A Two-Level Additive Schwarz Preconditioning Algorithm for the Weak

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 © 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 𝑃1 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 𝑂(ℎ−2 ), where ℎ denotes the mesh size. Recently,

Li and Xie [5] proposed a multigrid algorithm for the WG method using 𝑃1 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 𝑃1 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 ℎ and is of 𝑂((1 +

2

Mathematical Problems in Engineering

𝐻/𝛿)2 ), where 𝐻 denotes the coarse mesh size and 𝛿 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 𝜕Ω,

where Ω ⊂ R2 is bounded polygonal domain and 𝑓 ∈ 𝐿2 (Ω). Assume that the matrix A is symmetric positive definite (SPD); namely, there exist positive constants 𝛼, 𝛽 such that 𝑇

𝑇

2

𝛼𝜉 𝜉 ≤ 𝜉 A (𝑥) 𝜉 ≤ 𝛽𝜉 𝜉, ∀𝜉 ∈ R , 𝑥 ∈ Ω.

(2)

The variational formulations of (1) is to find 𝑢 ∈ 𝐻01 (Ω) such that 𝑎 (𝑢, V) = (𝑓, V)

∀V ∈ 𝐻01 (Ω) ,

(3)

where 𝑎(𝑢, V) = (A∇𝑢, ∇V). It follows from Lax-Milgram’s theorem that problem (3) has a unique solution. Let Tℎ be a quasiuniform triangulation of the domain Ω with the mesh size ℎ. On each 𝑇 ∈ Tℎ , the space 𝑊(𝑇) denotes the collections of weak functions, each of which is consisted of an interior part and a boundary part; that is, 2

𝑊 (𝑇) fl {V = {V0 , V𝑏 } : V0 ∈ 𝐿 (𝑇) , V𝑏 ∈ 𝐻

1/2

(𝜕𝑇)} . (4)

For any weak function V ∈ 𝑊(𝑇), the weak gradient of V, denoted by ∇𝑤 V ∈ 𝐻(div, 𝑇), satisfies ∫ ∇𝑤 V ⋅ q 𝑑𝑥 = − ∫ V0 ∇ ⋅ q 𝑑𝑥 + ∫ V𝑏 q ⋅ n 𝑑𝑠 𝑇

𝑇

𝜕𝑇

(5)

∀q ∈ 𝐻 (div, 𝑇) , where n is the unit outer normal of 𝜕𝑇. It is easy to see that ∇𝑤 is the classical gradient if it actions on a function 𝑢 ∈ 𝐻1 (𝑇). For each element 𝑇 ∈ Tℎ , we denote by 𝑇0 and 𝜕𝑇 the interior and the boundary of 𝑇. The notation F𝑇 stands for the set of edges of 𝑇. We also use Fℎ and F𝜕ℎ to denote the edges of Tℎ in Ω and on 𝜕Ω. Then the discrete WG space is defined as 𝑊ℎ fl {V : V|𝑇0 ∈ 𝑃𝑗 (𝑇0 ) ∀𝑇 ∈ Tℎ , V|𝐹 ∈ 𝑃𝑙 (𝐹) ∀𝐹 ∈ Fℎ , V|𝐹 = 0 ∀𝐹 ∈ F𝜕ℎ } ,

𝑇

𝑇

𝜕𝑇

(7)

where V0 and V𝑏 denote the values of V in the interior and on the boundary of 𝑇. Then the discrete problem for (3) is to seek 𝑢ℎ = {𝑢0 , 𝑢𝑏 } ∈ 𝑊ℎ such that 𝑎ℎ (𝑢ℎ , Vℎ ) = (𝑓, V0 ) , ∀Vℎ = {V0 , V𝑏 } ∈ 𝑊ℎ ,

−∇ ⋅ (A∇𝑢) = 𝑓, in Ω,

𝑇

∫ ∇𝑤 V ⋅ q 𝑑𝑥 = − ∫ V0 (∇ ⋅ q) 𝑑𝑥 + ∫ V𝑏 (q ⋅ n) 𝑑𝑠 ∀q ∈ 𝐺𝑗 (𝑇) ,

2. The Weak Galerkin Method for the Second-Order Elliptic Equation

𝑢 = 0,

where 𝑙 = 𝑗 or 𝑙 = 𝑗 + 1 and 𝑃𝑗 (𝑇0 ) denotes the set of polynomials on 𝑇0 with degree no more than 𝑗 ≥ 0. To define a discrete weak gradient, we use 𝐺𝑗 (𝑇) to denote the RT ̂𝑗 (𝑇)𝑥 if 𝑙 = 𝑗 and the BDM element element space [𝑃𝑗 (𝑇)]2 + 𝑃 2 ̂𝑗 (𝑇) is the homogeneous space [𝑃𝑗+1 (𝑇)] if 𝑙 = 𝑗 + 1, where 𝑃 polynomial with the degree 𝑗. For each V ∈ 𝑊ℎ , the discrete weak gradient ∇𝑤 V ∈ 𝐺𝑗 (𝑇) is defined as

(6)

(8)

where 𝑎ℎ (𝑢ℎ , Vℎ ) fl (A∇𝑤 𝑢ℎ , ∇𝑤 Vℎ ). For any Vℎ = {V0 , V𝑏 } and 𝑢ℎ = {𝑢0 , 𝑢𝑏 } in 𝑊ℎ , we use ((𝑢ℎ , Vℎ )) to indicate a special inner product defined as ((Vℎ , 𝑢ℎ )) fl ∑ [(V0 , 𝑢0 )𝑇 + ℎ (V0 − V𝑏 , 𝑢0 − 𝑢𝑏 )𝜕𝑇 ] , (9) 𝑇∈Tℎ

where (⋅, ⋅)𝑇 and (⋅, ⋅)𝜕𝑇 denote the 𝐿2 inner product on 𝑇 and 𝜕𝑇. Accordingly, a norm and seminorm are introduced for any Vℎ = {V0 , V𝑏 } ∈ 𝑊ℎ by 1/2

󵄩󵄩 󵄩󵄩 󵄩 󵄩2 󵄩󵄩Vℎ 󵄩󵄩0,ℎ fl ( ∑ 󵄩󵄩󵄩Vℎ 󵄩󵄩󵄩0,ℎ,𝑇 )

,

𝑇∈Tℎ

󵄨󵄨 󵄨󵄨 󵄨 󵄨2 󵄨󵄨Vℎ 󵄨󵄨1,ℎ fl ( ∑ 󵄨󵄨󵄨Vℎ 󵄨󵄨󵄨1,ℎ,𝑇 )

(10)

1/2

,

𝑇∈Tℎ

where ‖Vℎ ‖0,ℎ,𝑇 fl (‖V0 ‖20,𝑇 + ℎ‖V0 − V𝑏 ‖2𝜕𝑇 )1/2 and |Vℎ |1,ℎ,𝑇 fl (|V0 |21,𝑇 + ℎ−1 ‖V0 − V𝑏 ‖2𝜕𝑇 )1/2 . Let 𝐴 ℎ : 𝑊ℎ → 𝑊ℎ be an operator ((𝐴 ℎ 𝑢ℎ , Vℎ )) = (A∇𝑤 𝑢ℎ , ∇𝑤 Vℎ ) ,

∀𝑢ℎ , Vℎ ∈ 𝑊ℎ .

(11)

It can be proved that 𝐴 ℎ is symmetric and positive definite and the condition number is of 𝑂(ℎ−2 ) (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 Ω by 𝐽 overlapping subdomains Ω1 , Ω2 , . . . , Ω𝐽 such that each point in Ω belongs to no more than 𝑁𝐶 subdomains. We assume that the boundary of each subdomain does not cut through any elements in the triangulation Tℎ , and there


3

exist nonnegative 𝐶∞ functions 𝜃1 , 𝜃2 , . . . , 𝜃𝐽 satisfying the following properties: (1) 𝜃𝑗 = 0 in Ω \ Ω𝑗 ; (2) ∑𝐽𝑗=1 𝜃𝑗 = 1; (3) there exists a positive constant 𝛿, such that |∇𝜃𝑗 | ≤ 𝐶/𝛿, where 𝐶 is a constant independent of 𝛿, ℎ, and 𝐽. On each subdomain Ω𝑗 , the notation Tℎ,𝑗 stands for the triangulation inherited from Tℎ . The corresponding weak Galerkin subspace on Tℎ,𝑗 is defined as 𝑉𝑗 = {V ∈ 𝑊ℎ : V|Ω\Ω𝑗 = 0} .

(12)

We introduce an operator 𝐴 𝑗 : 𝑉𝑗 → 𝑉𝑗 by ((𝐴 𝑗 𝑢𝑗 , V𝑗 )) = (A∇𝑤 𝑢𝑗 , ∇𝑤 V𝑗 ) ,

∀𝑢𝑗 , V𝑗 ∈ 𝑉𝑗 .

(14)

((𝐴 ℎ V, V))

V∈𝑊ℎ ,V=0̸ min V=∑𝑗,V𝑗 ∈𝑉𝑗 𝐼𝑗 V𝑗

∑𝑗 ((𝐴 𝑗 V𝑗 , V𝑗 ))

, (19)

𝜆 min (𝐵ℎ 𝐴 ℎ ) ((𝐴 ℎ V, V))

V∈𝑊ℎ ,V=0̸ min V=∑𝑗,V𝑗 ∈𝑉𝑗 𝐼𝑗 V𝑗

∑𝑗 ((𝐴 𝑗 V𝑗 , V𝑗 ))

,

where the sum is taken over 𝑗 = 𝐻, 1, . . . , 𝐽. Lemma 2. For the intergrid transfer operator 𝐼𝐻, it holds for any V𝐻 ∈ 𝑉𝐻 that 󵄩󵄩 󵄩2 󵄨2 󵄨2 2󵄨 2󵄨 󵄩󵄩V𝐻 − 𝐼𝐻V𝐻󵄩󵄩󵄩0,ℎ + ℎ 󵄨󵄨󵄨𝐼𝐻V𝐻󵄨󵄨󵄨1,ℎ ≲ ℎ 󵄨󵄨󵄨V𝐻󵄨󵄨󵄨1 .

(20)

Proof. We only need consider the case that the functions in 𝑊ℎ are the constants in the interior and on the boundary of each element. It follows from the triangle inequality, the trace inequality, and the Poincaré-Friedrichs inequality that 󵄩2 󵄨2 󵄨󵄨 −1 󵄩 0 𝑏 󵄨󵄨𝐼𝐻V𝐻󵄨󵄨󵄨1,ℎ = ∑ ℎ 󵄩󵄩󵄩󵄩𝐼𝐻V𝐻 − 𝐼𝐻V𝐻󵄩󵄩󵄩󵄩0,𝜕𝑇 𝑇∈Tℎ

󵄩2 󵄩2 󵄩 0 󵄩 𝑏 ≲ ∑ ℎ−1 (󵄩󵄩󵄩󵄩𝐼𝐻 V𝐻 − V𝐻󵄩󵄩󵄩󵄩0,𝜕𝑇 + 󵄩󵄩󵄩󵄩𝐼𝐻 V𝐻 − V𝐻󵄩󵄩󵄩󵄩0,𝜕𝑇 )

(21)

𝑇∈Tℎ

(15)

0 𝑏 and 𝐼𝐻 satisfy where 𝐼𝐻

󵄨 󵄨2 = 󵄨󵄨󵄨V𝐻󵄨󵄨󵄨1 . Similarly, for the lower-order term, we have

1 󵄨 0 V𝐻)󵄨󵄨󵄨󵄨𝑇 = (𝐼𝐻 ∫ V 𝑑𝑥, |𝑇| 𝑇 𝐻 󵄨 𝑏 V𝐻)󵄨󵄨󵄨󵄨𝐹 (𝐼𝐻

= max

(13)

Note that the coarse space 𝑉𝐻 is a subspace of 𝑊ℎ if the degree of the piecewise polynomials in 𝑊ℎ is greater than 0, and we can choose a natural injection as the intergird transfer operator 𝐼𝐻 from 𝑉𝐻 to 𝑊ℎ . For the piecewise constant space case, the intergrid transfer operator 𝐼𝐻 : 𝑉𝐻 → 𝑊ℎ is defined as 0 𝑏 V𝐻, 𝐼𝐻 V𝐻} , ∀V𝐻 ∈ 𝑉𝐻, 𝐼𝐻V𝐻 = {𝐼𝐻

𝜆 max (𝐵ℎ 𝐴 ℎ )

= min

It is easy to see that 𝐴 𝑗 is symmetric and positive definite. Since 𝑉𝑗 is a subset of 𝑊ℎ , we use 𝐼𝑗 to denote a natural injection from 𝑉𝑗 to 𝑊ℎ . To define a coarse subproblem, we define a coarse triangulation T𝐻 with mesh size 𝐻 such that each element in Tℎ is a subdivision of the one in T𝐻. The notation 𝑉𝐻 ⊂ 𝐻01 (Ω) stands for 𝑃1 conforming finite element space associated with T𝐻. We also introduce an operator 𝐴 𝐻 : 𝑉𝐻 → 𝑉𝐻 satisfying (𝐴 𝐻𝑢𝐻, V𝐻) = (A∇𝑢𝐻, ∇V𝐻) , ∀𝑢𝐻, V𝐻 ∈ 𝑉𝐻.

Lemma 1 ([9, Theorem 7.1.20]). The eigenvalues of 𝐵ℎ 𝐴 ℎ are positive, and one has the following characterizations of the maximum and minimum eigenvalues:

1 = ∫ V 𝑑𝑠, |𝐹| 𝐹 𝐻

(16)

∀𝑢ℎ ∈ 𝑊ℎ , V𝑗 ∈ 𝑉𝑗 .

(17)

Then our two-level additive Schwarz preconditioner 𝐵ℎ is stated as 𝐽

𝑡 −1 𝑡 𝐵ℎ = 𝐼𝐻𝐴−1 𝐻 𝐼𝐻 + ∑ 𝐼𝑗 𝐴 𝑗 𝐼𝑗 .

󵄩2 󵄩2 󵄩 󵄩 0 0 𝑏 = ∑ (󵄩󵄩󵄩󵄩V𝐻 − 𝐼𝐻 V𝐻󵄩󵄩󵄩󵄩0,𝑇 + ℎ 󵄩󵄩󵄩󵄩𝐼𝐻 V𝐻 − 𝐼𝐻 V𝐻󵄩󵄩󵄩󵄩0,𝜕𝑇 ) 𝑇∈Tℎ

for any 𝑇 ∈ Tℎ and 𝐹 ∈ Fℎ . We also need the transpose of the intergrid transfer operators 𝐼𝑗𝑡 from 𝑊ℎ to the subspace 𝑉𝑗 for 𝑗 = 𝐻, 1, 2, . . . , 𝐽, defined as ((𝐼𝑗𝑡 𝑢ℎ , V𝑗 )) = ((𝑢ℎ , 𝐼𝑗 V𝑗 )) ,

󵄩2 󵄩󵄩 󵄩󵄩V𝐻 − 𝐼𝐻V𝐻󵄩󵄩󵄩0,ℎ

(18)

𝑗=1

3.2. Analysis. Our analysis is based on the standard Schwarz framework (see, e.g., [7, 9]).

(22)

󵄨 󵄨2 ≲ ℎ2 󵄨󵄨󵄨V𝐻󵄨󵄨󵄨1 , which completes the proof. Lemma 3. There exists an operator 𝑄𝐻 from 𝑊ℎ to 𝑉𝐻 such that for any Vℎ in 𝑊ℎ it holds that 󵄩2 󵄨 󵄨2 󵄩󵄩 󵄩󵄩Vℎ − 𝑄𝐻Vℎ 󵄩󵄩󵄩 + 𝐻2 󵄨󵄨󵄨𝑄𝐻Vℎ 󵄨󵄨󵄨 ≲ 𝐻2 󵄨󵄨󵄨󵄨Vℎ 󵄨󵄨󵄨󵄨21,ℎ . 󵄩0,ℎ 󵄨 󵄨1 󵄩

(23)

Proof. Let 𝑉ℎ be the 𝑃1 conforming finite element space defined on Tℎ . Then, for any Vℎ in 𝑊ℎ , we can construct a ̃Vℎ ∈ 𝑉ℎ satisfying ([6, Lemma 3.5]) 󵄩󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 󵄩󵄩Vℎ − ̃Vℎ 󵄩󵄩󵄩0,ℎ + ℎ 󵄨󵄨󵄨̃Vℎ 󵄨󵄨󵄨1,ℎ ≲ ℎ 󵄨󵄨󵄨Vℎ 󵄨󵄨󵄨1,ℎ .

(24)

4


Define 𝑄𝐻Vℎ = 𝑄𝐻̃Vℎ , where 𝑄𝐻 is the 𝐿2 projection operator from 𝑉ℎ to 𝑉𝐻. Using the standard properties of 𝑄𝐻 (see, e.g., [7]) and the inequality (24), we have 󵄨󵄨󵄨𝑄𝐻V 󵄨󵄨󵄨2 = 󵄨󵄨󵄨𝑄 ̃V 󵄨󵄨󵄨2 ≲ 󵄨󵄨󵄨̃V 󵄨󵄨󵄨2 ≲ 󵄨󵄨󵄨V 󵄨󵄨󵄨2 , 󵄨󵄨 ℎ 󵄨󵄨1 󵄨 𝐻 ℎ 󵄨1 󵄨 ℎ 󵄨1 󵄨 ℎ 󵄨1,ℎ 󵄩2 󵄩󵄩 󵄩󵄩Vℎ − 𝑄𝐻Vℎ 󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩Vℎ − 𝑄𝐻̃Vℎ 󵄩󵄩󵄩󵄩20,ℎ 󵄩0,ℎ 󵄩 (25) 󵄩2 󵄩2 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≲ 󵄩󵄩Vℎ − ̃Vℎ 󵄩󵄩0,ℎ + 󵄩󵄩𝑄𝐻̃Vℎ − ̃Vℎ 󵄩󵄩0,ℎ 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 ≲ ℎ2 󵄨󵄨󵄨Vℎ 󵄨󵄨󵄨1,ℎ + 𝐻2 󵄨󵄨󵄨̃Vℎ 󵄨󵄨󵄨1 ≲ 𝐻2 󵄨󵄨󵄨Vℎ 󵄨󵄨󵄨1,ℎ .

which, together with the triangle inequality and the assumption of 𝜃𝑗 , yields 󵄨2 󵄨 ((𝐴 𝑗 V𝑗 , V𝑗 )) ≲ ∑ (󵄨󵄨󵄨󵄨𝜃𝑗 𝑤0 󵄨󵄨󵄨󵄨1,𝑇 𝑇∈Tℎ,𝑗

󵄩 󵄩2 + ℎ−1 󵄩󵄩󵄩󵄩𝑄ℎ0 (𝜃𝑗 𝑤0 ) − 𝑄ℎ𝑏 (𝜃𝑗 𝑤𝑏 )󵄩󵄩󵄩󵄩0,𝜕𝑇 ) 󵄨 󵄨2 ≲ ∑ (∫ 󵄨󵄨󵄨󵄨(∇𝜃𝑗 ⋅ 𝑤0 + ∇𝑤0 ⋅ 𝜃𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑥 𝑇 𝑇∈Tℎ,𝑗

The proof is completed. Lemma 4. Given any Vℎ ∈ 𝑊ℎ , there exists a decomposition 𝐽

Vℎ = 𝐼𝐻V𝐻 + ∑ 𝐼𝑗 V𝑗 ,

(26)

𝑗=1

where V𝐻 ∈ 𝑉𝐻 and V𝑗 ∈ 𝑉𝑗 , such that

󵄩2 󵄩2 󵄩 󵄩 ≤ ∑ (2 󵄩󵄩󵄩󵄩∇𝜃𝑗 ⋅ 𝑤0 󵄩󵄩󵄩󵄩0,𝑇 + 2 󵄩󵄩󵄩󵄩∇𝑤0 ⋅ 𝜃𝑗 󵄩󵄩󵄩󵄩0,𝑇

(32)

𝑇∈Tℎ,𝑗

𝐽

󵄩 󵄩2 + ℎ−1 󵄩󵄩󵄩󵄩𝑄ℎ0 (𝜃𝑗 𝑤0 ) − 𝑄ℎ𝑏 (𝜃𝑗 𝑤𝑏 )󵄩󵄩󵄩󵄩0,𝜕𝑇 )

(𝐴 𝐻V𝐻, V𝐻) + ∑ ((𝐴 𝑗 V𝑗 , V𝑗 )) 𝑗=1

≲ (1 +

󵄩 󵄩2 + ℎ−1 󵄩󵄩󵄩󵄩𝑄ℎ0 (𝜃𝑗 𝑤0 ) − 𝑄ℎ𝑏 (𝜃𝑗 𝑤𝑏 )󵄩󵄩󵄩󵄩0,𝜕𝑇 )

(27)

2

𝐻 ) ((𝐴 ℎ Vℎ , Vℎ )) . 𝛿

≲ ∑ ( 𝑇∈Tℎ,𝑗

1 󵄩󵄩 󵄩󵄩2 󵄨 󵄨2 󵄩𝑤 󵄩 + 󵄨󵄨𝑤 󵄨󵄨 𝛿2 󵄩 0 󵄩0,𝑇 󵄨 0 󵄨1,𝑇

𝐻

Proof. Let V𝐻 = 𝑄 V, 𝑤 = Vℎ − 𝐼𝐻V𝐻, and V𝑗 = 𝑄ℎ (𝜃𝑗 𝑤) = {𝑄ℎ0 (𝜃𝑗 𝑤0 ), 𝑄ℎ𝑏 (𝜃𝑗 𝑤𝑏 )}, where 𝑄ℎ0 and 𝑄ℎ𝑏 are the piecewise 𝐿2 projection to the interior polynomial space 𝑃𝑗 (𝑇0 ) and the edge polynomial space 𝑃𝑙 (𝐹), respectively. It is easy to check that 𝐽

𝐽

𝑗=1

𝑗=1

𝐼𝐻V𝐻 + ∑ V𝑗 = 𝐼𝐻V𝐻 + ∑ 𝑄ℎ (𝜃𝑗 𝑤) 𝐽

= 𝐼𝐻V𝐻 + 𝑄ℎ ( ∑ (𝜃𝑗 𝑤))

(28)

(29)

󵄨 󵄨2 󵄨 󵄨2 ((𝐴 𝑗 V𝑗 , V𝑗 )) ≲ ∑ 󵄨󵄨󵄨󵄨V𝑗 󵄨󵄨󵄨󵄨1,ℎ,𝑇 = ∑ (󵄨󵄨󵄨󵄨𝑄ℎ0 (𝜃𝑗 𝑤0 )󵄨󵄨󵄨󵄨1,𝑇 𝑇∈Tℎ,𝑗

(30)

󵄩 󵄩2 + ℎ 󵄩󵄩󵄩󵄩𝑄ℎ0 (𝜃𝑗 𝑤0 ) − 𝑄ℎ𝑏 (𝜃𝑗 𝑤𝑏 )󵄩󵄩󵄩󵄩0,𝜕𝑇 ) . −1

2

By the inverse inequality, the stability of 𝐿 projection, the Poincaré inequality, and the scaling argument, we deduce that 󵄨 󵄨󵄨 0 󵄨2 󵄨2 󵄨󵄨𝑄ℎ (𝜃𝑗 𝑤0 )󵄨󵄨󵄨 = inf 󵄨󵄨󵄨𝑄ℎ0 (𝜃𝑗 𝑤0 − 𝑐)󵄨󵄨󵄨 󵄨 󵄨1,𝑇 𝑐∈R 󵄨 󵄨1,𝑇 󵄩 󵄩2 ≲ inf ℎ 󵄩󵄩󵄩󵄩𝑄ℎ0 (𝜃𝑗 𝑤0 − 𝑐)󵄩󵄩󵄩󵄩0,𝑇 𝑐∈R −1

󵄩 󵄨2 󵄩2 󵄨 ≤ inf ℎ−1 󵄩󵄩󵄩󵄩𝜃𝑗 𝑤0 − 𝑐󵄩󵄩󵄩󵄩0,𝑇 ≲ 󵄨󵄨󵄨󵄨𝜃𝑗 𝑤0 󵄨󵄨󵄨󵄨1,𝑇 , 𝑐∈R

󵄩2 󵄩 ≲ 󵄩󵄩󵄩󵄩𝑄ℎ0 (𝜃𝑗 𝑤0 ) − 𝜃𝑗 𝑤0 󵄩󵄩󵄩󵄩0,𝜕𝑇 󵄩 󵄩2 + 󵄩󵄩󵄩󵄩𝜃𝑗 𝑤0 − 𝑄ℎ𝑏 (𝜃𝑗 𝑤0 )󵄩󵄩󵄩󵄩0,𝜕𝑇

= 𝐼𝐻V𝐻 + 𝑄ℎ 𝑤 = Vℎ .

𝑇∈Tℎ,𝑗

For the last term, the triangle inequality gives 󵄩󵄩 0 󵄩2 󵄩󵄩𝑄ℎ (𝜃𝑗 𝑤0 ) − 𝑄ℎ𝑏 (𝜃𝑗 𝑤𝑏 )󵄩󵄩󵄩 󵄩 󵄩0,𝜕𝑇

𝑗=1

It follows from Lemma 3 that 󵄨 󵄨2 󵄨 󵄨2 (𝐴 𝐻V𝐻, V𝐻) ≲ 󵄨󵄨󵄨V𝐻󵄨󵄨󵄨1 ≲ 󵄨󵄨󵄨Vℎ 󵄨󵄨󵄨1,ℎ ≲ ((𝐴 ℎ Vℎ , Vℎ )) . On the other hand, we have for V𝑗 that

󵄩 󵄩2 + ℎ−1 󵄩󵄩󵄩󵄩𝑄ℎ0 (𝜃𝑗 𝑤0 ) − 𝑄ℎ𝑏 (𝜃𝑗 𝑤𝑏 )󵄩󵄩󵄩󵄩0,𝜕𝑇 ) .

(33)

󵄩 󵄩2 + 󵄩󵄩󵄩󵄩𝑄ℎ𝑏 (𝜃𝑗 𝑤0 ) − 𝑄ℎ𝑏 (𝜃𝑗 𝑤𝑏 )󵄩󵄩󵄩󵄩0,𝜕𝑇 = 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 𝑄ℎ0 , we have 󵄩2 󵄩 I = 󵄩󵄩󵄩󵄩𝑄ℎ0 (𝜃𝑗 𝑤0 ) − 𝜃𝑗 𝑤0 󵄩󵄩󵄩󵄩0,𝜕𝑇 󵄩 󵄩2 ≲ ℎ−1 󵄩󵄩󵄩󵄩𝑄ℎ0 (𝜃𝑗 𝑤0 ) − 𝜃𝑗 𝑤0 󵄩󵄩󵄩󵄩0,𝑇 󵄨2 󵄨2 󵄨 󵄨 + ℎ 󵄨󵄨󵄨󵄨𝑄ℎ0 (𝜃𝑗 𝑤0 ) − 𝜃𝑗 𝑤0 󵄨󵄨󵄨󵄨1,𝑇 ≲ ℎ 󵄨󵄨󵄨󵄨𝜃𝑗 𝑤0 󵄨󵄨󵄨󵄨1,𝑇

(31) ≲

ℎ 󵄩󵄩 󵄩󵄩2 󵄨 󵄨2 󵄩󵄩𝑤0 󵄩󵄩0,𝑇 + ℎ 󵄨󵄨󵄨𝑤0 󵄨󵄨󵄨1,𝑇 . 2 𝛿

(34)


5

Denote 𝜃𝑗 𝑤0 = (1/|𝑇|) ∫𝑇 𝜃𝑗 𝑤0 𝑑𝑥. The trace theorem, the 𝐿2 stability of 𝑄ℎ𝑏 , and the Poincaré inequality imply that 󵄩 󵄩2 II = 󵄩󵄩󵄩󵄩𝜃𝑗 𝑤0 − 𝑄ℎ𝑏 (𝜃𝑗 𝑤0 )󵄩󵄩󵄩󵄩0,𝜕𝑇 󵄩2 󵄩 󵄩 󵄩2 ≤ 󵄩󵄩󵄩󵄩𝜃𝑗 𝑤0 − 𝜃𝑗 𝑤0 󵄩󵄩󵄩󵄩0,𝜕𝑇 + 󵄩󵄩󵄩󵄩𝑄ℎ𝑏 (𝜃𝑗 𝑤0 − 𝜃𝑗 𝑤0 )󵄩󵄩󵄩󵄩0,𝜕𝑇 󵄩2 󵄨2 󵄩 󵄨 ≲ 󵄩󵄩󵄩󵄩𝜃𝑗 𝑤0 − 𝜃𝑗 𝑤0 󵄩󵄩󵄩󵄩0,𝜕𝑇 ≲ ℎ 󵄨󵄨󵄨󵄨𝜃𝑗 𝑤0 󵄨󵄨󵄨󵄨1,𝑇 ≲

(35)

ℎ 󵄩󵄩 󵄩󵄩2 󵄨 󵄨2 󵄩𝑤 󵄩 + ℎ 󵄨󵄨󵄨𝑤0 󵄨󵄨󵄨1,𝑇 . 𝛿2 󵄩 0 󵄩0,𝑇 󵄩2 󵄩 III = 󵄩󵄩󵄩󵄩𝑄ℎ𝑏 (𝜃𝑗 𝑤0 ) − 𝑄ℎ𝑏 (𝜃𝑗 𝑤𝑏 )󵄩󵄩󵄩󵄩0,𝜕𝑇 󵄩2 󵄩 󵄩2 󵄩 ≲ 󵄩󵄩󵄩󵄩𝜃𝑗 𝑤0 − 𝜃𝑗 𝑤𝑏 󵄩󵄩󵄩󵄩0,𝜕𝑇 ≤ 󵄩󵄩󵄩𝑤0 − 𝑤𝑏 󵄩󵄩󵄩0,𝜕𝑇 .

(36)

((𝐴 𝑗 V𝑗 , V𝑗 )) (37) 1 󵄩󵄩 󵄩󵄩2 󵄩2 󵄨 󵄨2 −1 󵄩 󵄩󵄩𝑤0 󵄩󵄩0,𝑇 + 󵄨󵄨󵄨𝑤0 󵄨󵄨󵄨1,𝑇 + ℎ 󵄩󵄩󵄩𝑤0 − 𝑤𝑏 󵄩󵄩󵄩0,𝜕𝑇 ] , 2 𝛿

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 𝐶, independent of 𝐻, ℎ, 𝛿, and 𝐽, such that 𝜆 max (𝐵ℎ 𝐴 ℎ ) 𝐻 2 ≤ 𝐶 (1 + ) . 𝛿 𝜆 min (𝐵ℎ 𝐴 ℎ )

(41)

Remark 7. According the theorem, the two-level additive preconditioner is optimal if 𝐻/𝛿 is bounded above by a constant. In particular, when 𝛿 = 𝑂(𝐻), we have 𝜅 (𝐵ℎ 𝐴 ℎ ) ⩽ 𝐶.

(42)

4. Numerical Experiments

which leads to 𝐽

𝐽

∑ ((𝐴 𝑗 V𝑗 , V𝑗 )) ≲ ∑ ∑ [

𝑗=1

𝑗=1 𝑇∈Tℎ,𝑗

1 󵄩󵄩 󵄩󵄩2 󵄨 󵄨2 󵄩𝑤 󵄩 + 󵄨󵄨𝑤 󵄨󵄨 𝛿2 󵄩 0 󵄩0,𝑇 󵄨 0 󵄨1,𝑇

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 󵄩 󵄩 󵄩2 󵄩2 󵄨 + ℎ−1 󵄩󵄩󵄩𝑤0 − 𝑤𝑏 󵄩󵄩󵄩0,𝜕𝑇 ] ≲ ( 2 󵄩󵄩󵄩Vℎ − 𝐼𝐻V𝐻󵄩󵄩󵄩0,ℎ + 󵄨󵄨󵄨Vℎ 𝛿 1 󵄩 1 󵄩 󵄨2 󵄩2 − 𝐼𝐻V𝐻󵄨󵄨󵄨1,ℎ ) ≤ ( 2 󵄩󵄩󵄩Vℎ − V𝐻󵄩󵄩󵄩0,ℎ + 2 󵄩󵄩󵄩V𝐻 𝛿 𝛿

−Δ𝑢 = 𝑓 (38)

𝐻2 󵄨 󵄨2 󵄩2 󵄨2 󵄨 − 𝐼𝐻V𝐻󵄩󵄩󵄩0,ℎ + 󵄨󵄨󵄨Vℎ − 𝐼𝐻V𝐻󵄨󵄨󵄨1,ℎ ) ≲ (1 + 2 ) 󵄨󵄨󵄨Vℎ 󵄨󵄨󵄨1,ℎ 𝛿 ≲ (1 +

1/8 10 14 60

𝜅 (𝐵ℎ 𝐴 ℎ ) =

Combining the above five inequalities, we find

𝑇∈Tℎ,𝑗

ℎ PCG (𝐻/𝛿 = 1) PCG (𝐻/𝛿 = 2) CG

As an immediate consequence, we have the following theorem.

Simiarily, we obtain

≲ ∑ [

Table 1: The iteration numbers of the PCG and CG.

𝐻 2 ) ((𝐴 ℎ Vℎ , Vℎ )) . 𝛿

Using (27) and (38), we achieve 𝐽

(𝐴 𝐻V𝐻, V𝐻) + ∑ ((𝐴 𝑗 V𝑗 , V𝑗 )) 𝑗=1

≲ (1 +

(39)

2

𝐻 ) ((𝐴 ℎ Vℎ , Vℎ )) . 𝛿

This ends the proof. Lemma 5. Let V𝐻 ∈ 𝑉𝐻 and V𝑗 ∈ 𝑉𝑗 , 1 ≤ 𝑗 ≤ 𝐽. For any Vℎ = 𝐼𝐻V𝐻 + ∑𝐽𝑗=1 V𝑗 , it is true that 𝐽

((𝐴 ℎ Vℎ , Vℎ )) ≲ (𝐴 𝐻V𝐻, V𝐻) + ∑ ((𝐴 𝑗 V𝑗 , V𝑗 )) . 𝑗=1

(40)

in Ω,

𝑢 = 0 on 𝜕Ω.

(43)

We choose 𝑓 to satisfy the exact solution 𝑢(𝑥1 , 𝑥2 ) = 𝑥1 (1−𝑥1 )𝑥2 (1−𝑥2 ) on Ω = [0, 1]×[0, 1]. Let T𝐻 be a uniform triangulation with the mesh size 𝐻 = 1/4, and let Tℎ be the refinement of T𝐻. All the tests are stopped when the relative error is less than 10−6 . 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 𝐻/𝛿 are fixed. If the overlap becomes small, that is, the rate 𝐻/𝛿 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 ℎ 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 10𝑖 (𝑖 = 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 (𝐻 = 1/4, 𝛿 = 1/8). The

6


Table 2: The iteration numbers with different mesh sizes ℎ and coefficient jumps (10𝑖 ). 𝑖 1 3 6

1/8 14 15 15

1/16 15 15 15

ℎ 1/32 15 15 15

1/64 16 15 15

1/128 16 16 15

iteration numbers, with different mesh sizes ℎ 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, “A weak Galerkin finite element method for second-order elliptic problems,” Journal of Computational and Applied Mathematics, vol. 241, pp. 103–115, 2013. [2] R. Zhang and Q. Zhai, “A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order,” Journal of Scientific Computing, vol. 64, no. 2, pp. 559– 585, 2015. [3] V. Ginting, G. Lin, and J. Liu, “On application of the weak Galerkin finite element method to a two-phase model for subsurface flow,” Journal of Scientific Computing, vol. 66, no. 1, pp. 225–239, 2016. [4] L. Chen, J. Wang, and X. Ye, “A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems,” Journal of Scientific Computing, vol. 59, no. 2, pp. 496–511, 2014. [5] B. Li and X. Xie, “A two-level algorithm for the weak Galerkin discretization of diffusion problems,” Journal of Computational and Applied Mathematics, vol. 287, pp. 179–195, 2015. [6] L. Chen, J. Wang, Y. Wang, and X. Ye, “An auxiliary space multigrid preconditioner for the weak Galerkin method,” Computers & Mathematics with Applications, vol. 70, no. 4, pp. 330–344, 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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