A New Multigrid Finite Element Method for the Transmission Eigenvalue Problems

0.22 MB·

by  Jiayu Han

A New Multigrid Finite Element Method for the
Transmission Eigenvalue Problems
6
1
0 Jiayu Han, Yidu Yang, Hai Bi
2

n
a School of Mathematics and Computer Science,
J
Guizhou Normal University,Guiyang, 550001, China

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[email protected], [email protected],[email protected]
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]
A
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Abstract : Numericalmethodsforthetransmission eigenvalueproblemsarehot
.
h topicsinrecentyears. BasedontheworkofLinandXie[Math. Comp.,84(2015),pp.
t 71-88],webuildamultigridmethodtosolvetheproblems. Withourmethod,weonly
a
m need to solve a series of primal and dual eigenvalue problems on a coarse mesh and
the associated boundary value problems on the finer and finer meshes. Theoretical
[
analysisandnumericalresultsshowthatourmethodissimpleandeasytoimplement

1 and is efficient for computingreal and complex transmission eigenvalues.
v Keywords : Transmission eigenvalues, Multigrid method, Nonsymmetric eigen-
1
valueproblems, Extended/Generalized finiteelement.
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3
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1 Introduction
0
.
1
Thetransmissioneigenvalueproblemshavetheoreticalimportanceintheunique-
0
ness and reconstruction in inverse scattering theory [1, 2]. Transmission eigen-
6
1 valuescanbe determinedfromthe far-fielddata ofthe scatteredwaveandused
: to obtain estimates for the material properties of the scattering object [3, 4].
v
Many literatures such as [2, 4, 5, 6, 7] studied the existence of transmission
i
X eigenvalues, and [4, 8, 9] et al. explored the upper and lower bounds for the
r index of refraction n(x).
a
Inrecentyears,numericalmethodsforthetransmissioneigenvalueproblems
have attractedthe attention from more and more researchers. The first numer-
ical treatment of the transmission eigenvalue problem appeared in [10] where
threefiniteelementmethodsareproposedfortheHelmholtztransmissioneigen-
values. Later on, many other numerical methods were developed to solve the
problems (see, e.g., [11, 12, 13, 14, 15]). In particular, Sun [11] proposed an
iterative method and gave a coarse error analysis. Furthermore, Ji et al. [12]
developedhisworkandprovedtheaccurateerrorestimatesforbotheigenvalues
and eigenfunctions by constructing an auxiliary problem as a bridge. An and
1
Shen[13]proposedaspectral-elementmethodtosolvethisproblemnumerically.
Afterwards using the linearized technique the authors in [14, 15] builded two
new weak formulations and the corresponding finite element discretizations.
The idea of multigrid methods for eigenvalue problems was developed orig-
inally from two grid methodology. In 2001, Xu and Zhou [16] proposed a two
grid method based on inverse iteration for elliptic eigenvalue problems, which
is, in a way, related to that in Lin and Xie [17]. After that, the two grid
method was further developed into multigrid method and local and parallel al-
gorithm in [18, 19, 20, 21] et al. In recent years, Lin and Xie [22, 23] proposed
a mutilevel correction method. This method can be regarded as the combina-
tion of two grid method and the extended/generalized finite element method.
The extended/generalized finite element method was developed in 1990s by
[24, 25, 26, 27] et al, which has important applications on problems in material
science [28, 29]. The method of Lin and Xie [22, 23] enriches the finite element
space at each correction step with the numerical eigenfunctions obtained from
the last step. So it is able to naturally reproduce the feature of eigenfunctions:
the discontinuity, singularity, boundary layer, etc. Such an embedding of the
problem’s feature into the finite element space can significantly improve con-
vergence rates and accuracy step by step, so that the multigrid method can
achieve the same accuracy as solving the eigenvalue problem directly but with
less computational work.
In this paper, based on the literatures [22, 23], we propose a new multi-
grid method to solve the transmission eigenvalue problems but based on the
new weak formulation (2.6) proposed in [15] which is a linear and nonsymmet-
ric eigenvalue problem. In this work, (1) we prove the error estimates of the
transmissioneigenvaluesandeigenfunctionsforourmultigridmethod. Ourthe-
oretical results are valid for arbitrary real and complex eigenvalues. (2) With
ourmethod, duetoadoptingthelinearizedweakformulation,wecantransform
the transmissioneigenvalue problem into a generalizedmatrix eigenvalue prob-
lem and can be solved efficiently by the sparse solver eigs in Matlab; (3) with
our multigrid method, the solution of eigenvalue problem on a fine mesh can
be reduced to a series of the solutions of the eigenvalue problem on a coarse
meshes and a series of solutions of the boundary value problems on the mul-
tilevel meshes. As numerical results indicate, this method is applicable to the
real and complex transmission eigenvalues.
2 Preliminaries
Let Hs(D) be a Sobolev space with norm k·k (s=1,2), and
s
∂v
H2(D)={v ∈H2(D):v| = | =0}.
0 ∂D ∂ν ∂D
ConsidertheHelmholtztransmissioneigenvalueproblem: Findk ∈C,ω,σ ∈
L2(D), ω−σ ∈H2(D) such that
∆ω+k2n(x)ω =0, in D, (2.1)
∆σ+k2σ =0, in D, (2.2)
ω−σ =0, on ∂D, (2.3)
∂ω ∂σ
− =0, on ∂D, (2.4)
∂ν ∂ν
2
where D ⊂ R2 or D ⊂ R3 is a bounded simply connected inhomogeneous
medium, ν is the unit outward normal to ∂D.
It is possible to write (2.1)-(2.4) as an equivalent eigenvalue problem for
u=ω−σ ∈H2(D). In particular,
0
(∆u+k2u)=∆ω+k2ω =k2(1−n)ω.
Dividing by n−1 and applying the operator (∆+k2n) to the above equality,
theeigenvalueproblem(2.1)-(2.4)canbestatedasfollows: Findk2 ∈C,k2 6=0,
nontrivial u∈H2(D) such that
0
1
( (∆u+k2u),∆v+k2n(x)v) =0, ∀v ∈ H2(D), (2.5)
n(x)−1 0 0
where (·,·) is the inner product of L2(D). As usual, we define λ = k2 as the
0
transmission eigenvalue in this paper. We suppose that the index of refraction
n∈L (D) satisfying either one of the following assumptions
∞
(C1) 1+δ ≤infn(x)≤n(x)≤supn(x)<∞,
D D
(C2) 0<infn(x)≤n(x)≤supn(x)<1−̺,
D D
for some constant δ >0 or ̺>0.
For simplicity, in the coming discussion we assume (C1) holds and n(x) is
propersmooth(forexamplen(x)∈W2, (D)). Forthecase(C2)theargument
∞
method is the same.
Define Hilbert space H = H02(D)×L2(D) with norm k(u,w)kH = kuk2+
kwk0, and define Hs = Hs(D) ×Hs−2(D) with norm k(u,w)kHs = kuks +
kwk , s=0,1.
s 2
−
From (2.5) we derive that
1 1
( ∆u,∆v) −λ(∇( u),∇v)
0 0
n−1 n−1
n n
−λ(∇u,∇( v)) +λ2( u,v) =0, ∀v ∈ H2(D).
n−1 0 n−1 0 0
Let w = λu, we arrive at a linear weak formulation: Find (λ,u,w) ∈ C ×
H2(D)×L2(D) such that
0
1 1
( ∆u,∆v) =λ(∇( u),∇v)
0 0
n−1 n−1
n n
+λ(∇u,∇( v)) −λ( w,v) , ∀v ∈ H2(D),
n−1 0 n−1 0 0
(w,z) =λ(u,z) , ∀z ∈L2(D).
0 0
We introduce the following sesquilinear forms
1
A((u,w),(v,z))=( ∆u,∆v) +(w,z) ,
0 0
n−1
B((u,w),(v,z))
1 n n
=(∇( u),∇v) +(∇u,∇( v)) −( w,v) +(u,z)
0 0 0 0
n−1 n−1 n−1
1 n n
=−( u,∇·∇v) −(u,∇·∇( v)) −( w,v) +(u,z) ,
0 0 0 0
n−1 n−1 n−1
3
then (2.5) can be rewritten as: Find λ∈C, nontrivial (u,w)∈H such that
A((u,w),(v,z))=λB((u,w),(v,z)), ∀(v,z)∈H. (2.6)
Let norm k·k be induced by the inner product A(·,·), then it is clear k·k is
A A
equivalent to k·kH.
Onecaneasilyverifythatforanygiven(f,g)∈H (s=0,1),B((f,g),(v,z))
s
is a continuous linear form on H:
B((f,g),(v,z)).k(f,g)kH k(v,z)kH, ∀(v,z)∈H.
s
Here and hereafter this paper, we use the symbols x . y to mean x ≤ Cy for
a constant C that is independent of mesh size and iteration times and may be
different at different occurrences.
The source problem associatedwith (2.6) is given by: Find (ψ,ϕ)∈H such
that
A((ψ,ϕ),(v,z))=B((f,g),(v,z)), ∀(v,z)∈H. (2.7)
FromtheLax-Milgramtheoremweknowthattheproblem(2.7)existsanunique
solution, therefore, we define the corresponding solution operator T : H →H
s
by
A(T(f,g),(v,z))=B((f,g),(v,z)), ∀(v,z)∈H. (2.8)
Then (2.6) has the equivalent operator form:
T(u,w)=λ 1(u,w). (2.9)
−
Consider the dual problem of (2.6): Find λ ∈ C, nontrivial (u ,w ) ∈ H
∗ ∗ ∗
such that
A((v,z),(u ,w ))=λ B((v,z),(u ,w )), ∀(v,z)∈H. (2.10)
∗ ∗ ∗ ∗ ∗
Note that the primal and dual eigenvalues are connected via λ=λ .
∗
Define the corresponding solution operator T :H →H by
∗ s
A((v,z),T∗(f,g))=B((v,z),(f,g)), ∀(v,z)∈H. (2.11)
Then (2.10) has the equivalent operator form:
T (u ,w )=λ 1(u ,w ). (2.12)
∗ ∗ ∗ ∗− ∗ ∗
Clearly, T is the adjoint operatorof T in the sense of inner product A(·,·).
∗
InordertodiscretizethespaceH,weneedtwofiniteelementspacestodiscretize
H2(D)andL2(D), respectively,but herewecanconstructonlyoneconforming
0
finite element space Sh ⊂H2(D) such that H :=Sh×Sh ⊂H2(D)×L2(D).
0 h 0
The conformingfinite element approximationof(2.6)is givenby the follow-
ing: Find λ ∈C, nontrivial (u ,w )∈H such that
h h h h
A((u ,w ),(v,z))=λ B((u ,w ),(v,z)), ∀(v,z)∈H . (2.13)
h h h h h h
4
We introduce the corresponding solution operator: T :H →H (s=0,1):
h s h
A(T (f,g),(v,z))=B((f,g),(v,z)), ∀(v,z)∈H . (2.14)
h h
Then (2.13) has the operator form:
Th(uh,wh)=λ−h1(uh,wh). (2.15)
Define the projection operators P1 :H2(D)→Sh and P2 :L2(D)→Sh by
h 0 h
1
( ∆(u−P1u),∆v) =0, ∀v ∈Sh, (2.16)
n−1 h 0
(w−P2w,z) =0, ∀z ∈Sh. (2.17)
h 0
Let
P (u,w)=(P1u,P2w), ∀(u,w)∈H.
h h h
Then P :H→H , and
h h
A((u,w)−P (u,w),(v,z))=0, ∀(v,z)∈H . (2.18)
h h
We need the following regularity assumption:
R(D). Foranyξ ∈H−s(D)(s=0,1),thereexistsψ ∈H2+rs(D)satisfying
1
∆( ∆ψ)=ξ, in D,
n−1
∂ψ
ψ = =0 on ∂D,
∂ν
and
kψk ≤C kξk , s=0,1 (2.19)
2+rs p −s
where r ∈ (0,1], r ∈ (0,2], C denotes the prior constant dependent on the
1 0 p
equation and D but independent of the right-hand side ξ of the equation.
It is easy to know that (2.19) is valid with r = 2−s when n ∈ W2,p(D)
s
(p is greater than but arbitrarily close to 2) and ∂D is appropriately smooth.
WhenD ⊂R2 isaconvexpolygon,fromTheorem2in[30],whenn∈W2,p(D),
we can get r =1 and that if the inner angle at each critical boundary point is
1
smaller than 126.283696...0then r =2.
0
The following Lemmas 2.1-2.3 come from [15]. They give in the sense of
lower norms the estimates of the finite element projection and the convergence
of T to T.
h
Lemma 2.1 [15, Lemma 3.4]. Suppose that n∈W2, (D) and R(D) is
∞
valid (s=0,1), then for (u,w)∈H,
k(u,w)−Ph(u,w)kHs .hrsk(u,w)−Ph(u,w)kH, s=0,1. (2.20)
The conforming finite element approximation of (2.10) is given by: Find
λ ∈C, (u ,w )∈H such that
∗h ∗h h∗ h
A((v,z),(u ,w ))=λ B((v,z),(u ,w )), ∀(v,z)∈H . (2.21)
∗h h∗ ∗h ∗h h∗ h
Note that the primal and dual eigenvalues are connected via λ =λ .
h ∗h
5
Define the solution operator T :H →H satisfying
h∗ s h
A((v,z),T (f,g))=B((v,z),(f,g)), ∀ (v,z)∈H . (2.22)
h∗ h
Naturally (2.21) has the following equivalent operator form
Th∗(u∗h,wh∗)=λ∗h−1(u∗h,wh∗). (2.23)
Lemma 2.2 [15, Theorem 3.1]. Let n∈W1, (D), then
∞
kT −ThkH →0, (2.24)
kT −ThkH1 →0, (2.25)
and let n∈W2, (D), then
∞
kT −ThkH0 →0. (2.26)
In this paper, let λ be the ith eigenvalue of (2.6) with the algebraic mul-
i
tiplicity q and the ascent 1. Then, according to spectral approximation theory
[31, 32], there are q eigenvalues λ (j = i,··· ,i+q−1) of (2.13) converging
j,h
to λ . Let M(λ ) be the space spanned by all eigenfunctions corresponding to
i i
the eigenvalue λ . Let M (λ ) be the space spanned by all generalized eigen-
i h i
functions corresponding to the numerical eigenvalues {λj,h}ji+=qi−1 of (2.13). As
for the dual problems (2.10) and (2.21), the definitions of M (λ ) and M (λ )
∗ ∗i h∗ ∗i
are made similarly to M(λ ) and M (λ ), respectively.
i h i
In what follows, to describe the approximation relation between the finite
elementspaceH andtheeigenfunctionspacesM(λ )andM (λ ),weintroduce
h i ∗ ∗i
the following quantities
δh(λi)= sup inf k(v,z)−(vh,zh)kH,
(v,z)∈M(λi)(vh,zh)∈Hh
k(v,z)kH=1
δh∗(λ∗i)= (v,z)s∈uMp∗(λ∗i)(vh,zihn)f∈Hhk(v,z)−(vh,zh)kH.
k(v,z)kH=1
Theoperatorconvergenceresults(2.24)-(2.26)arecriticalasabridgeofmak-
ing the error analysis for the discrete problem (2.13). From these results, we
yieldimmediatelythefollowinglemmausingthespectralapproximationtheory.
Lemma 2.3. Suppose n∈W2, (D). Let (u ,w ) (j =i,i+1,··· ,i+
∞ j,h j,h
q−1)beeigenfunctioncorrespondingtoλ andk(u ,w )k =1,thenthere
j,h j,h j,h A
exists eigenfunction (u ,w ) corresponding to λ such that
i i i
k(uj,h,wj,h)−(u,w)kH .δh(λi), (2.27)
k(uj,h,wj,h)−(u,w)kHs .hrsδh(λi), s=0,1, (2.28)
|λ −λ |.δ (λ )δ (λ ). (2.29)
i j,h h i h∗ ∗i
Remark 2.1. Thesimilarestimatesasabovearevalidforthedualproblem
(2.13) (see [15]).
3 A New Multigrid Method
In this section, based on the multilevel correction method proposed by Lin
and Xie [22, 23], we give the multigrid scheme for the weak form (2.6). Our
6
theoretical results are given in Theorems 3.1-3.2. Prior to our argument, we
give the following basic condition related to finite element spaces and their
approximationrelation to eigenfunction spaces.
We construct the finite element spaces such that H =H ⊂H ⊂···⊂
H h1 h2
H and
hn
1 1
δhm+1 ≈ βδhm, δh∗m+1 ≈ βδh∗m, (3.1)
where β >1 is a constant only dependent on the smoothness of eigenfunctions
corresponding to λ and the degree t of the piecewise polynomial space.
i
The above condition is readily satisfied on regular meshes. In particular,
if the meshes are obtained from a procedure of bisection mesh refinement and
M(λ ),M (λ ) ⊂ H2+r(D)×Hr(D) (r ≤ t−1), then we have approximately
i ∗ ∗i
β ≈2r.
Assumethatwehaveobtainedtheeigenpairapproximations(λ ,u ,w )
j,hm j,hm j,hm
(j =i,i+1,··· ,i+q−1)andthecorrespondingdualones(λ ,u ,w ).
∗j,hm ∗j,hm j∗,hm
First of all, we give one correction step of the multigrid scheme.
Algorithm 1. One Correction Step.
Step 1. Solve the following linear boundary value problems: For j = i,i+
1,··· ,i+q−1, find (u ,w )∈H such that
j,hm+1 j,hm+1 hm+1
b b
A((u ,w ),(v,z))=λ B((u ,w ),(v,z)), ∀(v,z)∈H ,
j,hm+1 j,hm+1 j,hm j,hm j,hm hm+1
b b
and find (u ,w )∈H such that
∗j,hm+1 j∗,hm+1 hm+1
b b
A((v,z),(u ,w ))=λ B((v,z),(u ,w )), ∀(v,z)∈H .
∗j,hm+1 j∗,hm+1 j,hm ∗j,hm j∗,hm hm+1
b b
Step2. ConstructanewfiniteelementspaceH ⊇H +span (u ,w ),
H,hm+1 H n j,hm+1 j,hm+1
b b
(u ,w ), j = i,i+1,··· ,i+q −1 and solve the following eigen-
∗j,hm+1 j∗,hm+1 o
vbalue probblems: For j = i,i +1,··· ,i + q − 1, find λj,hm+1 ∈ C, nontrivial
(u ,w )∈H such that
j,hm+1 j,hm+1 H,hm+1
A((u ,w ),(v,z))=λ B((u ,w ),(v,z)), ∀(v,z)∈H ,
j,hm+1 j,hm+1 j,hm+1 j,hm j,hm+1 H,hm+1
and find nontrivial (u ,w )∈H such that
∗j,hm+1 j∗,hm+1 H,hm+1
A((v,z),(u∗j,hm+1,wj∗,hm+1))=λj,hm+1B((v,z),(u∗j,hm+1,wj∗,hm+1)), ∀(v,z)∈HH,hm+1.
i+q 1 i+q 1
Weoutput λ − andabasis (u ,w ) − ofM (λ )
n j,hm+1oj=i n j,hm+1 j,hm+1 oj=i hm+1 i
i+q 1
withk(u ,w )k =1andabasis (u ,w ) − ofM (λ )
j,hm+1 j,hm+1 A n ∗j,hm+1 j∗,hm+1 oj=i h∗m+1 ∗i
with k(u ,w )k = 1. We define the above two steps as Procedure
∗j,hm+1 j∗,hm+1 A
Correction:
i+q 1
λ ,u ,w ,u ,w −
n j,hm+1 j,hm+1 j,hm+1 ∗j,hm+1 j∗,hm+1oj=i
i+q 1
=Correction H , λ ,u ,w ,u ,w − ,H
(cid:16) H n j,hm j,hm j,hm ∗j,hm j∗,hmoj=i hm+1(cid:17)
ImplementingProcedureCorrectionrepeatedlyleadstothefollowingScheme.
7
Algorithm 2. Mutigrid Scheme.
Step 1. Construct a series of nested finite element spaces H = H , H ,
H h1 h2
··· , H such that (3.1) holds.
hn
Step 2. For j = i,i+1,··· ,i+q−1, find (λ ,u ,w )∈ C×H such
j,h1 j,h1 j,h1 h1
that k(u ,w )k =1 and
j,h1 j,h1 A
A((u ,w ),(v,z))=λ B((u ,w ),(v,z)), ∀(v,z)∈H ,
j,h1 j,h1 j,h1 j,h1 j,h1 h1
and find (u ,w )∈H such that k(u ,w )k =1 and
∗j,h1 j∗,h1 h1 ∗j,h1 j∗,h1 A
A((v,z),(u∗j,h1,wj∗,h1))=λj,h1B((v,z),(u∗j,h1,wj∗,h1)), ∀(v,z)∈Hh1.
Step 3. For m=1,2,··· ,N −1
Obtain a new eigenpair approximation (λ ,u ,w ) by
j,hm+1 j,hm+1 j,hm+1
i+q 1
λ ,u ,w ,u ,w −
n j,hm+1 j,hm+1 j,hm+1 ∗j,hm+1 j∗,hm+1oj=i
i+q 1
=Correction H , λ ,u ,w ,u ,w − ,H
(cid:16) H n j,hm j,hm j,hm ∗j,hm j∗,hmoj=i hm+1(cid:17)
End
Before making the error analysis of multigrid scheme, we give the following
assumption.
i+q 1
(A)Quasi-biorthogonality. Supposethatthereare (u ,w ) −
n j,hm j,hm o
j=i
ei+q 1e
⊂Mhm(λi) with k(uj,hm, wj,hm)kH =1 and n(u∗j,hm,wj∗,hm)oj=i− ∈Mh∗m(λ∗i)
with k(u∗j,hm, wj∗,hme)kH =e1 such that e e
e e
|A((uj,hm,wj,hm),(u∗l,hm,wl∗,hm))|+|A((uj,hm,wj,hm),(u∗l,hm,wl∗,hm))|.Hr0,
e e e e
(j,l =i,i+1,··· ,i+q−1,j 6=l),
and |A((u ,w ),(u ,w ))|+|A((u ,w ),(u ,w ))| (j =
j,hm j,hm ∗j,hm j∗,hm j,hm j,hm ∗j,hm j∗,hm
i,i+1,··· ,i+q−1)hasapositivelowerbounduniformlywithrespectivetoh .
e e e e m
When λ is a simple eigenvalue (q = 1), it is clear that Assumption (A)
i
is valid; when q >1, we can prove the follow conclusion:
If the distance (in k·kH) from (uj,hm,wj,hm) (j =i,i+1,··· ,i+q−1) to
span (u ,w ),l =i,i+1,··· ,i+q−1,l6=j hasa positivelowerbound
n l,hm l,hm o
uniformly with respective to h , then Assumption (A) is valid.
m
The following theorem, an extension of the corresponding theorems in [22,
23, 33], indicates that the accuracy of numerical eigenpair can be apparently
improved after one correction step.
Theorem 3.1. Suppose that (A) is valid and the ascent of λ (j =
j,hm
i,··· ,i+q−1) is 1 , and there exist two eigenpairs (λ ,u ,w ) and(λ ,u ,w )
i i i ∗i ∗i i∗
such that the eigenpair approximations (λ , u , w ), (λ , u ,
j,hm j,hm j,hm ∗j,hm ∗j,hm
8
w )∈C×H have the following estimates
j∗,hm hm
k(uj,hm,wj,hm)−(uj,wj)kH . εhm(λi), (3.2)
k(uj,hm,wj,hm)−(uj,wj)kHs . Hrsεhm(λi), s=0,1, (3.3)
k(u∗j,hm,wj∗,hm)−(u∗j,wj∗)kH . ε∗hm(λi), (3.4)
k(u∗j,hm,wj∗,hm)−(u∗j,wj∗)kHs . Hrsε∗hm(λi), s=0,1, (3.5)
|λ −λ | . ε (λ )ε (λ ). (3.6)
i j,hm hm i ∗hm i
Thenafteronecorrectionstep,thereexiststwoeigenpairs(u ,w ),(u ,w )such
j j ∗j j∗
thattheresultantapproximations(λ ,u ,w ),(λ ,u ,
j,hm+1 j,hm+1 j,hm+b1 b ∗j,hmb+1 b∗j,hm+1
w )∈C×H has the following estimates
j∗,hm+1 hm+1
k(uj,hm+1,wj,hm+1)−(uj,wj)kH . εhm+1(λi), (3.7)
k(uj,hm+1,wj,hm+1)−(ubj,wbj)kHs . Hrsεhm+1(λi), s=0,1, (3.8)
k(u∗j,hm+1,wj∗,hm+1)−(bu∗j,bwj∗)kH . ε∗hm+1(λi), (3.9)
k(u∗j,hm+1,wj∗,hm+1)−(ub∗j,wbj∗)kHs . Hrsε∗hm+1(λi), s=0,1, (3.10)
|λ b−λb | . ε (λ )ε (λ ), (3.11)
i j,hm+1 hm+1 i ∗hm+1 i
where
εhm+1(λi):=δhm+1(λi)+Hr0εhm(λi)
and
ε∗hm+1(λ∗i):=δh∗m+1(λ∗i)+Hr0ε∗hm(λ∗i).
i+q 1 i+q 1
Proof. Since (u ,w ) − isabasisofM (λ ), (u ,w ) −
n j,hm+1 j,hm+1 oj=i hm i n j j oj=i
is a basis of M(λi). For any (v,z)∈M(λi) and k(v,z)kH =1 we denote
i+q 1
−
(v,z)= γ (u ,w ).
X j j j
j=i
For (u ,w )∈M (λ ) in Assumption (A), thanks to (2.27) there exists
∗l,hm l∗,hm hm i
(u ,w )∈M(λ ) satisfying
∗l el∗ e i
e e
k(u∗l,wl∗)−(u∗l,hm,wl∗,hm))kH .δh∗m(λ∗i).
e e e e
Then
i+q 1
−
A((v,z),(u ,w ))= γ A((u ,w ),(u ,w )).
∗l l∗ X j j j ∗l l∗
e e j=i e e
Hence
i+q 1
1 −
|γl| = A((u ,w ),(u ,w ))nA((v,z),(u∗l,wl∗))− X γjA((uj,wj),(u∗l,wl∗))o.
l l ∗l l∗ e e j=i,j=l e e
e e 6
Due to Assumption (A) and k(u∗l,wl∗)−(u∗l,hm,wl∗,hm))kH .δh∗m(λ∗i), since
e e e e
|A((u ,w ),(u ,w ))| ≤ |A((u ,w )−(u ,w ),(u ,w ))|
j j ∗l l∗ j j j,hm j,hm ∗l l∗
e e +|A((u ,w ),(u ,w )−e(ue ,w ))|
j,hm j,hm ∗l l∗ ∗l,hm l∗,hm
+|A((uj,hm,wj,hm),(ue∗l,hme,wl∗,hme))| e
. εhm(λi)+δh∗m(λ∗i)+eHr0, e
9
we have
i+q 1
1 −
|γl| . |A((u ,w ),(u ,w ))|n1+ X |γj|(εhm(λi)+ε∗hm(λ∗i)+Hr0)o.
l l ∗l l∗ j=i,j=l
e e 6
Since |A((u ,w ),(u ,w ))| has a positive lower bound uniformly
l,hm l,hm ∗l,hm l∗,hm
with respect to h , so does |A((u ,w ),(u ,w ))|. It is immediate that
m e e l l ∗l l∗
i+q 1 i+q 1
− −
X |γl| . q+q X |γj|(εhm(λi)+ε∗hm(λ∗i)+Hr0),
l=i j=i
from which it follows that
i+q 1
−
|γ | . 1.
X j
j=i
Wesetα =λ /λ . ByvirtueoftheorthogonalityofP andboundedness
j j j,hm hm+1
of B(·,·) we have
kαj(uj,hm+1,wj,hm+1)−Phm+1(uj,wj)k2H
.bA(α (u b ,w )−P (u ,w ),α (u ,w )−P (u ,w ))
j j,hm+1 j,hm+1 hm+1 j j j j,hm+1 j,hm+1 hm+1 j j
=A(α (ub ,wb )−(u ,w ),α (u b,w b)−P (u ,w ))
j j,hm+1 j,hm+1 j j j j,hm+1 j,hm+1 hm+1 j j
=B(λ b α (u b ,w )−λ (u ,w ),bα (u b ,w )−P (u ,w ))
j,hm j j,hm j,hm j j j j j,hm+1 j,hm+1 hm+1 j j
.k(uj,hm,wj,hm)−(uj,wj)kH0kαj(uj,hm+1,bwj,hm+1)b−Phm+1(uj,wj)kH,
b b
which together with (3.3) yields
kαj(uj,hm+1,wj,hm+1)−Phm+1(uj,wj)kH .Hr0εhm(λj).
b b
Thanks to Lemma 2.2, applying spectral approximation theory yields
k(uj,hm+1,wj,hm+1)−(uj,wj)kH .k(uj,wj)−Phm+1(uj,wj)kH
. (v,zs)∈uMp(λi)(vhm+1,zhbmi+n1fb)∈HH,hm+1kb(v,bz)−(vhm+1,zbhm+b1)kH
k(v,z)kH=1
i+q 1
−
.supk X γj((uj,wj)−αj(uj,hm+1,wj,hm+1))kH
γj j=i b b
i+q 1
−
. X k(uj,wj)−Phm+1(uj,wj)+Phm+1(uj,wj)−αj(uj,hm+1,wj,hm+1)kH
j=i b b
.δhm+1(λj)+Hr0εhm(λj)=εhm+1(λj),
and for s=0,1,
k(uj,hm+1,wj,hm+1)−(uj,wj)kHs .k(uj,wj)−Phm+1(uj,wj)kHs
.Hrsk(uj,wj)−Phbm+1b(uj,wj)kH b b b b
.Hrsε b b(λ ), b b
hm+1 j
wherewe haveused(2.20)inthe secondinequality above. The aboveargument
implies (3.7)-(3.8) hold. Similarly we can also prove (3.9)-(3.10). Using the
10