Superconvergence of the direct discontinuous Galerkin method for a time-fractional initial-boundary value problem Chaobao Huang
∗
Martin Stynes
†
October 9, 2018
Abstract A time-fractional reaction-diffusion initial-boundary value problem with periodic boundary condition is considered on Q := Ω × [0, T ], where Ω is the interval [0, l]. Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2. In the temporal direction we use the L1 approximation of the Caputo deriative on a suitably graded mesh. We prove that at each time level of the mesh, our L1-DDG solution is superconvergent of order k + 2 in L2 (Ω) to a particular projection of the exact solution. Moreover, the L1-DDG solution achieves superconvergence of order (k + 2) in a discrete L2 (Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H 1 (Q) seminorm at the Gauss points; numerical results show that these results are sharp.
Keywords: DDG method, Fractional reaction-diffusion equation, Periodic boundary condition, Superconvergence, Gauss points, Lobatto points
1
Introduction
Fractional derivatives appear in many recent models of physical processes, so the construction and analysis of numerical methods for their solutions is currently of great interest. In particular, initial-boundary value problems whose time derivative is fractional have received much attention. In this paper we consider such a problem, with a Caputo time derivative of order α ∈ (0, 1). To solve it numerically, we shall use the well-known L1 discretisation in time, combined with a DDG (direct discontinuous Galerkin) method in space. Our analysis appears to be the first superconvergence result for the DDG in a fractional-derivative problem. For 0 < α < 1, fixed T > 0, and Ω = (0, l), we shall examine the time-fractional initialboundary value problem Dtα u −
∂2u + c(x)u = f (x, t) ∀ (x, t) ∈ Q := Ω × (0, T ], ∂x2 u(x, 0) = u0 (x) for x ∈ Ω,
∗
(1a) (1b)
Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China;
[email protected] † Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China;
[email protected] The research of this author is supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF-U1530401. Corresponding author.
1
subject to the periodic boundary conditions u(0, t) = u(l, t) and ux (0, t) = ux (l, t) for 0 < t ≤ T.
(1c)
In (1a), Dtα is the standard Caputo fractional derivative operator, which is defined for all functions v(x, ·) that are absolutely continuous on [0, T ] by Dtα v(x, t)
1 = Γ(1 − α)
Z 0
t
(t − s)−α
∂v(x, s) ds. ∂s
(2)
¯ = [0, l] and Q ¯ := [0, l] × [0, T ]. We assume that c, u0 ∈ C(Ω) ¯ with c ≥ 0, and f ∈ C(Q). ¯ Set Ω Further regularity assumptions will be made later for these functions. Existence and uniqueness of a solution to (1) will follow from the analysis in Section 2. Problems similar to (1) have been considered in many numerical analysis papers; see [8, 9, 15] and their references. The structure of our paper is as follows. Section 2 discusses the solution of the initialboundary value problem (1). The L1-DDG numerical method and its basic properties are presented in Section 3. The heart of our paper is Section 4. In Section 4.1 we present some technical preliminaries including a novel Gronwall inequality, then in Section 4.2 we obtain our main results: at each discrete time level, one has supercloseness of our L1-DDG computed solution in L2 (Ω) to a particular projection of the exact solution, and consequently superconvergence of the solution in a discrete L2 (Q) norm computed at the Lobatto points and in a discrete H 1 (Q) seminorm at the Gauss points. Section 5 is a digression from our main study; in it we give a new shorter proof of a superconvergence result from [3] for a classical (integer-derivative) application of the DDG method. Finally, numerical results in Section 6 show that these superconvergence results at the Lobatto and Gauss points are sharp. Notation. We use C to denote a generic constant that is independent of the mesh; it can take different values in different places. The notation A . B indicates that A ≤ CB. Let N = {1, 2, 3, . . .} and N0 = {0, 1, 2, . . .}. We write k · k for the norm in L2 (Ω) and (·, ·) for its associated inner product. For each m ∈ N, the notation H m (Ω) is used for the standard Sobolev space with its associated norm k · km and seminorm | · |m . Furthermore, we write k · kL∞ (H m ) = sup0 k 1 − β1 (k − 1) + (k − 1) . (14) 3 Define a discrete energy norm k · kE by ( )1/2 M X β 0 kvkE = |v|21 + [v]2 ∀ v ∈ H 1 (Ω). h m m=1
Then as in [11, eq. (24)], one can use (14) to show that there exists γ ∈ (0, 1) such that A(vh , vh ) ≥ γkvh k2E ∀ vh ∈ Vh .
(15)
To prove that the L1-DDG method is well posed, one must show that for each n ≥ 1 the equation (13) has a unique solution unh ∈ V h . Equivalently, we prove that if f = u0h = u1h = · · · = un−1 = 0, then one must have the unique solution unh = 0 in (13). h α un , it Take vh = unh in (13). From c ≥ 0, inequality (15) and the explicit formula (9) for DN h follows that n−1
X dn,n dn,1 1 kunh k2 +γkunh k2E ≤ (f, unh )+ (u0h , unh )− (dn,i+1 −dn,i )(uhn−i , unh ) = 0 Γ(2 − α) Γ(2 − α) Γ(2 − α) i=1
since f = u0h = u1h = · · · = un−1 = 0. Hence unh = 0, as desired. h
4
Superconvergence of the computed solution
Our analysis imitates the approach of [3] for a classical (non-fractional) initial-boundary value problem: first the supercloseness of unh to the Gauss-Lobatto projection Ih u(·, tn ) of the exact solution u(·, tn ) is shown, and from this the superconvergence of unh to u(·, tn ) at the Gauss and Lobatto points follows. 6
4.1
Technical preliminary results
Recall the special projector Ph into Vh that is defined in [11, (29a)–(29(c)]: given a smooth function w, define Ph w ∈ Vh by Z (Ph w − w)v dx = 0 ∀ v ∈ P k−2 (Im ), (16a) Im
∂x\ (Ph w) := β0 h−1 [Ph w] + {∂x (Ph w)} + β1 h[∂x2 (Ph w)] xm = ∂x w(xm ), {Ph w} xm = w(xm ),
(16b) (16c)
where m = 1, 2, . . . , M in (16a) and m = 0, 1, . . . , M − 1 in (16b) and (16c). (For a piecewise smooth function w with w|Im ∈ H k+1 (Im ), the above definition should be modified by replacing the right-hand sides of (16b) and (16c) by w cx and {w}, respectively.) By [11, Lemma 4.1] the projection Ph w is well defined since (14) is satisfied. Its approximation properties are presented next. Lemma 4.1. [11, Lemma 4.2] If w ∈ H k+1 (Ω), then M X
kPh w − wk20,Im ≤ Ch2k+2 |w|2k+1,Ω
and
m=1
M X
|Ph w − w|2xm ≤ Ch2k+1 |w|2k+1,Ω ,
m=1
where C is a constant independent of h and w. Following [3], define the Gauss-Lobatto projector Ih by k X Ih v(x) Im := vm,j φm,j (x) for m = 1, 2, . . . , M, j=0
where − vm,0 := v(x+ m−1 ), vm,1 := v(xm ), vm,j :=
2j − 1 2
Z
v 0 (x)Lm,j−1 (x) dx for j = 2, 3 . . . ,
Im
and Lm,j and φm,j are, respectively, the Legendre and Lobatto polynomials of degree j on the interval Im . The superconvergence relationship between Ih v and Ph v can now be stated. Lemma 4.2. [3, Lemma 3.2] Let v ∈ H k+2 (Ω). If β1 = 1/ (2k(k + 1)), then kPh v − Ih vk . hk+2 kvkk+2 . We shall also need a stability result from [15] in our superconvergence analysis. As in [15], define the positive real numbers θn,j , for n = 1, 2, . . . , N and j = 1, 2, . . . , n − 1, by θn,n = 1,
θn,j =
n−j X
α τn−i (dn,i − dn,i+1 )θn−i,j .
(17)
i=1
Observe that (10) implies θn,j > 0 for all n, j. The weighted sum of the θn,j in the next result will be needed later. 7
Lemma 4.3. [15, Lemma 4.3] Let the parameter η satisfy η ≤ rα. Then for n = 1, 2, . . . , N , one has n X T α N −η τnα j −η θn,j ≤ . 1−α j=1
Now we prove a nonstandard Gronwall inequality, which is inspired by a related result in [13, Theorem 2.3]. n ∞ Lemma 4.4. Assume that the sequences {ξ n }∞ n=1 , {η }n=1 are nonnegative and the grid function n 0 { v : n = 0, 1, . . . , N } satisfies v ≥ 0 and α n n (DN v )v ≤ ξ n v n + (η n )2 for n = 1, 2, . . . , N.
(18)
Then n
0
v ≤v +
τnα Γ(2
− α)
n X
r j
θn,j ξ +
j=1
where we obey the standard conventions that
T α Γ(2 − α) max η j for n = 0, 1, . . . , N, 1≤j≤n 1−α P0
1···
(19)
= 0 and max1≤j≤0 · · · = 0.
Proof. For n = 1, . . . , N , set η∗n :=
p
ηn g n := ξ n + p . Γ(2 − α)T α /(1 − α)
Γ(2 − α)T α /(1 − α) max η j , 1≤j≤n
Our proof uses induction on n. When n = 0, the result is clearly true. Fix k ∈ {1, . . . , N }. Assume that (19) is valid for n = 0, 1, . . . , k − 1. If v k ≤ η∗k , then as v 0 , θn,j and ξj are all non-negative, the result for n = k follows immediately. Otherwise v k > η∗k , which implies p k v > Γ(2 − α)T α /(1 − α)η k ≥ 0 and the inequality (18) now gives us vk α k k (DN v )v ≤ ξ k v k + η k p , Γ(2 − α)T α /(1 − α) whence
ηk α k DN v ≤ ξk + p = gk , Γ(2 − α)T α /(1 − α)
i.e., k−1
X dk,1 dk,k 1 vk − v0 + (dk,i+1 − dk,i )v k−i ≤ g k . Γ(2 − α) Γ(2 − α) Γ(2 − α) i=1
This is equivalent to " k
v ≤
τkα
k
0
Γ(2 − α)g + dk,k v +
k−1 X i=1
8
# (dk,i − dk,i+1 )v
k−i
.
Combining this inequality with the inductive hypothesis yields n v k ≤ τkα Γ(2 − α)g k + dk,k v 0 r k−1 k−i α Γ(2 − α) X X T α + (dk,i − dk,i+1 ) v 0 + τk−i max η j Γ(2 − α) θk−i,j ξ j + 1≤j≤k−i 1−α i=1
j=1
k−1 k−i X X α ≤ τkα Γ(2 − α)ξ k + dk,k v 0 + (dk,i − dk,i+1 ) v 0 + τk−i Γ(2 − α) θk−i,j ξ j i=1
j=1
k−1 X (dk,i − dk,i+1 )
r
ηk
T α Γ(2 − α) max η j α 1≤j≤k−i 1 − α Γ(2 − α)T /(1 − α) i=1 k−1 k−i X X α ≤ τkα Γ(2 − α)ξ k + dk,1 v 0 + Γ(2 − α) τk−i (dk,i − dk,i+1 ) θk−i,j ξ j + τkα Γ(2 − α) p
+ τkα
i=1
r
j=1
k−1
r
T α Γ(2 − α) max η j 1≤j≤k 1−α i=1 k−j k−1 X X α (dk,i − dk,i+1 )θk−i,j ξj τk−i ≤ τkα Γ(2 − α)ξ k + dk,1 v 0 + Γ(2 − α) +
1 τkα
−α Tα
X T α Γ(2 − α) k η + τkα (dk,i − dk,i+1 ) 1−α
j=1
i=1
r T α Γ(2 − α) k T α Γ(2 − α) α + η + τk (dk,1 − dk,k ) max η j 1≤j≤k 1−α 1−α r k X T α Γ(2 − α) max η j ≤ v 0 + τkα Γ(2 − α) θk,j ξ j + 1≤j≤k 1−α j=1 r α T Γ(2 − α) α 1−α − dk,k max η j , + τk α 1≤j≤k T 1−α 1 τkα
−α Tα
r
(20)
where we used (17). To complete the inductive step, we show that (1 − α)/T α − dk,k ≤ 0. For T 1−α k r(1−α) − (k r − 1)1−α t1−α − (tk − t1 )1−α k dk,k = = τ1 N r(1−α) T N −r N rα (1 − α) k r(1−α) − (k r − 1)1−α = · Tα 1−α rα N (1 − α) r −α (1 − α) N rα ≥ (k ) = Tα Tα k 1−α ≥ . (21) Tα Combining (20) with (21) yields (19) for n = k. By the principle of induction, the lemma is proved. For any integrable function v, define the integral projector Dx−1 by Z x −1 v ds. Dx v(x) Im := xm
9
From [3, (3.15) and (3.17)] we obtain the useful properties kDx−1 vk . hkvk, Dx−1 (v
4.2
−
Ph v)(x− m+1 )
=
Dx−1 (v
(22a) −
Ph v)(x− m)
= 0 for m ∈ {1, 2, . . . , M } and k ≥ 2.
(22b)
Superconvergence analysis
This section contains our main results: superconvergence bounds for kuh − Ih uk and kuh − Ph uk at each discrete time level are derived in Theorem 4.5, then Corollary 4.6 gives superconvergence results for u − uh in two discrete L2 (Q) norms that are computed by summing over the Gauss and Lobatto points, respectively, at all discrete time levels. Theorem 4.5. Assume that k ≥ 2 and that ku0 kk+3 , kukL∞ (H k+1 ) and kDtα ukL∞ (H k+1 ) are finite. Assume also that the bound (5b) on the time derivatives of u is valid. Choose β1 = 1/ (2k(k + 1)). Let un and unh be the solutions of (6) and (11), respectively. Suppose that the computed initial value uh (x, 0) satisfies kuh (·, 0) − Ih u0 k . hk+2 ku0 kk+3 . Then kunh −Ph un k+kunh −Ih un k . N − min{2−α,rα} +hk+2 ku0 kk+3 + kukL∞ (H k+1 ) + kDtα ukL∞ (H k+1 ) . for n = 1, 2, . . . , N . Proof. Let n ∈ {1, . . . , N }. Set en = unh − Ph un and εn = un − Ph un . From (6) and (11) one has the error equation α n α n (DN e , vh ) + A(en , vh ) + (cen , vh ) = (DN ε , vh ) + A(εn , vh ) + (cεn , vh ) + (ϕn , vh ) ∀ vh ∈ Vh .
Choosing vh = en here, then invoking (15) and c ≥ 0, we obtain α n n α n n ε , e ) + A(εn , e) + (cεn , en ) + (ϕn , en ). (DN e , e ) + γken k2E ≤ (DN
But [3, (3.25)] gives A(εn , vh ) = 0 for all vh ∈ Vh . Thus α n n α n n (DN e , e ) + γken k2E ≤ (DN ε , e ) + (cεn , en ) + (ϕn , en ) α n = −(Dx−1 DN ε , ∂x en ) − (Dx−1 cεn , ∂x en ) + (ϕn , en )
≤
1 α n 2 kDx−1 DN ε k + kDx−1 cεn k2 + γ|en |21 + kϕn k ken k, 2γ
after integrating by parts on each mesh interval using (22b), then applying Cauchy-Schwarz inequalities. Hence (22a) and Lemma 4.1 give us α n n (DN e ,e ) ≤
1 2k+4 α h kDN u(·, tn )k2k+1 + ku(·, tn )k2k+1 + kϕn k ken k 2γ
1 2k+4 h kDtα u(·, tn )k2k+1 + kϕn k2 + ku(·, tn )k2k+1 + kϕn k ken k 2γ 1 2k+4 α h kDt u(·, tn )k2k+1 + n−2 min{2−α,rα} + ku(·, tn )k2k+1 + kϕn k ken k . 2γ .
.
1 2k+4 h kDtα u(·, tn )k2k+1 + 1 + ku(·, tn )k2k+1 + kϕn k ken k 2γ 10
(23)
where we used Lemma 3.1 and the definition of ϕn . But n−1
α n n (DN e ,e ) =
X dn,1 dn,n 1 (en , en ) − (e0 , en ) + (dn,i+1 − dn,i )(en−i , en ) Γ(2 − α) Γ(2 − α) Γ(2 − α) i=1
n−1
X dn,1 dn,n 1 ≥ ken k2 − ke0 k ken k + (dn,i+1 − dn,i )ken−i k ken k Γ(2 − α) Γ(2 − α) Γ(2 − α) i=1
α = (DN ken k) ken k.
Combining this with (23) yields α (DN ken k) ken k . h2(k+2) kDtα u(·, tn )k2k+1 + ku(·, tn )k2k+1 + 1 + kϕn k ken k for n = 1, . . . , N. Now apply Lemma 4.4. Recalling Lemmas 3.1 and 4.3, for n = 1, . . . , N one has ken k . ke0 k + τnα Γ(2 − α)
n X
θn,j kϕj k
j=1
r +
T α Γ(2
h i − α) max hk+2 (kDtα u(·, tj )kk+1 + ku(·, tj )kk+1 + 1) 1≤j≤n 1−α
0
. ke k + Γ(2 −
α)τnα
n X
θn,j j − min{2−α,rα}
j=1
r + . ke0 k + r +
h i T α Γ(2 − α) max hk+2 (kDtα u(·, tj )kk+1 + ku(·, tj )kk+1 + 1) 1≤j≤n 1−α Γ(2 − α)T α − min{2−α,rα} N 1−α h i T α Γ(2 − α) max hk+2 (kDtα u(·, tj )kk+1 + ku(·, tj )kk+1 + 1) . 1≤j≤n 1−α
Here ke0 k = kuh (·, 0) − Ph u0 k ≤ kuh (·, 0) − Ih u0 k + kIh u0 − Ph u0 k . hk+2 ku0 kk+3 , using the hypothesis on kuh (·, 0) − Ih u0 k and Lemma 4.2. Consequently Γ(2 − α)T α − min{2−α,rα} ken k . hk+2 ku0 kk+3 + N 1−α r h i T α Γ(2 − α) + max hk+2 (kDtα u(·, tj )kk+1 + ku(·, tj )kk+1 + 1) 1≤j≤n 1−α . N − min{2−α,rα} + hk+2 ku0 kk+3 + kukL∞ (H k+1 ) + kDtα ukL∞ (H k+1 ) . Combining this bound with Lemma 4.2 yields kunh − Ih un k ≤ ken k + kPh un − Ih un k . N − min{2−α,rα} + hk+2 ku0 kk+3 + kukL∞ (H k+1 ) + kDtα ukL∞ (H k+1 ) . Thus the theorem is proved. 11
In the statement and proof of Theorem 4.5, the term N − min{2−α,rα} is multiplied by a factor that depends on the time derivatives of u and is guaranteed to be finite by the hypothesis that the bound (5b) is valid. Whether the norms kukL∞ (H k+1 ) and kDtα ukL∞ (H k+1 ) are finite depends on the smoothness of the initial data u0 ; see [9, 10]. Remark 4.1. The hypothesis kuh (·, 0) − Ih u0 k . hk+2 ku0 kk+3 of Theorem 4.5 is clearly satisfied if we take uh (x, 0) = Ih u0 . If we choose β1 = 1/(2k(k + 1)), then by Lemma 4.2 this hypothesis is also satisfied if we take uh (x, 0) = Ph u0 . The k(·, tn )k error bounds of Theorem 4.5 imply error bounds at the Gauss and Lobatto points as follows. For each n ∈ {1, 2, . . . , N }, define a discrete H 1 (Q) seminorm and a discrete L2 (Q) norm of the error u − uh by eng :=
M k 1 XX ∂x (u − uh )2 (gm,i , tn ) Mk
!1/2
m=1 i=1
and enl
:=
!1/2 M X k+1 X 1 2 (u − uh ) (lm,i , tn ) , M (k + 1) m=1 i=1
where gm,i and lm,i are, respectively, the Gauss points of degree k and Lobatto points of degree k + 1 on the interval Im . Corollary 4.6. Suppose that the hypotheses of Theorem 4.5 hold. Then for n = 1, 2, . . . , N , one has eng . h−1 N − min{2−α,rα} + hk+1 ku0 kk+3 + kukL∞ (H k+1 ) + kDtα ukL∞ (H k+1 ) , enl . N − min{2−α,rα} + hk+2 ku0 kk+3 + kukL∞ (H k+1 ) + kDtα ukL∞ (H k+1 ) . Proof. Imitate the proof of [3, Corollary 3.5]. Remark 4.2. It seems impossible to imitate the analysis of [3, Section 3.C] to obtain an improved superconvergence result at the nodes of the mesh, because it involves terms like k(Dtα )m ukk+1 for m ≥ 1, which in general are unbounded as t → 0 for a fractional-derivative problem such as (1), as one can see from the sharp bounds of Lemma 2.3. Part of the proof of Theorem 4.5 resembles the proof of [3, Theorem 3.3], which we are able to simplify—see Section 5.
5
A new proof for a non-fractional problem
In this short section we digress from our fractional-derivative problem — we show that the proof technique of Theorem 4.5 gives a simpler proof of [3, Theorem 3.3], which is a superconvergence result for a semidiscrete DDG method applied to the classical parabolic initial-boundary value problem that is obtained by formally setting α = 1 in (1).
12
Theorem 5.1. [3, Theorem 3.3] Consider a DDG method that discretises the initial-boundary value in space but not in time. We use the same notation as previously, with k ≥ 2. Assume that f ∈ H k+1 (Ω). Choose β1 = 1/ (2k(k + 1)). Suppose that the initial value uh (x, 0) satisfies kuh (·, 0) − Ih u0 k . hk+2 ku0 kk+3 . Then k(uh − Ph u)(·, t)k + k(uh − Ih u)(·, t)k . hk+2 kuk2L∞ (H k+3 ) + kf k2L∞ (H k+1 ) . Proof. Let n ∈ {1, . . . , N }. Set en = unh − Ph un and εn = un − Ph un . Similarly to the proof of Theorem 4.5, one can derive the inequalities (∂t e, e) + γkek2E ≤ (∂t ε, e) + (cε, e) = (Dx−1 ∂t ε, ∂x e) + (Dx−1 cε, ∂x e) .
1 2(k+2) h k∂t uk2k+1 + kuk2k+1 + γ|e|21 , 2γ
where the term (ϕn , en ) no longer appears because we have not discretised in time. Hence ∂t kek2 . h2(k+2) k∂t uk2k+1 + kuk2k+1 . Integrating this inequality from 0 to t ∈ (0, T ] yields ke(·, t)k2 . ke0 (·)k2 + h2(k+2)
Z
t
k∂s u(·, s)k2k+1 + ku(·, s)k2k+1
ds
0
. h2(k+2) ku0 kk+3 + h2(k+2) T kut k2L∞ (H k+1 ) + kuk2L∞ (H k+1 ) , since ke(·, 0)k = kuh (·, 0) − Ph u0 k ≤ kuh (·, 0) − Ih u0 k + kIh u0 − Ph u0 k . hk+2 ku0 kk+3 by our hypothesis and Lemma 4.2. Finally, k∂t ukk+1 . kukk+3 +kf kk+1 follows from ∂t u−uxx +c(x)u = f , and we are done.
6
Numerical experiments
We compute numerical solutions for a typical example of problem (1). Example 6.1. Consider the problem Dtα u − uxx + u = f (x, t) for (x, t) ∈ (0, 4π) × (0, 1], u(x, 0) = 0 for x ∈ [0, 4π], with the periodic boundary conditions u(0, t) = u(4π, t) and ux (0, t) = ux (4π, t) for 0 < t ≤ 1. The function f (x, t) is chosen such that the exact solution of this problem is u(x, t) = 2(tα + t3 ) sin x, which has the same regularity as typical solutions of problem (1) (recall Lemma 2.3). Take k = 2, β0 = 4 and β1 = 1/12 in the DDG method, so the condition β1 = 1/(2k(k + 1)) of Theorem 4.5 and Corollary 4.6 is satisfied. Choose r = (2 − α)/α for the temporal mesh grading, in order to get the best possible rate of convergence in Theorem 4.5 and Corollary 4.6. We give the errors for the quantities eng and enl of Corollary 4.6 since they can be computed exactly; to measure the L2 (Ω) errors of Theorem 4.5 would require the use of a quadrature rule, which would be a possible further source of error. 13
N Table 1 displays the errors eN g and el , and their associated orders of convergence, at the time level tN = 1. In computing these quantities we have taken N = 104 so that the O(hk+2 ) spatial error is negligible and the temporal error dominates the results. The rates of convergence obtained agree with the theoretical rates of convergence stated in Corollary 4.6 for the eN g and N el errors, i.e., this corollary is sharp.
Table 1: Errors and orders of convergence on space direction for Example 6.1. Polynomial
P
2
M
eN g
order
eN l
order
5
9.8242E-2
-
6.2198E-2
-
10
1.0111E-2
3.2803
3.1129E-3
4.3205
15
2.6712E-3
3.2828
5.5225E-4
4.2650
20
1.0711E-3
3.1765
1.6764E-4
4.1439
25
5.3446E-4
3.1154
6.7768E-5
4.0591
30
3.0480E-4
3.0802
3.2873E-5
3.9678
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