Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 385463, 8 pages
http://dx.doi.org/10.1155/2013/385463
Research Article
Finite Element Multigrid Method for the Boundary Value
Problem of Fractional Advection Dispersion Equation
Zhiqiang Zhou and Hongying Wu
Department of Mathematics, Huaihua University, Huaihua 418008, China
Correspondence should be addressed to Zhiqiang Zhou; zhiqzhou@263.net
Received 2 March 2013; Accepted 14 May 2013
Academic Editor: Morteza Rafei
Copyright © 2013 Z. Zhou and H. Wu. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The stationary fractional advection dispersion equation is discretized by linear finite element scheme, and a full V-cycle multigrid
method (FV-MGM) is proposed to solve the resulting system. Some useful properties of the approximation and smoothing
operators are proved. Using these properties we derive the convergence results in both 𝐿2 norm and energy norm for FV-MGM.
Numerical examples are given to demonstrate the convergence rate and efficiency of the method.
1. Introduction
We investigate the finite element full V-cycle multigrid
method (FV-MGM) to the boundary value problem of linear
stationary fractional advection dispersion equation (FADE).
Problem 1. Given Ω = (0, 1), 𝑓 : Ω → R, find 𝑢 : Ω → R
such that
1
−𝛽
− 𝐷 ( 0 𝐷𝑥−𝛽 + 𝑥 𝐷1 ) 𝐷𝑢 + 𝑐 (𝑥) 𝑢 = 𝑓 (𝑥) , in Ω,
2
(1)
𝑢 = 0,
on 𝜕Ω,
where 0 < 𝛽 < 1, 𝑐(𝑥) ≥ 0, 𝐷 represents a single spatial
−𝛽
derivative, and 0 𝐷𝑥−𝛽 and 𝑥 𝐷1 represent Riemann-Liouville
left and right fractional integral operators, respectively [1–3].
FADE has been used in modeling physical phenomena
exhibiting anomalous diffusion, that is, diffusion not accurately modeled by the usual advection dispersion equation
[4–7]. For example, solutes moving through aquifers do
not generally follow a Fickian, second-order, and governing
equation because of large deviations from the stochastic
process of Brownian motion [8–10]. Many scholars developed
numerical methods, including finite difference method [11],
finite element method [12–14], spectral method [15] and
moving collocation method [16] to solve FADEs. Most of
them used Gauss elimination method or conjugate gradient
norm residual method to solve the resulting system, so the
computational complexity is O(𝑁3 ) or O(𝑁 log2 𝑁). To date
only a few of them considered MGM numerical methods. For
example, Pang and Sun [11] developed an MGM to solve the
linear system with Toeplitz-like structure, but the fractional
derivatives are defined in the Grünwald-Letnikov form and
the discretized system is obtained by difference scheme. It
motivates us to design a fast MGM algorithm to deal with FE
equations of FADE.
In this paper, we follow the ideas in [17, 18] to develop
a FV-MGM for solving the resulting system of Problem 1
discretized by linear finite element method. By selecting
appropriate iteration operator and iteration numbers, we
prove that FV-MGM has the same convergence rate as classic
FEM and the computational cost increases linearly with
respect to the increasing of unknown variables.
The remaining of this paper is organized as follows. The
next section recalls the variational formulation of the above
FADE and corresponding convergence results. In Section 3
we describe the multigrid algorithm, estimate the spectral
radius of FE equations, show the properties of approximation
and soothing operators, and prove the convergence theorems
for FV-MGM. Numerical examples demonstrating convergence rate and computational work are presented in Section 4.
Concluding remarks are given in the final section.
2
Journal of Applied Mathematics
2. Variational Formulation and
Convergence Results
Let 𝛼 := (2 − 𝛽)/2, so that 1/2 < 𝛼 ≤ 1. Define the associate
bilinear form 𝐵 : 𝐻0𝛼 (Ω) × 𝐻0𝛼 (Ω) → R as
𝐵 (𝑢, V) :=
1
⟨ 𝐷−𝛽 𝐷𝑢, 𝐷V⟩
2 0 𝑥
+
1
−𝛽
⟨ 𝐷 𝐷𝑢, 𝐷V⟩ + (𝑐 (𝑥) 𝑢, V) ,
2 𝑥 1
(2)
and linear functional 𝐹 : 𝐻0𝛼 (Ω) → R as
𝐹 (V) := ⟨𝑓, V⟩ ,
Theorem 3 (see [17, Corollary 4.3]). Let 𝑢 ∈ 𝐻0𝛼 (Ω) ⋂
𝐻𝑟 (Ω) (𝛼 ≤ 𝑟 ≤ 𝑚) solve (4) and 𝑢ℎ solve (9). Then there
exists a constant 𝐶3 > 0 such that 𝑒 = 𝑢 − 𝑢ℎ satisfies
‖𝑒‖𝐻𝛼 (Ω) ≤ 𝐶3 ℎ𝑘𝑟−𝛼 ‖𝑢‖𝐻𝑟 (Ω) .
(10)
Theorem 4 (see [17, Theorem 4.4]). Let 𝑢 ∈ 𝐻0𝛼 (Ω) ⋂
𝐻𝑟 (Ω) (𝛼 ≤ 𝑟 ≤ 𝑚) solve (4) and 𝑢ℎ solve (9), where 𝑚−1 is the
degree of Galerkin finite element model. Then, if the regularity
of the solution to the adjoint problem is satisfied, there exists a
constant 𝐶4 (or 𝐶4󸀠 ) such that the error 𝑒 = 𝑢 − 𝑢ℎ satisfies
3
𝛼 ≠ ,
4
‖𝑒‖𝐿2 (Ω) ≤ 𝐶4 ℎ𝑘𝛼 ‖𝑢‖𝐸 ,
(3)
‖𝑒‖𝐿2 (Ω) ≤
𝐶4󸀠 ℎ𝑘𝛼−𝜀 ‖𝑢‖𝐸 ,
3
1
𝛼 = , 0 < ∀𝜀 < .
4
2
(11)
where (⋅, ⋅) denotes the inner product on 𝐿2 (Ω) and ⟨⋅, ⋅⟩ the
𝜇
duality pairing of 𝐻−𝜇 (Ω) and 𝐻0 (Ω), 𝜇 ≥ 0.
Thus, the Galerkin variational solution of Problem 1 may
be defined as follows.
From Theorems 3 and 4 and the equivalence of ‖ ⋅ ‖𝐻𝛼 (Ω)
and ‖ ⋅ ‖𝐸 , we have the error estimates in the 𝐿2 norm.
Definition 2 (variational solution). A function 𝑢 ∈ 𝐻0𝛼 (Ω) is
a variational solution of Problem 1 provided that
Theorem 5. Under the same assumptions in Theorem 4, one
has
𝐵 (𝑢, V) = 𝐹 (V) ,
∀V ∈ 𝐻0𝛼 (Ω) .
(4)
Ervin and Roop ([17], 2006) proved that the bilinear form
𝐵(⋅, ⋅) satisfies the coercivity and continuity; that is, there exist
two positive constants 𝐶1 and 𝐶2 such that
𝐵 (𝑢, 𝑢) ≥ 𝐶1 ‖𝑢‖2𝐻𝛼 (Ω) ,
𝐵 (𝑢, V) ≤ 𝐶2 ‖𝑢‖𝐻𝛼 (Ω) ‖V‖𝐻𝛼 (Ω) ,
∀𝑢 ∈ 𝐻0𝛼 (Ω) ,
∀𝑢, V ∈ 𝐻0𝛼 (Ω) .
(5)
(6)
Hence, they claimed that there exists a unique solution to
(4). Note that from (5) and (6) we have norm equivalence of
‖ ⋅ ‖𝐻𝛼 (Ω) and energy norm ‖ ⋅ ‖𝐸 = 𝐵(⋅, ⋅)1/2 , that is,
√𝐶1 ‖V‖𝐻𝛼 (Ω) ≤ ‖V‖𝐸 ≤ √𝐶2 ‖V‖𝐻𝛼 (Ω) ,
∀V ∈ 𝐻0𝛼 (Ω) . (7)
Let 𝑆ℎ denote a partition of Ω such that Ω = {⋃ 𝐾 : 𝐾 ∈
𝑆ℎ }. Assume that there exists a positive constant 𝜌 such that
𝜌ℎ𝑘 ≤ ℎ𝐾 ≤ ℎ𝑘 , where ℎ𝑘 = max𝐾∈𝑆ℎ ℎ𝐾 . Let 𝑃𝑚 (𝐾) denote
the space of polynomials of degree less than or equal to 𝑚
on 𝐾 ∈ 𝑆ℎ . Associated with 𝑆ℎ , define the finite-dimensional
subspace 𝑋ℎ ⊂ 𝐻0𝛼 (Ω) as
𝑋ℎ := {V ∈ 𝐻0𝛼 (Ω) ∩ 𝐶0 (Ω) : V|𝐾 ∈ 𝑃𝑚−1 (𝐾) , ∀𝐾 ∈ 𝑆ℎ } .
(8)
Let 𝑢ℎ be the solution to the finite-dimensional variational
problem:
𝐵 (𝑢ℎ , Vℎ ) = 𝐹 (Vℎ ) ,
∀Vℎ ∈ 𝑋ℎ .
(9)
Note that the existence and uniqueness of solution to
(9) follow from the fact that 𝑋ℎ is a subset of the space
𝐻0𝛼 (Ω) ∩ 𝐻0𝑚 (Ω). Ervin and Roop have obtained convergence
estimate in the energy norm ‖ ⋅ ‖𝐸 . At the same time, under
the assumption concerning the regularity of the solution
to the adjoint problem of Problem 1 [17, Assumption 4.1],
convergence estimate in the 𝐿2 norm was proved.
3
𝛼 ≠ ,
4
‖𝑒‖𝐿2 (Ω) ≤ 𝐶5 ℎ𝑘𝑟 ‖𝑢‖𝐻𝑟 (Ω) ,
‖𝑒‖𝐿2 (Ω) ≤
𝐶5 ℎ𝑘𝑟−𝜀 ‖𝑢‖𝐻𝑟 (Ω) ,
3
1
𝛼 = , 0 < ∀𝜀 < ,
4
2
(12)
where 𝐶5 = √𝐶2 𝐶3 𝐶4 (or 𝐶5 = √𝐶2 𝐶3 𝐶4󸀠 ).
3. Multigrid Method for FADE
For the simpleness, we only discuss the case of linear finite
element, and it is necessary to restrict the mesh partition.
Let {T𝑘 } denote a sequence of partitions of Ω and ℎ𝑘 be the
mesh size of T𝑘 . From now on, we assume that {T𝑘 } is a
quasiuniform family, that is,
𝜌ℎ𝑘 ≤ 𝑥𝑖 − 𝑥𝑖−1 ≤ ℎ𝑘 ,
𝑖 = 1, 2, . . . , 𝑁𝑘 ,
(13)
with positive constant 𝜌. At the same time, let T𝑘 be obtained
from T𝑘−1 via a regular subdivision, that is, T𝑘−1 ⊂ T𝑘
for 𝑘 ≥ 2. Let 𝑉𝑘 denote 𝐶0 piecewise linear functions with
respect to T𝑘 that vanish on 𝜕Ω.
3.1. Mesh-Dependent Norms
Definition 6. The mesh-dependent inner product (⋅, ⋅)𝑘 on 𝑉𝑘
is defined by
𝑁𝑘 −1
(V, 𝑤)𝑘 = ℎ𝑘 ∑ V (𝑥𝑖 ) 𝑤 (𝑥𝑖 ) .
(14)
𝑖=1
Definition 7. The operator 𝐴 𝑘 : 𝑉𝑘 → 𝑉𝑘 is defined by
(𝐴 𝑘 V, 𝑤)𝑘 = 𝐵 (V, 𝑤) ,
∀V, 𝑤 ∈ 𝑉𝑘 .
(15)
In terms of the operator 𝐴 𝑘 , the discretized equation of
(9) can be written as
𝐴 𝑘 𝑢𝑘 = 𝑓𝑘 ,
(16)
Journal of Applied Mathematics
3
where 𝑓𝑘 ∈ 𝑉𝑘 satisfies
(𝑓𝑘 , V𝑘 ) = (𝑓, V) ,
∀V ∈ 𝑉𝑘 .
(17)
Remark 8. (i) From the Riesz representation theorem, we
point out that the operator 𝐴 𝑘 is defined uniquely. (ii)
Since 𝐵(V, 𝑤) is symmetric and satisfies the proposition of
coercivity, 𝐴 𝑘 is symmetric positive definite woth respect to
(⋅, ⋅)𝑘 .
where 𝐶8 > 0 is the constant of inverse estimate and 𝐶 =
𝐶2 𝐶62 𝐶82 .
Let 𝜆 be an eigenvalue of 𝐴 𝑘 with corresponding eigenvector 𝜙 ∈ 𝑉𝑘 . From Lemma 9, we have
𝐵 (𝜙, 𝜙) = (𝐴 𝑘 𝜙, 𝜙)𝑘
= 𝜆(𝜙, 𝜙)𝑘
The mesh-dependent norms are defined by
‖|V|‖𝑠,𝑘 = √(𝐴𝑠𝑘 V, V)𝑘 .
(18)
Observe that the energy norm ‖ ⋅ ‖𝐸 coincides with ‖| ⋅ |‖1,𝑘
norm on 𝑉𝑘 . Similarly, ‖| ⋅ |‖0,𝑘 is the norm associated with the
mesh-dependent inner product (14). The following lemma
shows that ‖| ⋅ |‖0,𝑘 is equivalent to the 𝐿2 norm.
Lemma 9. The norm ‖ ⋅ ‖𝐿2 (Ω) is equivalent to mesh-depentent
norm ‖| ⋅ |‖0,𝑘 , that is,
(23)
󵄩󵄨 󵄨󵄩2
= 𝜆󵄩󵄩󵄩󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄩󵄩󵄩0,𝑘
󵄩 󵄩2
= 𝜆󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐿2 (Ω) .
Combining (22) and (23), the following inequality holds:
−2𝛼 󵄩 󵄩2
𝐵 (𝜙, 𝜙) 𝐶ℎ𝑘 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐿2 (Ω)
𝜆 = 󵄩 󵄩2
≤
= 𝐶7 ℎ𝑘−2𝛼 ,
󵄩󵄩𝜙󵄩󵄩 2
󵄩󵄩󵄩𝜙󵄩󵄩󵄩2 2
󵄩 󵄩𝐿 (Ω)
󵄩 󵄩𝐿 (Ω)
(24)
where 𝐶7 = 𝐶2 𝐶62 𝐶82 .
(19)
3.2. The V-Cycle Multigrid Algorithm. Firstly, we give the
intergrid transfer operators which play a very important role
in convergence analysis.
Proof. The proof is analogous to Lemma 6.2.7 in reference
[18].
Definition 12 (intergrid transfer operator). The coarse-to-fine
𝑘
: 𝑉𝑘−1 → 𝑉𝑘 is taken to be the
intergrid transfer operator 𝐼𝑘−1
natural injection, that is,
𝜌‖|V|‖0,𝑘 ≤ ‖V‖𝐿2 (Ω) ≤ ‖|V|‖0,𝑘 ,
∀V ∈ 𝑉𝑘 .
Particularly, ‖| ⋅ |‖0,𝑘 = ‖ ⋅ ‖𝐿2 (Ω) for the uniform mesh.
In order to estimate the spectral radius, Λ(𝐴 𝑘 ), of 𝐴 𝑘 ,
we need the following norm interpolation lemma (see [19,
Theorem 1.3.7].
Lemma 10 (interpolation theorem). Assume that Ω is an open
set of R𝑑 with a Lipchitz continuous boundary. Let 𝑠1 < 𝑠2 be
two real numbers, and 𝜇 = (1 − 𝜃)𝑠1 + 𝜃𝑠2 , 0 ≤ 𝜃 ≤ 1. Then
there exists a constant 𝐶6 > 0 such that
‖𝑢‖𝐻𝜇 (Ω) ≤
𝜃
𝐶6 ‖𝑢‖1−𝜃
𝐻𝑠1 (Ω) ‖𝑢‖𝐻𝑠2 (Ω) .
(20)
Lemma 11 (spectral radius theorem). We have the estimation
for spectral radius Λ(𝐴 𝑘 ):
Λ (𝐴 𝑘 ) ≤ 𝐶7 ℎ𝑘−2𝛼 ,
(21)
where 𝐶7 is a positive constant independent of 𝑘.
󵄩 󵄩2
𝐵 (𝜙, 𝜙) ≤ 𝐶2 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐻𝛼 (Ω)
󵄩 󵄩2
= 𝐶ℎ𝑘−2𝛼 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐿2 (Ω) ,
(25)
The fine-to-coarse operator 𝐼𝑘𝑘−1 : 𝑉𝑘 → 𝑉𝑘−1 is defined by
𝑘
(𝐼𝑘𝑘−1 𝑤, V)𝑘−1 = (𝑤, 𝐼𝑘−1
V)𝑘 = (𝑤, V)𝑘 ,
∀V ∈ 𝑉𝑘−1 , 𝑤 ∈ 𝑉𝑘 .
(26)
Now, we describe the 𝑘th level of V-cycle MGM (VMGM) and full V-cycle MGM. Let 𝑚 be the smoothing
number, 𝑛𝑘 the iteration number of 𝑘th level V-MGM, and
Λ 𝑘 be a parameter dependent on 𝑘. Denote 𝑀𝐺(𝑘, 𝑧0 , 𝑔)
the approximate solution of the 𝐴 𝑘 𝑧 = 𝑔 obtained from
the 𝑘th level iteration with initial guess 𝑧0 . The discussion
of parameters Λ 𝑘 and 𝑛𝑘 is left to the next subsection and
Section 4.
BEGIN:
For 𝑘 = 1: 𝑀𝐺(1, 𝑧0 , 𝑔) = 𝐴−1
1 𝑔.
For 𝑘 > 1, 𝑀𝐺(𝑘, 𝑧0 , 𝑔) is obtained recursively in
three steps.
2
󵄩 󵄩1−𝛼 󵄩 󵄩𝛼
≤ 𝐶2 (𝐶6 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐻0 (Ω) ⋅ 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐻1 (Ω) )
󵄩 󵄩2
= 𝐶ℎ𝑘−2𝛼 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐻0 (Ω)
∀V ∈ 𝑉𝑘−1 .
Algorithm 13 (the 𝑘th level V-cycle multigrid algorithm).
Proof. Let 𝜙 ∈ 𝑉𝑘 . Applying the continuity (6) of 𝐵(⋅, ⋅), norm
interpolation Lemma 10, and inverse estimation, we have
2
󵄩 󵄩1−𝛼
󵄩 󵄩𝛼
≤ 𝐶2 (𝐶6 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐻0 (Ω) ⋅ 𝐶8 ℎ𝑘−𝛼 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐻0 (Ω) )
𝑘
𝐼𝑘−1
V = V,
(22)
Presmoothing Step. For 1 ≤ 𝑙 ≤ 𝑚, let 𝑧𝑙 = 𝑧𝑙−1 +
(1/Λ 𝑘 )(𝑔 − 𝐴 𝑘 𝑧𝑙−1 ).
Error Correction.
𝑔 = 𝐼𝑘𝑘−1 (𝑔 − 𝐴 𝑘 𝑧𝑚 ),
𝑞1 = 𝑀𝐺(𝑘 − 1, 0, 𝑔),
𝑘
𝑧𝑚+1 = 𝑧𝑚 + 𝐼𝑘−1
𝑞1 .
4
Journal of Applied Mathematics
Postsmoothing step. For 𝑚 + 1 ≤ 𝑙 ≤ 2𝑚 + 1, let
𝑧𝑙 = 𝑧𝑙−1 + (1/Λ 𝑘 )(𝑔 − 𝐴 𝑘 𝑧𝑙−1 ).
The output of the 𝑘th level iteration is
𝑀𝐺(𝑘, 𝑧0 , 𝑔) := 𝑧2𝑚+1 .
END.
which means that V is a finite element solution of variational
problem (4) on 𝑉𝑘 . Observing that
(𝑔, 𝑤)𝑘−1 = (𝐼𝑘𝑘−1 𝑔, 𝑤)𝑘−1
= (𝑔, 𝑤)𝑘 = ⟨𝑓, 𝑤⟩ ,
Algorithm 14 (the full V-cycle multigrid algorithm).
BEGIN:
∀𝑤 ∈ 𝑉𝑘−1 ,
we claim that 𝑞 = 𝑃𝑘−1 V is also a finite element solution of
variational problem (4) on 𝑉𝑘−1 . Therefore, noting 𝑉𝑘−1 ⊂ 𝑉𝑘
(since T𝑘−1 ⊂ T𝑘 ) and applying Theorem 4, we have
For 𝑘 = 1, 𝑢̂1 = 𝐴−1
1 𝑓1 .
For 𝑘 ≥ 2,
𝑘
𝑢0𝑘 = 𝐼𝑘−1
𝑢̂𝑘−1 ,
𝑘
, 𝑓𝑘 ),
𝑢𝑙𝑘 = 𝑀𝐺(𝑘, 𝑢𝑙−1
󵄩󵄩
󵄩
󵄩
𝛼󵄩
󵄩󵄩V − 𝑃𝑘−1 V󵄩󵄩󵄩𝐿2 (Ω) ≤ 𝐶4 ℎ𝑘 󵄩󵄩󵄩V − 𝑃𝑘−1 V󵄩󵄩󵄩𝐸 ,
1 ≤ 𝑙 ≤ 𝑛𝑘 ,
𝑢̂𝑘 = 𝑢𝑛𝑘𝑘 .
󵄩
󵄩
󵄩󵄩
󸀠 𝛼−𝜀 󵄩
󵄩󵄩V − 𝑃𝑘−1 V󵄩󵄩󵄩𝐿2 (Ω) ≤ 𝐶4 ℎ𝑘 󵄩󵄩󵄩V − 𝑃𝑘−1 V󵄩󵄩󵄩𝐸 ,
END.
3.3. Approximation and Smoothing Properties. In this subsection, we prove some properties of projection operator 𝑃𝑘−1
and smoothing operator 𝑅𝑘 , which are the key ingredients for
V-MGM and FV-MGM algorithm.
Definition 15. Let 𝑃𝑘 : 𝑉 → 𝑉𝑘 be the orthogonal projection
with respect to 𝐵(⋅, ⋅), that is,
𝐵 (V, 𝑤) = 𝐵 (𝑃𝑘 V, 𝑤) ,
∀𝑤 ∈ 𝑉𝑘 .
(27)
1
𝐴 .
Λ𝑘 𝑘
3
𝛼 ≠ ,
4
3
𝛼= .
4
󵄩󵄩󵄩󵄨󵄨󵄨(𝐼 − 𝑃𝑘−1 ) V󵄨󵄨󵄨󵄩󵄩󵄩
󵄩󵄨
󵄨󵄩0,𝑘
󵄩󵄨
󵄨󵄩
≤ 𝐶9 ℎ𝑘𝛼 󵄩󵄩󵄩󵄨󵄨󵄨(𝐼 − 𝑃𝑘−1 ) V󵄨󵄨󵄨󵄩󵄩󵄩1,𝑘 ,
󵄨󵄩
󵄩󵄩󵄨󵄨
󵄩󵄩󵄨󵄨(𝐼 − 𝑃𝑘−1 ) V󵄨󵄨󵄨󵄩󵄩󵄩0,𝑘
3
∀V ∈ 𝑉𝑘 , 𝛼 ≠ ,
4
3
1
∀V ∈ 𝑉𝑘 , 𝛼 = , 0 < 𝜀 < .
4
2
(33)
(28)
Throughout this paper, we assume that Λ 𝑘 denotes some
upper bound for the spectral radius of 𝐴 𝑘 satisfying Λ 𝑘 ≤
𝐶7 ℎ𝑘−2𝛼 .
Lemma 17. Let V ∈ 𝑉𝑘 , 𝑔 = 𝐼𝑘𝑘−1 𝐴 𝑘 V, and 𝑞 ∈ 𝑉𝑘−1 satisfy the
equation 𝐴 𝑘−1 𝑞 = 𝑔. Then 𝑞 = 𝑃𝑘−1 V.
Proof. We only prove the case 𝛼 ≠ 3/4, and the proof for the
other case is analogous. Applying Remark 18 and Lemma 9
we have
1󵄩
󵄨󵄩
󵄩
󵄩󵄩󵄨󵄨
󵄩󵄩󵄨󵄨(𝐼 − 𝑃𝑘−1 ) V󵄨󵄨󵄨󵄩󵄩󵄩0,𝑘 ≤ 󵄩󵄩󵄩(𝐼 − 𝑃𝑘−1 ) V󵄩󵄩󵄩𝐿2 (Ω)
𝜌
Proof. For 𝑤 ∈ 𝑉𝑘−1 ,
𝐶4 𝛼 󵄩󵄩
󵄩
ℎ 󵄩(V − 𝑃𝑘−1 V)󵄩󵄩󵄩𝐸
𝜌 𝑘󵄩
󵄩󵄨
󵄨󵄩
= 𝐶9 ℎ𝑘𝛼 󵄩󵄩󵄩󵄨󵄨󵄨(𝐼 − 𝑃𝑘−1 ) V󵄨󵄨󵄨󵄩󵄩󵄩1,𝑘 ,
≤
𝐵 (𝑞, 𝑤) = (𝐴 𝑘−1 𝑞, 𝑤)𝑘−1
= (𝑔, 𝑤)𝑘−1
= (𝐼𝑘𝑘−1 𝐴 𝑘 V, 𝑤)𝑘−1
(32)
Lemma 19. There exists a positive constant 𝐶9 such that
󵄩󵄨
󵄨󵄩
≤ 𝐶9 ℎ𝑘𝛼−𝜀 󵄩󵄩󵄩󵄨󵄨󵄨(𝐼 − 𝑃k−1 ) V󵄨󵄨󵄨󵄩󵄩󵄩1,𝑘 ,
Definition 16. Define the relaxation operator:
𝑅𝑘 := 𝐼 −
(31)
(29)
= (𝐴 𝑘 V, 𝑤)𝑘
(34)
where 𝐶9 = 𝐶4 /𝜌.
Lemma 20. There exists a positive constant 𝐶9 such that
= 𝐵 (V, 𝑤) .
Therefore, from the definition of 𝑃𝑘−1 we have 𝑞 = 𝑃𝑘−1 V.
󵄩󵄩󵄨󵄨
󵄨󵄩
𝛼
󵄩󵄩󵄨󵄨(𝐼 − 𝑃𝑘−1 ) V󵄨󵄨󵄨󵄩󵄩󵄩1,𝑘 ≤ 𝐶9 ℎ𝑘 ‖|V|‖2,𝑘 ,
Remark 18. Let 𝑔 = 𝐴 𝑘 V. From the Riesz representation
theorem there exists a unique 𝑓 ∈ 𝐻−𝛼 (Ω) such that
󵄨󵄩
󵄩󵄩󵄨󵄨
󵄩󵄩󵄨󵄨(𝐼 − 𝑃𝑘−1 ) V󵄨󵄨󵄨󵄩󵄩󵄩1,𝑘
(𝑔, 𝑤)𝑘 = ⟨𝑓, 𝑤⟩ ,
∀𝑤 ∈ 𝑉𝑘 ,
(30)
≤ 𝐶9 ℎ𝑘𝛼−𝜀 ‖|V|‖2,𝑘 ,
3
∀V ∈ 𝑉𝑘 , 𝛼 ≠ ,
4
3
1
∀V ∈ 𝑉𝑘 , 𝛼 = , 0 < 𝜀 < .
4
2
(35)
(36)
Journal of Applied Mathematics
5
Proof. Applying projection operator 𝑃𝑘−1 , generalized Cauchy-Schwarz inequality (see [18, Lemma 6.2.10]), and
Lemma 19
󵄨󵄩2
󵄩
󵄩2
󵄩󵄩󵄨󵄨
󵄩󵄩󵄨󵄨(𝐼 − 𝑃𝑘−1 )V󵄨󵄨󵄨󵄩󵄩󵄩1,𝑘 = 󵄩󵄩󵄩(𝐼 − 𝑃𝑘−1 )V󵄩󵄩󵄩𝐸
3.4. Convergence Analysis. In this subsection, we prove convergence theorems for V-cycle FE multigrid method.
Definition 23. The error operator of V-cycle multigrid 𝐸𝑘 :
𝑉𝑘 → 𝑉𝑘 is defined recursively by
𝐸1 = 0,
= 𝐵 (V − 𝑃𝑘−1 V, V − 𝑃𝑘−1 V)
= 𝐵 (V − 𝑃𝑘−1 V, V)
󵄨󵄩
󵄩󵄨
≤ 󵄩󵄩󵄩󵄨󵄨󵄨(𝐼 − 𝑃𝑘−1 ) V󵄨󵄨󵄨󵄩󵄩󵄩0,𝑘 ‖|V|‖2,𝑘
(37)
󵄩󵄨
󵄨󵄩
≤ 𝐶9 ℎ𝑘𝛼 󵄩󵄩󵄩󵄨󵄨󵄨(𝐼 − 𝑃𝑘−1 ) V󵄨󵄨󵄨󵄩󵄩󵄩1,𝑘 ‖|V|‖2,𝑘
if 𝛼 ≠ 3/4. Therefore, dividing through by ‖|(𝐼 − 𝑃𝑘−1 )V|‖1,𝑘
yields (35). The case 𝛼 = 3/4 is analogous.
Lemma 21. Let 𝑚 be the number of smoothing steps. Then
1
𝐵 ((𝐼 − 𝑅𝑘2𝑚 ) V, V) .
2𝑚
𝐶10
𝐵 ((𝐼 − 𝑅𝑘2𝑚 ) V, V) .
𝑚
(39)
Proof. Applying Lemmas 20, 11, and 21, we have
𝐵 (𝐸𝑘 V, 𝑤) = 𝐵 (V, 𝐸𝑘 𝑤) ,
∀V, 𝑤 ∈ 𝑉𝑘 ,
𝐵 (𝐸𝑘 V, V) ≥ 0.
(43)
∀V ∈ 𝑉𝑘 .
(44)
Proof. The proof is by induction. For 𝑘 = 1, (44) is trivially
true since 𝐸𝑘 = 0. Assume that (44) holds for 𝑘 − 1. Let 𝛾 =
𝐶10 /(𝑚 + 𝐶10 ); therefore 1 − 𝛾 = 𝛾𝑚/𝐶10 . Applying induction
hypothesis, Proposition 24, and Lemma 22, we have
= (1 − 𝛾) 𝐵 ((𝐼 − 𝑃𝑘−1 ) 𝑅𝑘𝑚 V, (𝐼 − 𝑃𝑘−1 ) 𝑅𝑘𝑚 V)
= 𝐶92 ℎ𝑘2𝛼 (𝐴2𝑘 𝑅𝑘𝑚 V, 𝑅𝑘𝑚 V)𝑘
= 𝐶92 ℎ𝑘2𝛼 Λ 𝑘 𝐵 ((𝐼 − 𝑅𝑘 ) 𝑅𝑘𝑚 V, 𝑅𝑘𝑚 V)
𝐶10
𝐵 (V, V) ,
𝑚 + 𝐶10
+ 𝛾𝐵 (𝑃𝑘−1 𝑅𝑘𝑚 V, 𝑃𝑘−1 𝑅𝑘𝑚 V)
󵄩󵄨
󵄨󵄩2
𝐶92 ℎ𝑘2𝛼 󵄩󵄩󵄩󵄨󵄨󵄨𝑅𝑘𝑚 V󵄨󵄨󵄨󵄩󵄩󵄩2,𝑘
= 𝐶92 ℎ𝑘2𝛼 𝐵 (𝐴 𝑘 𝑅𝑘𝑚 V, 𝑅𝑘𝑚 V)
𝐵 (𝐸𝑘 V, V) ≤
≤ 𝐵 ((𝐼 − 𝑃𝑘−1 ) 𝑅𝑘𝑚 V, (𝐼 − 𝑃𝑘−1 ) 𝑅𝑘𝑚 V)
󵄩󵄨
󵄨󵄩2
= 󵄩󵄩󵄩󵄨󵄨󵄨(𝐼 − 𝑃𝑘−1 ) 𝑅𝑘𝑚 V󵄨󵄨󵄨󵄩󵄩󵄩1,𝑘
−
(ii) 𝐸𝑘 is symmetric positive semidefinite with respect to
𝐵(⋅, ⋅) for 𝑘 ≥ 1, that is,
+ 𝐵 (𝐸𝑘−1 𝑃𝑘−1 𝑅𝑘𝑚 V, 𝑃𝑘−1 𝑅𝑘𝑚 V)
󵄩2
󵄩
= 󵄩󵄩󵄩(𝐼 − 𝑃𝑘−1 )𝑅𝑘𝑚 V󵄩󵄩󵄩𝐸
≤
(42)
𝐵 (𝐸𝑘 V, V) = 𝐵 (𝑅𝑘𝑚 V, 𝑅𝑘𝑚 V) − 𝐵 (𝑃𝑘−1 𝑅𝑘𝑚 V, 𝑃𝑘−1 𝑅𝑘𝑚 V)
𝐵 ((𝐼 − 𝑃𝑘−1 ) 𝑅𝑘𝑚 V, (𝐼 − 𝑃𝑘−1 ) 𝑅𝑘𝑚 V)
𝐶92 𝐶7 𝐵 ((𝐼
𝐸𝑘 (𝑧 − 𝑧0 ) = 𝑧 − 𝑀𝐺 (𝑘, 𝑧0 , 𝑔) ,
Theorem 25. Let 𝑚 be the number of smoothing steps. Then
Lemma 22. Let 𝑚 be the number of smoothing steps. Then
there exists a positive constant 𝐶10 such that
≤
(41)
Proposition 24. The operator 𝐸𝑘 has the following propositions (see [18, Lemma 6.6.2 and Proposition 6.6.9]):
(38)
Proof. See Lemma 6.6.7 in [18].
𝐵 ((𝐼 − 𝑃𝑘−1 ) 𝑅𝑘𝑚 V, (𝐼 − 𝑃𝑘−1 ) 𝑅𝑘𝑚 V) ≤
𝑘 > 1.
(i) if 𝑧, 𝑔 ∈ 𝑉𝑘 satisfy 𝐴 𝑘 𝑧 = 𝑔, then
󵄩󵄨
󵄨󵄩
= 𝐶9 ℎ𝑘𝛼 󵄩󵄩󵄩󵄨󵄨󵄨(𝐼 − 𝑃𝑘−1 ) V󵄨󵄨󵄨󵄩󵄩󵄩1,𝑘 ‖|V|‖2,𝑘
𝐵 ((𝐼 − 𝑅𝑘 ) 𝑅𝑘2𝑚 V, V) ≤
𝐸𝑘 = 𝑅𝑘𝑚 [𝐼 − (𝐼 − 𝐸𝑘−1 ) 𝑃𝑘−1 ] 𝑅𝑘𝑚 ,
+ 𝛾𝐵 ((𝐼 − 𝑃𝑘−1 ) 𝑅𝑘𝑚 V, (𝐼 − 𝑃𝑘−1 ) 𝑅𝑘𝑚 V)
+ 𝛾𝐵 (𝑃𝑘−1 𝑅𝑘𝑚 V, 𝑃𝑘−1 𝑅𝑘𝑚 V)
(40)
𝑅𝑘 ) 𝑅𝑘𝑚 V, 𝑅𝑘𝑚 V)
= 𝐶92 𝐶7 𝐵 ((𝐼 − 𝑅𝑘 ) 𝑅𝑘2𝑚 V, V)
(45)
≤
(1 − 𝛾) 𝐶10
+ 𝛾𝐵 (𝑅𝑘𝑚 V, 𝑅𝑘𝑚 V)
𝑚𝐵 ((𝐼 − 𝑅𝑘2𝑚 ) V, V)
= 𝛾𝐵 ((𝐼 − 𝑅𝑘2𝑚 ) V, V) + 𝛾𝐵 (𝑅𝑘𝑚 V, 𝑅𝑘𝑚 V)
= 𝛾𝐵 (V, V) ,
≤
𝐶92 𝐶7
(2𝑚) 𝐵 ((𝐼 − 𝑅𝑘2𝑚 ) V, V)
which ends the proof.
=
𝐶10
𝑚𝐵 ((𝐼 − 𝑅𝑘2𝑚 ) V, V)
Corollary 26. Let 𝑚 be the number of smoothing steps. Then
one has the estimation for spectral radius Λ(𝐸𝑘 ):
if 𝛼 ≠ 3/4, where 𝐶10 = 𝐶92 𝐶7 /2. Note that (39) also holds for
𝛼 = 3/4 since 𝜀 > 0.
Λ (𝐸𝑘 ) ≤
𝐶10
.
𝑚 + 𝐶10
(46)
6
Journal of Applied Mathematics
Theorem 27 (convergence of the 𝑘th level iteration for
V-cycle). Let 𝑚 be the number of smoothing steps. Then
𝐶10 󵄩󵄩
󵄩
󵄩
󵄩󵄩
󵄩𝑧 − 𝑧0 󵄩󵄩󵄩𝐸 .
󵄩󵄩𝑧 − 𝑀𝐺 (𝑘, 𝑧0 , 𝑔)󵄩󵄩󵄩𝐸 ≤
𝑚 + 𝐶10 󵄩
Corollary 29. Under the assumptions in Theorem 28, the
convergence rate of multigrid solution 𝑢̂𝑘 in 𝐿2 norm is 𝑟; that
is, there exists a constant 𝐶12 > 0 such that
(47)
Hence, the 𝑘th level iteration for any 𝑚 is a contraction with
the contraction number independent of 𝑘.
󵄩󵄩 ̂
󵄩
𝑟
󵄩󵄩𝑢𝑘 − 𝑢󵄩󵄩󵄩𝐿2 (Ω) ≤ 𝐶12 ℎ𝑘 ‖𝑢‖𝐻𝑟 (Ω) ,
󵄩
󵄩󵄩 ̂
𝑟−𝜀
󵄩󵄩𝑢𝑘 − 𝑢󵄩󵄩󵄩𝐿2 (Ω) ≤ 𝐶12 ℎ𝑘 ‖𝑢‖𝐻𝑟 (Ω) ,
Proof. From (42) in Proposition 24, it suffices to show
𝐶10
󵄩󵄩 󵄩󵄩
‖V‖ ,
󵄩󵄩𝐸𝑘 V󵄩󵄩𝐸 ≤
𝑚 + 𝐶10 𝐸
(48)
which can be proved by Corollary 26.
Theorem 28 (full multigrid convergence). Let 𝑢 ∈
𝐻𝑟 (Ω) (𝛼 ≤ 𝑟 ≤ 2) solve (4). If the iteration numbers in full
multigrid algorithm 𝑛𝑘 are large enough, there exists a positive
constant 𝐶11 such that
󵄩󵄩
󵄩
𝑟−𝛼
󵄩󵄩𝑢𝑘 − 𝑢̂𝑘 󵄩󵄩󵄩𝐸 ≤ 𝐶11 ℎ𝑘 ‖𝑢‖𝐻𝑟 (Ω) .
(49)
Proof. For the simpleness we assume that 𝑛𝑘 ≥ 𝑛 ≥ 1. Define
𝑒̂𝑘 = 𝑢𝑘 − 𝑢̂𝑘 . In particular, 𝑒̂1 = 0. Let 𝛾 = 𝐶10 /(𝑚 + 𝐶10 ) (𝐶10
is the constant in Theorem 27). We have
󵄩
󵄩󵄩 ̂ 󵄩󵄩
𝑛󵄩
󵄩󵄩𝑒𝑘 󵄩󵄩𝐸 ≤ 𝛾 󵄩󵄩󵄩𝑢𝑘 − 𝑢̂𝑘−1 󵄩󵄩󵄩𝐸
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
≤ 𝛾𝑛 (󵄩󵄩󵄩𝑢𝑘 − 𝑢󵄩󵄩󵄩𝐸 + 󵄩󵄩󵄩𝑢 − 𝑢𝑘−1 󵄩󵄩󵄩𝐸 + 󵄩󵄩󵄩𝑢𝑘−1 − 𝑢̂𝑘−1 󵄩󵄩󵄩𝐸 )
We end this section with a proposition that the work
involved in the full multigrid algorithm is O(𝑑𝑘 ), where 𝑑𝑘 =
𝑁𝑘 − 2 = dim 𝑉𝑘 (see [18, Proposition 6.7.4]).
In this section we employ the FV-MGM listed as Algorithm 14 in Section 3 to solve FADEs, which demonstrate the
convergence rate and involved work in Theorem 28 and its
Corollary 29. The parameter 𝑚 is taken 5, Λ 𝑘 = 𝐶7 ℎ𝑘−2𝛼 , and
𝑛𝑘 is controlled by the stopping criterion:
󵄩󵄩
󵄩
󵄩󵄩𝐴 𝑘 𝑢̂𝑛𝑘 − 𝑔󵄩󵄩󵄩 2 ≤ O (ℎ𝑘𝛼 ) .
𝑘
󵄩
󵄩𝐿 (Ω)
󵄩
󵄩
𝑟−𝛼
+√𝐶2 𝐶3 ℎ𝑘−1
‖𝑢‖𝐻𝑟 (Ω) + 󵄩󵄩󵄩𝑒̂𝑘−1 󵄩󵄩󵄩𝐸 ]
󵄩
󵄩
= 𝛾𝑛 [3√𝐶2 𝐶3 ℎ𝑘𝑟−𝛼 ‖𝑢‖𝐻𝑟 (Ω) + 󵄩󵄩󵄩𝑒̂𝑘−1 󵄩󵄩󵄩𝐸 ] .
(54)
Example 30. Let 𝑐(𝑥) = 1. It can be verified that 𝑢(𝑥) = 𝑥2 −𝑥3
is the exact solution to the boundary value problem:
󵄩󵄩 ̂ 󵄩󵄩
𝑛 𝑟−𝛼
󵄩󵄩𝑒𝑘 󵄩󵄩𝐸 ≤ 3√𝐶2 𝐶3 𝛾 ℎ𝑘 ‖𝑢‖𝐻𝑟 (Ω)
1
−𝛽
− 𝐷 ( 0 𝐷−𝛽
𝑥 + 𝑥 𝐷1 ) 𝐷𝑢 + 𝑢 = 𝑓,
2
𝑢 (0) = 0,
(55)
𝑢 (1) = 0,
where
1
1
𝑓 (𝑥) = 𝑥2 − 𝑥3 + 𝑓1 (𝑥) + 𝑓2 (𝑥) ,
2
2
𝑟−𝛼
+ 3√𝐶2 𝐶3 𝛾2𝑛 ℎ𝑘−1
‖𝑢‖𝐻𝑟 (Ω) + ⋅ ⋅ ⋅
(51)
𝛾𝑛
ℎ𝑟−𝛼 ‖𝑢‖𝐻𝑟 (Ω)
1 − 2𝛾𝑛 𝑘
= 𝐶11 ℎ𝑘𝑟−𝛼 ‖𝑢‖𝐻𝑟 (Ω) ,
𝑛
1 2𝛼
(ℎ Λ (𝐴 1 ) + ℎ22𝛼 Λ (𝐴 2 )) .
2 1
Here we calculate 𝐶7 = 0.05, and, from experimental experience, let the right side of stopping criterion be 0.085ℎ𝑘𝛼 . All
numerical experiments are run in MATLAB 7.0.0.19920(R14)
on a PC with the configuration: Intel(R)Core(TM)i3-2100
CPU @ 3.2 GHz and 1.88 GB RAM.
In the following analysis, we assume that 2𝛾𝑛 < 1 (therefore
𝑛 > log1/2
). By iterating the above inequality and from
𝛾
continuity (6) and Theorem 3, we have
≤ 3√𝐶2 𝐶3
(53)
We note that the positive constants 𝐶7 can be estimated by
testing examples on the 1th and 2th level meshes, that is,
𝐶7 =
(50)
+ 3√𝐶2 𝐶3 𝛾𝑘𝑛 ℎ1𝑟−𝛼 ‖𝑢‖𝐻𝑟 (Ω)
3
1
𝛼 = , 0 < ∀𝜀 < .
4
2
(52)
4. Numerical Examples
𝐻0𝛼 (Ω) ⋂
≤ 𝛾𝑛 [√𝐶2 𝐶3 ℎ𝑘𝑟−𝛼 ‖𝑢‖𝐻𝑟 (Ω)
3
𝛼 ≠ ,
4
𝑛
where 𝐶11 = 3√𝐶2 𝐶3 𝛾 /(1 − 2𝛾 ).
The following Corollary is a natural conclusion of Theorems 5 and 28.
𝑓1 (𝑥) = −
2𝑥𝛽
6𝑥1+𝛽
+
,
Γ (1 + 𝛽) Γ (2 + 𝛽)
𝛽(1 − 𝑥)𝛽−1 4 (𝛽 + 1) (1 − 𝑥)𝛽
𝑓2 (𝑥) = −
+
Γ (1 + 𝛽)
Γ (2 + 𝛽)
−
(56)
6 (𝛽 + 2) (1 − 𝑥)𝛽+1
.
Γ (3 + 𝛽)
As 𝑢 ∈ 𝐻02 (Ω), Corollary 29 predicts a rate of convergence
of 2 in 𝐿2 norm. Table 1 includes numerical results over
Journal of Applied Mathematics
7
Table 1: Experimental error ‖ 𝑢 − 𝑢̂𝑘 ‖𝐿2 (Ω) for different 𝛽.
𝛽 = 0.3
ℎ𝑘
1/4
1/8
1/16
1/32
1/64
1/128
Error
9.3587𝑒 − 3
2.1751𝑒 − 3
5.0679𝑒 − 4
1.1967𝑒 − 4
2.9496𝑒 − 5
7.0818𝑒 − 6
𝛽 = 0.6
Conv. rate
Error
8.3460𝑒 − 3
1.8444𝑒 − 3
4.1362𝑒 − 4
9.4937𝑒 − 5
2.2629𝑒 − 5
5.4727𝑒 − 6
2.1388
2.1016
2.0823
2.0205
2.0583
𝛽 = 0.9
Conv. rate
2.1779
2.1568
2.1233
2.0688
2.0478
Error
7.6044𝑒 − 3
1.6348𝑒 − 3
3.6349𝑒 − 4
8.3872𝑒 − 5
2.0343𝑒 − 5
5.0852𝑒 − 6
Conv. rate
2.2177
2.1692
2.1157
2.0436
2.0002
Table 2: Comparisons for solving Example 30 with 𝛽 = 0.7 by the GE method, the CGNR method, and the FV-MGM, respectively.
ℎ𝑘
1/128
1/256
1/512
1/1024
1/2048
1/4096
FVMGM
Error
2.0270𝑒 − 5
1.3811𝑒 − 5
1.1507𝑒 − 5
1.1101𝑒 − 5
1.1002𝑒 − 5
1.0849𝑒 − 5
GE
CPU(s)
0.0313
0.0625
0.1719
0.5156
2.5000
15.484
𝑅cpu
0.995
1.451
1.580
2.274
2.627
Iter
29
46
72
112
174
270
a uniform partition of [0, 1], which support the predicted
rates of convergence for different values of 𝛽.
As comparisons, we also carry out the Gauss elimination (GE) and conjugate gradient normal residual (CGNR)
method with the same stopping criterion of the FV-MGM,
that is,
CGNR
CPU(s)
0.0000
0.0156
0.0313
0.2187
1.5625
9.2031
𝑅cpu
0.999
2.796
2.833
2.554
Iter = 𝑛𝑘
4
5
5
3
1
1
FVMGM
CPU(s)
0.0000
0.0156
0.0469
0.1250
0.2688
0.5719
󵄩
󵄩󵄩
󵄩󵄩𝐴 𝑘 𝑢̂𝑛𝑘 − 𝑔󵄩󵄩󵄩 2 ≤ 10−𝑛 ,
𝑘
󵄩𝐿 (Ω)
󵄩
𝑅cpu
1.579
1.417
1.103
1.088
(57)
to solve the corresponding system. For escaping “out of
memory,” we define that the data type of 𝐴 𝑘 is short float in
our program. Table 2 includes CPU time (without including
stiffness matrix calculating time) and iteration numbers for
each of the numerical methods with 𝑛 = 4. The rate of the
increasing CPU time is defined by
𝑅cpu
=
log [CPU time on finer mesh/CPU time on coarser mesh]
.
log [dim of finer mesh/ dim of coarser mesh]
From the table, we see that the numbers of iterations by
our FV-MGM are independent of ℎ𝑘 . In contrast, the CGNR
method (with the initial value 𝑧0 = 0) needs more iterations
when ℎ𝑘 decrease. Similarly, the CPU time needed for GE and
CGNR method increases much faster than that of the FVMGM.
5. Concluding Remarks
In general the discretized system of FADE has a full and
ill-conditioned coefficient matrix, so FV-MGM is a highefficient algorithm for solving these equations. Theorems and
examples in this paper show that the convergence rate of FVMGM is the same as classic FEM under the stopping criterion
(53), and the computational work is only O(dim 𝑉𝑘 ) while
the stopping criterion is taken (57). Different from integerorder equations, all the convergence analysis is on fractional
(58)
Sobolev spaces 𝐻𝛼 (Ω). The nonsymmetric form of FADE will
be studied in the future.
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