Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 2685659, 6 pages http://dx.doi.org/10.1155/2016/2685659
Research Article A Two-Level Additive Schwarz Preconditioning Algorithm for the Weak Galerkin Method for the Second-Order Elliptic Equation Fangfang Qin, Min Zha, and Feng Wang Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China Correspondence should be addressed to Feng Wang;
[email protected] Received 16 December 2015; Accepted 9 March 2016 Academic Editor: Yann Favennec Copyright Š 2016 Fangfang Qin et al. 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. This paper proposes a two-level additive Schwarz preconditioning algorithm for the weak Galerkin approximation of the secondorder elliptic equation. In the algorithm, a đ1 conforming finite element space is defined on the coarse mesh, and a stable intergrid transfer operator is proposed to exchange the information between the spaces on the coarse mesh and the fine mesh. With the framework of the Schwarz method, it is proved that the condition number of the preconditioned system only depends on the rate of the coarse mesh size and the overlapping size. Some numerical experiments are carried out to verify the theoretical results.
1. Introduction The weak Galerkin (WG) finite element method (FEM) is a new efficient numerical method for solving partial differential equations. It was first introduced for the second-order elliptic problem in [1]. The central idea of the WG method is using weak derivatives in place of strong derivatives that define the weak formulation for the underlying partial differential equations. Due to the weak derivative, the degrees of freedom can be discontinuous from the element interior to the element boundary, which makes the weak Galerkin method have more choices of the finite elements than the standard FEMs. The WG FEM also has some other properties, such as high-order accuracy and multiphysics capability. The idea of the weak derivative has been generalized to some other problems, such as the biharmonic equation [2] and the twophase subsurface flow problems [3]. Chen et al. introduced a posteriori error estimates for the WG method for the secondorder elliptic problems in [4]. Although many works focus on developing WG methods for different problems, there are few works on efficiently solving the discretized systems, of which the condition number is đ(ââ2 ), where â denotes the mesh size. Recently,
Li and Xie [5] proposed a multigrid algorithm for the WG method using đ1 conforming element on the coarser meshes. In [6], Chen et al. also constructed a fast auxiliary space multigrid preconditioner for the WG method, as well as a corresponding reduced system involving only the degrees of freedom on edges/faces. It is proved that the condition number of the preconditioned system is independent of the mesh size. Due to its high parallelizability and scalability, the domain decomposition method is one of the most efficient algorithms for the numerical solutions of partial differential equations (see, e.g., [7]). In this paper, we study an overlapping domain decomposition algorithm and propose a two-level additive Schwarz preconditioner for the WG method for the secondorder elliptic equation. The preconditioner is consisted of overlapping local subproblems on a fine mesh and a global subproblem on a coarse mesh. The coarse subproblem is defined by the đ1 conforming element method, and an intergrid transfer operator, which has stability and approximability, is introduced to transmit information between the coarse mesh and the fine mesh. Under the Schwarz framework, we prove that the condition number of the preconditioned system is independent of the fine mesh size â and is of đ((1 +
2
Mathematical Problems in Engineering
đť/đż)2 ), where đť denotes the coarse mesh size and đż stands for the overlapping size. The rest of this paper is organized as follows. In Section 2, we recall the WG method and give some notations. In Section 3, we propose an intergrid transfer operator and a two-level additive Schwarz preconditioner. Moreover, we prove the stability and approximation of the intergrid transfer operator. Then we estimate the upper bound of the maximum eigenvalue and the lower bound of minimum eigenvalue, respectively. Section 4 is devoted to numerical experiments, which are carried out to confirm our theoretical results.
For the sake of simplicity, we consider the model problem as follows: (1)
on đΊ,
where Ί â R2 is bounded polygonal domain and đ â đż2 (Ί). Assume that the matrix A is symmetric positive definite (SPD); namely, there exist positive constants đź, đ˝ such that đ
đ
2
đźđ đ ⤠đ A (đĽ) đ ⤠đ˝đ đ, âđ â R , đĽ â Ί.
(2)
The variational formulations of (1) is to find đ˘ â đť01 (Ί) such that đ (đ˘, V) = (đ, V)
âV â đť01 (Ί) ,
(3)
where đ(đ˘, V) = (Aâđ˘, âV). It follows from Lax-Milgramâs theorem that problem (3) has a unique solution. Let Tâ be a quasiuniform triangulation of the domain Ί with the mesh size â. On each đ â Tâ , the space đ(đ) denotes the collections of weak functions, each of which is consisted of an interior part and a boundary part; that is, 2
đ (đ) fl {V = {V0 , Vđ } : V0 â đż (đ) , Vđ â đť
1/2
(đđ)} . (4)
For any weak function V â đ(đ), the weak gradient of V, denoted by âđ¤ V â đť(div, đ), satisfies ⍠âđ¤ V â
q đđĽ = â ⍠V0 â â
q đđĽ + ⍠Vđ q â
n đđ đ
đ
đđ
(5)
âq â đť (div, đ) , where n is the unit outer normal of đđ. It is easy to see that âđ¤ is the classical gradient if it actions on a function đ˘ â đť1 (đ). For each element đ â Tâ , we denote by đ0 and đđ the interior and the boundary of đ. The notation Fđ stands for the set of edges of đ. We also use Fâ and Fđâ to denote the edges of Tâ in Ί and on đΊ. Then the discrete WG space is defined as đâ fl {V : V|đ0 â đđ (đ0 ) âđ â Tâ , V|đš â đđ (đš) âđš â Fâ , V|đš = 0 âđš â Fđâ } ,
đ
đ
đđ
(7)
where V0 and Vđ denote the values of V in the interior and on the boundary of đ. Then the discrete problem for (3) is to seek đ˘â = {đ˘0 , đ˘đ } â đâ such that đâ (đ˘â , Vâ ) = (đ, V0 ) , âVâ = {V0 , Vđ } â đâ ,
ââ â
(Aâđ˘) = đ, in Ί,
đ
⍠âđ¤ V â
q đđĽ = â ⍠V0 (â â
q) đđĽ + ⍠Vđ (q â
n) đđ âq â đşđ (đ) ,
2. The Weak Galerkin Method for the Second-Order Elliptic Equation
đ˘ = 0,
where đ = đ or đ = đ + 1 and đđ (đ0 ) denotes the set of polynomials on đ0 with degree no more than đ ⼠0. To define a discrete weak gradient, we use đşđ (đ) to denote the RT Ěđ (đ)đĽ if đ = đ and the BDM element element space [đđ (đ)]2 + đ 2 Ěđ (đ) is the homogeneous space [đđ+1 (đ)] if đ = đ + 1, where đ polynomial with the degree đ. For each V â đâ , the discrete weak gradient âđ¤ V â đşđ (đ) is defined as
(6)
(8)
where đâ (đ˘â , Vâ ) fl (Aâđ¤ đ˘â , âđ¤ Vâ ). For any Vâ = {V0 , Vđ } and đ˘â = {đ˘0 , đ˘đ } in đâ , we use ((đ˘â , Vâ )) to indicate a special inner product defined as ((Vâ , đ˘â )) fl â [(V0 , đ˘0 )đ + â (V0 â Vđ , đ˘0 â đ˘đ )đđ ] , (9) đâTâ
where (â
, â
)đ and (â
, â
)đđ denote the đż2 inner product on đ and đđ. Accordingly, a norm and seminorm are introduced for any Vâ = {V0 , Vđ } â đâ by 1/2
óľŠóľŠ óľŠóľŠ óľŠ óľŠ2 óľŠóľŠVâ óľŠóľŠ0,â fl ( â óľŠóľŠóľŠVâ óľŠóľŠóľŠ0,â,đ )
,
đâTâ
óľ¨óľ¨ óľ¨óľ¨ óľ¨ óľ¨2 óľ¨óľ¨Vâ óľ¨óľ¨1,â fl ( â óľ¨óľ¨óľ¨Vâ óľ¨óľ¨óľ¨1,â,đ )
(10)
1/2
,
đâTâ
where âVâ â0,â,đ fl (âV0 â20,đ + ââV0 â Vđ â2đđ )1/2 and |Vâ |1,â,đ fl (|V0 |21,đ + ââ1 âV0 â Vđ â2đđ )1/2 . Let đ´ â : đâ â đâ be an operator ((đ´ â đ˘â , Vâ )) = (Aâđ¤ đ˘â , âđ¤ Vâ ) ,
âđ˘â , Vâ â đâ .
(11)
It can be proved that đ´ â is symmetric and positive definite and the condition number is of đ(ââ2 ) (see, e.g., [8]), which brings difficulty to solve the discrete problem when the mesh size is small. In the next section, we will present a preconditioner to overcome this difficulty.
3. The Overlapping Domain Decomposition Method 3.1. A Two-Level Additive Schwarz Preconditioner. To introduce our preconditioner, we first divide the domain Ί by đ˝ overlapping subdomains Ί1 , Ί2 , . . . , Ίđ˝ such that each point in Ί belongs to no more than đđś subdomains. We assume that the boundary of each subdomain does not cut through any elements in the triangulation Tâ , and there
Mathematical Problems in Engineering
3
exist nonnegative đśâ functions đ1 , đ2 , . . . , đđ˝ satisfying the following properties: (1) đđ = 0 in Ί \ Ίđ ; (2) âđ˝đ=1 đđ = 1; (3) there exists a positive constant đż, such that |âđđ | ⤠đś/đż, where đś is a constant independent of đż, â, and đ˝. On each subdomain Ίđ , the notation Tâ,đ stands for the triangulation inherited from Tâ . The corresponding weak Galerkin subspace on Tâ,đ is defined as đđ = {V â đâ : V|Ί\Ίđ = 0} .
(12)
We introduce an operator đ´ đ : đđ â đđ by ((đ´ đ đ˘đ , Vđ )) = (Aâđ¤ đ˘đ , âđ¤ Vđ ) ,
âđ˘đ , Vđ â đđ .
(14)
((đ´ â V, V))
Vâđâ ,V=0̸ min V=âđ,Vđ âđđ đźđ Vđ
âđ ((đ´ đ Vđ , Vđ ))
, (19)
đ min (đľâ đ´ â ) ((đ´ â V, V))
Vâđâ ,V=0̸ min V=âđ,Vđ âđđ đźđ Vđ
âđ ((đ´ đ Vđ , Vđ ))
,
where the sum is taken over đ = đť, 1, . . . , đ˝. Lemma 2. For the intergrid transfer operator đźđť, it holds for any Vđť â đđť that óľŠóľŠ óľŠ2 óľ¨2 óľ¨2 2óľ¨ 2óľ¨ óľŠóľŠVđť â đźđťVđťóľŠóľŠóľŠ0,â + â óľ¨óľ¨óľ¨đźđťVđťóľ¨óľ¨óľ¨1,â Ⲡâ óľ¨óľ¨óľ¨Vđťóľ¨óľ¨óľ¨1 .
(20)
Proof. We only need consider the case that the functions in đâ are the constants in the interior and on the boundary of each element. It follows from the triangle inequality, the trace inequality, and the Poincar´e-Friedrichs inequality that óľŠ2 óľ¨2 óľ¨óľ¨ â1 óľŠ 0 đ óľ¨óľ¨đźđťVđťóľ¨óľ¨óľ¨1,â = â â óľŠóľŠóľŠóľŠđźđťVđť â đźđťVđťóľŠóľŠóľŠóľŠ0,đđ đâTâ
óľŠ2 óľŠ2 óľŠ 0 óľŠ đ Ⲡâ ââ1 (óľŠóľŠóľŠóľŠđźđť Vđť â VđťóľŠóľŠóľŠóľŠ0,đđ + óľŠóľŠóľŠóľŠđźđť Vđť â VđťóľŠóľŠóľŠóľŠ0,đđ )
(21)
đâTâ
(15)
0 đ and đźđť satisfy where đźđť
óľ¨ óľ¨2 = óľ¨óľ¨óľ¨Vđťóľ¨óľ¨óľ¨1 . Similarly, for the lower-order term, we have
1 óľ¨ 0 Vđť)óľ¨óľ¨óľ¨óľ¨đ = (đźđť ⍠V đđĽ, |đ| đ đť óľ¨ đ Vđť)óľ¨óľ¨óľ¨óľ¨đš (đźđť
= max
(13)
Note that the coarse space đđť is a subspace of đâ if the degree of the piecewise polynomials in đâ is greater than 0, and we can choose a natural injection as the intergird transfer operator đźđť from đđť to đâ . For the piecewise constant space case, the intergrid transfer operator đźđť : đđť â đâ is defined as 0 đ Vđť, đźđť Vđť} , âVđť â đđť, đźđťVđť = {đźđť
đ max (đľâ đ´ â )
= min
It is easy to see that đ´ đ is symmetric and positive definite. Since đđ is a subset of đâ , we use đźđ to denote a natural injection from đđ to đâ . To define a coarse subproblem, we define a coarse triangulation Tđť with mesh size đť such that each element in Tâ is a subdivision of the one in Tđť. The notation đđť â đť01 (Ί) stands for đ1 conforming finite element space associated with Tđť. We also introduce an operator đ´ đť : đđť â đđť satisfying (đ´ đťđ˘đť, Vđť) = (Aâđ˘đť, âVđť) , âđ˘đť, Vđť â đđť.
Lemma 1 ([9, Theorem 7.1.20]). The eigenvalues of đľâ đ´ â are positive, and one has the following characterizations of the maximum and minimum eigenvalues:
1 = ⍠V đđ , |đš| đš đť
(16)
âđ˘â â đâ , Vđ â đđ .
(17)
Then our two-level additive Schwarz preconditioner đľâ is stated as đ˝
đĄ â1 đĄ đľâ = đźđťđ´â1 đť đźđť + â đźđ đ´ đ đźđ .
óľŠ2 óľŠ2 óľŠ óľŠ 0 0 đ = â (óľŠóľŠóľŠóľŠVđť â đźđť VđťóľŠóľŠóľŠóľŠ0,đ + â óľŠóľŠóľŠóľŠđźđť Vđť â đźđť VđťóľŠóľŠóľŠóľŠ0,đđ ) đâTâ
for any đ â Tâ and đš â Fâ . We also need the transpose of the intergrid transfer operators đźđđĄ from đâ to the subspace đđ for đ = đť, 1, 2, . . . , đ˝, defined as ((đźđđĄ đ˘â , Vđ )) = ((đ˘â , đźđ Vđ )) ,
óľŠ2 óľŠóľŠ óľŠóľŠVđť â đźđťVđťóľŠóľŠóľŠ0,â
(18)
đ=1
3.2. Analysis. Our analysis is based on the standard Schwarz framework (see, e.g., [7, 9]).
(22)
óľ¨ óľ¨2 Ⲡâ2 óľ¨óľ¨óľ¨Vđťóľ¨óľ¨óľ¨1 , which completes the proof. Lemma 3. There exists an operator đđť from đâ to đđť such that for any Vâ in đâ it holds that óľŠ2 óľ¨ óľ¨2 óľŠóľŠ óľŠóľŠVâ â đđťVâ óľŠóľŠóľŠ + đť2 óľ¨óľ¨óľ¨đđťVâ óľ¨óľ¨óľ¨ Ⲡđť2 óľ¨óľ¨óľ¨óľ¨Vâ óľ¨óľ¨óľ¨óľ¨21,â . óľŠ0,â óľ¨ óľ¨1 óľŠ
(23)
Proof. Let đâ be the đ1 conforming finite element space defined on Tâ . Then, for any Vâ in đâ , we can construct a ĚVâ â đâ satisfying ([6, Lemma 3.5]) óľŠóľŠ óľŠ óľ¨ óľ¨ óľ¨ óľ¨ óľŠóľŠVâ â ĚVâ óľŠóľŠóľŠ0,â + â óľ¨óľ¨óľ¨ĚVâ óľ¨óľ¨óľ¨1,â Ⲡâ óľ¨óľ¨óľ¨Vâ óľ¨óľ¨óľ¨1,â .
(24)
4
Mathematical Problems in Engineering
Define đđťVâ = đđťĚVâ , where đđť is the đż2 projection operator from đâ to đđť. Using the standard properties of đđť (see, e.g., [7]) and the inequality (24), we have óľ¨óľ¨óľ¨đđťV óľ¨óľ¨óľ¨2 = óľ¨óľ¨óľ¨đ ĚV óľ¨óľ¨óľ¨2 Ⲡóľ¨óľ¨óľ¨ĚV óľ¨óľ¨óľ¨2 Ⲡóľ¨óľ¨óľ¨V óľ¨óľ¨óľ¨2 , óľ¨óľ¨ â óľ¨óľ¨1 óľ¨ đť â óľ¨1 óľ¨ â óľ¨1 óľ¨ â óľ¨1,â óľŠ2 óľŠóľŠ óľŠóľŠVâ â đđťVâ óľŠóľŠóľŠ = óľŠóľŠóľŠóľŠVâ â đđťĚVâ óľŠóľŠóľŠóľŠ20,â óľŠ0,â óľŠ (25) óľŠ2 óľŠ2 óľŠ óľŠ óľŠ óľŠ óľŠ óľŠ â˛ óľŠóľŠVâ â ĚVâ óľŠóľŠ0,â + óľŠóľŠđđťĚVâ â ĚVâ óľŠóľŠ0,â óľ¨ óľ¨2 óľ¨ óľ¨2 óľ¨ óľ¨2 Ⲡâ2 óľ¨óľ¨óľ¨Vâ óľ¨óľ¨óľ¨1,â + đť2 óľ¨óľ¨óľ¨ĚVâ óľ¨óľ¨óľ¨1 Ⲡđť2 óľ¨óľ¨óľ¨Vâ óľ¨óľ¨óľ¨1,â .
which, together with the triangle inequality and the assumption of đđ , yields óľ¨2 óľ¨ ((đ´ đ Vđ , Vđ )) Ⲡâ (óľ¨óľ¨óľ¨óľ¨đđ đ¤0 óľ¨óľ¨óľ¨óľ¨1,đ đâTâ,đ
óľŠ óľŠ2 + ââ1 óľŠóľŠóľŠóľŠđâ0 (đđ đ¤0 ) â đâđ (đđ đ¤đ )óľŠóľŠóľŠóľŠ0,đđ ) óľ¨ óľ¨2 Ⲡâ (⍠óľ¨óľ¨óľ¨óľ¨(âđđ â
đ¤0 + âđ¤0 â
đđ )óľ¨óľ¨óľ¨óľ¨ đđĽ đ đâTâ,đ
The proof is completed. Lemma 4. Given any Vâ â đâ , there exists a decomposition đ˝
Vâ = đźđťVđť + â đźđ Vđ ,
(26)
đ=1
where Vđť â đđť and Vđ â đđ , such that
óľŠ2 óľŠ2 óľŠ óľŠ â¤ â (2 óľŠóľŠóľŠóľŠâđđ â
đ¤0 óľŠóľŠóľŠóľŠ0,đ + 2 óľŠóľŠóľŠóľŠâđ¤0 â
đđ óľŠóľŠóľŠóľŠ0,đ
(32)
đâTâ,đ
đ˝
óľŠ óľŠ2 + ââ1 óľŠóľŠóľŠóľŠđâ0 (đđ đ¤0 ) â đâđ (đđ đ¤đ )óľŠóľŠóľŠóľŠ0,đđ )
(đ´ đťVđť, Vđť) + â ((đ´ đ Vđ , Vđ )) đ=1
Ⲡ(1 +
óľŠ óľŠ2 + ââ1 óľŠóľŠóľŠóľŠđâ0 (đđ đ¤0 ) â đâđ (đđ đ¤đ )óľŠóľŠóľŠóľŠ0,đđ )
(27)
2
đť ) ((đ´ â Vâ , Vâ )) . đż
Ⲡâ ( đâTâ,đ
1 óľŠóľŠ óľŠóľŠ2 óľ¨ óľ¨2 óľŠđ¤ óľŠ + óľ¨óľ¨đ¤ óľ¨óľ¨ đż2 óľŠ 0 óľŠ0,đ óľ¨ 0 óľ¨1,đ
đť
Proof. Let Vđť = đ V, đ¤ = Vâ â đźđťVđť, and Vđ = đâ (đđ đ¤) = {đâ0 (đđ đ¤0 ), đâđ (đđ đ¤đ )}, where đâ0 and đâđ are the piecewise đż2 projection to the interior polynomial space đđ (đ0 ) and the edge polynomial space đđ (đš), respectively. It is easy to check that đ˝
đ˝
đ=1
đ=1
đźđťVđť + â Vđ = đźđťVđť + â đâ (đđ đ¤) đ˝
= đźđťVđť + đâ ( â (đđ đ¤))
(28)
(29)
óľ¨ óľ¨2 óľ¨ óľ¨2 ((đ´ đ Vđ , Vđ )) Ⲡâ óľ¨óľ¨óľ¨óľ¨Vđ óľ¨óľ¨óľ¨óľ¨1,â,đ = â (óľ¨óľ¨óľ¨óľ¨đâ0 (đđ đ¤0 )óľ¨óľ¨óľ¨óľ¨1,đ đâTâ,đ
(30)
óľŠ óľŠ2 + â óľŠóľŠóľŠóľŠđâ0 (đđ đ¤0 ) â đâđ (đđ đ¤đ )óľŠóľŠóľŠóľŠ0,đđ ) . â1
2
By the inverse inequality, the stability of đż projection, the Poincar´e inequality, and the scaling argument, we deduce that óľ¨ óľ¨óľ¨ 0 óľ¨2 óľ¨2 óľ¨óľ¨đâ (đđ đ¤0 )óľ¨óľ¨óľ¨ = inf óľ¨óľ¨óľ¨đâ0 (đđ đ¤0 â đ)óľ¨óľ¨óľ¨ óľ¨ óľ¨1,đ đâR óľ¨ óľ¨1,đ óľŠ óľŠ2 Ⲡinf â óľŠóľŠóľŠóľŠđâ0 (đđ đ¤0 â đ)óľŠóľŠóľŠóľŠ0,đ đâR â1
óľŠ óľ¨2 óľŠ2 óľ¨ â¤ inf ââ1 óľŠóľŠóľŠóľŠđđ đ¤0 â đóľŠóľŠóľŠóľŠ0,đ Ⲡóľ¨óľ¨óľ¨óľ¨đđ đ¤0 óľ¨óľ¨óľ¨óľ¨1,đ , đâR
óľŠ2 óľŠ â˛ óľŠóľŠóľŠóľŠđâ0 (đđ đ¤0 ) â đđ đ¤0 óľŠóľŠóľŠóľŠ0,đđ óľŠ óľŠ2 + óľŠóľŠóľŠóľŠđđ đ¤0 â đâđ (đđ đ¤0 )óľŠóľŠóľŠóľŠ0,đđ
= đźđťVđť + đâ đ¤ = Vâ .
đâTâ,đ
For the last term, the triangle inequality gives óľŠóľŠ 0 óľŠ2 óľŠóľŠđâ (đđ đ¤0 ) â đâđ (đđ đ¤đ )óľŠóľŠóľŠ óľŠ óľŠ0,đđ
đ=1
It follows from Lemma 3 that óľ¨ óľ¨2 óľ¨ óľ¨2 (đ´ đťVđť, Vđť) Ⲡóľ¨óľ¨óľ¨Vđťóľ¨óľ¨óľ¨1 Ⲡóľ¨óľ¨óľ¨Vâ óľ¨óľ¨óľ¨1,â Ⲡ((đ´ â Vâ , Vâ )) . On the other hand, we have for Vđ that
óľŠ óľŠ2 + ââ1 óľŠóľŠóľŠóľŠđâ0 (đđ đ¤0 ) â đâđ (đđ đ¤đ )óľŠóľŠóľŠóľŠ0,đđ ) .
(33)
óľŠ óľŠ2 + óľŠóľŠóľŠóľŠđâđ (đđ đ¤0 ) â đâđ (đđ đ¤đ )óľŠóľŠóľŠóľŠ0,đđ = I + II + III. We estimate I, II, and III, respectively, as follows. From the trace theorem, the scaling argument, and the stability and approximation of đâ0 , we have óľŠ2 óľŠ I = óľŠóľŠóľŠóľŠđâ0 (đđ đ¤0 ) â đđ đ¤0 óľŠóľŠóľŠóľŠ0,đđ óľŠ óľŠ2 Ⲡââ1 óľŠóľŠóľŠóľŠđâ0 (đđ đ¤0 ) â đđ đ¤0 óľŠóľŠóľŠóľŠ0,đ óľ¨2 óľ¨2 óľ¨ óľ¨ + â óľ¨óľ¨óľ¨óľ¨đâ0 (đđ đ¤0 ) â đđ đ¤0 óľ¨óľ¨óľ¨óľ¨1,đ Ⲡâ óľ¨óľ¨óľ¨óľ¨đđ đ¤0 óľ¨óľ¨óľ¨óľ¨1,đ
(31) â˛
â óľŠóľŠ óľŠóľŠ2 óľ¨ óľ¨2 óľŠóľŠđ¤0 óľŠóľŠ0,đ + â óľ¨óľ¨óľ¨đ¤0 óľ¨óľ¨óľ¨1,đ . 2 đż
(34)
Mathematical Problems in Engineering
5
Denote đđ đ¤0 = (1/|đ|) âŤđ đđ đ¤0 đđĽ. The trace theorem, the đż2 stability of đâđ , and the Poincar´e inequality imply that óľŠ óľŠ2 II = óľŠóľŠóľŠóľŠđđ đ¤0 â đâđ (đđ đ¤0 )óľŠóľŠóľŠóľŠ0,đđ óľŠ2 óľŠ óľŠ óľŠ2 ⤠óľŠóľŠóľŠóľŠđđ đ¤0 â đđ đ¤0 óľŠóľŠóľŠóľŠ0,đđ + óľŠóľŠóľŠóľŠđâđ (đđ đ¤0 â đđ đ¤0 )óľŠóľŠóľŠóľŠ0,đđ óľŠ2 óľ¨2 óľŠ óľ¨ â˛ óľŠóľŠóľŠóľŠđđ đ¤0 â đđ đ¤0 óľŠóľŠóľŠóľŠ0,đđ Ⲡâ óľ¨óľ¨óľ¨óľ¨đđ đ¤0 óľ¨óľ¨óľ¨óľ¨1,đ â˛
(35)
â óľŠóľŠ óľŠóľŠ2 óľ¨ óľ¨2 óľŠđ¤ óľŠ + â óľ¨óľ¨óľ¨đ¤0 óľ¨óľ¨óľ¨1,đ . đż2 óľŠ 0 óľŠ0,đ óľŠ2 óľŠ III = óľŠóľŠóľŠóľŠđâđ (đđ đ¤0 ) â đâđ (đđ đ¤đ )óľŠóľŠóľŠóľŠ0,đđ óľŠ2 óľŠ óľŠ2 óľŠ â˛ óľŠóľŠóľŠóľŠđđ đ¤0 â đđ đ¤đ óľŠóľŠóľŠóľŠ0,đđ ⤠óľŠóľŠóľŠđ¤0 â đ¤đ óľŠóľŠóľŠ0,đđ .
(36)
((đ´ đ Vđ , Vđ )) (37) 1 óľŠóľŠ óľŠóľŠ2 óľŠ2 óľ¨ óľ¨2 â1 óľŠ óľŠóľŠđ¤0 óľŠóľŠ0,đ + óľ¨óľ¨óľ¨đ¤0 óľ¨óľ¨óľ¨1,đ + â óľŠóľŠóľŠđ¤0 â đ¤đ óľŠóľŠóľŠ0,đđ ] , 2 đż
1/16 11 15 119
1/32 11 15 231
1/64 11 16 436
1/128 12 16 864
Proof. The inequality can be obtained directly by using the triangle inequality and the assumption on the finite cover of subdomains.
Theorem 6. There exists a positive constant đś, independent of đť, â, đż, and đ˝, such that đ max (đľâ đ´ â ) đť 2 ⤠đś (1 + ) . đż đ min (đľâ đ´ â )
(41)
Remark 7. According the theorem, the two-level additive preconditioner is optimal if đť/đż is bounded above by a constant. In particular, when đż = đ(đť), we have đ
(đľâ đ´ â ) ⊽ đś.
(42)
4. Numerical Experiments
which leads to đ˝
đ˝
â ((đ´ đ Vđ , Vđ )) Ⲡâ â [
đ=1
đ=1 đâTâ,đ
1 óľŠóľŠ óľŠóľŠ2 óľ¨ óľ¨2 óľŠđ¤ óľŠ + óľ¨óľ¨đ¤ óľ¨óľ¨ đż2 óľŠ 0 óľŠ0,đ óľ¨ 0 óľ¨1,đ
In this section, we give some numerical results to demonstrate the efficiency of our preconditioner. For convenience, we consider a simple two-dimensional Poisson equation with homogeneous boundary as follows:
1 óľŠ óľŠ óľŠ2 óľŠ2 óľ¨ + ââ1 óľŠóľŠóľŠđ¤0 â đ¤đ óľŠóľŠóľŠ0,đđ ] Ⲡ( 2 óľŠóľŠóľŠVâ â đźđťVđťóľŠóľŠóľŠ0,â + óľ¨óľ¨óľ¨Vâ đż 1 óľŠ 1 óľŠ óľ¨2 óľŠ2 â đźđťVđťóľ¨óľ¨óľ¨1,â ) ⤠( 2 óľŠóľŠóľŠVâ â VđťóľŠóľŠóľŠ0,â + 2 óľŠóľŠóľŠVđť đż đż
âÎđ˘ = đ (38)
đť2 óľ¨ óľ¨2 óľŠ2 óľ¨2 óľ¨ â đźđťVđťóľŠóľŠóľŠ0,â + óľ¨óľ¨óľ¨Vâ â đźđťVđťóľ¨óľ¨óľ¨1,â ) Ⲡ(1 + 2 ) óľ¨óľ¨óľ¨Vâ óľ¨óľ¨óľ¨1,â đż Ⲡ(1 +
1/8 10 14 60
đ
(đľâ đ´ â ) =
Combining the above five inequalities, we find
đâTâ,đ
â PCG (đť/đż = 1) PCG (đť/đż = 2) CG
As an immediate consequence, we have the following theorem.
Simiarily, we obtain
Ⲡâ [
Table 1: The iteration numbers of the PCG and CG.
đť 2 ) ((đ´ â Vâ , Vâ )) . đż
Using (27) and (38), we achieve đ˝
(đ´ đťVđť, Vđť) + â ((đ´ đ Vđ , Vđ )) đ=1
Ⲡ(1 +
(39)
2
đť ) ((đ´ â Vâ , Vâ )) . đż
This ends the proof. Lemma 5. Let Vđť â đđť and Vđ â đđ , 1 ⤠đ ⤠đ˝. For any Vâ = đźđťVđť + âđ˝đ=1 Vđ , it is true that đ˝
((đ´ â Vâ , Vâ )) Ⲡ(đ´ đťVđť, Vđť) + â ((đ´ đ Vđ , Vđ )) . đ=1
(40)
in Ί,
đ˘ = 0 on đΊ.
(43)
We choose đ to satisfy the exact solution đ˘(đĽ1 , đĽ2 ) = đĽ1 (1âđĽ1 )đĽ2 (1âđĽ2 ) on Ί = [0, 1]Ă[0, 1]. Let Tđť be a uniform triangulation with the mesh size đť = 1/4, and let Tâ be the refinement of Tđť. All the tests are stopped when the relative error is less than 10â6 . In Table 1, we list the iteration numbers of the preconditioning conjugate gradient (PCG) method and the conjugate gradient (CG) method with different meshes. From the second and third rows of the table, we see that the iteration numbers of the PCG method are almost the same if the overlapping factors đť/đż are fixed. If the overlap becomes small, that is, the rate đť/đż is increasing, one needs a little more steps to achieve the tolerance. This indicates that the condition number of the preconditioned system is almost a constant which is independent of the mesh size â and only depends on the rate of the coarse mesh size and the overlap. Finally, we will do some experiments to show the efficiency of our algorithm for problems with checkboard distributed coefficient, which equals 1 and 10đ (đ = 1, 3, 6) on adjacent subdomains. We assume that the coefficient is constant on each coarser element, and the coarser mesh size and the overlap size are fixed (đť = 1/4, đż = 1/8). The
6
Mathematical Problems in Engineering
Table 2: The iteration numbers with different mesh sizes â and coefficient jumps (10đ ). đ 1 3 6
1/8 14 15 15
1/16 15 15 15
â 1/32 15 15 15
1/64 16 15 15
1/128 16 16 15
iteration numbers, with different mesh sizes â and jumps in the coefficient, are reported in Table 2, from which, we conclude that our preconditioner also works well for problems with discontinuous coefficients at least in two dimensions.
Competing Interests The authors declare that they have no competing interests.
Acknowledgments This work was supported by NSFC under the Grants 11371199, 11301275, 11371198, 11526097, and 11401294, the Opening Fund of Jiangsu Key Lab for NSLSCS under the Grant 201402, the Doctoral Fund of Ministry of Education of China under the Grant 20123207120001.
References [1] J. Wang and X. Ye, âA weak Galerkin finite element method for second-order elliptic problems,â Journal of Computational and Applied Mathematics, vol. 241, pp. 103â115, 2013. [2] R. Zhang and Q. Zhai, âA weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order,â Journal of Scientific Computing, vol. 64, no. 2, pp. 559â 585, 2015. [3] V. Ginting, G. Lin, and J. Liu, âOn application of the weak Galerkin finite element method to a two-phase model for subsurface flow,â Journal of Scientific Computing, vol. 66, no. 1, pp. 225â239, 2016. [4] L. Chen, J. Wang, and X. Ye, âA posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems,â Journal of Scientific Computing, vol. 59, no. 2, pp. 496â511, 2014. [5] B. Li and X. Xie, âA two-level algorithm for the weak Galerkin discretization of diffusion problems,â Journal of Computational and Applied Mathematics, vol. 287, pp. 179â195, 2015. [6] L. Chen, J. Wang, Y. Wang, and X. Ye, âAn auxiliary space multigrid preconditioner for the weak Galerkin method,â Computers & Mathematics with Applications, vol. 70, no. 4, pp. 330â344, 2015. [7] A. Toselli and O. Widlund, Domain Decomposition Methods Algorithms and Theory, Springer, New York, NY, USA, 2005. [8] L. Wang and X. Xu, The Mathematical Basis of the Finite Element Method, Science Press, Beijing, China, 2005 (Chinese). [9] S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, NY, USA, 2008.
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