New error estimates of linear triangle finite elements for the Steklov eigenvalue problem

0.17 MB·

by  Hai Bi

NEW ERROR ESTIMATES OF LINEAR
TRIANGLE FINITE ELEMENTS FOR THE
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1
0 STEKLOV EIGENVALUE PROBLEM
2

n
a Hai Bi, Yidu Yang, Yuanyuan Yu
J

9
School of Mathematical Sciences, Guizhou Normal University,

] Guiyang, 550001, China
A
[email protected],[email protected],[email protected]
N
.
h
t Abstract
a
m
In this paper we make a further discussion on the finite elements ap-
[
proximation fortheStekloveigenvalueproblemonconcavepolygonaldo-

1 main. We make full use of the regularity estimate and the characteristic
v ofedgeaverageinterpolationoperatorofnonconformingCrouzeix-Raviart
3
element,which isdifferentfrom theexistingproofargument,andprovea
1
1 newand optimal errorestimate in k·k0,∂Ω for theeigenfunction of linear
conforming finite element and the nonconforming Crouzeix-Raviart ele-
2
ment,whichisanimprovementofthecurrentresults. Finally,wepresent
0
some numerical experimentsto support thetheoretical analysis.
.
1
0 Key words. Steklov eigenvalue problem, Concave polygonal domain, Lin-
7
ear conforming finite element, Nonconforming Crouzeix-Raviartelement, Error
1
estimates.
:
v
i
X
1 Introduction
r
a
Steklov eigenvalue problems have important physical background and many
applications. Forinstance,theyappearintheanalysisofstabilityofmechanical
oscillatorsimmersedinaviscousfluid(see[12]andthereferencestherein),inthe
studyofsurfacewaves(see[7]),inthestudyofthevibrationmodesofastructure
incontactwithanincompressiblefluid(see [6]), inthe analysisofthe antiplane
shearing on a system of collinear faults under slip-dependent friction law (see
[10]),etc. Thusthenumericalmethodsforsolvingtheseproblemshaveattracted
moreandmorescholars’attention. Tillnow,systematicalandprofoundstudies
ontheconformingfiniteelementsapproximationforStekloveigenvalueproblems
have been made on polygonal domain such as [2, 3, 4, 6, 9, 15, 16, 20, 21]).
Recently,thenonconformingfiniteelementsforSteklovproblemshavealsobeen
1
considered, e.g., see [1, 8, 17, 18, 22]. The aim of this paper is to discuss the
errorestimatesoflineartrianglefiniteelements,includingthelinearconforming
finiteelementandthenonconformingCrouzeix-Raviartelement,approximation
for Steklov eigenvalue problems with variable coefficients on concave polygonal
domain.
We consider the following Steklov eigenvalue problem
∂u
−div(α∇u)+βu=0 in Ω, α =λu on ∂Ω, (1.1)
∂n
where Ω⊂R2 is a polygonaldomain with ω being the largestinner angle of Ω,
and ∂u is the outward normal derivative.
∂n
Having in mind that Hs(Ω) denotes the Sobolev space with real order s on
Ω, k·k is the norm on Hs(Ω) and H0(Ω) = L (Ω), and Hs(∂Ω) denotes the
s 2
Sobolev space with real order s on ∂Ω with the norm k·k .
s,∂Ω
Suppose that the coefficients α=α(x) and β =β(x) are bounded by above
and below by positive constants. We assume that α∈C1(Ω¯).
The weak form of (1.1) is given by: Find λ ∈ R, u ∈ H1(Ω),kuk = 1,
0,∂Ω
such that
a(u,v)=λb(u,v), ∀v ∈H1(Ω), (1.2)
where
a(u,v)= (α∇u·∇v+βuv)dx, b(u,v)= uvds.
Z Z
Ω ∂Ω
It is easy to know that a(·,·) is a symmetric, continuous and H1(Ω)-elliptic
bilinear form on H1(Ω)×H1(Ω).
Intheexistingliteratures,theerrorestimateoflineartriangleelementseigen-
function,includingconformingelementandnonconformingCrouzeix-Raviartel-
ement (hereafter termed C-R element for simplicity), in k·k0,∂Ω is all O(hr+r2)
where r is the regularity exponent of the eigenfunction (see Lemma 2.1). It is
obvious that this estimate is not optimal since it doesn’t achieve the order of
interpolationerror. Inthispaper,weimprovethisestimatewheneigenfunctions
are singular (i.e., r <1) and prove that in this case the error estimate of linear
triangle elements eigenfunction can achieve O(hr+21). Comparing the proof ar-
guments of existing estimates (e.g., see [3, 9, 17, 22]), we make full use of the
regularityestimateandthecharacteristicofedgeaverageinterpolationoperator
of C-R element, especially in the analysis for conforming finite elements, and
obtain the improved error estimates (2.25) and (3.6) which are optimal.
Throughout this paper, C denotes a positive constant independent of h,
which may not be the same constant in different places.
2
2 The nonconforming Crouzeix-Raviart element
approximation for the Steklov eigenvalue prob-
lem
Considerthe sourceproblem(2.1)associatedwith(1.1): Findw ∈H1(Ω), such
that
a(w,v)=b(f,v), ∀v ∈H1(Ω). (2.1)
As for the source problem (2.1), there hold the following regularity results.
Lemma 2.1. If f ∈L2(∂Ω), then w ∈H1+r2(Ω) and
kwk1+r2 ≤CΩkfk0,∂Ω; (2.2)
if f ∈H21(∂Ω), then w ∈H1+r(Ω) and
kwk1+r ≤CΩkfk21,∂Ω; (2.3)
if f ∈Hε(∂Ω), ε∈(0,r−1/2), then w∈H32+ε(Ω) and
kwk23+ε ≤CΩkfkε,∂Ω. (2.4)
Here r = 1 when ω < π, and r < π which can be arbitrarily close to π when
ω ω
ω >π, and C is a priori constant.
Ω
Proof. See [14]. (cid:3)
Note that a(·,·) is coercive, using the source problem (2.1) associated with
(1.2) we can define the operator A:L2(∂Ω)→H32(Ω)⊂H1(Ω), satisfying
a(Af,v) = b(f,v), ∀v ∈H1(Ω).
Define the operator T :L (∂Ω)→H1(∂Ω) satisfying
2
Tf =(Af),
′
where denotes the restriction to ∂Ω.
′
Bramble and Osborn [9] proved that (1.2) has the operator form:
Tw=µw. (2.5)
Namely, if (µ,w) is an eigenpair of (2.5), then (λ,Aw) is an eigenpair of (1.2),
λ= 1. Conversely, if (λ,u) is an eigenpair of (1.2), then (µ,u) is an eigenpair
µ ′
of (2.5), µ= 1.
λ
Let λ be the j-th eigenvalue of T. We arrangeeigenvalues by the increasing
order with each eigenvalue counted according to its algebraic multiplicity. And
let M(λ) denote the space spanned by eigenfunctions of (1.2) corresponding to
the eigenvalue λ.
3
Let π = {K} be a regular triangulation of Ω in the sense of the minimal
h
internalangle condition (see [11], pp. 131). We denote h=max h where
h is the diameter of element K. Let Sh be the C-R element sKp∈aπche (Ksee [13])
K nc
defined on π :
h
Sh = {v ∈ L (Ω) : v | ∈ span{1,x ,x }, v is continuous at the midpoints
nc 2 K 1 2
of the edges of elements}.
TheC-Relementapproximationof(1.2)is: Findλ ∈R,u ∈Sh ,ku k =
h h nc h 0,∂Ω
1, such that
a (u ,v)=λ b(u ,v), ∀v ∈Sh , (2.6)
h h h h nc
where
a (u ,v)= (α∇u ·∇v+βu v)dx.
h h Z h h
X
K∈πhK
Define kvkh = ( kvk21,K)21, kvk21,K = K(|∂∂xv1|2+|∂∂xv2|2+|v|2)dx. Evi-
dently,k·k isthenKoPr∈mπhonSh anditissimplRetoshowthata (·,·)isuniformly
h nc h
Sh -elliptic.
nc
The C-R element approximation of (2.1) is: Find w ∈Sh , such that
h nc
a (w ,v)=b(f,v), ∀v ∈Sh . (2.7)
h h nc
Denote the consistency term of the C-R element by
E (w,v)=a (w,v)−b(f,v). (2.8)
h h
And based on the standard method (see, for example [1, 8, 17]), the following
consistency error estimate can be proved.
Lemma 2.2. Suppose that w ∈H1+t(Ω) with t ∈[r,1] is the weak solution of
2
(2.1), then
E (w,v)≤Chtkwk kvk , ∀v ∈Sh +H1(Ω), (2.9)
h 1+t h nc
kw−w k ≤Chtkwk , (2.10)
h h 1+t
kw−whk0,∂Ω ≤Cht+r2kwk1+t. (2.11)
Proof. See, e.g., Lemma 2.2 in [8], or Theorem 2.1 in [17]. (cid:3)
Define the interpolation operator I :H1(Ω)→Sh :
h nc
I uds= uds ∀l, ∀u∈H1(Ω), (2.12)
Z h Z
l l
where l is an edge of arbitrary element in π .
h
According to the interpolation theory (see [11]), we have
ku−I uk ≤ Ch1+rkuk , (2.13)
h 0 1+r
ku−I uk ≤ Chrkuk . (2.14)
h h 1+r
4
Theorem 2.3. Let w ∈ H1+t(Ω) with t ∈ (1,1] be the weak solution of (2.1),
2
then
kw−whk ε,∂Ω ≤Cht+12+εkwk1+t (2.15)
−
where t= 1 +ε if f ∈Hε(∂Ω), 0<ε<r− 1, and t=r if f ∈H21(∂Ω).
2 2
Proof. For each g ∈ Hε(∂Ω), let ϕ be the unique solution of the following
variational problem:
a(v,ϕ)=b(g,v), ∀v ∈H1(Ω).
From(2.4)weknowthatϕ∈H32+ε(Ω). Letϕh =Ihϕ∈Shbetheinterpolation
of ϕ, then
b(f,ϕ−ϕ )=b(f,ϕ)−b(f,ϕ )
h h
=a (w,ϕ)−a (w ,ϕ )
h h h h
=a (w,ϕ)−a (w ,ϕ)+a (w ,ϕ)−a (w ,ϕ )
h h h h h h h h
=a (w−w ,ϕ)+a (w −w,ϕ−ϕ )+a (w,ϕ−ϕ ).
h h h h h h h
By the definition of consistency term we have
a (w−w ,ϕ)+a (w,ϕ−ϕ )
h h h h
=a (w−w ,ϕ)−b(w−w ,g)+b(w−w ,g)
h h h h
+a (w,ϕ−ϕ )−b(f,ϕ−ϕ )+b(f,ϕ−ϕ )
h h h h
=E (ϕ,w−w )+b(w−w ,g)+E (w,ϕ−ϕ )+b(f,ϕ−ϕ ).
h h h h h h
Combining the above two relationships, we get
b(w−w ,g)=−E (ϕ,w−w )−E (w,ϕ−ϕ )−a (w −w,ϕ−ϕ ),
h h h h h h h h
then
|b(g,w−w )|=|b(w−w ,g)|
h h
≤|E (ϕ,w−w )|+|E (w,ϕ−ϕ )|+|a (w −w,ϕ−ϕ )|.(2.16)
h h h h h h h
From (2.9), (2.3), (2.4), (2.10) and the error estimate of interpolation, we can
deduce that
|Eh(ϕ,w−wh)|≤Ch12+εkϕk23+εkw−whkh ≤Cht+21+εkwk1+tkgkε,∂Ω,
|Eh(w,ϕ−ϕh)|≤Chtkwk1+tkϕ−ϕhkh ≤Cht+12+εkwk1+tkgkε,∂Ω,
|ah(wh−w,ϕ−ϕh)|≤Ckw−whkhkϕ−ϕhkh ≤Cht+21+εkwk1+tkgkε,∂Ω.
And substituting the above three estimates into (2.16), we obtain
|b(g,w−wh)|≤Cht+21+εkwk1+tkgkε,∂Ω.
5
By the definition of negative norm, we have
|b(g,w−w )|
h
kw−w k = sup
h ε,∂Ω
− g Hε(∂Ω) kgkε,∂Ω
∈
≤ sup Cht+21+εkwk1+tkgkε,∂Ω ≤Cht+21+εkwk1+t, (2.17)
kgk
g Hε(∂Ω) ε,∂Ω
∈
namely, (2.15) is true. (cid:3)
Let us denote by δSh the functions defined on ∂Ω, which are restriction of
nc
functions in Sh to ∂Ω. From [1], pp.189 we know that
nc
1
δSh ⊂Hε(∂Ω), for any ε∈[0, ). (2.18)
2
Since a (·,·) is uniformly elliptic with respect to h, the approximate source
h
problem (2.7) associated with (2.6) is uniquely solvable. Thus, we can define
the operator A :L (∂Ω)→Sh , satisfying
h 2 nc
a (A f,v) = b(f,v), ∀v ∈Sh .
h h nc
Define T :L (∂Ω)→δSh ⊂Hε(∂Ω)⊂L (∂Ω), satisfying
h 2 nc 2
Thf =(Ahf)′.
[22] proved that (2.6) has the operator form:
T w =µ w . (2.19)
h h h h
Namely, if (µ ,w ) is an eigenpair of (2.19), then (λ ,A w ) is an eigenpair of
h h h h h
(2.6), λ = 1 . Conversely, if (λ ,u ) is an eigenpair of (2.6), then (µ ,u ) is
h µh h h h ′h
an eigenpair of (2.19), µ = 1 .
h λh
We prove the following interpolation estimates.
Lemma 2.4. Let u∈H1+r(Ω), then the following estimates hold:
||u−Ihu||0,∂Ω ≤Chr+21kuk1+r, (2.20)
||u−Ihu|| ε,∂Ω ≤Chr+21+εkuk1+r. (2.21)
−
Proof. Letl⊂∂ΩbetheedgeoftheelementK,thenbythetraceinequality
( see Lemma 7.1.1 in [19]) we have
||u−Ihu||0,l ≤C(h−K12ku−Ihuk0,K +hK12 ku−Ihuk1,K)≤ChKr+21kuk1+r,K,
thus (2.20) is valid.
For any g ∈Hε(∂Ω) (ε∈(0,r− 1)), let I g be the piecewise constant interpo-
2 0
lation of g on ∂Ω. From the definition of I and the interpolation estimates we
h
have
|b(g,u−I u)| = |b(g−I g,u−I u)|
h 0 h
≤ Ckg−I gk ku−I uk
0 0,∂Ω h 0,∂Ω
≤ Chr+21+εkuk1+rkgkε,∂Ω,
and using the definition of negative norm we know that (2.21) holds. (cid:3)
6
Lemma 2.5. Suppose that M(λ) ⊂ H1+r(Ω) and λ be the j-th eigenvalue
of (1.2). Let λ be the j-th eigenvalue of (2.6) and u be an eigenfunction
h h
corresponding to λ with ku k = 1. Then there exists u ∈ M(λ) with
h h 0,∂Ω
kuk =1, such that
0,∂Ω
|λ −λ|≤Ch2rkuk2 , (2.22)
h 1+r
ku−u k ≤Chrkuk , (2.23)
h h 1+r
ku−uhk0,∂Ω ≤Ch23rkuk1+r. (2.24)
Proof. See [1, 17, 22]. (cid:3)
Lemma 2.5 is an existing conclusion. Next we will improve the estimate
(2.24).
Theorem 2.6. Under the conditions of Lemma 2.5, further assume that π is
h
a quasi-uniform mesh (see pp.135 in [11]), then
ku−uhk0,∂Ω ≤ Chr+21kuk1+r. (2.25)
Proof. Since Au and A u are solutions of (2.1) and (2.7) with f = u,
h
respectively, then from (2.15) we know that
kTu−Thuk ε,∂Ω ≤Chr+21+εkuk1+r. (2.26)
−
Using (2.21) we obtain
kTu−IhTuk ε,∂Ω ≤Chr+21+εkuk1+r. (2.27)
−
From (2.26) and (2.27), we have
kThu−IhTuk ε,∂Ω ≤Chr+12+εkuk1+r. (2.28)
−
By the definition of negative norm and the inverse estimates, we have
kT u−I Tuk2 ≤kT u−I Tuk kT u−I Tuk
h h 0,∂Ω h h ε,∂Ω h h ε,∂Ω
−
≤kT u−I Tuk h εkT u−I Tuk ,
h h ε,∂Ω − h h 0,∂Ω
−
thus
kThu−IhTuk0,∂Ω ≤Chr+12kuk1+r. (2.29)
By using (2.29) and (2.20), we get
kTu−T uk ≤ kT u−I Tuk +kTu−I Tuk
h 0,∂Ω h h 0,∂Ω h 0,∂Ω
≤ Chr+12kuk1+r. (2.30)
Ithasbeenprovedin[17,8]thatkT−T k →0(h→0),thus,fromTheorem
h 0,∂Ω
7.4 in [5] we get
ku −uk ≤Ck(T −T )uk . (2.31)
h 0,∂Ω h 0,∂Ω
Substituting (2.30) into (2.31), we obtain (2.25). (cid:3)
Remark 2.1. Ifr ∈(1,1),i.e.,Ω isconcave,itisclearthatthe estimate(2.25)
2
is better than (2.24).
7
3 The conforming element approximation for the
Steklov eigenvalue problem
Let Sh ⊂ C(Ω) be a space of piecewise linear polynomials defined on π . The
c h
conforming element approximation of (1.2) is: Find λ ∈ R, u ∈ Sh with
h h c
ku k =1, such that
h 0,∂Ω
a(u ,v)=λ b(u ,v), ∀v ∈Sh. (3.1)
h h h c
As for the conforming finite element approximation (3.1), the following re-
sults are valid (see [3, 9]).
Lemma3.1. Let(λ ,u )bethej-theigenpair of (3.1), λbethej-theigenvalue
h h
of (1.2), and M(λ)⊂H1+r(Ω). Then there exists u∈M(λ) such that
|λ−λ |≤Ch2rkuk , (3.2)
h 1+r
ku−u k ≤Chrkuk , (3.3)
h 1 1+r
ku−uhk0,∂Ω ≤Ch32rkuk1+r, (3.4)
where the principle to determine r see Lemma 2.1.
Now, let P :H1(Ω)→Sh be the Ritz projection defined by
h c
a(w−P w,v)=0, ∀v ∈Sh.
h c
We can define the operator A :L (∂Ω)→Sh, satisfying
h 2 c
a(A f,v) = b(f,v), ∀v ∈Sh.
h c
It is easy to know that A =P A.
h h
Let δSh be the space of functions defined on ∂Ω, which are restriction of
c
functions in Sh to ∂Ω. Define T :L (∂Ω)→δSh ⊂H1(∂Ω), satisfying
c h 2 c
T f =(A f).
h h ′
It has been proved in [5, 9] that kT −T k → 0 (h → 0), and (3.1) has the
h 0,∂Ω
operator form:
1
T w = w . (3.5)
h h h
λ
h
Next we will give a new error estimate for the conforming finite element.
Theorem 3.2. Under the conditions of Lemma 3.1, further assume that π is
h
quasi-uniform mesh, then
ku−uhk0,∂Ω ≤Chr+21kuk1+r. (3.6)
8
Proof. For each g ∈ Hε(∂Ω), let ϕ be the unique solution of the following
variational problem:
a(v,ϕ)=b(g,v), ∀v ∈H1(Ω).
From (2.4) we know that Ag =ϕ∈H32+ε(Ω), and
b(u−P u,g)=a(u−P u,Ag)
h h
=a(u−Phu,Ag−PhAg)≤Chr+12+εkuk1+rkgkε,∂Ω,
thus, by the definition of negative norm, we have
ku−Phuk ε,∂Ω ≤Chr+21+εkuk1+r. (3.7)
−
Let I u be the interpolation of u defined by (2.12). By using the inverse esti-
h
mates, (3.7), (2.21) and (2.20), we get
ku−P uk ≤kP u−I uk +ku−I uk
h 0,∂Ω h h 0,∂Ω h 0,∂Ω
≤Ch−εkPhu−Ihuk ε,∂Ω+ku−Ihuk0,∂Ω
−
≤Ch ε(kP u−uk +ku−I uk )+ku−I uk
− h ε,∂Ω h ε,∂Ω h 0,∂Ω
− −
≤Chr+21kuk1+r. (3.8)
By using the spectral approximation theory, we get
ku−u k ≤ CkTu−T uk =CkAu−A uk
h 0,∂Ω h 0,∂Ω h 0,∂Ω
1
= CkAu−P Auk =C ku−P uk . (3.9)
h 0,∂Ω h 0,∂Ω
λ
Substituting (3.8) into (3.9), we obtain (3.6). (cid:3)
Remark 3.1. Comparing (3.4) and (3.6), we can see that when eigenfunctions
are singular, i.e., r<1, the error estimate in k·k is improved.
0,∂Ω
When we prove the improved estimates (2.25) and (3.6), we make full use
ofthe regularityestimate (2.4)to analyzethe negativenormestimate, thenuse
the negative norm estimate and the interpolation of C-R element, especially in
theanalysisforconformingelements,toobtaintheoptimalestimatesinL2(∂Ω);
while the existing workis to analyzedirectly the errorin L2(∂Ω)by using (2.2)
which leads to the lost of error order.
Remark 3.2. We prove the estimates (2.25) and (3.6) under the condition
thatπ is quasi-uniform. Infact, this conditionis nota restriction. Since when
h
π˜ is a regular partition derived from π by local refinement, the approximate
h h
eigenfunction u˜ computed on π˜ generally satisfies ku˜ −uk ≤ Cku −
h h h 0,∂Ω h
uk , then, for such regular meshes (2.25) and (3.6) are still valid.
0,∂Ω
4 Numerical Experiments
Consider the problem (1.1), where α(x) = β(x) = 1, Ω ⊂ R2, Ω = [0,1]×
[0,1] [0,1]×[1,1] is a L-shaped domain with the largestinner angle ω = 3π,
2 2 2 2
S
9
orΩ=[0,1]×[0,1]\{x=(x ,x ): 1 ≤x ≤1,x = 1} isthe unitsquarewith
1 2 2 1 2 2
a slit which the largest inner angle ω =2π.
We adopt a uniform isosceles right triangulation π . We use the formula
h
ratio(λh) = lg(λλhh/2−−λλ)/lg2 and ratio(uh) = lg(kkuuhh/2−−uukk0,0∂,∂ΩΩ)/lg2 to compute
theconvergenceorderofapproximationsoflinearconformingelementtovalidate
our analysis.
By calculationwe find thatthe eigenfunction associatedwith λ is singular.
2
So in our numerical experiments we compute the approximation of the second
eigenvalueλ andthecorrespondingeigenfunctionu . Sincetheexacteigen-
2,h 2,h
pairsoftheproblem(1.1)areunknown,weusetheadaptivemethodtocompute
a high-precision approximation λ ∈ [0.89364476,0.89364690]for the L-shaped
2
domain and λ ∈[0.734554376,0.73455822]for the unit square with a slit, and
2
use them as the exact values, and the corresponding eigenfunction u is taken
as the approximation computed on the uniform mesh with the mesh diameter
h = √2 . The numerical results on the L-shaped domain and the slit domain
1024
are listed in Table 1 and Table 2, respectively.
Table 1: The results by using linear conforming element on the
L-shaped domain
h λ2,h ratio(λ2,h) ku2,h−uk0,∂Ω ratio(u2,h)
√2 0.92115806 1.40979290 0.02800065 1.12103345
8
√2 0.90400049 1.39401631 0.01287370 1.12243165
16
√2 0.89758582 1.37720866 0.00591313 1.13940548
32
√2 0.89516258 1.36395810 0.00268425 1.16362010
64
√2 0.89423511 1.35431173 0.00119822 1.20908566
128
√2 0.89387631 0.00051828
256
Table 2: The results by using linear conforming element on the unit
square with a slit
h λ2,h ratio(λ2,h) ku2,h−uk0,∂Ω ratio(u2,h)
√2 0.79372467 1.05162089 0.04741663 0.85372994
8
√2 0.76310065 1.03053027 0.02623810 0.88264565
16
√2 0.74852962 1.01703653 0.01423081 0.91530246
32
√2 0.74146094 1.00934037 0.00754564 0.95510109
64
√2 0.73798634 1.00505527 0.00389208 1.01983389
128
√2 0.73626532 0.00191947
256
For the L-shaped domain r = 2, 2r = 4. From Table 1 we can see that
3 3
the convergence order of λ is approximately equal to 2r = 4 ≈ 1.333333. It
2,h 3
alsocanbe seenfromTable 1thatthe convergenceorderofu is verycloseto
2,h
r+ 1 = 7 ≈1.166667,which is coincide with the theoretical result (3.6); while
2 6
the convergence order of u according to the previous conclusion (3.4) should
2,h
be 3r =1.
2
For the unit square with a slit r = 1. From Table 2 we can see that the
2
convergence order of λ is approximately equal to 2r = 1. We can also see
2,h
10