A Superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and non-smooth kernels

0.24 MB·English

by  Kassem Mustapha

MATHEMATICSOFCOMPUTATION
Volume00,Number0,Pages000–000
S0025-5718(XX)0000-0
A SUPERCONVERGENT DISCONTINUOUS GALERKIN
METHOD FOR VOLTERRA INTEGRO-DIFFERENTIAL
EQUATIONS, SMOOTH AND NON-SMOOTH KERNELS
3
1
0
KASSEMMUSTAPHA
2

n
a
J

8 Abstract. We study the numerical solution for Volerra integro-differential
2 equationswithsmoothandnon-smoothkernels. Weuseah-versiondiscontin-
uous Galerkin (DG) method and derive nodal error bounds that are explicit

] inthe parameters ofinterest. In thecase of non-smooth kernel, itisjustified
A that the start-up singularities can be resolved at superconvergence rates by
N using non-uniformly graded meshes. Our theoretical results are numerically
validatedinasampleoftestproblems.
.
h Integro-differential equation, weakly singular kernel, smooth kernel, DG time-
t
a stepping, error analysis, variable time steps
m
[ 1. Introduction

1 In this paper, we study the discontinuous Galerkin (DG) for a nonlocal time
v dependent Volterra integro-differential equation of the form
8
7 (1.1) u′(t)+a(t)u(t)+ u(t)=f(t), 0<t<T with u(0)=u0,
B
7
where is the Volterra operator:
6
B
. t
1
(1.2) u(t)= β(t,s)u(s)ds,
0 B
3 Z0
1 such that,
v: (1.3) β(t,s)=(t s)α−1b(s) for all 0<s<t T
− ≤
i
X with either α (0,1) (weakly singular kernel) or α N := 1,2,3, (smooth
0
∈ ∈ { ···}
r kernel). Here a, b and f are continuous real valued functions on [0,T]. We assume
a
that there exist µ > 0 such that a(t) µ for all t [0,T]. As a consequence of
∗ ∗
≥ ∈
thisandthecontinuityassumptionsonthefunctionsaandb;thereexistµ ,µ∗ >0
∗
such that
(1.4) µ a(t) µ∗ and b(t) µ∗ for all t [0,T].
∗
≤ ≤ | |≤ ∈
For any u R, problem (1.1) has a unique solution u which is continuously
0
∈
differentiable, see for example [1]. However for α (0,1), even if the functions a,
∈
b and f in (1.1)–(1.3) are smooth, the second derivative of u is not bounded at
t = 0 (see [3] and related references therein), and behaves like u′′(t) Ctα−1.
| | ≤
The singular behavior of u near t=0 may lead to suboptimal convergence rates if
SupportoftheKFUPMthroughtheprojectSB101020isgratefullyacknowledged.
(cid:13)c2006AmericanMathematicalSociety
1
2 KASSEMMUSTAPHA
we work with quasi-uniform time meshes. To overcome this problem, we employ a
family of non-uniform meshes, where the time-steps are concentrated near t=0.
Various numerical methods had been studied for problem (1.1). For instance,
collocation methods for (1.1) with a weakly singular kernel were investigated by
many authors where an O(kp+1) (k is the maximum time-step size and p is the
degree of the approximate solution) global convergencerate had been achieved us-
ing a non-uniform graded mesh of the form (2.10), see for example [1, 3, 21] and
references therein. Spectral methods and the corresponding error analysis were
providedin[7,22]assumingthatα=1andthesolutionuof(1.1)issmooth. How-
ever, for 0<α<1 (that is, the kernel is weakly singular), the spectral collocation
method were recently studied in [23] where the convergence analysis was carried
out assuming again that the solution u is smooth. For other numerical tools, refer
to [23] and references therein.
In the present paper we shall study the nodal error analysis for the DG time-
stepping method (with a fixed approximation order) applied to problem (1.1).
Indeed, the DG time-stepping method for (1.1) when α (0,1) has been intro-
∈
duced in [2], where a uniform optimal O(kp+1) convergence rate had been shown
assuming that u is sufficiently regular. In this work, we show that a faster conver-
gence than O(kp+1) is possible at the nodal points. For a weakly singular kernel
(α (0,1)), we prove that by using non-uniformly refined time-steps, start-up
∈
singularities near t = 0 can be resolved at O(kmin{p,α+1}+p+1) superconvergence
rates. Such convergence rates can not be obtained by using the approach given
in [2]. Very briefly, our proof technique will be carried out in two steps; deriving
firstthe globalconvergenceresults ofthe DG method for the dual problemof (1.1)
(whichis essentialforthe nodalerrorbut irrelevantforthe globalerrorestimates),
see Theorem 4.1. Then, we use these results with the orthogonal property of the
DG scheme for (1.1) very appropriately (see (5.1) and Theorem 5.1) to achieve
nodal superconvergence estimates. For smooth kernels (α N ), we appropriately
0
∈
modifyourearlieranalysestoshownodalsuperconvergenceratesoforderO(k2p+1)
assuming that the functions a, b and f are sufficiently regular (see Theorem 6.2).
The origins of the DG methods can be traced back to the seventies where they
hadbeenproposedasvariationalmethodsfornumericallysolvinginitial-valueprob-
lems and transport problems [10, 18, 4, 6, 8] and the references therein. In the
eighties, DG time-stepping methods were successfully applied to parabolic prob-
lems; see for example,[5], wherea nodal O(k2p+1) superconvergencerate hadbeen
proved. Subsequently,in[9],apiecewiselineartime-steppingDGmethodhadbeen
proposed and studied for a parabolic integro-differential equation:
(1.5) u +Au+ Au=f in (0,T] Ω with u(0)=v(x) on Ω for α (0,1),
t
B × ∈
where Ω Rd is a beounded convex domain, A is a linear self-adjoint, positive-
⊂
definite operator (spatial), with compact inverse, defined in D(A), and where A
dominates the spatial operator A. A nodal O(k3) superconvergence rate had been
derivedassumingthatb(s)=1in(1.3),wheretheerroranalysistherewasbasedon
the fact that on each time interveal, the DG solution takes its maximum values on
oneoftheendpoints. However,thisisnottrueinthecaseofDGmethodsofhigher
orderp. Thehighordertime-steppingDGfor(1.5)wasinvestigatedin[15]wherea
globaloptimalO(kp+1)convergenceratehadbeenproved,assumingthatthemesh
isnon-uniformlygraded. (Forothernumericalmethodsfor(1.5),see[12,14,16,17]
DGM FOR VOLETRRA INTEGRO-DIFFERENTIAL EQUATIONS 3
and related references therein.) Indeed, our convergence analysis can in principle
be extendedto coverthe nodalerrorestimates fromthe DG time-stepping method
of order p, applied to (1.5).
The outline of the paper is as follows. In Section 2, we introduce the DG
time-stepping method with a fixed approximationdegree p (typically low) on non-
uniformly refined time-steps with p 1. In Section 3, we give a globalformulation
≥
of the DG scheme, introduce our projection operator, and also provide some tech-
nicallemmas. InSection4,wedefine the dualofthe problem(1.1)andthenderive
the error estimates from the discretization by the DG method when α (0,1); see
∈
Theorem 4.1. In Section 5, we prove our main nodal error bounds. For α (0,1),
∈
an error Un u(t ) of order O(kmin{p,α+1}+p+1) (i.e., superconvergent of order
| − − n |
k3 for p = 1 and kp+2+α for p 2) has been shown provided that the solution u
≥
of(1.1) satisfies (2.7)and the meshgradingparameterγ >(p+1)/σ; see Theorem
5.1. In Section 6, we consider the case α N (in (1.3)) and thus the kernel is
0
∈
smooth. We show a nodalerroroforderO(k2p+1) (overa uniform mesh)assuming
thatthe solutionuof(1.1)is sufficiently regular,refer toTheorem6.2. We present
a series of numerical examples to validate our theoretical results in Section 7.
2. Discontinuous Galerkin time-stepping
To describe the DG method, we introduce a (possibly non-uniform) partition of
the time interval [0,T] given by the points
(2.1) 0=t <t < <t =T.
0 1 N
···
We set I =(t ,t ) and k =t t for 1 n N. The maximum step-size
n n−1 n n n n−1
− ≤ ≤
isdefinedask =max k . We nowintroducethe discontinuousfiniteelement
1≤n≤N n
space
(2.2) = v :J R : v P , 1 n N ,
Wp { N → |In ∈ p ≤ ≤ }
where J = N I , and P denotes the space of polynomials of degree p where
N ∪n=1 n p ≤
p is a positive integer 1. We denote the left-hand limit, right-hand limit and
≥
jump at t by vn =v(t−), vn =v(t+) and [v]n =vn vn, respectively.
n − n + n +− −
The DG approximation U is now obtained as follows: Given U(t) for
p
t I with 1 j n 1, the∈apWproximation U P on the next time-step I is
j p n
∈ ≤ ≤ − ∈
determined by requesting that
(2.3)
tn tn
Un−1Xn−1+ U′+a(t)U(t)+ U(t) Xdt=Un−1Xn−1+ fXdt
+ + B − +
Ztn−1h i Ztn−1
for all test functions X P . This time-stepping procedure starts from U0 = u ,
∈ p − 0
and after N steps it yields the approximate solution U for t J .
p N
∈W ∈
Remark 2.1. For the piecewise-constant case p = 0, since U′(t) = 0 and U(t) =
Un = Un−1 =: Un for t I , the DG method (2.3) amounts to a generalized
− + ∈ n
4 KASSEMMUSTAPHA
backward-Eulerscheme
Un Un−1 1 tn
− +Un a(t)dt+ω k Un
nn n
k k
n n Ztn−1
1 tn 1 tn n−1 min(t,tj)
= f(t)dt Uj (t s)α−1b(s)dsdt.
k − k −
n Ztn−1 n Ztn−1 j=1 Ztj−1
X
In this case, the nodal and global errors have the same rate of convergence which
is O(k), see [2, Theorem 3.8].
For our error analysis,it will be convenient to reformulate the DG scheme (2.3)
in terms of the global bilinear form
(2.4)
N−1 N tn
G (U,X)=U0 X0 + [U]nXn+ U′(t)+a(t)U(t)+ U(t) Xdt.
N + + + B
nX=1 nX=1Ztn−1h i
By summing up (2.3) over all the time-steps and using U0 = u , the DG method
− 0
can now equivalently be written as: Find U such that
p
∈W
tN
(2.5) G (U,X)=u X0 + fXdt X .
N 0 + ∀ ∈Wp
Z0
Since the solution u is continuous, it follows that
tN
G (u,X)=u X0 + fXdt X .
N 0 + ∀ ∈Wp
Z0
Thus, the following Galerkin orthogonality property holds:
(2.6) G (U u,X)=0 X .
N p
− ∀ ∈W
Before stating the regularity property of the solution u of (1.1), we display in the
next remark an alternative form of G which will be used in our error analysis.
N
Remark 2.2. Integrationbypartsyieldsthefollowingalternativeexpressionforthe
bilinear form G in (2.4):
N
N−1
G (U,X)=UNXN Un[X]n
N − − − −
n=1
X
N tn
+ [ U(t)X′+a(t)U(t)X + U(t)X] dt.
− B
n=1Ztn−1
X
Throughout the paper, we assume that the solution u of (1.1) satisfies:
(2.7) u(j)(t) Ctσ−j for 1 j p+1 where 1 σ α+1
| |≤ ≤ ≤ ≤ ≤
where the constant C depends on j. For instance, if in (1.1) the function f =
tκ1f +tκ2f forsomeκ , κ 0andthefunctionsa, b, f andf areinCj−1[0,T]
1 2 1 2 1 2
≥
for 1 j p, then (2.7) holds for σ = 1+min κ ,κ ,α , see [1, Section 7.1] for
1 2
≤ ≤ { }
more details.
We notice from (2.7) that u(j)(t) is not bounded near t = 0 for j 2. Hence,
| | ≥
to compensate the singular behavior of u near t = 0, we employ a family of non-
uniformmeshes,wherethetime-stepsareconcentratednearzero. Thus,weassume
DGM FOR VOLETRRA INTEGRO-DIFFERENTIAL EQUATIONS 5
that, for a fixed γ 1,
≥
(2.8) k C kt1−1/γ and t C t for 2 n N,
n ≤ γ n n ≤ γ n−1 ≤ ≤
with
(2.9) c kγ k C kγ.
γ 1 γ
≤ ≤
For instance, one may choose
(2.10) t =(n/N)γT for 0 n N.
n
≤ ≤
Under the assumptions (2.7)–(2.9), we show in Theorem 5.1 that the error Un
| −−
u(t ) isoforderkγσ+min{p,1+α},for1 n N. So,wehaveasuperconvergenceof
n
| ≤ ≤
order kp+1+min{p,1+α} provided γ >(p+1)/σ. However, for a quasi-uniform mesh
(i.e., γ =1) our bound yields a poorer convergence rate of order kσ+min{p,1+α}.
3. Projection operator and technical lemmas
In this section we introduce a projection operator that has been used various
times in the analysis of DG time-stepping methods; see [24], and state some pre-
liminary results that are needed in our convergence analysis in the forthcoming
sections.
For a given function u C[0,T], we define the interpolant Π−u by
p
∈ ∈W
tn
(3.1) Π−u(t−)=u(t ) and (u Π−u)vdt=0 v P (I )
n n − ∀ ∈ p−1 n
Ztn−1
and for 1 n N. From [19, Lemma 3.2] it follows that Π− is well-defined.
≤ ≤
To state the approximation properties of Π−, we introduce the notation
φ = sup φ(t) for any φ C(t ,t ).
k kIn | | ∈ n−1 n
t∈In
Theorem 3.1. There exists a constant C, which depends on p such that:
(i) For any 0 q p and u Hq+1(I ), there holds
≤ ≤ |In ∈ n
tn tn
Π−u u2dt Ck2q+2 u(q+1) 2dt for 1 n N.
| − | ≤ n | | ≤ ≤
Ztn−1 Ztn−1
(ii) For any 0 q p and u Hq+1(I ) C(I ), there holds
≤ ≤ |In ∈ n ∩ n
tn
Π−u u 2 Ck2q+1 u(q+1) 2dt for 1 n N.
k − kIn ≤ n | | ≤ ≤
Ztn−1
Proof. For the proof of the first bound, we refer to [19, Section 3] or [24, Chapter
12, Page 214]. For the second bound, see [20, Theorem 3.9 and Corollary 3.10] or
[24, Equation (12.10)]. (cid:3)
The following two technical lemmas are needed in our derivation of the error
estimates. The first lemma has been proved in [9, Lemma 6.3].
Lemma 3.2. If g L (0,T) and α (0,1) then
2
∈ ∈
T t 2 Tα T t
(t s)α−1g(s)ds dt (T t)α−1 g2(s)dsdt.
− ≤ α −
Z0 (cid:18)Z0 (cid:19) Z0 Z0
The next lemma is the following Gronwall inequality; see [9, Lemma 6.4].
6 KASSEMMUSTAPHA
Lemma 3.3. Let a N and b N be sequences of non-negative numbers with
{ j}j=1 { j}j=1
0 b b b . Assume that there exists a constant K 0 such that
1 2 N
≤ ≤ ≤···≤ ≥
n tj
a b +K a (t t)α−1dt for 1 n N and α (0,1).
n n j n
≤ − ≤ ≤ ∈
j=1 Ztj−1
X
Assume further that δ = Kkα < 1. Then for n = 1, ,N, we have a Cb
α ··· n ≤ n
where C is a constant that solely depends on K, T, α and δ.
Throughout the rest of the paper, we shall always implicitly assume that the
maximum step-size k is sufficiently small sothatthe conditionδ <1inLemma3.3
is satisfied. More precisely, following Lemma 4.2, we shall require that
µ∗ 2
4Tα kα <1.
αµ
(cid:18) ∗(cid:19)
4. Error analysis of the dual problem
This section is devoted to deriving error estimates for the DG method applied
to the dual problem of the Volterra integro-differential equation (1.1). The main
results ofthis section(more precisely,Theorem4.1)play a crucialrolein the proof
of the superconvergence error estimate in section 5.
Let z be the solution of the dual problem
(4.1) z′+a(t)z(t)+ ∗z(t)=0 for 0 t<T, with z(T)=z ,
T
− B ≤
where ∗v(t)= T β(s,t)v(s)ds ( ∗ is the dual of the integral operator ).
B t B B
Since z has no jumps and since
R
T
v(t)z′(t)+a(t)v(t)z(t)+ v(t)z(t) dt
− B
Z0
(cid:2) T (cid:3)
= v(t)( z′(t)+a(t)z(t)+ ∗z(t))dt=0,
− B
Z0
the alternative expression of G given in Remark 2.2 yields the identity
N
(4.2) G (v,z)=vNz forall v C[0,T].
N − T ∈
(C(0,T] denotes the space of continuous functions on [0,T]). Let Z denote
p
∈ W
the approximate solution of (4.1) given by
(4.3) G (V,Z)=VNz V .
N − T ∀ ∈Wp
Hence, the following Galerkin orthogonality property holds:
(4.4) G (V,Z z)=0 V .
N p
− ∀ ∈W
At this stage, the main aim is to estimate the error Z z in the L -norm. First
2
−
it is good to notice that (4.4) is a discrete backward analogue of (2.6). Since it is
moreconvenienttodealwithadiscreteforwardproblem,weintroducethefunctions
z˜(t)=z(t t) and Z˜(t)=Z(t t) and then, (4.4) can be rewritten as;
N N
− −
(4.5) G˜ (Z˜ z˜,V)=0 V ;
N p
− ∀ ∈W
where G˜ is defined as in (2.4) but with a˜(t) := a(t t) in place of a(t) and
N fN −
β(t s,t t) in place of β(t,s). The finite dimensional space is defined as
N N p
− − W
f
DGM FOR VOLETRRA INTEGRO-DIFFERENTIAL EQUATIONS 7
but on the reverse mesh: 0 = t˜ < t˜ < < t˜ , where t˜ = t˜ +k˜ with
p 0 1 N i i−1 i
W ···
k˜ =k .
i N+1−i
Setting ζ = Π˜−z˜ z˜ and θ = Z˜ Π˜−z˜ where Π˜− is the interpolant operator
− −
defined as in (3.1), but on the reverse mesh. Then (4.5) implies that
(4.6) G˜ (θ,V)= G˜ (ζ,V) V .
N N p
− ∀ ∈W
By the construction of the interpolant we have ζ(t˜n) = 0 for all n 1 and hence,
− ≥
usingthealternativeexpressionforG giveninRemark2.2and t˜n ζ(t)V′(t)dt=
N t˜n−1
0 (by definition of the operator Π−),
R
N t˜n
(4.7) G˜ (ζ,V)= a˜(t)ζ(t)V(t)+ ˜ζ(t)V(t) dt
N
n=1Zt˜n−1 B
X (cid:2) (cid:3)
where
t
˜ζ(t)= β(t s,t t)ζ(s)ds.
N N
B − −
Z0
In the next theorem we estimate the error between z and Z.
Theorem 4.1. If z is the solution of the backward VIE (4.1), and if Z is
p
∈ W
the approximate solution defined by (4.3), then
tN
z Z 2dt Ck2α+2 z 2
T
| − | ≤ | |
Z0
provided that
tN tN
(4.8) θ(t)2dt C ζ(t)2dt.
| | ≤ | |
Z0 Z0
Proof. From the decomposition: Z˜ z˜= ζ +θ, the triangle inequality, and (4.8),
−
we have
tN tN tN
(4.9) z Z 2dt= z˜ Z˜ 2dt C ζ 2dt.
| − | | − | ≤ | |
Z0 Z0 Z0
Thus, the task reduces to bound the right-hand side of (4.9). Starting from the
relationz˜(t)=z(t t) and recalling that z satisfies (4.1), it is clear that z˜solves
N
−
the VIE:
t
z˜′+a(t t)z˜(t)+ β(t s,t t)z˜(s)ds=0 for 0<t<T,
N N N
− − −
Z0
with z˜(0) = z . Hence, an application of (2.7) for σ = α+1 with z˜ in place of u
T
gives
(4.10) z˜′(t) +t1−α z˜′′(t) +t2−α z˜′′′(t) C z .
T
| | | | | |≤ | |
8 KASSEMMUSTAPHA
Now, Theorem 3.1 on the reverse mesh (with ζ in place of Π−u u) and (4.10)
−
yield
(4.11)
N t˜n N t˜n N t˜n
ζ(t)2dt C k˜4 z˜′′(t)2dt C k˜4 t2α−2 z 2dt
n=2Zt˜n−1| | ≤ n=2 nZt˜n−1| | ≤ n=2 nZt˜n−1 | T|
X X X
N N
C z 2 k˜5t˜2α−2 =C z 2 k˜3+2α(k˜ /t˜ )2−2α
≤ | T| n n−1 | T| n n n−1
n=2 n=2
X X
N
C z 2 k˜3+2α Ck2α+2 z 2
≤ | T| n ≤ | T|
n=2
X
and on (0,t˜), we notice for 1/2<α 1 that
1
≤
t˜1 t˜1 t˜1
ζ(t)2dt Ck˜4 z˜′′(t)2dt Ck4 t2α−2 z 2dt C z 2k3+2α,
| | ≤ 1 | | ≤ N | T| ≤ | T| N
Z0 Z0 Z0
and for 0<α 1/2 that
≤
t˜1 t˜1 t˜1
ζ(t)2dt Ck˜2 z˜′(t)2dt Ck2 z 2dt
(4.12) | | ≤ 1 | | ≤ N | T|
Z0 Z0 Z0
C z 2k3 C z 2k2+2α.
≤ | T| N ≤ | T| N
Finally, combine (4.9) and (4.11)–(4.12), we obtain the desired result. (cid:3)
In the next lemma we prove the applicability of the assumption (4.8).
Lemma 4.2. For 1 n N, we have
≤ ≤
t˜n t˜n
θ(t)2dt C ζ(t)2dt
| | ≤ | |
Z0 Z0
Proof. Choosing V =θ on (0,t˜ ) and zero elsewhere in (4.6) and (4.7), then using
n
the alternative definition of G in Remark 2.2 and θ′θ =(d/dt)θ 2/2, we observe
N
| |
that
n−1 t˜n
θ(t˜n)2+ θ(t˜+)2+ [θ]j 2+2 a˜(t)θ(t)2dt
| − | | 0 | | | | |
j=1 Z0
X
t˜n
= 2 a˜(t)ζ(t)+B˜ζ(t)+ ˜θ(t) θ(t)dt.
− B
Z0 h i
So
t˜n t˜n t˜n
a˜(t)θ(t)2dt a˜(t)θ(t) ζ(t) dt+ ˜ζ(t)+ ˜θ(t) θ(t)dt.
| | ≤ | || | |B B || |
Z0 Z0 Z0
We use the geometric-arithmetic mean inequality xy εx2 + y2 (valid for any
| | ≤ 2 2ε
ε>0) we find that
t˜n t˜n
a˜(t)θ(t) ζ(t) dt √µ∗ a˜(t)θ(t) ζ(t) dt
| || | ≤ | || |
Z0 Z0
p 1 t˜n t˜n
a˜(t)θ(t)2dt+µ∗ ζ(t)2dt
≤ 4 | | | |
Z0 Z0
DGM FOR VOLETRRA INTEGRO-DIFFERENTIAL EQUATIONS 9
and thus
3 t˜n t˜n t˜n
(4.13) a˜(t)θ(t)2dt µ∗ ζ(t)2dt + ˜ζ(t) + ˜θ(t) θ(t)dt.
4 | | ≤ | | |B B || |
Z0 Z0 Z0
WeemploytheCauchy-Schwarzinequality,againthegeometric-arithmeticmean
inequality, and Lemma 3.2 (with T =t˜ ):
n
t˜n t˜n t
ζ(t)θ(t)dt µ∗ (t s)α−1 ζ(s) θ(t) dsdt
|B | ≤ − | || |
Z0 Z0 Z0
µ∗ t˜n t
a˜(t)1/2 θ(t) (t s)α−1a˜(s)1/2 ζ(s) dsdt
≤ µ | | − | |
∗ Z0 Z0
µ∗ t˜n t 2 1/2 t˜n 1/2
(t s)α−1a˜(s)1/2 ζ(s) ds dt a˜(t)θ(t)2dt
≤ µ∗ Z0 (cid:18)Z0 − | | (cid:19) ! Z0 | | !
µ∗ 2 t˜n t 2 1 t˜n
(t s)α−1a˜(s)1/2 ζ(s) ds dt+ a˜(t)θ(t)2dt
≤ µ − | | 4 | |
(cid:18) ∗(cid:19) Z0 (cid:18)Z0 (cid:19) Z0
t˜α µ∗ 2 t˜n t 1 t˜n
n (t˜ t)α−1 a˜(s)ζ(s)2dsdt+ a˜(t)θ(t)2dt
n
≤ α µ − | | 4 | |
(cid:18) ∗(cid:19) Z0 Z0 Z0
t˜αµ∗ 2 t˜n 1 t˜n
n a˜(s)ζ(s)2ds+ a˜(t)θ(t)2dt.
≤ αµ | | 4 | |
(cid:18) ∗(cid:19) Z0 Z0
Similarly, we notice that
t˜n
θ(t)θ(t)dt
|B |
Z0
t˜α µ∗ 2 t˜n t 1 t˜n
n (t˜ t)α−1 a˜(s)θ(s)2dsdt+ a˜(t)θ(t)2dt.
n
≤ α µ − | | 4 | |
(cid:18) ∗(cid:19) Z0 Z0 Z0
Inserting the above bounds in (4.13) implies that
t˜n
a˜(t)θ(t)2dt
| |
Z0
t˜n t˜α µ∗ 2 n t˜j t˜j
C ζ(t)2dt+4 n (t˜ t)α−1dt a˜(t)θ(t)2dt.
n
≤ Z0 | | α (cid:18)µ∗(cid:19) j=1Zt˜j−1 − Z0 | |
X
Therefore,thedesiredresultnowimmediatelyfollowsafterapplyingoftheGronwall
inequality inLemma 3.3 andusing the assumption(1.4) onthe function a˜ (instead
of a). (cid:3)
5. Superconvergence results
In this section, we study the nodal error analysis of the DG solution U defined
by (2.3) with U0 = u . We derive error estimate of the DG solution, giving rise
− 0
to superconvergencealgebraicrates. Our analysispartiallyreliesonthe techniques
introduced in [24, Chapter 12] for parabolic problems.
Theorem 5.1. Let α (0,1) in (1.3). Let the solution u of problem (1.1) satisfy
∈
theregularityproperty (2.7)andletU betheDGapproximatesolutiondefined
p
∈W
10 KASSEMMUSTAPHA
by (2.3)withp 1. Inaddition tothemeshassumption (2.8)and (2.9), weassume
≥
that k k for 1 n N. Then
n n−1
≥ ≤ ≤
for p=1,
•
kγσ, 1 γ 2/σ
max Un u(t ) Ck ≤ ≤
1≤n≤N| −− n |≤ ×(k2, γ 2/σ
≥
and for p 2, we have
• ≥
kγσ, 1 γ (p+1)/σ
max Un u(t ) Cmax 1,logn kα+1 ≤ ≤
1≤n≤N| −− n |≤ { } ×(kp+1, γ (p+1)/σ.
≥
Proof. From(4.3),(4.2),(2.6)and(4.4)(recallthatη =Π−u u),weobservethat
−
(UN u(t ))z =G (U,Z) G (u,z)
(5.1) − − N T N − N
=G (u,Z z)=G (η,z Z).
N N
− −
The alternative expression for G given in Remark 2.2 and the equality η(tn)=0
N −
show that
(5.2) G (η,z Z)=δ +δ ,
N 1N 2N
−
where
N tn tN
δ = η(z Z)′dt and δ = (a(t)η(t)+ η(t))(z Z)(t)dt.
1N 2N
− − B −
j=1Ztn−1 Z0
X
Toboundδ andδ ,westartfromtheregularityproperty(4.10)andtherelation
1N 2N
z(t)=z˜(t t), and get
N
−
(5.3) z′(t) +(t t)1−α z′′(t) +(t t)2−α z′′′(t) C z .
N N T
| | − | | − | |≤ | |
For p=1, the orthogonality property of Π− yields
N tn N tn
δ = η(t)z′(t)dt= η(t)[z′(t) z′(t )]dt
1N n
− −
j=1Ztn−1 j=1Ztn−1
X X
N tn tn
= η(t) z′′(s)dsdt
j=1Ztn−1 Zt
X
and hence, with the help of (5.3) we have
N tn
δ C η k z′′(t) dt
| 1N|≤ k kJN n | |
(5.4) n=1 Ztn−1
X
tN
Ck η (t t)α−1 z dt=Ck η z tα/α.
≤ k kJN N − | T| k kJN| T| N
Z0
For p 2, again the orthogonality property of Π− gives
≥
N tn N−1 tn
δ = η(t)z′(t)dt= η(t)[Π+z′(t) z′(t)]dt
1N
− −
j=1Ztn−1 j=1 Ztn−1
X X
tN tn
+ η(t) z′′(s)dsdt
ZtN−1 Zt