
l
(
2
)
CPU
k
,
l
(
2
)
1
5
4369
0.18296573
0.46
4369
0.18296573
0.45
1
12
35119
0.18296446
2.00
35119
0.18296446
1.97
1
18
227719
0.18296426
14.90
227719
0.18296426
14.80
1
21
557370
0.18296425
39.77
557370
0.18296425
40.10
1
22
738108
0.18296425
55.04
738108
0.18296425
55.43
1
23
1105016
0.18296424
78.27
1105016
0.18296424
78.83
3
5
3690
1.68889444
0.49
3690
1.68889444
0.52
3
12
31089
1.68863207
1.79
31089
1.68863207
1.84
3
18
183089
1.68860633
11.84
183089
1.68860633
12.08
3
24
1098418
1.68860134
81.42
1098418
1.68860134
83.23
3
25
1485695
1.68860109
112.52
1485695
1.68860109
114.26
3
26
2002315
1.68860097
155.64
2002315
1.68860097
156.76Figure 2
The curves of error and the a posteriori error estimators of two algorithms for the 1st (a) and 3rd (b) eigenvalues on theL-shaped domain.
(a)
(b)Example 3.
We use Algorithms1 and 2 to compute the approximations of the 1st and the 5th eigenvalue of (1) with the triangle linear finite element on Ω
=
{
(
x
1
,
x
2
)
:
x
1
+
x
2
<
1
}
∖
{
(
x
1
,
x
2
)
:
0
≤
x
1
≤
1
,
x
2
=
0
}. The numerical results are presented in Table 3.
Since the exact eigenvalues are unknown, we compute the approximations of two exact eigenvalues of (1): λ
1
≈
0.23957338768 and λ
5
≈
1.41238071918 by the standard adaptive algorithm (see, e.g., [22]) with the degrees of freedom of more than 5000000. We show the curves of the error and the a posteriori estimators obtained by two algorithms for λ
1 and λ
5 in Figure 3. We can see from Figure 3 that the error curves are approximately parallel to the line with slope -
1, which indicates that Algorithm 2 achieves the optimal convergence rate of O
(
N
l
-
1
) as well as Algorithm 1.
From the numerical results in Table3, we can conclude that Algorithm 2 is also an efficient approach like Algorithm 1 for solving the Steklov eigenvalue problem.Table 3
The 1st and 5th eigenvalues of Example3 obtained by two algorithms with H
=
2
/
32.
k
l
N
k
,
l
(
1
)
λ
k
h
l
(
1
)
CPU
k
,
l
(
1
)
N
k
,
l
(
2
)
λ
k
h
l
(
2
)
CPU
k
,
l
(
2
)
1
5
9391
0.23957586
0.66
9391
0.23957586
0.75
1
10
42645
0.23957397
2.45
42645
0.23957397
2.56
1
15
189550
0.23957352
12.52
189550
0.23957352
12.72
1
19
634556
0.23957342
48.00
634556
0.23957342
47.09
1
20
860490
0.23957341
66.32
860490
0.23957341
65.38
1
21
1131274
0.23957340
91.04
1131274
0.23957340
90.05
5
5
9842
1.41254843
0.70
9842
1.41254843
0.74
5
12
77005
1.41241115
4.53
77005
1.41241115
4.68
5
18
453543
1.41238410
31.53
453543
1.41238410
31.74
5
21
1092642
1.41238245
82.23
1092642
1.41238245
82.74
5
22
1497488
1.41238165
112.92
1497488
1.41238165
113.82
5
23
1993327
1.41238104
155.36
1993327
1.41238104
156.14Figure 3
The curves of error and the a posteriori error estimators of two algorithms for the 1st (a) and 5th (b) eigenvalues on slit domain.
(a)
(b)Example 4.
We use the method in [10] (see Algorithms 4.1 and 7.2 there) to compute the numerical eigenvalues of (1) on [
0,1
]
×
[
0,1
],  (
[
0,1
]
×
[
0
,
1
/
2
]
)
⋃
‍
(
[
0
,
1
/
2
]
×
[
1
/
2
,
1
]
), and {
(
x
1
,
x
2
)
:
x
1
+
x
2
<
1
}
∖
{
(
x
1
,
x
2
)
:
0
≤
x
1
≤
1
,
x
2
=
0
}, respectively, and list the associated results in Tables 4–6 which are denoted by λ
k
h
l
(
3
) and CPU
k
,
l
(
3
).
From Tables4–6 we can see that, with the same degrees of freedom N
k
,
l, our method uses less CPU time to obtain the same accurate approximations, especially for multiple eigenvalue λ
2 on [
0,1
]
×
[
0,1
], comparing with the one in [10].Table 4
The results of Example4 on Ω
=
[
0,1
]
×
[
0,1
].
k
l
N
k
,
l
λ
k
h
l
(
3
)
CPU
k
,
l
(
3
)
λ
k
h
l
(
2
)
CPU
k
,
l
(
2
)
1
1
1089
0.240088481
0.08
0.240088481
0.18
1
2
4225
0.240081438
0.17
0.240081438
0.22
1
3
16641
0.240079674
0.48
0.240079674
0.43
1
4
66049
0.240079233
1.92
0.240079233
1.45
1
5
263169
0.240079122
8.92
0.240079122
6.48
1
6
1050625
0.240079095
38.36
0.240079095
32.00
2
1
1089
1.492905398
0.11
1.492905378
0.19
2
2
4225
1.492454269
0.23
1.492454267
0.24
2
3
16641
1.492340958
0.65
1.492340958
0.46
2
4
66049
1.492312593
2.67
1.492312593
1.47
2
5
263169
1.492305499
12.19
1.492305499
6.57
2
6
1050625
1.492303726
56.05
1.492303726
32.00Table 5
The results of Example4 on Ω
=
(
[
0,1
]
×
[
0
,
1
/
2
]
)
∪
(
[
0
,
1
/
2
]
×
[
1
/
2
,
1
]
).
k
l
N
k
,
l
λ
k
h
l
(
3
)
CPU
k
,
l
(
3
)
λ
k
h
l
(
2
)
CPU
k
,
l
(
2
)
1
1
833
0.182975157
0.04
0.182975157
0.19
1
2
3201
0.182966980
0.11
0.182966980
0.22
1
3
12545
0.182964924
0.31
0.182964924
0.36
1
4
49665
0.182964409
1.25
0.182964409
1.06
1
5
197633
0.182964280
5.88
0.182964280
4.60
1
6
788481
0.182964248
26.02
0.182964248
21.61
3
1
833
1.690165085
0.05
1.690165013
0.20
3
2
3201
1.688996545
0.12
1.688996536
0.22
3
3
12545
1.688700132
0.34
1.688700131
0.36
3
4
49665
1.688625481
1.27
1.688625481
1.04
3
5
197633
1.688606742
5.95
1.688606742
4.34
3
6
788481
1.688602046
26.11
1.688602046
20.20Table 6
The results of Example4 on Ω
=
(
x
1
,
x
2
)
:
x
1
+
x
2
<
1∖(
x
1
,
x
2
)
:
0
≤
x
1
≤
1
,
x
2
=
0.
k
l
N
k
,
l
λ
k
h
l
(
3
)
CPU
k
,
l
(
3
)
λ
k
h
l
(
2
)
CPU
k
,
l
(
2
)
1
1
2145
0.239589697
0.12
0.239589697
0.22
1
2
8385
0.239577621
0.28
0.239577621
0.31
1
3
33153
0.239574482
0.96
0.239574482
0.77
1
4
131841
0.239573668
4.03
0.239573668
2.99
1
5
525825
0.239573457
17.43
0.239573457
13.80
1
6
2100225
0.239573403
78.31
0.239573403
71.35
5
1
2145
1.413086665
0.12
1.413086553
0.20
5
2
8385
1.412557485
0.28
1.412557472
0.29
5
3
33153
1.412424609
0.91
1.412424608
0.68
5
4
131841
1.412391332
3.92
1.412391332
2.77
5
5
525825
1.412383004
17.35
1.412383004
12.79
5
6
2100225
1.412380921
79.20
1.412380921
64.99



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*Source: 4691759-2016-07-05.xml*