Domain Decomposition Methods for Space Fractional Partial Differential Equations

0.24 MB·English

by  Yingjun Jiang

Domain Decomposition Methods for Space Fractional Partial
Differential Equations∗
Yingjun Jiang†and Xuejun Xu‡
Abstract
5
1 In this paper, a two-level additive Schwarz preconditioner is proposed for solving the al-
0 gebraic systems resulting from the finite element approximations of space fractional partial
2
differential equations (SFPDEs). It is shown that the condition number of the preconditioned

n system is bounded by C(1+H/δ), where H is the maximum diameter of subdomains andδ is
a theoverlapsizeamongthesubdomains. Numericalresultsaregiventosupportourtheoretical
J
findings.
4
1
Keywords. fractionaldifferentialequations, overlappingdomaindecompositions,preconditioners

]
A
1 Introduction
N
.
h Space fractional partial differential equations have been wildly used to describe the supper-
t
a diffusion processes in the natural world (see [19]). Let Ω denote a polyhedral domain in Rd and
m M˜(z) denotes a probability density function on Sd−1, where Sd−1 = {z ∈ Rd;||z|| = 1} and ||·||
2 2
[
denotes the standard Euclidean norm. In this paper, we consider the following multi-dimensional

2 SFPDE( [15])
v
5
6 − D2αu(x)M˜(z)dz +cu(x) = f(x), x ∈ Ω, (1.1)
z
9 ZSd−1
2
0 where 1/2 < α < 1, c ≥ 0 and D2α, which will be given later, denotes the directional derivative
z
1. of order 2α in the direction z. We assume M˜ is symmetric about origin, i.e., M˜(z) = M˜(z′) if
0 z,z′ ∈ Sd−1 satisfy z+z′ = 0, which means that the above SFPDE is symmetric.
5
Actually, the equation (1.1) is an appropriate extension from one dimensional problem
1
:
v −(p D2α+q D2α)u+cu = f, (1.2)
i −∞ x x ∞
X
r anditscorrespondingdevelopingequationcanbeusedtodescribeageneralsuper-diffusionprocess
a
(see [15] for details), where D2α, D2α denote Riemann-Liouville fractional derivatives. One
−∞ x x ∞
∗The work of the first author is supported by National Natural Science Foundation of China (No. 10901027).
The work of second author is supported by the National Basic Research Program under the Grant 2011CB30971
and National Natural Science Foundation of China (No. 11171335, 11225107).
†DepartmentofMathematicsandScientificComputing,ChangshaUniversityofScienceandTechnology,Chang-
sha, 410076, China ([email protected]).
‡InstituteofComputationalMathematics andScientific/EngineeringComputing,AcademyofMathematics and
SystemsScience, Chinese Academy of Sciences, P.O.Box 2719, Beijing, 100190, P.R. China([email protected]).
1
special case of (1.1) is
d
− (p D2α+q D2α)u+cu= f (1.3)
i −∞ xi i xi ∞
i=1
X
with p ,q ∈ R satisfying p = q and p +q = 1. We may find that equation (1.3) can be obtained
i i i i i i
from (1.1) by taking
d
M˜ = p δ(z −e )+q δ(z +e ), (1.4)
i i i i
i=1
X
where e is the ith column of identity matrix in Rd×d and δ denotes the Dirac function on Sd−1.
i
Extensive numerical methods have already been developed for SFPDEs, like finite difference
methods (see e.g., [2,17,18,25,26]), finite element methods (see e.g., [6–8,14,22]), and spectral
methods [11]. Due to the nonlocal properties of fractional differential operators, the most impor-
tant issue for numerical computation of SFPDEs is how to reduce the computation costs. Some
methods for the reduction have already been designed, like alternating-direction implicit methods
(ADI) [18,30,31], special iterative methods [12,20,31–34] and multigrid methods [20,36].
The discrete systems Ax = b of SFPDEs usually have the following characteristics: 1). the
condition number of A increases fast, as the mesh becomes fine; 2). the coefficient matrix A is
dense. As for iterative methods, two issues need to be concerned for efficiency: one is how to
construct good preconditioners for the discrete system Ax = b, which may help us to save the
iterative steps; the other is how to reduce the computation cost of each iterative step, for which
somepapersemploythemultiplication ofToeplitzmatrices andvectors withnlog(n)computation
complexity (see e.g., [12,20,29,31–34]). If the iterative step can be carried out in parallel, the
efficiency of solving SFPDEs may be significantly improved. The parallelizable algorithms have
been wildly used in numerical solutions for PDEs (see e.g. [28]). Whereas, to the best of our
knowledge, no parallelizable algorithms have been designed for SFPDEs.
In this paper, we shall construct a two-level additive preconditioner for the discrete system
Ax = b resulting from the finite element approximation of (1.1), and then use the preconditioned
conjugate method (PCG) to solve it. The preconditioner we construct is almost optimal, i.e.,
the condition number of the preconditioned system is bounded by C(1+H/δ), where H is the
maximum diameter of subdomains and δ is the overlap size among the subdomains. Moreover,
the preconditioner may be employed in parallel or each step of the PCG can be carried out in
parallel. As a result, the whole numerical solution processes, including the generation of A,b and
the multiplication of matrices and vectors, may be conducted in parallel.
Without loss of generality, we focus on the case d = 2, namely, we consider the problem
(1.1) in R2. For Λ ⊂ R2, denote L2(Λ) the space of all measurable function v on Λ satisfying
(v(x))2dx < ∞,andC∞(Λ)thespaceofinfinitelydifferentiablefunctionswithcompactsupport
Λ 0
in Λ. Set
R
1/2
(v,w) = vwdxdy, ||v|| = (v,v) ,
Λ Λ Λ
ZZΛ
and they are abbreviated as (v,w) and ||v|| respectively if Λ =R2.
To simplify ourstatement, wemake aconvention here: function v definedon a domain Λ⊂ R2
alsodenotesitsextensiononR2whichextendsvbyzerooutsideΛ. TheconstantC withorwithout
subscript shall denote a generic positive constant which may take on different values in different
places. These constants shall always be independent of mesh sizes and numbers of subdomains.
Following [35], we also use symbols .,& and ≈ in this paper. That a . b , a & b and a ≈ b
1 1 2 2 3 3
2
mean that a ≤ C b , a ≥ C b and C b ≤ a ≤ C′b for some positive constants C ,C ,C and
1 1 1 2 2 2 3 3 3 3 3 1 2 3
C′.
3
The rest of the paper is organized as follows: in section 2, the variational problem of (1.1) and
its finite element discretization are described; in section 3, the two-level additive preconditioner
for the SFPDEs is presented; in section 4, it is proved that the preconditioner is almost optimal;
finally, in section 5, the numerical results shall be given to support our theoretical findings.
2 The model problem and its discretization
In this section, we shall describe the SFPDEs in details, and then introduce its variational formu-
lation and finite element discretization.
2.1 The model problem
Definition 2.1. [8] Let µ > 0, θ ∈ R. The µth order fractional integral in the direction z =
(cosθ,sinθ) is defined by
∞ τµ−1
D−µv(x,y) := D−µv(x,y) = v(x−τ cosθ,y−τ sinθ)dτ,
z θ Γ(µ)
Z0
where Γ is the Gamma function.
Definition 2.2. [8] Let n be a positive integer, and θ ∈ R. The nth order derivative in the
direction of z = (cosθ,sinθ) is given by
n
∂ ∂
Dnv(x,y) := cosθ +sinθ v(x,y).
θ ∂x ∂y
(cid:18) (cid:19)
Definition 2.3. [8] Let µ > 0, θ ∈ R. Let n be the integer such that n−1 ≤ µ < n, and define
σ = n−µ. Then the µth order directional derivative in the direction of z = (cosθ,sinθ) is defined
by
Dµv(x,y) := Dµv(x,y) = DnD−σv(x,y).
z θ θ θ
µ µ
If v is viewed as a function in x, D , D are just the left and the right Riemamm-Liouville
0 π
derivatives (see Appendix).
Definition 2.4. [8] Assume that v : R2 → R, µ > 0. The µth order fractional derivative with
respect to the measure M˜ is defined as
Dµ v(x,y) := Dµv(x,y)M˜(θ)dθ,
M˜ θ
ZS1
whereS1 = [0+ν,2π+ν) with a suitablescalar ν, andM˜(θ), which satisfies 2π+νM˜(θ)dθ = 1, is
ν
a periodic function with period 2π. Usually we take ν = 0, if it causes no unreasonable expression
R
(see (1.4)).
For u :R2 → R, define differential operator L in R2 as
α
L u= −D2αu+cu.
α M˜
3
Denote Ω a polygonal domain in R2, set 1/2 < α < 1. The model problem of this paper is to find
u: Ω¯ → R such that
L u= f, in Ω,
α (2.1)
u= 0, on ∂Ω,
(cid:26)
where f is a source term and we assume that M˜(θ) satisfies M˜(θ) = M˜(θ +π) for θ ∈ R, i.e.,
(2.1) is a symmetric problem. Here, we recall the convection made in Section 1: u also denotes
its extension by zero outside Ω.
2.2 The variational formulation and finite element discretization
Definition 2.5. [27] Let µ ≥ 0, Fv(ξ ,ξ ) be the Fourier transform of v(x,y), |ξ| = ξ2+ξ2.
1 2 1 2
Define norm
p
||v||Hµ(R2) := (1+|ξ|2)µ/2|Fv| .
(cid:13) (cid:13)
Let Hµ(R2):= {v ∈ L2(R2);||v||Hµ(R2) < ∞}.(cid:13)(cid:13) (cid:13)(cid:13)
For v ∈ H0µ(Ω), we also denote ||v||Hµ(R2) by ||v||Hµ(Ω). It is known that Hµ(R2) is a Hilbert
space equipped with the inner product(v,w)Hµ(R2) = ((1+|ξ|2)µFv,Fw) and C0∞(R2) is dense in
Hµ(R2)(see[27]). Sinceweemploythefiniteelementdiscretization,theweakfractionaldirectional
derivative need to be introduced. Let L1 (R2) denote the set of locally integrable functions on
loc
R2.
Definition 2.6. [9] Given µ > 0, θ ∈ R, let v ∈ L2(R2). If there is a function v ∈ L1 (R2) such
µ loc
that
(v,Dµ w) = (v ,w), ∀w ∈ C∞(R2),
θ+π µ 0
µ
then v is called the weak µth order derivative in the direction of θ for v, denoted by D v, i.e.,
µ θ
µ
v = D v.
µ θ
µ
The weak derivative D v is unique if it exists and the weak derivative coincides with the
θ
correspondent derivative defined in Definition 2.3 if v ∈ C∞(R2). In the following, we use Dµv to
0 θ
denote the weak derivative.
Lemma 2.7. [8,9] Let µ > 0. For any v ∈ Hµ(R2), 0 < s ≤ µ and θ ∈ R, the weak derivative
Dsv exists and satisfies
θ
FDsv(ξ ,ξ ) = (2πiξ cosθ+2πiξ sinθ)sFv(ξ ,ξ ), (2.2)
θ 1 2 1 2 1 2
||Dθsv|| ≤ C||v||Hµ(R2). (2.3)
Define the bilinear form B˜ : Hα(Ω)×Hα(Ω) → R as
0 0
2π
B˜(u,v) := − (Dαu,Dα v)M˜(θ)dθ+c(u,v).
θ θ+π
Z0
Because M˜(θ)= M˜(θ+π) for θ ∈ R, it is easy to check that B˜(v,w) is a symmetric bilinear form,
i.e., B˜(v,w) = B˜(w,v) for v,w ∈ Hα(Ω). The variational formulation of (2.1) (see [8,9]) is to find
0
u∈ Hα(Ω) such that
0
B˜(u,v) = (f,v), ∀v ∈ Hα(Ω). (2.4)
0
4
Here M˜ is taken such that (2.4) admits a unique solution in Hα(Ω) (for the details, we refer
0
to [8,9]).
We construct a quasi-uniform triangulation Γ = {Ω }J of Ω with
H i i=1
the diameter of Ω ≈ O(H).
i
Divide each Ω into smaller simplices τ of diameter O(h), such that Γ = {τ } form a finer
i j h j
triangulation of Ω. Denote V and V piecewise linear finite element function spaces defined on
H h
the triangulations Γ and Γ respectively. It is known that V ⊂H1(Ω)⊂ Hα(Ω).
H h h 0 0
The finite element approximation for (2.4) (the details please see [9]) is to find u ∈ V such
h h
that
B(u ,v) = (f,v), ∀v ∈ V , (2.5)
h h
whereB(v,w) =− 2π(Dαv,Dα w)M(θ)dθ+c(v,w),M(θ)isequaltoadiscreteform L p δ(θ−
0 θ θ+π k=1 k
θ ) such that B(·,·) is a symmetric bilinear form, and
k
R P
B(v,v) & ||v||2 , B(v,w) . ||v|| ||w|| , v,w ∈ Hα(Ω). (2.6)
Hα(Ω) Hα(Ω) Hα(Ω) 0
Meanwhile
2π
M(θ)dθ . 1.
Z0
Remark 2.8. The direct finite element discretization of (2.4) is
B˜(u ,v) = (f,v), v ∈V . (2.7)
h h
But the discretization is hardly computed. So we use (2.5) instead of (2.7), where B(·,·) is
understood as the approximation to B˜(·,·). For the details, please refer to [9].
3 A two-level additive Schwarz preconditioner
Take f ∈ V such that (f ,v) = (f,v), ∀v ∈ V and define a linear operator A : V → V
h h h h h h
satisfying
(Av,w) =B(v,w), ∀v,w ∈ V . (3.1)
h
Since B(v,w) is a symmetric bilinear form, by (2.6), we know that A : V → V is symmetric
h h
positive definite with respect to (·,·), i.e.,
(Av,w) = (v,Aw), v,w ∈ V ; (Av,v) >0, 0 6= v ∈ V .
h h
Then bilinear form
(v,w) := (Av,w)
A
also induces an inner product on V . Set norm
h
||v|| = (Av,v)1/2, v ∈ V .
A h
By (2.6), we have
||v|| ≈ ||v|| , ∀v ∈ V . (3.2)
A Hα(Ω) h
5
The problem (2.5) can be restated as to find u ∈ V such that
h h
Au = f . (3.3)
h h
For the above equation, we shall construct our two-level Schwarz preconditioner and then use
PCG method to solve it.
Our preconditioner is designed by making use of the following overlapping domain decompo-
sition Ω = ∪J {Ω′}, where the subdomain Ω′ contains coarse subdomain Ω , and satisfies
i=1 i i i
the diameter of Ω′ ≈ O(H).
i
Meanwhile the boundary of Ω′ align with the mesh of triangulation Γ , and the distance from
i h
∂Ω′∩Ωto Ω isgreater thanδ , whichis apositive constant measuringtheoverlapping size among
i i
the subdomians. Define subspaces V of V as
i h
V = {v ∈ V |v(x) = 0,x ∈ Ω\Ω′}, i =1,2,...,J,
i h i
and let V = V .
0 H
For our analysis, we regroup the subregions in terms of the coloring strategy (see e.g. [28]).
By a minimal or good coloring, we group the subregions {Ω′} into JC classes, each of which has
i
some disjoint subregions and can be regarded as one subregion. Exactly, decompose the index set
{1,2,...,J} = ∪JC I with I satisfying that Ω′∩Ω′ = ∅ for any l,k ∈ I (l 6= k); for i= 1,...,JC,
i=1 i i l k i
define new subregions Ω˜ = ∪ Ω′ and new subspaces V˜ = V .
i j∈Ii j i j∈Ii j
For each k ∈ {0,1,2,...,J}, we define some projectors Q ,P : V → V by
k k h k
L
(Q v,w) = (v,w), (P v,w) = (v,w) , ∀w ∈ V ,v ∈ V ,
k k A A k h
and define the linear operator A :V → V by
k k k
(A v,w) = (Av,w), v,w ∈ V .
k k
It is not hard to verify that
A P = Q A, k = 0,1,...,J. (3.4)
k k k
To help our analysis, we define some projectors P˜ :V → V˜, i = 1,...,JC, by
i h i
(P˜v,w) = (v,w) , ∀w ∈ V˜,v ∈ V .
i A A i h
Remark 3.1. Different from the integer order PDEs, it is interesting to see that P˜ 6= P .
i j∈Ii j
Now, we are ready to present our two-level additive Schwarz pre-conditioner, i.e.,
P
J
B = A−1Q .
h i i
i=0
X
By (3.4), we have
J J J
B A= A−1Q A= A−1A P = P .
h i i i i i i
i=0 i=0 i=0
X X X
Define P := J P , and then the preconditioned system is
h i=0 i
P Phuh = Bhfh. (3.5)
In the next section, we shall prove the condition number of P is bounded by C(1+ H/δ),
h
where the constant C is independent of mesh size and the numbers of subdomains,but dependent
of JC.
6
4 Condition number estimate
We first introduce two interpolation norms and relevant Sobolev spaces (see e.g., [27]). Let Λ be
a domain in R2. For integer m, denote by ||·|| the Sobolev norm of integer order m, i.e.,
H˜m(Λ)
1/2
||v|| := ||Dlv||2 ,
H˜m(Λ)  L2(Λ)
|l|≤m
X
 
with l = (l ,l ), |l| = l +l and Dl = ( d )l1( d )l2. Let µ > 0 be a non-integer and 0 < s < 1, n
1 2 1 2 dx dy
is a non-negative integer such that n < µ < n+1. We introduce the interpolation norms
∞ 1/2 ∞ 1/2
||v|| := K˜(v,t)t−2µ−1dt ,||v|| := Kˆ(v,t)t−2s−1dt , (4.1)
H˜µ(Λ) Hˆs(Λ)
(cid:18)Z0 (cid:19) (cid:18)Z0 (cid:19)
where
K˜(v,t) := inf ||v−w||2 +t2||w||2 ,
w∈H˜n+1(Λ) H˜n(Λ) H˜n+1(Λ)
(cid:16) (cid:17)
Kˆ(v,t) := inf ||v−w||2 +t2||w||2 .
w∈H˜1(Λ) L2(Λ) H˜1(Λ)
0 (cid:16) (cid:17)
Relevant Sobolev spaces are
H˜µ(Λ) := {v ∈L2(Λ);||v|| < ∞}, Hˆs(Λ) := {v ∈ L2(Λ);||v|| < ∞}. (4.2)
H˜µ(Λ) Hˆs(Λ)
For µ > 0, it is known that H˜µ(R2) coincides with Hµ(R2). The following norms relation is
useful in our analysis.
Lemma 4.1. [13,27] Let 0 < µ < 1 with µ 6= 1/2 and Λ be a domain in R2 with Lipschitz
boundary. Then Hˆµ(Λ) coincides with H˜µ(Λ) with equivalent norms.
0
In the following, we shall give some useful results.
Lemma 4.2. If dist(Ω′,Ω′)≥ lH for integer l ≥ 1, we have
i j
1
(Dαv,Dα w) . ||v|| ||w|| , v ∈ V ,w ∈ V . (4.3)
θ θ+π l0.5+α A A i j

Figure 1:
7
Proof. Without loss of generality, we prove that (4.3) holds under the situation as the figure 1
shows. Cartesian coordinate x′Oy′ is obtained by rotating xOy θ angle counterclockwise. We
set that p ∈ Ω¯ and p ∈ Ω¯ such that the length of line segment p p is equal to dist(Ω′,Ω′),
i i j j i j i j
the graphs of functions x′ = Γ (y′) and x′ = Γ (y′) are parts of boundary ∂Ω′ such that Ω′ =
i1 i2 i i
{(x′,y′): Γ (y′) < x′ <Γ (y′)}, thegraphsoffunctionsΓ (y′)andΓ (y′)areparts ofboundary
i1 i2 j1 j2
∂Ω′ such that Ω′ = {(x′,y′) : Γ (y′) < x′ < Γ (y′)}, and the graph of x′ = Γ (y′) is the
j j j1 j2 0
perpendicular bisector of p p . Then we have Dα = Dα, Dα = Dα , where Dα and
i j θ −∞ x′ θ+π x′ ∞ −∞ x′
Dα are the left and right Riemann-Liouville fractional derivative operator (see Appendix). We
x′ ∞
in the following only prove (4.3) for the case that there exists a ray in (cosθ,sinθ) direction going
through both Ω′ and Ω′ from Ω′ to Ω′. Indeed, when the other case is true, (Dαv,Dα w) = 0
i j i j θ θ+π
and (4.3) naturally holds.
Denote
M = max y′, m = min y′,
y′ y′
((xx′′′,,yy′′))∈∈ΩΩ′i′j ((xx′′′,,yy′′))∈∈ΩΩ′i′j
M = max x′, m = min x′.
x′ x′
(x′,y′)∈Ω′j (x′,y′)∈Ω′i
The shaded area is
Λ= {(x′,y′)|Γ (y′) ≤ x′ ≤ Γ (y′),m ≤ y′ ≤ M }.
i,1 j,2 y′ y′
Let Λ = {(x′,y′)|x′ ≤ Γ (y′)}∩Λ and Λ = {(x′,y′)|x′ ≥ Γ (y′)}∩Λ. Then we have
i 0 j 0
(Dαv,Dα w) = (Dαv,Dα w) = (Dαv,Dα w) +(Dαv,Dα w)
θ θ+π θ θ+π Λ θ θ+π Λi θ θ+π Λj
≤ ||Dαv|| ||Dα w|| +||Dαv|| ||Dα w|| . (4.4)
θ Λi θ+π Λi θ Λj θ+π Λj
For (x,y) ∈ Λ whose coordinate under x′Oy′ is (x′,y′), noting that supp(v) ⊂ Ω′, we have
j i
d
Dαv(x,y) = Dαv(x′,y′) = D−(1−α)v(x′,y′)
θ −∞ x′ dx′−∞ x′
1 d x′
= (x′−s)−αv(s,y′)ds
Γ(1−α)dx′
Z−∞
1 d Γi2(y′)
= (x′−s)−αv(s,y′)ds
Γ(1−α)dx′
ZΓi1(y′)
1 Γi2(y′)
= (x′−s)−(1+α)v(s,y′)ds
Γ(−α)
ZΓi1(y′)
Γi2(y′)
. (lH)−(1+α) |v(s,y′)|ds
ZΓi1(y′)
Γi2(y′) 1/2
. (lH)−(1+α)H1/2 v2(s,y′)ds , (4.5)
ZΓi1(y′) !
where in the fifth equality we have used the relation Γ(1−α) = −αΓ(−α), the first inequality is
8
by (x′−s)≥ lH/2 when s≤ Γ (y′) and the last by the Cauchy-Schwarz inequality. Then
i2
Mx′ My′ Γi2(y′)
||Dαv||2 . (lH)−2(1+α)H dx′ dy′ v2(s,y′)ds
θ Λj
Zmx′ Zmy′ ZΓi1(y′)
My′ Γi2(y′)
. (lH)−(1+2α)H dy′ v2(s,y′)ds
Zmy′ ZΓi1(y′)
≤ (lH)−(1+2α)H v2dx′dy′ = l−(1+2α)H−2α||v||2 , (4.6)
Ω′
ZZΩ′i i
where the second inequality is by Mx′ dx′ . lH. Denote d as the diameter of Ω′, and define a
mx′ i i
function in x′ as
R
0, if x′ > d ;
H (x′)= i
i x′(α−1) if 0≤ x′ ≤ d .
( Γ(α) i
For (x,y) ∈ Ω′ (whose coordinate is (x′,y′) under x′Oy′), by (A.4), we have
i
v(x,y) =D−αDαv(x,y) = D−α Dαv(x′,y′) = H ∗ Dαv(·,y′), (4.7)
θ θ −∞ x′ −∞ x′ i −∞ x′
where v∗w denote the convolution product(see e.g., [1]). Then by the Young Theorem (Theorem
4.30 in [1]),
||v(·,y′)||[Γi1,Γi2] ≤ ||v(·,y′)||R ≤ ||Hi||L1(R)||−∞Dxα′v(·,y′)||R . Hα||−∞Dxα′v(·,y′)||R. (4.8)
Furthermore we have
My′
||v||2 = v2dx′dy′ = ||v(·,y′)||2 dy′
Ω′i ZZΩ′i Zmy′ [Γi1(y′),Γi2(y′)]
. H2α|| Dαv||2 = H2α||Dαv||2
−∞ x′ R2 θ R2
. H2α||v||2 ≈ H2α||v||2, (4.9)
Hα(Ω) A
where the second inequality is by Lemma 2.7, the last equality is by (3.2). Combining (4.9) with
(4.6), we obtain
1
||Dαv|| . ||v|| . (4.10)
θ Λj l1/2+α A
Similarly we may also obtain
1
||Dα w|| . ||w|| . (4.11)
θ+π Λi l1/2+α A
By Lemma 2.7,
||Dα w|| . ||w|| ≈ ||w|| , ||Dαv|| . ||v|| ≈ ||v|| . (4.12)
θ+π Λj Hα(Ω) A θ Λi Hα(Ω) A
Combining with (4.4), (4.10) and (4.11), we may obtain (4.3). ✷
Lemma 4.3. Let S ⊂ {1,2,...,J} denote an index set such that Ω′ ∩Ω′ = ∅ for any i,j ∈ S,
i j
i 6= j. Then for v = v with v ∈ V , we have
i∈S i i i
P
(v,v) . (v ,v ) . (4.13)
A i i A
i∈S
X
9
Proof. Note that
2π
(v,v) =B(v,v) = − (Dαv,Dα v)M(θ)dθ+c(u,v),
A θ θ+π
Z0
it is easy to see that the lemma follows after we prove that
|(Dαv,Dα v)| ≤ C (v ,v ) (4.14)
θ θ+π i i A
i∈S
X
holds for any θ. So next, we give a proof of (4.14). It is easy to see that
|(Dαv,Dα v)| ≤ |(Dαv ,Dα v )|. (4.15)
θ θ+π θ k θ+π j
k,j∈S
X
In fact, we may prove
|(Dαv ,Dα v )| ≤ C (v ,v ) (4.16)
θ k θ+π j i i A
k,j∈S i∈S
X X
through using the following two kind of inequalities:
(Dαv ,Dα v ) ≤ ||Dαv ||||Dα v ||
θ k θ+π j θ k θ+π j
≤ C||v || ||v || ≤ C||v ||2 +C||v ||2, if dist(Ω′,Ω′)< H (4.17)
k A j A k A j A k j
(by Lemma 2.7 and (3.2));
C C
(Dαv ,Dα v ) ≤ ||v || ||v || ≤ (||v ||2 +||v ||2), if dist(Ω′,Ω′)≥ lH (4.18)
θ k θ+π j l0.5+α k A j A l0.5+α k A j A k j
(by Lemma 4.2).
Fixing i ∈ S, we write
S = {j ∈ S : dist(Ω′,Ω′)< H}, S′ = {j ∈ S : dist(Ω′,Ω′)≥ H}. (4.19)
i i j i i j
It is easy to see that
the sum of the terms containing v on the left hand of (4.16)
i
≤ |(Dαv ,Dα v )|+ |(Dαv ,Dα v )|
θ i θ+π j θ j θ+π i
j∈S j∈S
X X
= |(Dαv ,Dα v )|+ |(Dαv ,Dα v )|
θ i θ+π j θ j θ+π i
jX∈Si jX∈Si
+ |(Dαv ,Dα v )|+ |(Dαv ,Dα v )|. (4.20)
θ i θ+π j θ j θ+π i
j∈S′ j∈S′
Xi Xi
We know that
card(S )≤ C, (4.21)
i
where card(S) denotes the number of elements contained in S, C is a positive constant dependent
of JC. By (4.21) and (4.17), we have
|(Dαv ,Dα v )|+ |(Dαv ,Dα v )| ≤ C||v ||2 +terms which do not contain v . (4.22)
θ i θ+π j θ j θ+π i i A i
jX∈Si jX∈Si
10