DOI: 10.2478/auom-2014-0047
An. Şt. Univ. Ovidius Constanţa
Vol. 22(3),2014, 13–35
A posteriori analysis of the spectral element
discretization of heat equation
Nejmeddine Chorfi, Mohamed Abdelwahed, Ines Ben Omrane
Abstract
In this paper, we present a posteriori analysis of the discretization
of the heat equation by spectral element method. We apply Euler’s
implicit scheme in time and spectral method in space. We propose two
families of error indicators both of them are built from the residual of
the equation and we prove that they satisfy some optimal estimates. We
present some numerical results which are coherent with the theoretical
ones.
1
Introduction
The a posteriori error analysis and mesh adaptivity methods have received considerable attention by mathematicians and engineers in the last two decades
[2] However, the majority of works dealing with this theory are limited to finite
element method see Verfürth [12] [13] [14] [15] and still insufficient in spectral
methods. The spectral element method consists on approaching the solution
of a partial differential equation by polynomial functions of high degree on
each element of a decomposition. The main results consist on optimizing the
discretization of the heat equation. The later relies on a spectral element
method with respect to space variables and Euler’s implicit scheme with respect to time. The parameter of discretization is a K-tuple formed by the
maximum degrees Nk of polynomial on each element. However, like the h − p
Key Words: Heat equation, Spectral elements discretization, Error indicators.
2010 Mathematics Subject Classification: Primary 46G05, 46L05; Secondary 47A30,
47B47.
Received: May, 2013.
Revised: June, 2013.
Accepted: July 2013.
13
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
14
version of finite element see [1][9], it can also involve in this parameter a quantity hk related to the diameter of elements.
In this paper we are interested in the a posteriori analysis of the spectral
element discretization of heat equation. This work is an extension in spectral
element method of some results obtained by Bergam and al. [3] in the case of
finite element method. More precisely, we introduce two kinds of indicators,
both of them of residual type. The first family, which is similar to that introduced in [8], is global with respect to the space variables but local with respect
to the time discretization. Thus, at each time, the error indicator provides appropriate information for the choice of the next time step. The second family
is local with respect to both the time and space variables, and the idea is that
at each time is an efficient tool for the mesh adaptivity. These indicators are
local quantities which can be computed explicitly as a function of the discrete
solution and the data of the problem. They are said to be optimal if their
Hilbertian sum is equivalent to the error such that the equivalence constants
are independent of the parameter of discretization.
The outline of the paper is as follows :
• In Section 2, we present the linear heat equation and we describe the
time semi-discret problem and its space discretization.
• Section 3 is devoted to the construction of error indicators for the heat
equation and for the proof of upper and lower bounds based on time and space
indicators.
• Section 4 deals with some numerical experiments which confirm the interest of the discretization.
2
Time and space discretisation of the heat equation
Let Ω be a connected and bounded open set in R and T be a fixed positive
integer. We consider the one dimensional heat equation: Find u the solution
of

∂2u

 ∂t u −
= f in Ω× ]0, T [,
∂x2
(1)
u
=
0
on ∂Ω× ]0, T [,


u|t=0 = u0
in Ω.
The data f and the function u0 are given. It is readily checked that the
equation (1) admits the equivalent variational formulation
Find u ∈ C 0 (0, T ; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) such that for all t in ]0, T [,
(∂t u(t), v) + (
∂u
∂v
(t),
) = (f (t), v),
∂x
∂x
∀v ∈ H01 (Ω).
(2)
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
15
u|t=0 = u0
in Ω,
(3)
It is well-known that, for all f in L2 (0, T ; H −1 (Ω)) and u0 in L2 (Ω), problem
(2)-(3) admits a unique solution.(see [10] [11]).
First by taking v equal to u(t) in (2) and integrating on the interval ]0, T [,
we derive the following estimate, for all t in [0, T ],
ku(t)k2L2 (Ω)
t
Z
|u(s)|2H 1 (Ω) ds ≤ ku0 k2L2 (Ω) + kf k2L2 (0,T ;H −1 (Ω)) .
+
(4)
0
We introduce the norm
Z t
12
[[v]](t) = kv(t)k2L2 (Ω) +
|v(s)|2H 1 (Ω) ds .
(5)
0
then the (4) is written
[[v]](t) ≤ ku0 k2L2 (Ω) + kf k2L2 (0,T ;H −1 (Ω)) .
(6)
On the other hand if the function f belongs to L2 (0, T ; L2 (Ω)) and u0 belongs
to H 1 (Ω), by replacing v in (2) by ∂t u(t), we obtain for all t in [0, T ],
|u(t)|2H 1 (Ω)
Z
+
t
k∂t u(s)k2L2 (Ω) ds ≤ |u0 |2H 1 (Ω) + kf k2L2 (0,T ;L2 (Ω)) .
(7)
0
2.1
Time discretisation
We suppose that f is a continuous function on t with values in H −1 (Ω). We
introduce a partition of the interval [0, T ] into subintervals [tn−1 , tn ], 1 ≤ n ≤
N , such that 0 = t0 < t1 < .... < tN = T . We denote τn = tn − tn−1 , by τ
the N -tuple (τ1 , ...., τN ) and by |τ | = max τn . We also define the regularity
1≤n≤N
parameter
στ = max
2≤n≤N
τn
τn−1
(8)
With each family (v n )0≤n≤N , we agree to associate the function vτ on [0, T ]
which is affine on each interval [tn−1 , tn ], 1 ≤ n ≤ N , and equal to v n at
tn , 0 ≤ n ≤ N .
For simplicity, we introduce the notation
f n = f (tn ).
The semi-discrete problem issued from Euler’s implicit scheme now writes
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
16
 n
u − un−1
∂ 2 un



= fn
−
τn
∂x2
un = 0


 0
u = u0
in Ω, 1 ≤ n ≤ N ,
on ∂Ω, 1 ≤ n ≤ N ,
in Ω.
(9)
Equivalently, its admits the variational formulation Find
(un )0≤n≤N ∈ L2 (Ω) × (H01 (Ω))N
satisfying for 1 ≤ n ≤ N,
(un , v) + τn (
∂v
∂un
(t),
) = (un−1 , v) + τn (f n , v),
∂x
∂x
u0 = u0
∀v ∈ H01 (Ω),
in Ω.
(10)
(11)
The existence and uniqueness of a solution (un )0≤n≤N for any data f ∈
C 0 (0, T ; H −1 (Ω)) and u0 ∈ L2 (Ω) is a simple consequence of the Lax-Milgram
Lemma. Moreover, let us introduce the ”local” norm on each v n in H01 (Ω)
[[v n ]] = kv n k2L2 (Ω) + τn |v n |2H 1 (Ω)
12
.
(12)
By Taking v equal to un in (10), we easily derive the estimate
[[un ]] ≤ kun−1 k2L2 (Ω) + τn kf n k2H −1 (Ω) .
(13)
The global norm is now defined on whole sequences (um )0≤m≤n by
[[(v m )]]n = (kv n k2L2 (Ω) +
n
X
1
τn |v m |2H 1 (Ω) ) 2 .
(14)
m=1
By summing up estimate (13) on n, we derive the semi-discrete analogue of
(4)
[[(um )]]n ≤ (ku0 k2L2 (Ω) +
n
X
1
τm kf m k2H −1 (Ω) ) 2 .
(15)
m=1
It can be observed that the norm [[(um )]]n involved in this estimate is not
equal to the norm [[(uτ )]](tn ). However, when u0 is supposed to be in H 1 (Ω),
there are equivalent, as proven in [3] which is of great use in what follows
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
17
Lemma 2.1. For any family (v n )0≤n≤N ∈ H 1 (Ω)N +1 , we have
1
1
τ1
[[(v m )]]2n ≤ [[vτ ]]2 (tn ) ≤ (1 + στ ) [[(v m )]]2n +
k∇v 0 k2 .
4
2
2
(16)
In order to state the a priori error estimate, we observe that the family
(en )0≤n≤N , with en = u(tn ) − un , satisfies e0 = 0 and also, by integrating ∂t u
between tn−1 and tn and using equation (10) at time t = tn ,
∀ v ∈ H01 (Ω),
(en , v) + τn (∇en , ∇v) = (en−1 , v) + τn (εn , v),
where the consistency error εn is given by
Z tn
1
n
(∂t u)(s)ds − (∂t u)(tn ), v).
(ε , v) = (
τn tn−1
By applying (15) to this new problem and evaluating the consistency error
thanks to a Taylor expansion, we derive the estimate: if the solution u is such
that ∂t2 u belongs to L2 (0, T ; H −1 (Ω)) and for 1 ≤ n ≤ N ,
[[u(tm ) − um ]]n ≤ ( max τm )k∂t2 ukL2 (0,tn ;H −1 (Ω)) .
1≤m≤n
2.2
(17)
Space discretisation
We now describe the space discretization of problem (10)-(11). Let Λ the
interval ] − 1, 1[. For each n, 0 ≤ n ≤ N , we introduce a family of reals
numbers ak such that
−1 = a0 ≤ a1 ≤ ......... ≤ aK−1 ≤ aK = 1.
We denote by Λk the interval ]ak−1 , ak [, 1 ≤ k ≤ K, and hk his length. With
each interval Λk , 1 ≤ k ≤ K, we agree to associate positive integer Nk ≥ 2.
The parameter of discretization δ is a K-tuple of couples (hk , Nk ). To define
the discrete form associated with problem (1), we construct on each interval
Λk a quadrature formula.
First of all, we recall the formulas which we will use. Let ξ0 < ... < ξN be
0
the zeros of the polynomial (1 − x2 )LN and ρj the associated weights. The
Gauss-Lobatto quadrature formula on the interval Λ =] − 1, 1[ is written
Z
1
∀φ ∈ P2N −1 (Λ);
φ(x)dx =
−1
N
X
j=0
φ(ξjN )ρN
j
(18)
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
18
where PN (Λ) is the space of polynomials, defined on Λ, with degree ≤ N .
We introduce an approximation of the scalar product in L2 (Λ), so we define
a discrete scalar product on any u and v continuous on Λ by
(u, v)N =
N
X
u(ξjN )v(ξjN )ρN
j .
j=0
By translation and homothety, we define on each Λk =]ak−1 , ak [ a Gaussk
Lobatto quadrature formula of Nk + 1 nodes ξjNk and of weights ρN
j , then we
set:
Nk
K X
X
(u, v)δ =
k
u(ξjNk )v(ξjNk )ρN
j .
(19)
k=1 j=0
We note by iδ the Lagrange interpolation operator on the set of nodes ξjNk
with values in
n
o
Yδ = vδ ∈ H 1 (Λ); vδ|Λk ∈ PNk (Λk ), 1 ≤ k ≤ K .
Equivalently, for each function ϕ continuous on Λk , iδ (ϕ)|Λk in PNk (Λk ) and
verify
iδ (ϕ)|Λk (ξjNk ) = ϕ|Λk (ξjNk ).
(20)
We recall the following property, which is useful in what follows
kuδ k2L2 (Λ) ≤ (uδ , uδ )δ ≤ 3kuδ k2L2 (Λ) .
∀ uδ ∈ Yδ ,
For each δ, the discrete spaces are defined by
n
o
Xδ = vδ ∈ H01 (Λ); vδ|Λk ∈ PNk (Λk ), 1 ≤ k ≤ K .
(21)
(22)
We suppose that f is continuous on Λ × [0, T ] and u0 continuous on Λ, the
fully discrete problem now reads
N
Y
n
Find (uδ )0≤n≤N in Yδ ×
Xδ such that for 1 ≤ n ≤ N ,
n=1
(unδ , vδ )δ + τn (
∂unδ
∂vδ
)δ = (un−1
, vδ )δ + τn (f n , vδ )δ
δ
∂x ∂x
,
u0δ = iδ u0
in Ω,
∀vδ ∈ Xδ ,
(23)
(24)
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
19
The form aδ (., .) is defined by
aδ (unδ , vδ ) = (unδ , vδ )δ + τn (
∂unδ ∂vδ
,
)δ .
∂x ∂x
(25)
The existence and uniqueness of a solution (unδ )0≤n≤N for any data f in
C 0 (0, T ; H −1 (Λ)) and u0 ∈ C(Λ), follows from the Lax-Milgram Lemma.
Moreover exactly the same arguments as for (15) leads to the estimate
n
21
X
2
m 2
+
.
[[um
]]
≤
c
ki
u
k
τ
ki
f
k
2
2
n
δ
0
m
δ
δ
L (Λ)
L (Λ)
(26)
m=1
3
Error indicators for the linear heat equation
We are now interested in exhibiting error indicators and studying their equivalence with the error. We first describe the two types of indicators. Next we
prove an upper bound for the error as a function of the Hilbertian bound for
each indicator.
3.1
The error indicator
As already hinted, we work with two types of indicators, the first one being
linked to time discretization and the second ones to space discretization. The
first ones are local in time but global in space while the second ones are local
with respect to both the time and space variables.
For each n, 1 ≤ n ≤ N , we define the time error indicator
ηn = (
τn 1 ∂ n
) 2 k (uδ − un−1
)kL2 (Λ) .
δ
3
∂x
(27)
We refer to [8] for analogous time error indicators, however leading to estimates
in different norms.
For the technical reasons, we need to introduce the space
n
o
Xδ− = vδ ∈ H01 (Λ); vδ|Λk ∈ PNk −1 (Λk ), 1 ≤ k ≤ K .
(28)
For each n, 1 ≤ n ≤ N and each interval Λk in Λ, we define the space error
indicator
1
1
un − uδn−1
∂ 2 un
δ
ηn,k = Nk−1 k iδ f n − δ
(x) (x − ak−1 ) 2 (ak − x) 2 kL2 (Λk ) .
+
2
τn
∂x
(29)
These indicators are local with respect to both the time and spaces variables
and be computed explicitly.
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
20
3.2
An upper bound for the error
We now intend to bound the norm introduced in (5) by the error indicators
and some further terms involving the data (where of course uδτ denotes the
piecewise affine function equal to unδ in each tn . Here we use the triangular
inequality
[[u − uδτ ]](tn ) ≤ [[u − uτ ]](tn ) + [[uτ − uδτ ]](tn ),
and we begin by evaluating [[u − uτ ]](tn ). The proof of the estimate is rather
technical.
Let πτ denote the interpolation operator with values in piecewise constant
functions on [0, T ] defined as follows: for any function v continuous on [0, T ],
Πτ v is constant on each interval [tn−1 , tn ], 1 ≤ n ≤ N , equal to v(tn ).
Proposition 3.1. Assume that the data f is continuous on [0, T ] with values
in H −1 (Λ) and the function u0 belongs to H 1 (Λ). The following a posteriori
error estimate holds between the solution u of problem (1) and the solution
(un )0≤n≤N of problem (9), for all tn , 1 ≤ n ≤ N ,
n
X
1
2 12
ηm
)
[[u − uτ ]](tn ) ≤ c (1 + στ ) 2 [[uτ − uδτ ]](tn ) + (
m=1
+kf − πτ f kL2 (0,tn ,H −1 (Λ)) .
(30)
Proof:
When ”applying” equation (2) to the function uτ , we obtain for all t ∈
[tn−1 , tn ] and v ∈ H01 (Λ)
(∂t uτ (t), v)+(∂x uτ (t), ∂x v) = (
un − un−1
, v)+ ∂x (uτ (t)−un ), ∂x v +(∂x un , ∂x v),
τn
whence, from (9),
(∂t uτ (t), v) + (∂x uτ (t), ∂x v) = (f n , v) + ∂x (uτ (t) − un ), ∂x v .
Thus, subtracting this line from equation (2) leads to
(∂t (u − uτ )(t), v) + (∂x (u − uτ )(t), ∂x v) = (f (t) − f n , v) + (∂x (un − uτ (t)), ∂x v).
We now take v equal to (u − uτ )(t), we obtain
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
21
∂
∂
∂t (u − uτ )(t), (u − uτ )(t) +
(u − uτ )(t),
(u − uτ )(t)
∂x
∂x
n
= f (t) − f , (u − uτ )(t)
∂
∂
(uτ (t) − un ),
(u − uτ )(t) .
∂x
∂x
Integrate this line on [tn−1 , tn ] and sum up on the n. By noting that u − uτ
vanishes at t = 0, this yields
−
n Z
X
m=1
tm
tm−1
n Z tm
X
∂
1
∂t k(u − uτ )(s)k2L2 (Λ) ds +
k (u − uτ )(s)k2L2 (Λ) ds
2
∂x
m=1 tm−1
n Z tm
X
=
f (s) − f m , (u − uτ )(s) ds
t
m=1
Z tm m−1
∂
∂
−
(uτ (s) − un ),
(u − uτ )(s) ds .
∂x
tm−1 ∂x
Then, we have
Z tn
1
1
∂
k(u − uτ )(tn )k2L2 (Λ) − k(u − uτ )(0)k2L2 (Λ) +
k (u − uτ )(s)k2L2 (Λ) ds
2
2
∂x
0
n Z tm
X
=
f (s) − f m , (u − uτ )(s) ds
tm−1
Z tm
m=1
−
tm−1
∂
∂
(uτ (s) − un ),
(u − uτ )(s) ds .
∂x
∂x
We obtain
n Z tm
X
1
2
[[u − uτ ]] (tn ) ≤
f (s) − f m , (u − uτ )(s) ds
2
tm−1
m=1
Z tm
∂
∂
−
(uτ (s) − un ),
(u − uτ )(s) ds .
∂x
tm−1 ∂x
We now evaluate separately each term in the right-hand side of this equation.
1) It follows from the definition of πτ that
Z tm
f (s) − f m , (u − uτ )(s) ds
tm−1
Z tm
12 Z tm
12
∂
≤
k(f − Πτ f )(s)k2H −1 (Λ) ds
k (u − uτ )(s)k2L2 (Λ) ds .
tm−1
tm−1 ∂x
Moreover, note that
n Z
X
m=1
tm
tm−1
k
12
∂
(u − uτ )(s)k2L2 (Λ) ds
≤ [[u − uτ ]](tn ).
∂x
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
22
2) Concerning the second term, we have
Z tm ∂
∂
(uτ (s) − un ),
(u − uτ )(s) ds
∂x
∂x
tm−1
12 Z tm
12
Z tm
∂
∂
≤
k (uτ (s) − um )k2L2 (Λ) ds
k (u − uτ )(s)k2L2 (Λ) ds .
tm−1 ∂x
tm−1 ∂x
By using the definition of the function uτ , we obtain
t − s ∂
∂
m
(uτ − um ) = −
(um − um−1 ),
∂x
τm
∂x
(31)
then,
Z
tm
k
tm−1
∂
(uτ (s) − um )k2L2 (Λ) ds
∂x
∂ m
(u − um−1 )k2L2 (Λ)
∂x
Z
tm
=
k
=
τm ∂ m
k (u − um−1 )k2L2 (Λ) .
3 ∂x
tm−1
(s − tm )2
ds
2
τm
By using again (31), we obtain
12
Z tm
τ 21 ∂
∂
m
k (uτ (s) − um )k2L2 (Λ) ds
≤
k (um − um
δ )kL2 (Λ)
3
∂x
tm−1 ∂x
τ 12 ∂
τ 12 ∂
m
m
m−1
2 (Λ) +
+
k (um
k (um−1 − um−1
)kL2 (Λ) .
−
u
)k
L
δ
δ
3
∂x δ
3
∂x
By summing the previous line on n, we have
n Z tm
X
∂
k (uτ (s) − um )k2L2 (Λ) ds
∂x
m=1 tm−1
n
n
X
X
τm
∂
∂ m−1 m−1 2
2
2
ηm
+
≤ c
k (um −um
(u
−uδ )kL2 (Λ) .
δ )kL2 (Λ) +k
3
∂x
∂x
m=1
m=1
Using (16) yields that the sum over n of the square of the last two terms can
be bounded by [[uτ − uτ δ ]]2 (tn ) times a constant depending on the regularity
parameter στ , we have
n
X
τm ∂ m
∂ m−1
2
k (u − um
(u
− um−1
)k2L2 (Λ)
δ )kL2 (Λ) + k
δ
3
∂x
∂x
m=1
n
X
τ1 ∂
τm ∂ m
2
≤ k (u0 − u0δ )k2L2 (Λ) +
k (u − um
δ )kL2 (Λ)
3 ∂x
3
∂x
m=1
n
X
τm−1 στ ∂ m−1
+
k (u
− um−1
)k2L2 (Λ) .
δ
3
∂x
m=2
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
23
then, we have
n
X
τm ∂ m
2
k (u − um
δ )kL2 (Λ)
3
∂x
m=1
∂
+k ∂x
(um−1 − um−1
)k2L2 (Λ)
δ
≤ c(1 + στ )[[uτ − uτ δ ]]2 (tn ).
Combining all this yields the desired result.
Now, we evaluate the norms [[uτ − uτ δ ]](tn ),1 ≤ n ≤ N . For this, we will
need to introduce the following property for the operator Π1,0
N in this lemma,
where TΠ1,0
is
the
orthogonal
projection
operator
from
H01 (Λ) to
N
1
PN (Λ) H0 (Λ) ([5] [7]).
Lemma 3.1. Let s a real ≥ 1. For each function v ∈ H s (Λ) ∩ H01 (Λ), we
have the following estimate
Z 1
12
2
2 −1
(v − Π1,0
v)
(ζ)(1
−
ζ
)
dζ
≤ c N −s |v|H s (Ω) .
(32)
N
−1
Proposition 3.2. Assume the data f continuous on [0, T ] with values in
H −1 (Λ) and u0 ∈ H 1 (Λ). Then, the following a posteriori error estimate holds
between the solution (un )0≤n≤N of problem (9) and the solution (unδ )0≤n≤N of
problem(23)-(24), for all tn , 1 ≤ n ≤ N ,
P
21
PK
n
2
m
m 2
τ
[[uτ − uτ δ ]](tn ) ≤ c
(η
+
kf
−
i
f
k
)
2
m
δ
n,k
m=1
k=1
L (Λk )
+ku0 − iδ u0 kL2 (Λ) .
(33)
Proof:
By taking v equal to vδ ∈ Xδ− in (10) we have
∂un ∂vδ
,
) = (un−1 , vδ ) + τn (f n , vδ ).
∂x ∂x
in (24), the property of exactness of the quadrature
(un , vδ ) + τn (
By taking vδ ∈ Xδ−
formula implies,
(unδ , vδ ) + τn (
∂unδ ∂vδ
,
) = (un−1
, vδ ) + τn (f n , vδ )δ .
δ
∂x ∂x
The difference of the last two equations yields
(un −unδ , vδ )+τn (
∂ n n ∂vδ
(u −uδ ),
) = (un−1 −un−1
, vδ )+τn (f n , vδ )−τn (f n , vδ )δ .
δ
∂x
∂x
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
24
∂
By addition and substraction of the terms (un − unδ , v) and τn ( (un −
∂x
∂v
unδ ),
), we obtain
∂x
∂ n
∂v
(u − unδ ),
)
∂x
∂x
∂
∂
(v − vδ ))
= (un − unδ , v − vδ ) + τn ( (un − unδ ),
∂x
∂x
n−1
n−1
n
n
+(u
− uδ , vδ ) + τn (f , vδ ) − τn (f , vδ )δ .
(un − unδ , v)
+τn (
Then,
a(un − unδ , v) = a(un − unδ , v − vδ ) + (un−1 − un−1
, vδ )
δ
, vδ ) + τn (f n , vδ ) − τn (f n , vδ )δ .
+ (un−1 − un−1
δ
(34)
We interested now to the term a(un − unδ , v − vδ ), for that we can write
a(un − unδ , v − vδ )
=
a(un , v − vδ ) − a(unδ , v − vδ )
=
(un−1 , v − vδ ) + τn (f n , v − vδ ) − (unδ , v − vδ )
∂un ∂
(v − vδ )).
−τn ( δ ,
∂x ∂x
Adding and subtracting the term (un−1
, v − vδ ), we note that
δ
a(un − unδ , v − vδ ) =
(un−1 − un−1
, v − vδ ) − (unδ − un−1
, v − vδ )
δ
δ
n
∂u
∂
(v − vδ )).
+τn (f n , v − vδ ) − τn ( δ unδ ,
∂x
∂x
So
a(un − unδ , v − vδ ) = (un−1 − un−1
, v − vδ ) − (unδ − un−1
, v − vδ )
δ
δ
K Z
X
ak
∂
∂unδ
(x) (v − vδ )(x)dx
∂x
∂x
k=1 ak−1
By integrating by parts the last term of the previous equation, we obtain
+τn (f n , v − vδ ) − τn
a(un −unδ , v−vδ ) = (un−1 −un−1
, v−vδ )−(unδ −un−1
, v−vδ )+τn (f n , v−vδ )
δ
δ
−τn
K−1
Xh
k=1
K
X
∂unδ i
(ak )(v−vδ )(ak )+τn
∂x
So,
a(un − unδ , v − vδ ) = (un−1 − un−1
, v − vδ )
δ
k=1
Z
ak
ak−1
∂ 2 unδ
(x)(v−vδ )(x)dx.
∂x2
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
25
+τn
K Z
X
ak
fn −
unδ − un−1
∂ 2 unδ
δ
(x)
(x)(v − vδ )(x)dx
+
τn
∂x2
k=1 ak−1
K−1
X h ∂un i
δ
−τn
(ak )(v
− vδ )(ak )
∂x
By inserting the previous equation to (34), we obtain
k=1
a(un − unδ , v) = (un−1 − uδn−1 , v)
K Z ak X
un − un−1
∂ 2 unδ
δ
(x)
(x)(v − vδ )(x)dx
+
+τn
fn − δ
τn
∂x2
k=1 ak−1
K−1
X h ∂un i
δ
−τn
(ak )(v − vδ )(ak ) + τn (f n , vδ ) − τn (f n , vδ )δ .
∂x
k=1
By using (19), we have
a(un − unδ , v)
= (un−1 − uδn−1 , v)
K Z ak X
un − un−1
∂ 2 unδ
δ
+τn
(x)
(x)(v − vδ )(x)dx
fn − δ
+
τn
∂x2
k=1 ak−1
Nk
K Z ak
X
X
k
+τn
f n (x)vδ (x)dx −
f n (ξjNk )vδ (ξjNk )ρN
j
−τn
ak−1
k=1
K−1
X h ∂un i
δ
k=1
∂x
j=1
(ak )(v − vδ )(ak ).
By using (20), we can write
K Z ak X
un − uδn−1 ∂ 2 unδ
n
n
a(u − uδ , v) = τn
(x)
(x)(v − vδ )(x)dx
+
fn − δ
τn
∂x2
ak−1
k=1
+(un−1 − uδn−1 , v) − τn
K−1
Xh
k=1
+ τn
K Z
X
k=1
ak
∂unδ i
(ak )(v − vδ )(ak )
∂x
f n − iδ f n (x)vδ (x)dx. (35)
ak−1
The idea is now to associate for each v in H 1 (Λ) the function (see [4])
wδ =
K
X
k=1
K
X
Π1,0
v
−
v(a
)ϕ
−
v(a
)ϕ
+
v(ak )ϕk
k−1
k−1
k
k
Nk −1
k=0
(36)
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
26
where ϕk are continuous functions, affixes on each Λk , equal to 1 in ak and to
0 in the other ak0 , k 0 6= k. The function wδ is in Yδ (in Xδ when v in H01 (Λ)).
By taking vδ equal to wδ of the formula (36) in (35), we note that the last
term disappears
a(un − unδ , v) = (un−1 − un−1
, v) + τn
δ
K Z
X
k=1
+ τn
K Z
X
k=1
ak
iδ f n −
ak−1
ak
f n − iδ f n (x)v(x)dx
ak−1
unδ − un−1
∂ 2 unδ
δ
(x)
(x)(v − vδ )(x)dx. (37)
+
τn
∂x2
1
1
By apparition of the expression (x − ak−1 ) 2 (ak − x) 2 in the equation (37), we
have
(un − unδ , v)
∂
∂ n
(u − unδ ),
v)
∂x Z
∂x
K
X ak un − un−1
∂ 2 unδ δ
= τn
iδ f n − δ
+
(x)
τn
∂x2
ak−1
+τn (
k=1
1
1
1
1
(x − ak−1 ) 2 (ak − x) 2 (x − ak−1 )− 2 (ak − x)− 2 (v − wδ )(x)dx
K Z ak X
n−1
n−1
+(u
− uδ , v) + τn
f n − iδ f n (x)v(x)dx.
k=1
ak−1
By applying Cauchy-Schwarz inequality, we obtain
∂ n
∂
(u − unδ ),
v) ≤ (un−1 − un−1
, v)
δ
∂x
∂x
K Z ak
12 Z ak
21
X
n
n 2
+τn
(f − iδ f ) (x)dx
v 2 (x)dx
(un − unδ , v) + τn (
+τn
ak−1
k=1
Z
K
a
k
X
(iδ f n −
k=1
ak−1
Z ak
ak−1
unδ
un−1
δ
−
τn
+
21
∂ 2 unδ 2
)
(x)(x
−
a
)(a
−
x)dx
k−1
k
∂x2
(v − wδ )2 (x)(x − ak−1 )−1 (ak − x)−1 dx
ak−1
By applying Lemma 3.1,we obtain
(un − unδ , v) + τn (
∂ n
∂
(u − unδ ),
v) ≤ (un−1 − un−1
, v)
δ
∂x
∂x
21
.
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
27
+ τn
K
X
K
ηn,k k
k=1
X
∂v
kL2 (Ωk ) + τn
kf n − iδ f n kL2 (Ωk ) kvkL2 (Ωk ) .
∂x
(38)
k=1
By taking v equal to un − unδ and using ab ≤
b2
a2
+
we have
2
2
∂ n
(u − unδ )k2L2 (Λ)
∂x
kun−1 − un−1
k2L2 (Λ)
kun − unδ k2L2 (Λ)
δ
≤
+
2
2
K
X
∂
+τn
ηn,k +kf n −iδ f n kL2 (Ωk ) k (un −unδ )k2L2 (Λ) .
∂x
k=1
By applying Cauchy-Schwarz inequality, we obtain
kun − unδ k2L2 (Λ) + τn k
kun−1 − un−1
k2
δ
2
K
X
12
+τn
(ηn,k + kf n − iδ f n kL2 (Ωk ) )2
∂
kun − unδ k2
+ τn k (un − unδ )k2 ≤
2
∂x
k=1
K
X
k=1
12
∂ n
(u − unδ )k2 .
∂x
2
2
We again use ab ≤
k
b
a
+ , we obtain
2
2
kun−1 − un−1
k2
kun − unδ k2
∂
δ
+ τn k (un − unδ )k2 ≤
2
∂x
2
K
K
τn X ∂ n n 2
τn X
(ηn,k +kf n −iδ f n kL2 (Ωk ) )+
k (u −uδ )k .
+
2
2
∂x
k=1
k=1
So,
2
kun −un
δk
2
+
τn ∂
n
2 k ∂x (u
− unδ )k2
kun−1 − un−1
k2
δ
+
2
K
X
2
c τn
ηn,k
+ kf n − iδ f n k2L2 (Ωk ) .
≤
k=1
By summing up this inequality on n and using (16), we obtain the desired
result.
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
28
Combining the results of propositions 3.1 and 3.2 leads to the full a posteriori estimate.
Theorem 3.1. Assume the data f continuous on [0, T ] with values in H −1 (Λ)
and u0 ∈ H 1 (Λ). If moreover the regularity parameter στ is bounded by a
constant independent of τ , the following a posteriori error holds between the
solution u of problem (1) and the solution (unδ )0≤n≤N of problem (23)-(24),
for all tn , 1 ≤ n ≤ N ,
n
K
X
X
21
2
2
[[u − uδτ ]](tn ) ≤ c
ηm
+ τm
(ηn,k
+ kf m − iδ f m k2L2 (Λk ) )
m=1
k=1
+c ku0 − iδ u0 kL2 (Λ) + kf − Πf kL2 (0,tn ,H −1 (Λ)) .
3.3
An upper bound for the error indicators.
The idea is now to prove separate bounds for each indicators ηn and ηn,k . We
begin with the ηn .
Proposition 3.3. Assume the data f continous on [0, T ] with values in H −1 (Λ)
and the function u0 ∈ H01 (Λ). The following estimate holds for the indicator
ηn defined in (27), 1 ≤ n ≤ N
1
1
]]+ c (τn ) 2 kf −πτ f kL2 (tn−1 ,tn ;H −1 (Λ))
ηn ≤ [[un −unδ ]]+(στ ) 2 [[un−1 −un−1
δ
1
+ c (τn ) 2 k∂t (u − uτ )kL2 (tn−1 ,tn ;H −1 (Λ)) + k∂x (u − uτ )kL2 (tn−1 ,tn ;L2 (Λ) .
Proof:
As previously, we use the triangular inequality
τn 1 ηn ≤ ( ) 2
k∂x (unδ − un )kL2 (Λ) + k∂x (un − un−1 )kL2 (Λ)
3
2 (Λ) .
+k∂x (un−1 − un−1
)k
L
δ
(39)
By using (12), we have
21
[[un − unδ ]] = kun − unδ k2L2 (Λ) + τn k∂x (un − unδ )k2L2 (Λ) .
So, we derive
(
τn 1
) 2 k∂x (un − unδ )kL2 (Λ) ≤ [[un − unδ ]].
3
(40)
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
29
Similary, we use (12), we obtain
12
[[un−1 − un−1
]] = kun−1 − un−1
k2L2 (Λ) + τn−1 k∂x (un−1 − un−1
)k2L2 (Λ) .
δ
δ
δ
We also have
τn k∂x (un−1 − un−1
)k2L2 (Λ) ≤ τn−1 στ k∂x (un−1 − un−1
)k2L2 (Λ) .
δ
δ
So, we have
1
τn 1
kL2 (Λ) ≤ (στ ) 2 [[un−1 − un−1
]].
(41)
( ) 2 k∂x (un−1 − un−1
δ
δ
3
τn 1
For estimate the term ( ) 2 k∂x (un − un−1 kL2 (Λ) , we can use the same tech3
nical of proposition 3.1
When ”applying” equation (2) to the function uτ , we obtain for all t ∈
[tn−1 , tn ] and v ∈ H01 (Λ)
(∂t uτ (t), v) + (∂x uτ (t), ∂x v) =
un − un−1
, v) + ∂x (uτ (t) − un ), ∂x v
τn
+(∂x un , ∂x v),
(
By injection the equation (10) in the last equation, we obtain
(∂t uτ (t), v) + (∂x uτ (t), ∂x v) = (f n , v) + ∂x (uτ (t) − un ), ∂x v .
Thus, subtracting this line from equation (2) leads to
(∂t (u − uτ )(t), v) + (∂x (u − uτ )(t), ∂x v) = (f (t) − f n , v) + (∂x (un − uτ (t)), ∂x v).
By taking v equal un − un−1 and integrating between tn−1 and tn , we obtain
Z tn
Z tn
n
n−1
(∂t (u − uτ )(t), u − u
)ds +
(∂x (u − uτ )(t), ∂x (un − un−1 ))ds
tn−1
Z
tn−1
tn
=
tn−1
(f (s) − f n , un − un−1 )ds −
Z
tn
(∂x (uτ (t) − un , ∂x (un − un−1 ))ds. (42)
tn−1
By using (31), the last term of (42) can be written
Z tn
−
(∂x (uτ (t) − un ), ∂x (un − un−1 ))ds
tn−1
Z tn
tn − s
k∂x (un − un−1 )k2L2 (Λ) ds
=
τn
tn−1
Z tn
tn − s
n
n−1 2
= k∂x (u − u
)kL2 (Λ)
ds
τn
tn−1
τn
= k∂x (un − un−1 )k2L2 (Λ) .
2
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
30
Then the equation (42) becomes
τn
k∂x (un − un−1 )k2L2 (Λ) =
2
Z
tn
(∂t (u − uτ )(t), un − un−1 )ds
tn−1
Z tn
+
Ztn−1
tn
−
(∂x (u − uτ )(t), ∂x (un − un−1 ))ds
(f − πτ f )(s), un − un−1 )ds.
tn−1
(43)
Next, in order to evaluate the terms in the right of the equation (43), we use
divers Cauchy-Schwarz inequalities, we obtain
1
τn
k∂x (un − un−1 )k2 ≤ (τn ) 2 k∂t (u − uτ )kL2 (tn−1 ,tn ;H −1 (Λ))
2
+ k∂x (u − uτ )kL2 (tn−1 ,tn ;L2 (Λ)) + kf − πτ f kL2 (tn−1 ,tn ;H −1 (Λ)) . (44)
Combining all this yields the desired result.
We evaluate now the indicator η. We need to introduce the following
Lemma (see [4]).
Lemma 3.2.
For all ϕN ∈ PN (Λ), we have
Z 1
Z
2 2
2
ϕ02
(ζ)(1
−
ζ
)
dζ
≤
c
N
N
−1
1
ϕ2N (ζ)(1 − ζ 2 )dζ
(45)
−1
and
Z
1
−1
ϕ2N (ζ)dζ ≤ c N 2
Z
1
ϕ2N (ζ)(1 − ζ 2 )dζ.
(46)
−1
Proposition 3.4. Assume the data f continuous on [0, T ] with values in
H −1 (Λ) and the function u0 ∈ H01 (Λ). The following estimate holds for the
indicator ηn,k defined in (29) for all 1 ≤ k ≤ K and 1 ≤ n ≤ N :
∂
(un − unδ ) − (un−1 − un−1
)
δ
kH −1 (Λk )
ηn,k ≤ c k (un − unδ )kL2 (Λk ) + k
∂x
τn
+ Nk−1 hk kf n − iδ f n kL2 (Λk ) . (47)
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
31
Proof:
We use equation (35) with vδ = 0 and v chosen as

un − un−1
∂ 2 unδ 
δ
(x − ak−1 )(ak − x)
v = iδ f n − δ
+
τn
∂x2

v=0
sur Λk
sur Λ \ Λk
We obtain
unδ − un−1
∂ 2 unδ 2
δ
(x − ak−1 )(ak − x)dx
+
τn
∂x2
ak−1
Z
ak
un−1 − un−1
un − unδ
δ
= a(
, v) −
(f n − iδ f n )v(x)dx − (
, v)
τn
τn
ak−1
Z
ak
∂
∂v =
(un − unδ ),
−
(f n − iδ f n )v(x)dx
∂x
∂x
ak−1
(un − un ) − (un−1 − un−1 ) δ
δ
,v .
+
τn
By using Cauchy-schwarz inequality, we have
Z
ak
iδ f n −
unδ − un−1
∂ 2 unδ 2
δ
+
(x − ak−1 )(ak − x)dx
τn
∂x2
ak−1
(un − unδ ) − (un−1 − un−1
)
∂v
δ
kH −1 (Λk ) k kL2 (Λk )
≤k
τn
∂x
Z
ak
+k
iδ f n −
∂ n
∂v
(u − unδ )kL2 (Λk ) k kL2 (Λk ) + kf n − iδ f n kL2 (Λk ) kvkL2 (Λk ) . (48)
∂x
∂x
Using the inequality (a + b)2 ≤ 2a2 + 2b2 yields
∂
un − uδn−1 ∂ 2 unδ 2
(iδ f n − δ
) (x−ak−1 )2 (ak −x)2 dx
+
2
∂x
τ
∂x
n
ak−1
Z ak n−1
n
u
−
u
∂ 2 unδ 2
δ
+ 2
+
(ak + ak−1 − 2x)2 dx
iδ f n − δ
τn
∂x2
ak−1
By applying (45) et (46), we obtain
k
∂v
kL2 (Λk ) ≤ 2
∂x
k
Z
ak
un − un−1
1
1
∂ 2 unδ
∂v
δ
kL2 (Ωk ) ≤ c Nk k(iδ f n − δ
+
)(x−ak−1 ) 2 (ak −x) 2 kL2 (Ωk ) .
∂x
τn
∂x2
Similarly, we have
kvkL2 (Ωk ) ≤ c hk k(iδ f n −
unδ − un−1
1
1
∂ 2 unδ
δ
+
)(x − ak−1 ) 2 (ak − x) 2 kL2 (Ωk ) .
τn
∂x2
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
32
By substitution of the last two inequalities in (48), we derive
unδ − un−1
1
1
∂ 2 unδ
δ
)(x − ak−1 ) 2 (ak − x) 2 k2L2 (Λk )
+
2
τn
∂x
(un − unδ ) − (un−1 − un−1
)
δ
≤ c Nk k
kH −1 (Λk )
τn
n−1
un − uδ
1
1
∂ 2 unδ
)(x − ak−1 ) 2 (ak − x) 2 kL2 (Λk )
k(iδ f n − δ
+
2
τn
∂x
unδ − un−1
1
1
∂ 2 unδ
n
δ
)(x − ak−1 ) 2 (ak − x) 2 kL2 (Λk )
+ c Nk k(iδ f −
+
2
τn
∂x
∂ n
n
k (u − uδ )kL2 (Λk )
∂x
un − un−1
1
1
∂ 2 unδ
δ
+ c hk k(iδ f n − δ
+
)(x − ak−1 ) 2 (ak − x) 2 kL2 (Λk )
τn
∂x2
kf n − iδ f n kL2 (Λk ) .
k(iδ f n −
unδ − un−1
1
1
∂ 2 unδ
δ
+
)(x − ak−1 ) 2 (ak − x) 2 kL2 (Λk ) and
τn
∂x2
multiplying by Nk−1 , we derive the desired bound.
Simplifying by k(iδ f n −
4
Numerical experiments
We present some numerical experiments in order to test the obtained time
and space indicators. In the following η(t) and η(N ) denote respectively the
time and the space errors indicators defined in (27) and (29). We consider the
domain Λ =] − 1, 1[ and the solution u(x, t) = et sin(π x). All presented figures
are plotted in logarithmic scale for the error axis.
Figure 1 (a) plots the time error and the time indicator (N fixed and t
variable). We remark that we obtain the same slope of convergence and that
the error is upper bounded by the time indicator η(t) which is in coherence
with proposition 3.3.
In figure 1 (b) we show the time error and the space indicator (t fixed
and N variable). We remark that we obtain the same slope of convergence
and that the error is lower bounded by the space indicator η(N ) which is in
coherence with theorem 3.1.
In the second test we study the influence of time t and space N on the
indicators η(t) and η(N ). For N fixed equal to 30 and t varying between 0.1
and 10−5 , figure 2 (a) presents the error ku − utN kH 1 (Ω) (plain red line), the
error indicator η(t) (dashed dotted blue line) and η(N ) (dashed blach line).
It can thus be checked that the error indicator η(N ) is fully independent of t.
Moreover the error indicator η(t) descreases with exactly the same slope as the
error until the discretization error becomes larger than the error. Similarly,
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
33
(a)
(b)
Figure 1: Error and indicators slope.
(a)
Figure 2: Influence of time and space on the indicators.
(b)
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
34
for t fixed equal to 10−5 and N varying between 10 and 35. Figure 2 (b)
presents the full error ku − utN kH 1 (Ω) (plain red line), the error indicator η(t)
(dashed black line) and η(N ) (dashed dotted blue line). Here, the error η(t)
are completely independent of N .
5
Conclusion
In this paper, we are interested in the posteriori analysis of the discretization
of the one-dimensional heat equation by spectral element method which is the
very efficient tool for mesh adaptivity. More precisely, we are constructed two
kinds of indicators of residual type and we proved optimal upper and lower
error bounds, in the sense of [6]. As usual and up to some terms concerning
the approximation of the data, the global error between the solution of the
exact equation and the solution of the time-space discrete problem at each
discrete time is bounded by the Hilbertian sum of the indicators while each
indicator is bounded by the local error.
Acknowledgements
This project was supported by King Saud University, Deanship of Scientific
Research, College of Science Research Center.
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Nejmeddine Chorfi, Mohamed Abdelwahed
Department of Mathematics,
College of Science, King Saud University,
P.O. Box 2455, Riyadh 11451, Saudi Arabia.
Email: nchorfi@ksu.edu.sa, mabdelwahed@ksu.edu.sa
Ines Ben Omrane
Département de Mathématique,
Faculté des Sciences de Tunis,
Campus Universitaire 1060 Tunis, Tunisie.
Email: ines.benomrane@fst.rnu.tn.
A POSTERIORI ANALYSIS OF THE SPECTRAL ELEMENT DISCRETIZATION
OF HEAT EQUATION
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