IMA Journal of Numerical Analysis (2011) 31, 1038−1061
doi:10.1093/imanum/drp054
Advance Access publication on April 28, 2010
A quadratic finite volume element method for parabolic problems on
quadrilateral meshes
M IN YANG∗
Department of Mathematics, Yantai University, Yantai, Shandong 264005, China
∗ Corresponding author: yang@ytu.edu.cn
AND
J IANGGUO L IU
Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874, USA
liu@math.colostate.edu
In this paper we utilize affine biquadratic elements and a two-step temporal discretization to develop a
finite volume element method for parabolic problems on quadrilateral meshes. The method is proved to
have an optimal order convergence rate in L 2 (0, T ; H 1 (Ω)) under the ‘asymptotically parallelogram’
mesh assumption. Numerical experiments that corroborate the theoretical analysis are also presented.
Keywords: error estimation; finite volume element methods; parabolic equations; quadrilateral meshes;
second-order accuracy.
1. Introduction
In this paper we develop a second-order finite-volume method for the following model parabolic initial
boundary-value problem:

u t − ∇ ∙ (a(x)∇u) = f (x, t),



u = 0,



u(x, 0) = u 0 (x),
(x, t) ∈ Ω × (0, T ],
(x, t) ∈ ∂Ω × (0, T ],
(1.1)
x ∈ Ω,
where Ω is a bounded polygonal domain in R2 with boundary ∂Ω and x = (x, y). It is assumed that
f (x, t) ∈ L 2 (Ω) for t ∈ [0, T ] and a(x) is Lipschitz continuous and bounded almost everywhere with
positive lower and upper bounds a∗ and a ∗ . For the purpose of error analysis, we further assume that the
initial data u 0 ∈ H 3 (Ω) and the solution of (1.1) satisfies
u t ∈ L 1 (0, T ; H 3 (Ω)),
u tt ∈ L ∞ (0, T ; L 2 (Ω)),
u ttt ∈ L 1 (0, T ; L 2 (Ω)).
(1.2)
Finite-volume methods have been widely used in sciences and engineering, for example, computational fluid mechanics and petroleum reservoir simulations. The methods can be formulated in the
finite-difference framework, known as cell-centred methods, or in the Petrov–Galerkin framework, categorized as finite volume element methods. We refer to the monographs Eymard et al. (2000) and
Li et al. (2000) for general presentations of these methods and to the papers Bank & Rose (1987),
c The author 2010. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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[Received on 26 April 2009; revised on 17 September 2009]
A QUADRATIC FINITE VOLUME ELEMENT METHOD FOR PARABOLIC PROBLEMS
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Hackbusch (1989), Morton (1998), Ewing et al. (2002), Carstensen et al. (2005), Nicaise & Djadel
(2005), Ye (2006) and Sinha & Geiser (2007) (also the references therein) for more details. Compared
to finite-difference and finite-element methods, finite-volume methods are usually easier to implement
and offer flexibility in handling complicated domain geometries. More importantly, the methods ensure
local mass conservation, a highly desirable property in many applications.
Piecewise linear finite-volume methods have been extensively studied for various parabolic problems. The main idea is to treat the linear finite-volume methods as ‘small perturbations’ of the linear
finite-element methods. Consequently, techniques (such as a posteriori error estimates and superconvergence) for the finite elements can also be used for the finite-volume methods. In Chou & Li (2000)
a unified approach was presented to derive error estimates in the L 2 -, H 1 - and L ∞ -norms. Later, error
estimates and superconvergence results in the L p -norm (2 6 p < ∞) were obtained by Chou et al.
(2003). Finite volume element methods for parabolic problems in convex polygonal domains were studied by Chatzipantelidis et al. (2004) and error estimates in the H 1 -, L 2 - and L ∞ -norms under limited
regularities of exact solutions were established. In order to solve the discrete equations more efficiently,
Rui (2002) and Ma et al. (2003) constructed several symmetric finite-volume schemes. Ohlberger &
Rohde (2002) obtained a posteriori error estimates in the L 1 -norm for finite-volume approximations of
weakly coupled convection-dominated parabolic systems.
Compared to linear finite-volume methods, quadratic and higher-order elements differ considerably
from the corresponding finite-element methods. The development of higher-order finite-volume methods
is a challenging task and has been attracting many researchers’ attention in recent years. Plexousakis
& Zouraris (2004) performed an analysis of higher-order finite-volume methods for one-dimensional
elliptic problems. Cai et al. (2003) presented several higher-order finite-volume schemes from mixed
finite-element methods over rectangular meshes for elliptic problems. Based on different dual partitions,
two types of quadratic simplicial finite-volume schemes were discussed in Liebau (1996) and Li et al.
(2000) for two-dimensional elliptic problems, and both schemes were shown to be second-order accurate
in the H 1 -norm. Xu & Zou (2009) combined the above two schemes together and gave an analysis under
some weak conditions on grids. Error estimation of quadratic quadrilateral finite-volume methods for
elliptic problems was studied in Yang (2006). The analysis was then extended to three-dimensional right
quadrangular prism meshes in Yang et al. (2009).
The aforementioned papers on higher-order finite-volume schemes focused on stationary problems.
To the best of the authors’ knowledge, little progress has been made on time-dependent problems up to
now. The main difficulty is due to the fact that the inner product (∙, Ih∗ ∙) and the weak form ah (∙, Ih∗ ∙)
(see Section 2) of higher-order finite-volume methods are not symmetric even for constant coefficient
problems. Therefore it is difficult to handle temporal terms in theoretical analysis. How to control nonsymmetry of the related bilinear forms is generally still not clear.
It is well known that quadrilateral meshes have simpler mesh data structures than triangular meshes
but are more flexible than rectangular meshes in handling complicated domain geometries. Shown in
Fig. 1 are two examples of domains for which quadrilateral meshes are preferred: a compression channel
and a triangular obstacle in a flow (see Schroll & Svensson, 2006).
The purpose of this paper is to develop a second-order finite volume element method on quadrilateral
meshes for parabolic problems. It is a continuation of our efforts in Yang (2006) and Yang et al. (2009)
for two-dimensional and three-dimensional elliptic problems. However, the dual partition used in Yang
(2006) for elliptic problems no longer works for parabolic problems (Remark 3.1). Here we introduce a
new dual partition for two-dimensional parabolic problems. By exploring properties of affine mappings,
we manage to control nonsymmetry of the bilinear form (∙, Ih∗ ∙) (Lemma 3.5) and ensure coercivity of
the bilinear form ah (∙, Ih∗ ∙) (Lemma 3.8). Our new method also features a well-balanced combination
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M. YANG AND J. LIU
FIG. 1. Two examples that favour quadrilateral meshes.
2. A fully discrete finite-volume scheme on quadrilateral meshes
2.1
Notation
We shall use the standard notation for the Sobolev spaces W m, p (Ω) with the norm k ∙ km, p,Ω and the
seminorm | ∙ |m, p,Ω . We also denote W m,2 (Ω) by H m (Ω) and omit the index p = 2 and the domain
Ω when there is no ambiguity, i.e., kukm, p = kukm, p,Ω and kukm = kukm,2,Ω . The same convention
is adopted for the seminorms.
Let Ωh = {Q} be a quadrilateral partition of Ω, where any two closed quadrilaterals share a common
b = (0, 1)2 be the reference element in the x̂ ŷ-plane. For each element
edge, vertex or nothing. Let Q
b → Q satisfying (see Fig. 2)
Q ∈ Ωh there exists a bijective bilinear mapping FQ : Q
FQ ( P̂i ) = Pi ,
1 6 i 6 4.
Let JFQ be the Jacobian matrix of FQ at x̂ and J FQ = detJFQ . Accordingly, JF −1 is the Jacobian
Q
matrix of FQ−1 at x and J F −1 = detJF −1 . Based on the partition Ωh , we define Sh as the standard
Q
Q
conforming finite-element space of piecewise affine biquadratic functions
Sh = {v ∈ H01 (Ω): v| Q = v̂ ◦ FQ−1 , v̂| Qb is biquadratic ∀ Q ∈ Ωh ; v|∂Ω = 0}.
(2.1)
Before developing the finite volume element method, we make some assumptions on quadrilateral
meshes. For any Q ∈ Ωh let h Q be its diameter, let h 0Q be the smallest length of the edges and let θ Q be
any interior angle. We set h = max Q∈Ωh h Q .
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of a quadratic spatial approximation and a two-step temporal discretization. As an implicit method, it is
unconditionally stable (Theorem 4.1), allows relatively large time steps (Δt = O(h)) and can generate
numerical solutions that have a second-order accuracy in the energy norm (Theorem 4.2).
The rest of this paper is organized as follows. In Section 2 we introduce some notation and establish
a finite volume element scheme for the parabolic initial boundary-value problem (1.1). In Section 3
we prove some auxiliary results about quadrilateral meshes and the bilinear forms. A second-order error
estimate in the energy norm is proved in Section 4 under certain regularity conditions. Section 5 presents
numerical experiments to illustrate the theoretical analysis. The paper is concluded with some remarks
in Section 6.
Throughout this paper we use C (with or without subscripts) to denote a generic positive constant
that is independent of discretization parameters.
A QUADRATIC FINITE VOLUME ELEMENT METHOD FOR PARABOLIC PROBLEMS
1041
b → Q.
FIG. 2. An invertible bilinear mapping FQ : Q
Mesh assumption A. The partition Ωh = {Q} is regular, that is, there exist two positive constants σ
and γ such that
h Q / h 0Q 6 σ,
•
| cos θ Q | 6 γ < 1 ∀ Q ∈ Ωh .
(2.2)
Mesh assumption B. The quadrilateral meshes are ‘asymptotically parallelogram’, that is, for Q ∈
Ωh one has
ϑ Q = O(h Q ),
(2.3)
where ϑ Q = max(|π − θ1 |, |π − θ2 |), θ1 is the angle between the outward normals of two opposite
sides of Q and θ2 is the angle between the outward normals of the other two sides.
R EMARK 2.1 The asymptotically parallelogram assumption has been adopted by many authors, although it takes several different forms in the literature. In Arnold et al. (2002) it was called ‘asymptotically parallelogram’ and defined through the maximal angle difference of the outward normals. It is
referred to as ‘h 2 -parallelogram’ and defined through the edge deviation in Ewing et al. (1999). A definition based on the distance of the two diagonal midpoints was adopted by Süli (1992). A detailed
analysis on the equivalence of these different forms can be found in Chou & He (2002).
R EMARK 2.2 It should be pointed out that ‘h 2 -uniform quadrilateral meshes’, a closely related but
stronger assumption, was also used in Ewing et al. (1999) to prove some superconvergence results. The
‘h 2 -uniformness’ requires that any two adjacent quadrilaterals form an h 2 -parallelogram. This stronger
assumption was needed to obtain a good cancellation property and then the superconvergence results
(see Ewing et al., 1999, p. 782).
R EMARK 2.3 We may adopt a more general Assumption B that ϑ Q = O(h τQ ) for some τ > 0. Then
the main difference will appear in Lemma 3.5, i.e., |(u h , Ih∗ vh ) − (vh , Ih∗ u h )| 6 Ch τ ku h k0 kvh k0 , which
will be used in Theorems 4.1 and 4.2. Consequently, we shall assume that Δt = O(h τ ) to obtain an
optimal order convergence rate. Since the expected error is O(h 2 +Δt 2 ), a linear rate in h Q is a practical
assumption that avoids unnecessary complexities.
In order to describe the finite volume element scheme, we introduce a dual partition Ωh∗ , whose
elements are called control volumes. As shown in Fig. 3, each edge of Q ∈ Ωh is partitioned into
three segments so that the ratio of these segments is 1:4:1. We connect these partition points with line
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•
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M. YANG AND J. LIU
FIG. 3. A generic quadrilateral Q is partitioned into nine subregions.
R EMARK 2.4 The dual partition introduced here is different to the one used in Yang (2006), where the
sub-quadrilaterals were constructed based on the partition ratio 1:2:1. This new dual partition allows
us to control nonsymmetry of the bilinear form (∙, Ih∗ ∙) (see Lemma 3.5) and ensure coercivity of the
bilinear form ah (∙, Ih∗ ∙) (see Lemma 3.8). Both forms will be defined later.
2.2
Finite volume element scheme
Now we formulate the finite volume element method for the model problem (1.1). Given an interpolation
node z ∈ Zh0 , integrating the first equation in (1.1) over the associated control volume Vz and applying
Green’s formula, we obtain
Z
Z
Z
u t dx −
a∇u ∙ n ds =
f (x, t)dx,
(2.4)
Vz
∂ Vz
Vz
where n denotes the unit outer normal vector on ∂ Vz . The above formulation also states that we have an
integral conservation form on the control volume.
The integral form (2.4) can be further written in a variational form similar to the finite-element
method, with the help of a transfer operator Ih∗ : Sh → Sh∗ from the trial space to the test space defined
in Chou & Li (2000) by
X
v(z)Ψz ,
(2.5)
Ih∗ v =
z∈Zh0
where
n
Sh∗ = v ∈ L 2 (Ω): v|Vz is constant ∀ z ∈ Zh0 ; v|Vz = 0
and Ψz is the characteristic function of the control volume Vz .
∀ z ∈ ∂Ω
o
(2.6)
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segments to the corresponding points on the opposite edge. This way, each quadrilateral of Ωh is divided
into nine sub-quadrilaterals Q z , with z ∈ Zh (Q), where Z
Sh (Q) is the set of the vertices, the midpoints
of the edges and the centre of Q. For each node z ∈ Zh = Q∈Ωh Zh (Q) we associate a control volume
Vz , which is the union of the subregions Q z containing the node z. Therefore we obtain a collection of
control volumes covering the domain Ω. This is the dual partition Ωh∗ of the primal partition Ωh . We
denote the set of interior nodes of Zh by Zh0 .
A QUADRATIC FINITE VOLUME ELEMENT METHOD FOR PARABOLIC PROBLEMS
1043
We multiply (2.4) by vh (z) and sum over all z ∈ Zh0 to obtain
(u t , Ih∗ vh ) + ah (u, Ih∗ vh ) = ( f, Ih∗ vh )
∀ vh ∈ Sh ,
(2.7)
where the bilinear form ah (∙, Ih∗ ∙) is defined as follows. For any u ∈ H01 (Ω) and vh ∈ Sh we have
Z
X
∗
ah (u, Ih vh ) = −
vh (z)
a∇u ∙ n ds.
(2.8)
z∈Zh0
∂ Vz
Let N be a positive integer. For simplicity of presentation, we consider a uniform time step Δt =
T /N and set tn = nΔt (0 6 n 6 N ). Let
u n − u n−1
3 n
1 n−2
n
n
n
n−1
ˉ
ˉ
u = u(∙, tn ), ∂u =
, Du =
Δt.
+ u
u − 2u
Δt
2
2
ˉ n , Ih∗ vh ) + ah (u n , Ih∗ vh ) = ( f (tn ), Ih∗ vh ),
( Du
h
h
(2.9)
with the initial approximations u 0h and u 1h given by
u 0h = Rh u 0 ,
(2.10)
ˉ 1h , Ih∗ vh ) + ah (u 1h , Ih∗ vh ) = ( f (t1 ), Ih∗ vh )
(∂u
∀ vh ∈ Sh ,
(2.11)
where Rh : H01 (Ω) ∩ H 3 (Ω) → Sh is the elliptic (Ritz) projection defined by
ah (Rh u, Ih∗ vh ) = ah (u, Ih∗ vh )
∀ vh ∈ Sh .
(2.12)
R EMARK 2.5 The scheme (2.9) can be written as
3 n ∗
1 n−2 ∗
∗
∗
(u , I vh ) + Δtah (u nh , Ih∗ vh ) = 2(u n−1
h , Ih vh ) − (u h , Ih vh ) + Δt ( f (tn ), Ih vh ),
2 h h
2
n > 2.
Set A(∙, ∙) = 32 (∙, Ih∗ ∙) + Δtah (∙, Ih∗ ∙). From Lemmas 3.4, 3.6 and 3.8, we know that the bilinear form
A(∙, ∙) is bounded and coercive on Sh for sufficiently small h. The right-hand side of the above equation
n−2
defines a continuous linear functional on Sh , for given u n−1
and f (tn ). Therefore there exists a
h , uh
n
unique solution u h at each time step by the Lax–Milgram theorem (see Brenner & Scott, 2002).
3. Properties of quadrilateral meshes and the bilinear forms
In this section we prove some lemmas regarding properties of quadrilateral meshes and the bilinear
forms defined in Section 2. Let P1 and P2 be two points. We use P1 P2 to denote the line segment,
−−→
|P1 P2 | to denote its length and P1 P2 to denote the vector from point P1 to point P2 .
According to Lemmas 3.1–3.4 in Yang (2006), we have the following results.
L EMMA 3.1 Let Q = P1 P2 P3 P4 ∈ Ωh . Mesh assumption B is equivalent to the following condition:
−−→ −−→
| P1 P2 + P3 P4 | = O(h 2Q ).
(3.1)
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A fully discrete finite-volume scheme for (1.1) is formulated as follows. Find u nh ∈ Sh (2 6 n 6 N )
such that, for any vh ∈ Sh , we have
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M. YANG AND J. LIU
L EMMA 3.2 Suppose that Ωh satisfies Mesh assumption B, Q = P1 P2 P3 P4 ∈ Ωh and
|P3 P6 |
|P2 P5 |
=
= d,
|P2 P1 |
|P3 P4 |
where 0 6 d 6 1 is an arbitrary constant. Then we have
−−→ −−→
| P6 P5 + P1 P4 | = O(h 2Q ).
(3.2)
L EMMA 3.3 Under the same assumptions as in Lemma 3.2, we have
6
|6 P5 P6 P4 − 6 P4 P1 P5 | = O(h Q ),
(3.3)
P5 P6 P4 + 6 P6 P4 P1 = π + O(h Q ).
(3.4)
|u h |21,h =
=
where
X
Q∈Ωh
|kvh k|20 = (vh , Ih∗ vh ),
(3.5)
|u h |21,h,Q

X  X

Q∈Ωh
kvh k20,h = (Ih∗ vh , Ih∗ vh ),
X
δx u h xi+νi , j+ν j
νi = 12 ,1 ν j =0, 12 ,1
2
+
X
X
δ y u h xi+νi , j+ν j
νi =0, 12 ,1 ν j = 12 ,1

2 
 , (3.6)
δx u h xi+νi , j+ν j = u h xi+νi , j+ν j − u h xi+νi − 1 , j+ν j ,
2
δ y u h xi+νi , j+ν j = u h xi+νi , j+ν j − u h xi+νi , j+ν j − 1 .
2
The following lemma indicates that the discrete norms are equivalent to the corresponding L 2 -norm
or H 1 -seminorm.
L EMMA 3.4 Assume that Ωh satisfies Mesh assumption A. Then there exist positive constants C0 and
C1 , independent of h, such that, for any u h ∈ Sh , we have
C0 ku h k0 6 |ku h k|0 6 C1 ku h k0 ,
(3.7)
C0 ku h k0 6 ku h k0,h 6 C1 ku h k0 ,
(3.8)
C0 |u h |1 6 |u h |1,h 6 C1 |u h |1 .
(3.9)
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It is assumed that there exist a pair of integers n x and n y such that the cardinality of Ωh is equal to
n x n y , and we can assign each Q ∈ Ωh a pair of integers (i, j), where 0 6 i 6 n x −1 and 0 6 j 6 n y −1.
Thus we label Q with the subscript (i, j) and denote its vertices by xi, j , xi+1, j , xi+1, j+1 and xi, j+1 ,
corresponding to P1 , P2 , P3 and P4 in Fig. 2. Let νi , ν j = 0 or 12 . Then the midpoints of the edges of
Q are denoted by xi+νi , j+ν j , where νi + ν j = 12 ,and the centre of Q is denoted by xi+ 1 , j+ 1 .
2
2
We now define some discrete norms on Sh . For any u h ∈ Sh we have
A QUADRATIC FINITE VOLUME ELEMENT METHOD FOR PARABOLIC PROBLEMS
1045
Proof. Since the partition is regular, we have
1/2
6
kû h k L 2 ( Q)
−1
J
b
F L ∞ (Q)
Q
1/2
ku h k L 2 (Q) 6 J FQ L ∞ ( Q)
b ,
b kuˆh k L 2 ( Q)
where
J F −1 Q
Therefore
For
Q
L ∞ (Q)
JF 6 Ch −2
Q ,
Q
b
L ∞ ( Q)
6 Ch 2Q .
C −1 h Q kuˆh k L 2 ( Q)
b 6 ku h k L 2 (Q) 6 Ch Q kuˆh k L 2 ( Q)
b .
u h Ih∗ u h dx we have a similar estimate as follows:
Z
Z
Z
C −1 h 2Q
Ih∗ u h dx̂ 6
Ih∗ u h dx̂.
uˆh 2 [
u h Ih∗ u h dx 6 Ch 2Q
uˆh 2 [
b
Q
b
Q
Q
ϕ1 (x) = (x − 1)(2x − 1),
ϕ2 (x) = 4x(1 − x),
(3.10)
(3.11)
ϕ3 (x) = x(2x − 1)
be the local quadratic basis in one dimension. Let ψi (x), where 1 6 i 6 3, be the characteristic
functions associated with the partitions [0, 1/6], [1/6, 5/6] and [5/6, 1], respectively. Then, using the
standard tensor-product basis and the resulting interpolation form of uˆh on the reference element, we
immediately obtain
Z
T
Ih∗ u h dx̂ = u Q (G2 ⊗ G2 )uTQ ,
=
u
(G
⊗
G
)u
,
uˆh [
kuˆh k2L 2 ( Q)
Q 1
1 Q
b
b
Q
where u Q ∈ R9 is a vector consisting of the nodal values of u h on Q

2
4
"Z
#3
1
1 
 2 16
ϕi ϕ j dx
=
G1 =
30 
0
i, j=1
−1 2
G2 =
"Z
1
ϕi ψ j dx
0
#3
i, j=1
=

83
32
and

−1

2
,
4
1 
 32 368
648 
−7 32

−7

32 
.
83
Since the matrices G1 and G2 are symmetric and positive definite, it is not difficult to see that kuˆh k2L 2 ( Q)
b
R
∗
[
and Qb uˆh Ih u h dx̂ are equivalent. Applying (3.10) and (3.11) and summing the result over Ωh yields
estimate (3.7). Estimate (3.8) can be obtained similarly. Estimate (3.9) was proved in Yang (2006). The bilinear form (∙, Ih∗ ∙) is generally nonsymmetric. The next lemma measures how far it is from
being symmetric.
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Let
R
ku h k L 2 (Q) ,
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M. YANG AND J. LIU
L EMMA 3.5 Assume that Ωh satisfies Mesh assumptions A and B. For any u h , vh ∈ Sh we have
|(u h , Ih∗ vh ) − (vh , Ih∗ u h )| 6 Chku h k0 kvh k0 .
(3.12)
Proof. For any Q ∈ Ωh a change of variables in multiple integrals gives us
Z
Z
∗
∗
u h Ih vh dx =
uˆh Id
h vh J FQ dx̂.
Q̂
Q
A similar calculation to that in the proof of Lemma 3.4 yields
Z
T
∗
uˆh Id
m(Q)
h vh dx̂ = m(Q)v Q (G2 ⊗ G2 )u Q ,
Q̂
Q̂
(3.13)
Q̂
Direct calculations yield (see Fig. 2)
J FQ = 2A124 + 2(A123 − A124 )x̂ + 2(A134 − A124 ) ŷ,
0 6 x̂, ŷ 6 1,
where Ai jk denotes the area of the triangle with vertices Pi , P j and Pk . It is obvious that m(Q) =
J FQ 12 , 12 . Therefore, from Lemmas 3.1 and 3.3, we have
J F − m(Q) 6 |A123 − A124 | + | A134 − A124 |
Q
1
6 |P1 P2 | ||P2 P3 | sin(6 P1 P2 P3 ) − |P4 P1 | sin(6 P4 P1 P2 )|
2
1
+ |P4 P1 | ||P1 P3 | sin(6 P4 P1 P3 ) − |P3 P4 | sin(6 P3 P4 P1 )|
2
6 Ch 3Q .
Combining the above estimates, we obtain
Z
Z
Z
Z
∗
∗
∗
d
u h I ∗ vh dx −
vh Ih u h dx 6 u h Ih vh dx − m(Q)
uˆh Ih vh dx̂
h
Q
Q
Q
Q̂
Z
Z
Ih∗ u h dx̂
vˆh [
+ vh Ih∗ u h dx − m(Q)
Q
6 Ch 3Q
By (3.10), we have
Z
Q̂
Z
Q̂
Q̂
∗
|uˆh Id
h vh |dx̂ +
Z
Q̂
Ih∗ u h |dx̂ .
|vˆh [
2
∗
|uˆh Id
h vh |dx̂ 6 Ckuˆh k L 2 ( Q̂) kvˆh k L 2 ( Q̂) 6 C/ h Q ku h k L 2 (Q) kvh k L 2 (Q) ,
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where m(Q) is the measure of Q. Since the matrix G2 is symmetric, we have
Z
Z
∗ v dx̂ − m(Q)
uˆh Id
vˆh [
m(Q)
Ih∗ u h dx̂ = 0.
h h
A QUADRATIC FINITE VOLUME ELEMENT METHOD FOR PARABOLIC PROBLEMS
and hence
1047
Z
Z
∗
u h I ∗ vh dx −
vh Ih u h dx 6 Ch Q ku h k L 2 (Q) kvh k L 2 (Q) .
h
Q
Therefore
Q
|(u h , Ih∗ vh ) − (vh , Ih∗ u h )| 6
Z
X Z
∗
u h I ∗ vh dx −
vh Ih u h dx
h
Q∈Ωh
6C
X
Q∈Ωh
Q
Q
h Q ku h k L 2 (Q) kvh k L 2 (Q)
6 Chku h k0 kvh k0
by the Cauchy–Schwarz inequality.
which is not symmetric. Therefore
Z
Z
∗
d
m(Q)
Ih∗ u h dx̂ = m(Q)v Q (G2 ⊗ G2 − G2T ⊗ G2T )uTQ .
uˆh Ih vh dx̂ − m(Q)
vˆh [
Q̂
Q̂
If the maximum eigenvalue of the matrix G2 ⊗ G2 − G2T ⊗ G2T is O(h Q ) then we would obtain an almost
symmetric result as in Lemma 3.5. But a simple calculation gives


1
0
1
0
0 −8 0 −8 −6


8
0
8
0 −22 0 −1 0 −1




 0 −8 0
1
−6 −8 0
1
0





8
0
−1
0
−22
0
8
0
−1




1
T
T

6
22
6
22
0
22
6
22
6
G2 ⊗ G2 − G2 ⊗ G2 =
,

576 

−1 0
8
0 −22 0 −1 0
8




0
1
0 −8 −6
1
0 −8 0 





−1 0 −1 0 −22 0
8
0
8


0
1
0
1
−6 −8 0 −8 0
which could not produce a factor O(h Q ) as we expected! Then it is very difficult to measure how far
the bilinear form (∙, Ih∗ ∙) is from being symmetric based on the 1:2:1 dual partition ratio.
Downloaded from imajna.oxfordjournals.org at Colorado State University on July 22, 2011
R EMARK 3.1 As shown in the proof of Lemma 3.5, the symmetry of the matrix G2 is needed for (3.13)
to hold. The symmetry relies on the dual partition ratio 1:4:1 introduced in this paper. If the ratio of the
dual segments is set as 1:2:1 then one can verify that


1 −1
8

1 
 5 22 5  ,
G2 =

48 
−1 1
8
1048
M. YANG AND J. LIU
Let Ih : H01 (Ω) ∩ H 3 (Ω) → Sh be the usual nodal interpolation operator satisfying the approximation property (see, e.g., Brenner & Scott, 2002)
ku − Ih ukr 6 Ch 3−r kuk3 ,
0 6 r 6 2.
(3.14)
Ih∗ ∙).
Next we consider the continuity of the bilinear form ah (∙,
The proof is very similar to that for
Lemma 3.6 in Yang (2006). We provide a short one for completeness.
L EMMA 3.6 There exists a constant C > 0, independent of h, such that
|ah (u h , Ih∗ vh )| 6 Cku h k1 kvh k1
|ah (u − Ih u, Ih∗ vh )| 6 Ch 2 kuk3 kvh k1
∀ u h , vh ∈ Sh ,
(3.15)
∀ u ∈ H 3 (Ω), vh ∈ Sh .
(3.16)
Q∈Th z 1 ,z 2 ∈Z ∩Q
h
1
2
where z 1 and z 2 are chosen in Q with no repetition. It is obvious from (3.6) that
|vh (z 1 ) − vh (z 2 )| 6 |vh |1,h,Q .
It follows from the trace inequality (see, e.g., Theorem 1.6.6 in Brenner & Scott, 2002) that
Z
1/2
a∇w ∙ n ds 6 Ca ∗ h Q kwk H 1 (∂ Vz ∩∂ Vz )
1
2
∂ Vz ∩∂ Vz
1
2
1/2
1/2
1/2
6 Ch Q kwk H 1 (Q) kwk H 2 (Q) .
If w = u h then we use the inverse estimate ku h k H 2 (Q) 6 Ch −1
Q ku h k H 1 (Q) (see Brenner & Scott, 2002)
to get
Z
a∇u h ∙ n ds 6 Cku h k H 1 (Q) .
∂ Vz ∩∂ Vz
1
2
If w = u − Ih u then we use the approximation property (3.14) to get
Z
a∇(u − Ih u) ∙ n ds 6 Ch 2Q ku h k H 3 (Q) .
∂ Vz ∩∂ Vz
1
2
Combining the above estimates and using Lemma 3.4 gives
X
|ah (u h , Ih∗ vh )| 6 C
ku h k H 1 (Q) |vh |1,h,Q 6 Cku h k1 |vh |1
Q∈Th
and
|ah (u − Ih u, Ih∗ vh )| 6 C
as desired.
X
Q∈Th
h 2Q kuk H 3 (Q) |vh |1,h,Q 6 Ch 2 kuk3 |vh |1 ,
Downloaded from imajna.oxfordjournals.org at Colorado State University on July 22, 2011
Proof. Let w denote u h or u − Ih u. In view of definition (2.8), we reorder by edges to get
Z
X
X
∗
a∇w ∙ n ds ,
|ah (w, Ih vh )| =
(vh (z 1 ) − vh (z 2 ))
∂ Vz ∩∂ Vz
0
A QUADRATIC FINITE VOLUME ELEMENT METHOD FOR PARABOLIC PROBLEMS
1049
The following lemma about partitioned matrices was proved in Yang (2006) and will be used in this
paper for the proof of the coercivity of ah (∙, Ih∗ ∙).
A1 A2
B1 B2
and B =
be two partitioned matrices and let κ 6= 0 be
L EMMA 3.7 Let A =
B2 B1
A2T A1
A
κB
is positive definite if and only if the matrices
a constant. Then the matrix
κBT κ 2 A
"
B1 + B1T
A1 +
2
#
1
± [(A2 + A2T ) + (B2 + B2T )]
2
"
B1 + B1T
A1 −
2
#
1
± [(A2 + A2T ) − (B2 + B2T )]
2
and
L EMMA 3.8 Assume that Ωh satisfies Mesh assumptions A and B. There exists a constant C0 > 0,
independent of h, such that, for sufficiently small h, we have
ah (u h , Ih∗ u h ) > C0 ku h k21
∀ u h ∈ Sh .
(3.17)
Proof. We first study some properties of the auxiliary bilinear form
Z
X
aˉ h (u, Ih∗ vh ) = −
vh (z)
(a∇u)
ˉ
∙ n ds ∀ u ∈ H01 (Ω), vh ∈ Sh ,
(3.18)
∂ Vz
z∈Zh0
where a|
ˉ Q = a xi+ 1 , j+ 1 . We can rewrite (3.18) as
2
2
aˉ h (u, Ih∗ vh ) =
=
X
Q∈Ωh
aˉ Q,h (u, Ih∗ vh )
X Q∈Ωh
−
X
z∈Zh (Q)
vh (z)aˉ
Z
∂ Vz ∩Q
∇u ∙ n ds .
The Piola transformation maps a vector-valued function on Q̂ to one on Q by
v=
1
J FQ
JFQ v̂ ◦ FQ−1 .
This transformation has the following well-known property (see Brezzi & Fortin, 1991):
Z
Z
v ∙ n ds = v̂ ∙ n̂ dŝ,
e
ê
(3.19)
Downloaded from imajna.oxfordjournals.org at Colorado State University on July 22, 2011
are positive definite.
1050
M. YANG AND J. LIU
where s and ŝ denote the arc lengths along the edges e and ê, respectively, with n and n̂ as the unit
normal vectors. For any u h ∈ Sh let α = (ux,Q , u y,Q ) be a vector with ux,Q , u y,Q ∈ R6 defined by
ux,Q = δx u h xi+ 1 , j , δx u h xi+ 1 , j+ 1 , δx u h xi+ 1 , j+1 , . . . , δx u h (xi+1, j+1 ) ,
2
2
2
2
u y,Q = δ y u h xi, j+ 1 , δ y u h xi+ 1 , j+ 1 , δ y u h xi+1, j+ 1 , . . . , δ y u h (xi+1, j+1 ) .
2
2
2
2
Based on the Piola transformation, we integrate equation (3.19) along the edges of the reference element
to obtain
1
1
aˉ Q,h (u h , Ih∗ u h ) = αA α T = α(A + A T )α T ,
(3.20)
2
a xi+ 1 , j+ 1
2
2
which involves the partition matrix
A1
A2
A2
A1
#
.
Here the matrices (A1 )6×6 and (A 1 )6×6 represent the contraction distortion, while the matrices (A2 )6×6
and (A 2 )6×6 characterize the rotational distortion. Let
ϕ1 (x) = (x − 1)(2x − 1),
ϕ2 (x) = 4x(1 − x),
ϕ3 (x) = x(2x − 1)
be the local quadratic basis in one dimension. The entries of these matrices are specified as follows:
 Z
ϕ j ( ŷ)

|P12 P43 |2 ,
1 6 i, j 6 3,
 73

1

J
FQ 6 , ŷ
I

i


Z


ϕ j−3 ( ŷ)

2
1

1 6 i, j − 3 6 3,

− 3 I JF 1 , ŷ |P12 P43 | ,
Q 6
i
Z
(A1 )i j =

ϕ j ( ŷ)
1

|P21 P34 |2 , 1 6 i − 3, j 6 3,

−

3
1

,
ŷ
J
F

I
6

 Z i−3 Q



ϕ j−3 ( ŷ)

|P21 P34 |2 ,
 73
4 6 i, j 6 6,
1
Ii−3 J FQ
6 , ŷ
 Z
−−−→
−−−→
(4 ŷ−3)(1−ŷ) −−→ −
ŷ −−→ −

P1 P2 ∙ P12 P43 + (4 ŷ−3)
P4 P3 ∙ P12 P43 ,
Cj


1
1

J FQ 6 , ŷ
Ii J FQ 6 , ŷ



Z


−−−→
−−−→

(4 ŷ−1)(1−ŷ) −−→ −
(4 ŷ−1) ŷ −−→ −


−C j−3 I JF 1 , ŷ P1 P2 ∙ P12 P43 + JF 1 , ŷ P4 P3 ∙ P12 P43 ,
Q 6
Q 6
i
Z
(A2 )i j =

−
−
→
−
−
−
→
−−−→
−
(4
ŷ−1)(1−
(4
ŷ −−→ −
ŷ)

P1 P2 ∙ P21 P34 + ŷ−1)

P4 P3 ∙ P21 P34 ,
−C j

5
5

J
J
,
ŷ
,
ŷ
FQ 6
FQ 6

Ii−3


Z


−
−
−
→
−
−
−
→
−−−→

(4 ŷ−3)(1−ŷ)
ŷ −−→ −

C j−3
P1 P2 ∙ P21 P34 + (4 ŷ−3)
P4 P3 ∙ P21 P34 ,
5
5
Ii−3
J FQ
6 , ŷ
J FQ
6 , ŷ
1 6 i, j 6 3,
1 6 i, j − 3 6 3,
1 6 i − 3, j 6 3,
4 6 i, j 6 6,
where C1 = C2 = 5/9 and C3 = −1/9. The integral intervals are I1 = [0, 1/6], I2 = [1/6, 5/6] and
I3 = [5/6, 1]. The matrices A 1 and A 2 take a similar form to those above.
Downloaded from imajna.oxfordjournals.org at Colorado State University on July 22, 2011
A=
"
A QUADRATIC FINITE VOLUME ELEMENT METHOD FOR PARABOLIC PROBLEMS
1051
FIG. 4. A figure for Lemmas 3.1–3.3.
with
A˜1 =
"
A˜12
A˜11
A˜12
A˜11
#
,
A˜2 =
"
A˜21
A˜22
A˜22
A˜21
#
and
A˜11 =

83
32
7 
 32 368
1944
−7 32



−7

32  ,
83
−20 −20
4

83
32
−1 
A˜12 =
 32 368
1944
−7 32
cos θ Q 

A˜21 =
−30 −30 6  ,
81
5
5
−1
The minimum principal major of the matrices

5

32  ,
83
5
cos θ Q 
A˜22 =
−30 −30
81
−20 −20
T
A˜21 + A˜21
A˜11 +
2
!
T
A˜22 + A˜22
± A˜12 +
2
!
T
A˜21 + A˜21
A˜11 −
2
!
T
A˜22 + A˜22
± A˜12 −
2
!
and

−7
is
50
50
85
−
| cos(θ Q )| −
| cos2 (θ Q )|.
729 729
2916

−1

6 .
4
Downloaded from imajna.oxfordjournals.org at Colorado State University on July 22, 2011
We first investigate an auxiliary matrix A˜ for a parallelogram. Without loss of generality, we choose
θ Q = 6 P4 P1 P2 (see Fig. 4). Let κ = |P1 P4 |/|P1 P2 |. Note that the Jacobian determinant J FQ is a
constant for a parallelogram. Then
"
#
A˜1
κ A˜2
|P1 P2 |2 ˜
˜
˜
A=
M, M =
m(Q)
κ A˜2 κ 2 A˜1
1052
M. YANG AND J. LIU
Let
F(α) =
50
50
85 2
−
α−
α .
729 729
2916
It is not difficult to verify that F(α) is monotone decreasing for α ∈ [0, 1] and has a root α0 ∈ (0, 1)
(approximately 0.75667). We can choose γ < α0 in (2.2) so that F(| cos(θ Q )|) > 0. Then the principal
minors of the above matrices are all positive, and hence these matrices are positive definite. By Lemma
3.7, we know that the matrix M˜ itself is positive definite as well. The minimum eigenvalue λγ > 0 of
the matrix M˜ depends only on the constant γ . From Mesh assumption A we have
!
2λγ
A˜ + A˜T
|P1 P2 |2
λmin
> λγ
>
.
(3.21)
2
m(Q)
σ
˜
˜T
Now we consider the difference between the matrix A +2A on a parallelogram and the matrix
we have
T
˜
˜T
on an asymptotically parallelogram. Let D = A +2A − A +2A . By Lemmas 3.2 and 3.3,
|P12 P43 | = |P21 P34 | = |P1 P4 | + O(h 2Q ),
|P14 P23 | = |P41 P32 | = |P1 P2 | + O(h 2Q )
(3.22)
and
−−→ −−−−→ −−→ −−−−→ −−→ −−−−→ −−→ −−−−→
P1 P2 ∙ P12 P43 = P4 P3 ∙ P12 P43 = P1 P2 ∙ P21 P34 = P4 P3 ∙ P21 P34
−−→ −−−−→ −−→ −−−−→ −−→ −−−−→ −−→ −−−−→
= P1 P4 ∙ P14 P23 = P2 P3 ∙ P14 P43 = P1 P4 ∙ P41 P32 = P2 P3 ∙ P41 P32
= |P1 P2 ||P1 P4 | cos θ Q cos h Q + O(h 3Q ).
By (2.3), (3.22) and (3.23), we can verify that, when h Q is small enough, we have
|(D)i j | 6 C
(1 − cos h Q )h 2Q + h 3Q
, 1 6 i, j 6 12.
m(Q)
p
Mesh assumption A implies that h 2Q /m(Q) 6 2σ/ 1 − γ 2 . So we have
|(D)i j | 6 C(1 − cos h Q + h Q ),
1 6 i, j 6 12,
and hence
λmax (D) 6 kDk∞ 6 12C(1 − cos h Q + h Q ) 6 Ch Q 6 Ch.
Combining the above results with (3.20), we obtain
A + AT
αT
aˉ Q,h (u h , Ih∗ u h ) = a xi+ 1 , j+ 1 α
2
2
2
= a xi+ 1 , j+ 1 α[(A˜ + A˜T )/2 + D]α T
2
2
(3.23)
Downloaded from imajna.oxfordjournals.org at Colorado State University on July 22, 2011
A +A T
2
A QUADRATIC FINITE VOLUME ELEMENT METHOD FOR PARABOLIC PROBLEMS
1053
> a∗ [λmin ((A˜ + A˜T )/2) − λmax (D)]αα T
= a∗ [λmin ((A˜ + A˜T )/2) − λmax (D)]|u h |21,h,Q
> a∗ (2λγ /σ − Ch)|u h |21,h,Q .
Since a(x) is Lipschitz continuous, we have (see Yang, 2006)
|a(u h , Ih∗ u h ) − a(u
ˉ h , Ih∗ u h )| 6 Ch|u h |21 .
(3.24)
Applying Poincaré’s inequality, we obtain for sufficiently small h that
X
ˉ h , Ih∗ u h )]
aˉ Q,h (u h , Ih∗ u h ) + [a(u h , Ih∗ u h ) − a(u
ah (u h , Ih∗ u h ) =
Q∈Ωh
X
Q∈Ωh
(2λγ /σ − Ch)|u h |21,h,Q − Ch|u h |21
> C0 ku h k21 ,
which completes the proof.
4. Stability and convergence analysis
In this section we present a stability and convergence analysis for the finite volume element method.
First, we derive some elementary inequalities that will be used in the proofs of the two main theorems.
For any u h , vh ∈ Sh one has
0 6 |ku h − vh k|20 = (u h − vh , Ih∗ (u h − vh ))
= |ku h k|20 + |kvh k|20 − (u h , Ih∗ vh ) − (vh , Ih∗ u h ).
(4.1)
Together with Lemma 3.5, this implies that
1
(u h , Ih∗ vh ) 6 (|ku h k|20 + |kvh k|20 + (u h , Ih∗ vh ) − (vh , Ih∗ u h ))
2
Ch
1
ku h k0 kvh k0 .
6 (|ku h k|20 + |kvh k|20 ) +
2
2
(4.2)
From (4.1) we have
|ku h + vh k|20 = |ku h k|20 + |kvh k|20 + (u h , Ih∗ vh ) + (vh , Ih∗ u h )
6 2(|ku h k|20 + |kvh k|20 ).
(4.3)
The following theorem indicates that the finite volume element method is stable in the L 2 -norm.
T HEOREM 4.1 Let u nh be the numerical solution of the finite-volume scheme (2.9)–(2.11). There exists
a positive constant C, independent of the discretization parameters, such that
!
M
X
M
T (h/Δt+1)
0
ku h k0 6 C e
k f (tn )k0 , 1 6 M 6 N .
ku h k0 + Δt
(4.4)
n=1
Downloaded from imajna.oxfordjournals.org at Colorado State University on July 22, 2011
> a∗
1054
M. YANG AND J. LIU
ˉ n =
Proof. Introducing two difference operators δi u nh = u nh − u n−i
for i = 1, 2, we rewrite Δt Du
h
h
n
n
1
2δ1 u h − 2 δ2 u h . Note that, for i = 1, 2, we have
n−i
∗ n
2(δi u nh , Ih∗ u nh ) = δi |ku nh k|20 + |kδi u nh k|20 + [(u nh , Ih∗ u n−i
h ) − (u h , Ih u h )],
2
where δi |ku nh k|20 = |ku nh k|20 − |ku n−i
h k|0 . So we have
ˉ n , Ih∗ u n ) = δ1 |ku n k|20 − 1 δ2 |ku n k|20 + |kδ1 u n k|20 − 1 |kδ2 u n k|20
Δt ( Du
h
h
h
h
h
h
4
4
1 n ∗ n−2
n−1 ∗ n
n−2 ∗ n
+ [(u nh , Ih∗ u n−1
h ) − (u h , Ih u h )] − [(u h , Ih u h ) − (u h , Ih u h )].
4
(4.5)
It is clear that
n=2
1
δ1 |ku nh k|20 − δ2 |ku nh k|20
4
=
3 M 2 1 M−1 2 3 1 2 1 0 2
|ku k| − |ku
k|0 − |ku h k|0 + |ku h k|0 .
4 h 0 4 h
4
4
(4.6)
Since δ2 u nh = δ1 u nh + δ1 u n−1
h , it follows from (4.3) that
M X
n=2
M
1
1X
2
(|kδ1 u nh k|20 − |kδ1 u n−1
|kδ1 u nh k|20 − |kδ2 u nh k|20 >
h k|0 )
4
2
n=2
1
= (|kδ1 u hM k|20 − |kδ1 u 1h k|20 ).
2
(4.7)
By Lemma 3.5, we have
M X
n=2
1 n ∗ n−2
n−1 ∗ n
n−2 ∗ n
)
−
(u
,
I
u
)]
−
,
I
u
)
−
(u
,
I
u
)]
[(u nh , Ih∗ u n−1
[(u
h h
h h
h
h
h
4 h h h
6 Ch
M
X
n=2
n−2
n
(ku n−1
h k0 + ku h k0 )ku h k0 .
(4.8)
Since |ku hM−1 |k20 6 2|ku hM k|20 + 2|kδ1 u hM k|20 , we have from (4.5)–(4.8) that
Δt
M
X
n=2
ˉ n , Ih∗ u n ) > 3 |ku hM k|20 − 1 |ku M−1 k|20 − 3 |ku 1h k|20 + 1 |ku 0 k|20
( Du
h
h
4
4 h
4
4 h
M
X
1
n−2
n
+ (|kδ1 u hM k|20 − |kδ1 u 1h k|20 ) − Ch
(ku n−1
h k0 + ku h k0 )ku h k0
2
n=2
1
3
1
1
> |ku hM k|20 − |ku 1h k|20 + |ku 0h k|20 − |kδ1 u 1h k|20
4
4
4
2
− Ch
M
X
n=2
n−2
n
(ku n−1
h k0 + ku h k0 )ku h k0 .
(4.9)
Downloaded from imajna.oxfordjournals.org at Colorado State University on July 22, 2011
M X
A QUADRATIC FINITE VOLUME ELEMENT METHOD FOR PARABOLIC PROBLEMS
1055
On the other hand,
ˉ 1h , Ih∗ u 1h ) > 1 (|ku 1h k|20 − |ku 0 k|20 ) + 1 |kδ1 u 1h k|20 − Chku 0 k0 ku 1h k0 .
Δt (∂u
h
h
2
2
(4.10)
Combining (4.9) and (4.10) yields
ˉ 1h , Ih∗ u 1h ) + Δt
Δt (∂u
M
X
n=2
ˉ n , Ih∗ u n ) > 1 |ku hM k|20 − 1 |ku 1h k|20 − 1 |ku 0h k|20
( Du
h
h
4
4
4
− Ch
M
X
n=1
n
ku n−1
h k0 ku h k0 − Ch
M
X
n=2
n
ku n−2
h k0 ku h k0 . (4.11)
From (2.9) and (2.11) we have
M
X
n=2
ˉ n , Ih∗ u n ) + Δt
( Du
h
h
6 Δt
M
X
n=1
M
X
ah (u nh , Ih∗ u nh )
n=1
k f (tn )k0 kIh∗ u h k0 6 C1 Δt
M
X
n=1
k f (tn )k0 ku nh k0 .
Applying (3.7) and (4.11), we obtain
M
M
n=1
n=1
X
X
1 M 2
1
1
n−2
n
|ku h k|0 + Δt
ah (u nh , Ih∗ u nh ) 6 |ku 1h k|20 + |ku 0h k|20 + Ch
(ku n−1
h k0 + ku h k0 )ku h k0
4
4
4
+ C1 Δt
M
X
n=1
k f (tn )k0 ku nh k0
M
X
1
1
n
6 |ku 1h k|20 + |ku 0h k|20 + Ch
|ku n−1
h k|0 |ku h k|0
4
4
n=1
+ Ch
M
X
n=2
n
|ku n−2
h k|0 |ku h k|0 + CΔt
M
X
n=1
k f (tn )k0 |ku nh k|0 .
Let J be such that |ku hJ k|0 = max16n 6 M |ku nh k|0 . Then
|ku hJ k|20
6
|ku hM k|0
6
|ku 1h k|0
+ |ku 0h k|0
+ Ch
M
X
n=1
|ku n−1
h k|0
+ CΔt
M
X
n=1
!
k f (tn )k0 |ku hJ k|0 .
Therefore
|ku hJ k|0
6
|ku 1h k|0
+ |ku 0h k|0
+ Ch
M
X
n=1
|ku n−1
h k|0
+ CΔt
M
X
n=1
k f (tn )k0 .
Downloaded from imajna.oxfordjournals.org at Colorado State University on July 22, 2011
ˉ 1h , Ih∗ u 1h ) + Δt
Δt (∂u
1056
M. YANG AND J. LIU
Using (4.2), we have
|ku 1h k|0 6 (1 + Ch)|ku 0h k|0 + CΔtk f (t1 )k0 .
Applying the discrete Gronwall inequality yields
|ku hM k|0 6 C e M(h+Δt) |ku 0h k|0 + Δt
6Ce
T(h/Δt+1)
|ku 0h k|0
M
X
n=1
+ Δt
k f (tn )k0
M
X
n=1
!
!
1 6 M 6 N.
k f (tn )k0 ,
The proof is completed by another use of (3.7).
u nh
ku M − u hM k0 + Δt
M
X
n=0
ku n − u nh k21
! 12
6 C(Δt 2 + h 2 ),
0 6 M 6 N,
(4.12)
for sufficiently small h and Δt, where the positive constant C is independent of the discretization
parameters.
Proof. We decompose the error as u nh − u n = ξ n − ηn , where ξ n = u nh − Rh u n and ηn = u n − Rh u n ,
with Rh being the elliptic projection defined in (2.12). By (2.7), (2.9) and (2.11), we have the following
error equations:
ˉ n , Ih∗ vh ) + ah (ξ n , Ih∗ vh ) = (ωn , Ih∗ vh ),
( Dξ
n > 2,
ˉ 1 , Ih∗ vh ) + ah (ξ 1 , Ih∗ vh ) = (ω1 , Ih∗ vh ),
(∂ξ
where
ˉ n + u nt − Du
ˉ n,
ωn = Dη
n > 2,
ˉ 1 + u 1t − ∂u
ˉ 1.
ω1 = ∂η
By Lemma 3.8, definition (2.12) and the Galerkin orthogonality, we have
C0 kIh u − Rh uk21 6 ah (Ih u − Rh u, Ih∗ (Ih u − Rh u))
= ah (Ih u − u, Ih∗ (Ih u − Rh u)).
It follows from Lemma 3.6 that
kIh u − Rh uk1 6 Ch 2 kuk3 .
(4.13)
Using (3.14) and a triangle inequality, we obtain
kηn k1 6 ku n − Ih u n k1 + kIh u n − Rh u n k1 6 Ch 2 ku n k3 .
(4.14)
Downloaded from imajna.oxfordjournals.org at Colorado State University on July 22, 2011
be the numerical solution of the finite-volume
T HEOREM 4.2 Let u be the solution of (1.1) and let
scheme (2.9)–(2.11). Assume that (1.2) and Mesh assumptions A and B hold. If Δt = O(h) then we
have the following error estimate in the energy norm:
A QUADRATIC FINITE VOLUME ELEMENT METHOD FOR PARABOLIC PROBLEMS
1057
Suppose that J is such that |kξ J k|0 = max16n 6 M |kξ n k|0 , where 1 6 M 6 N . Since ξ 0 = 0, a
similar argument as that in the proof of Theorem 4.1 leads to
M
M
M
n=1
n=1
n=1
X
X
X
1 J 2
|kξ k|0 + C0 Δt
kξ n k21 6 Ch
kξ n−1 k0 kξ n k0 + CΔt
kωn k0 |kξ J k|0 + CΔt 2 kω1 k20
4
M
X
6 C(h + Δt)
n=1
|kξ n k|20
+ 8C Δt
M
X
n=1
n
kω k0
!2
1
+ |kξ J k|20 + CΔt 2 kω1 k20 .
8
(4.15)
Applying Taylor’s expansion with the remainder term in integral form, we get
M
X
n=2
kωn k0 6 Δt
6C
M
X
n=2
ˉ n k0 + ku nt − Du
ˉ n k0 )
(k Dη
M Z
X
n=2
6C
Z
6C h
tn
tn−2
kηt k0 dτ + Δt 2
T
kηt k0 dτ + Δt
0
2
Z
2
Z
T
0
ku t k3 dτ + Δt
Z
tn
tn−2
ku ttt k0 dτ
T
ku ttt k0 dτ
0
2
Z
T
0
ku ttt k0 dτ .
(4.16)
Similarly, we have
ˉ 1 k0 + Δtku 1t − ∂u
ˉ 1 k0
Δtkω1 k0 6 Δtk∂η
Z Δt
Z Δt
6 Ch 2
ku t k3 dτ + Δt
ku tt k0 dτ.
0
(4.17)
0
Combining (4.15)–(4.17), we obtain
|kξ
J
k|20
+ Δt
M
X
n=1
kξ n k21
6 C(h + Δt)
M
X
n=1
|kξ n k|20 + Cu (Δt 4 + h 4 ),
where Cu = C(ku t k L 1 (0,T ;H 3 (Ω)) + ku tt k L ∞ (0,T ;L 2 (Ω)) + ku ttt k L 1 (0,T ;L 2 (Ω)) )2 .
To balance spatial and temporal errors it is sufficient to have h = O(Δt), which allows us to use
(3.7) and the discrete Gronwall inequality to derive
kξ M k20
+ Δt
M
X
n=1
kξ n k21
6
C0−2 |kξ J k|20
+ Δt
M
X
n=1
kξ n k21 6 C(Δt 4 + h 4 ),
M > 1,
Downloaded from imajna.oxfordjournals.org at Colorado State University on July 22, 2011
Δt
1058
M. YANG AND J. LIU
for sufficiently small h and Δt. By (4.14), we obtain
kη (t)k1 6 Ch ku k3 6 Ch
n
2
n
2
ku 0 k3 +
Z
t
0
ku t k3 dτ .
Combining the above two estimates, we get the desired estimate.
5. Numerical experiments
xi, j =
π
π
i+
sin( j)rand(),
M
4M
yi, j =
π
π
j+
sin(i)rand(),
M
4M
1 6 i, j 6 M − 1,
where sin(i) and sin( j) use the radian unit for angles and rand() generates a random number in (0, 1). An
implementation of the random function rand() depends on the computer platform used but is available
in the standard library of most C/C++ compilers.
Given a spatial mesh parameter h = π/M, the time step size is chosen as Δt = T /(2π )h. We
use the first-order backward Euler as a starter scheme (see, e.g., Thomée, 2006). The preconditioned
bi-conjugate gradient stabilized method is employed to solve the nonsymmetric discrete linear systems.
FIG. 5. An example for asymptotically parallelogram quadrilateral meshes (used in our numerical experiments).
Downloaded from imajna.oxfordjournals.org at Colorado State University on July 22, 2011
In this section we present numerical experiments to illustrate the theoretical results presented in the
previous sections. The domain is Ω = [0, π ]2 , the permeability coefficient a(x, y) = (x + 1)2 + y 2 ,
the exact solution u(x, y, t) = e−0.1t sin(x) sin(y) and the final time T = 0.2. The right-hand side
f (x, y, t) is computed accordingly.
We test the finite-volume method first on a family of rectangular meshes (see Fig. 5) that have
M = 4, 8, 16, 32 or 64 partitions in both the x- and y-directions and then on a set of quadrilateral
meshes generated by
A QUADRATIC FINITE VOLUME ELEMENT METHOD FOR PARABOLIC PROBLEMS
1059
TABLE 1 Errors and convergence rates on the rectangular meshes
M
4
8
16
32
64
kIh u N − u hN kl ∞
8.563 × 10−3
2.074 × 10−3
5.138 × 10−4
1.281 × 10−4
3.200 × 10−5
|Ih u N − u hN |1,h
Rate
2.202 × 10−2
—
2.04
2.01
2.00
2.00
ku nh k E
Rate
—
1.98
1.99
1.99
2.00
5.546 × 10−3
1.388 × 10−3
3.471 × 10−4
8.676 × 10−5
Rate
2.032 × 10−2
—
1.95
1.99
2.00
1.99
ku nh k E
Rate
5.226 × 10−3
1.312 × 10−3
3.281 × 10−4
8.241 × 10−5
TABLE 2 Errors and convergence rates on the quadrilateral meshes
M
8.089 × 10−3
2.274 × 10−3
5.483 × 10−4
1.400 × 10−4
3.426 × 10−5
|Ih u N − u hN |1,h
Rate
Rate
1.945 × 10−2
—
1.83
2.05
1.96
2.03
—
1.73
2.00
1.97
1.99
5.883 × 10−3
1.466 × 10−3
3.718 × 10−4
9.316 × 10−5
2.126 × 10−2
6.397 × 10−3
1.596 × 10−3
4.072 × 10−4
1.024 × 10−4
—
1.73
2.00
1.97
1.99
The tolerance for residuals is set as 10−11 and simple diagonal preconditioning is used. The numerical
order of convergence is then measured by comparing the computed errors on two successive mesh levels.
From the proof of Theorem 4.2 we know that the second-order error estimate in the energy norm is
essentially determined by the second-order convergence in Δt and h of the following quantity:
ku hN k E
= |kIh u
N
− u hN k|0
+ Δt
N
X
n=1
|Ih u
n
− u nh |21,h
!1/2
,
due to the approximation property (3.14), the norm equivalences (3.7) and (3.9) and the estimate (4.13).
We thus measure ku nh k E in the numerical experiments in lieu of the energy norm defined in (4.12). The
numerical results listed in Tables 1 and 2 clearly reflect the second-order convergence of ku nh k E for both
the rectangular and the quadrilateral meshes, which agrees very well with our theoretical findings. As a
by-product, we can also obtain second-order accuracy in the max-norm kIh u N − u hN kl ∞ and the discrete
H 1 -norm |Ih u N − u hN |1,h .
6. Concluding remarks
In this paper we presented a quadratic finite volume element method for parabolic problems on quadrilateral meshes. A suitable dual partition was introduced to control the nonsymmetry of the bilinear forms.
A second-order convergence in the L 2 (0, T ; H 1 (Ω))-norm was derived and verified in numerical experiments. However, there are technical difficulties in deriving error estimates in L ∞ (0, T ; H 1 (Ω)) and
L ∞ (0, T ; L 2 (Ω)) similar to those for the finite-element methods. The main reason for this deficiency is
that we are not able to derive the optimal L 2 -norm error estimate for the Ritz projection for the bilinear
form ah (∙, Ih∗ ∙). Such a deficiency exists on triangular and quadrilateral meshes (see Liebau, 1996; Li
et al., 2000; Yang, 2006; Xu & Zou, 2009). How to obtain the optimal L 2 -norm error estimates for finite
volume element methods based on multidimensional higher-order elements is still an open problem.
Downloaded from imajna.oxfordjournals.org at Colorado State University on July 22, 2011
4
8
16
32
64
kIh u N − u hN kl ∞
1060
M. YANG AND J. LIU
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