The Cascadic Multigrid Method of the Weak Galerkin Method for Second-Order Elliptic Equation

The weak Galerkin (WG) finite element method (FEM) is a recently developed numerical method for solving various types of partial differential equations. A new concept of the discrete weak gradient is introduced, which is the most significant feature of the weak Galerkin method. Due to the definition of weak gradient, the weak Galerkin finite element method is flexible in numerical approximation. There have been some studies and applications of the weak Galerkin finite element method. The method was first introduced by Wang and Ye in [1] for second-order elliptic problems. The corresponding numerical analysis of the weak Galerkin method based on Raviart-Thomas (RT) elements and Brezzi-Douglas-Marini (BDM) elements is given in [2]. A stabilization technique was presented and applied to the weak Galerkin finite element method, and the resulting weak Galerkin finite element method is no longer limited to RT and BDM elements [3]. In [4], the weak Galerkin mixed finite element method for biharmonic equations has been developed. For the applications of the weak Galerkin finite element method for other types of partial differential equations, the readers are referred to [5–8]. In this paper, we consider the cascadic multigrid method for solving the linear system generated by the weak Galerkin finite element method for second-order elliptic problems. Multigrid methods [9] have been shown to be very effective in solving large scale system theoretically and numerically. The cascadic multigridmethod [10, 11] is a one-waymultigrid method and easy to be implemented since it requires no coarse grid corrections at all. Much effort has been made to the analysis of cascadic multigrid method (see, e.g., [12, 13]). Following the idea of [13], we can establish the error estimate in energy norm and the computational complexity estimate of the proposed cascadic multigrid method. The rest of this paper is organized as follows. In Section 2, we introduce the weak Galerkin finite element method for second-order elliptic problems. In Section 3, a cascadic multigrid algorithm based on the weak Galerkin finite element discretization is proposed and analyzed, and the error estimates in energy norm and computational complexity are obtained. Numerical experiments are conducted to confirm our theoretical results in Section 4. Finally, we give the conclusion in Section 5.


Introduction
The weak Galerkin (WG) finite element method (FEM) is a recently developed numerical method for solving various types of partial differential equations.A new concept of the discrete weak gradient is introduced, which is the most significant feature of the weak Galerkin method.Due to the definition of weak gradient, the weak Galerkin finite element method is flexible in numerical approximation.
There have been some studies and applications of the weak Galerkin finite element method.The method was first introduced by Wang and Ye in [1] for second-order elliptic problems.The corresponding numerical analysis of the weak Galerkin method based on Raviart-Thomas (RT) elements and Brezzi-Douglas-Marini (BDM) elements is given in [2].A stabilization technique was presented and applied to the weak Galerkin finite element method, and the resulting weak Galerkin finite element method is no longer limited to RT and BDM elements [3].In [4], the weak Galerkin mixed finite element method for biharmonic equations has been developed.For the applications of the weak Galerkin finite element method for other types of partial differential equations, the readers are referred to [5][6][7][8].
In this paper, we consider the cascadic multigrid method for solving the linear system generated by the weak Galerkin finite element method for second-order elliptic problems.Multigrid methods [9] have been shown to be very effective in solving large scale system theoretically and numerically.The cascadic multigrid method [10,11] is a one-way multigrid method and easy to be implemented since it requires no coarse grid corrections at all.Much effort has been made to the analysis of cascadic multigrid method (see, e.g., [12,13]).Following the idea of [13], we can establish the error estimate in energy norm and the computational complexity estimate of the proposed cascadic multigrid method.
The rest of this paper is organized as follows.In Section 2, we introduce the weak Galerkin finite element method for second-order elliptic problems.In Section 3, a cascadic multigrid algorithm based on the weak Galerkin finite element discretization is proposed and analyzed, and the error estimates in energy norm and computational complexity are obtained.Numerical experiments are conducted to confirm our theoretical results in Section 4. Finally, we give the conclusion in Section 5.

Model Problem and Its WG Finite Element Approximation
Consider the following second-order elliptic problem: where Ω is a convex polygonal domain with boundary . Furthermore, assume that A is symmetric uniformly positive definite and uniformly bounded-above diffusion; namely, there exist positive constants  and  such that Here and thereafter, for any subset  ⊆ R 2 , we use the standard notations for the Sobolev spaces   () and   0 () with  ≥ 0. The inner-product, norm, and seminorm in   () are denoted by (⋅, ⋅) , , ‖ ⋅ ‖ , and | ⋅ | , , respectively, and we skip the subscript  when  = Ω.
Since the domain Ω is convex, the unique solution  of problem (1) exists and satisfies the full regularity assumption [14] ‖‖ 2 ≤          0 .
Let  be a polygonal domain with interior  0 and boundary .Denote by () the space of weak functions on ; that is, For any V ∈ (), the weak gradient of V is defined as where (div, ) = {q : q ∈ [ 2 ()] 2 , ∇ ⋅ q ∈  2 ()}.
The discrete weak gradient, denoted by ∇ ,, V ∈ [  ()] 2 , is defined as follows: Let  ℎ be a shape-regular, quasi-uniform triangular mesh of the domain Ω, with the mesh size ℎ. denotes a generic positive constant independent of the mesh size ℎ throughout this paper.Denote the weak function space on  ℎ by ; that is, For any given integer  ≥ 1, define   () as the discrete weak function space consisting of polynomials of degree  in  and piecewise polynomials of degree  on ; that is, The weak Galerkin finite element spaces are defined as follows: It follows from [3] that For the discrete weak gradient, we will drop the subscript −1 in the notation ∇ ,−1 for simplicity.The weak Galerkin finite element method can be written as to find  ℎ = { 0 ,   } ∈  0 ℎ such that where for any  = { 0 ,   }, V = {V 0 , V  } ∈ .
For each element  ∈  ℎ , denote by  0 the  2 projection from  2 () onto   (). ℎ is the set of all edges in  ℎ .For each edge  ∈  ℎ , let   be the  2 projection from  2 () onto Denote by Q ℎ the  2 projection onto the local discrete gradient space [ −1 ()] 2 .Lemma 5.1 in [3] shows that, on each  ∈  ℎ , For  ℎ and Q ℎ defined above, the following lemmas provide some estimates.

Lemma 4.
Let  ∈  +1 (Ω) be the exact solution of problem (1), and let  ℎ = { 0 ,   } be the weak Galerkin finite element solution of problem (11); then we have Proof.Apparently, for any edge  ∈  ℎ , we have is an element with  as an edge.For any function  ∈  1 (), the following trace inequality is well known Using the trace inequalities ( 26) and (15), we have Then it follows from ( 14) and ( 16) that (28) Combining the above three inequalities, we obtain the aimed result (24).

Cascadic Multigrid Algorithm
In this section, the error estimate and computational complexity of the cascadic multigrid method are analyzed.Assume that  ℎ  ( ≥ 0) is a triangular partition of Ω with the mesh size ℎ  ,  ℎ  is the set of all edges in  ℎ  , and  ℎ  is the corresponding weak discrete space on mesh  ℎ  .Noting that  +1 is obtained by connecting the midpoints of three edges of all triangles in   , we have ℎ  = 2 − ℎ 0 , where ℎ 0 is the mesh size of  0 .For simplicity, define The weak Galerkin finite element approximation of problem (11) on level  can be rewritten as to find   = { 0 ,   } ∈   such that Define an intergrid transfer operator   :  −1 →   , for any V ∈  −1 (1) If  ∈  0  and the element   ∈   is obtained by refining  −1 ∈  −1 , then (2) If  ∈   and edge   ∈   locates in the interior of  −1 ∈  −1 , then (3) If  ∈   and edge   ∈   is part of edge  −1 ∈  −1 , then Then the cascadic multigrid method can be written as follows.
Step 2. For  = 1, . . ., , set  0  =    * −1 , and do iterations Step where   represents the number of iteration steps on level  and the parameter 0 ≤  ≤ 1.As a matter of fact, the assumptions above hold for the Richardson, Jacobi, and Gauss-Seidel iterations with  = 1/2 and for conjugate gradient iteration with  = 1.We refer to [9,13] for details on these results.
The following two lemmas can be proved based on the definition of   .Lemma 5.   is the intergrid transfer operator.For any Thus, we have For any  ∈  −1 , from the definition of weak gradient (5), we have If  is a polynomial in , from trace inequality (26) and the standard inverse inequality, we have Setting q = ∇V 0 , with (44), we have This implies     ∇V 0    0, ≤  |||V|||  .
Note that  consisted of some elements in   ; that is, there exist   ∈   ,  = 1, . . ., , such that  = ∪  =1   .Application of (44) yields By the definition of ∇  V, (46) and (47), we have we obtain               V which completes the proof of this lemma.
which completes the proof of this lemma.
Let   −1 be the standard finite element space associated with  −1 .Define the projection operator It is easy to check that The following lemma is needed in the convergence analysis.

Lemma 7.
For the projection operator   , we have Proof.Since Ω is a convex polygonal domain, for a given V ∈  −1 , we introduce an auxiliary problem, that is, to find  ∈  2 (Ω) such that The solution  satisfies the following inequality: Let  = { 0 ,   } =   V −  −1 V.By the definition of   , we have Then (58), the definition of weak gradient, and (14) give Since we get from (18) that For any   −1 ∈   −1 , we have which implies Thus, combining ( 16) and (18), we have It is easy to show that Combining ( 68) and (70), we obtain which completes the proof of this lemma.
The following two theorems are the main results of this paper, which can be proved in a similar way of [13]  and the computational complexity where   denotes the dimension of the space   on level .
Theorem 9.If we take the Richardson, Jacobi, and Gauss-Seidel iterations as the smoother, the number of iterations on the level  is the minimum integer satisfying while the number of iterations on the level  is the minimum integer satisfying with some fixed  * ≥ 1, then the cascadic multigrid method is quasi-optimal: that is, the error

Numerical Examples
In this section, we give some numerical experiments of the cascadic multigrid algorithm based on the weak Galerkin finite element method to verify the theoretical results proved in Section 3.
For simplicity, we choose  = 1; that is, the weak Galerkin finite element space is and the weak gradient space is For the model problem (1) in the domain Ω = (0, 1) 2 , we give the following numerical examples.
The numerical results are reported in Table 3.
The right-hand function  in the performed numerical examples is chosen to match the given  and A. We take the parameters of the cascadic multigrid method as  = 3 and   = 30.The numerical experiments are conducted in the computer of Intel(R) Core(TM) i5-2500 CPU 3.30 GH, 12.0 GB, Windows 7 (x64).The time each experiment takes is also listed in the tables.The numerical results obtained by using the conjugate gradient iteration to solve these examples directly are also provided at the rows with  = 1.We can observe that the proposed cascadic multigrid method is optimal for both convergence rate and computation complexity, which confirms our theoretical results.

Conclusion
This paper gives error and complexity estimates of the cascadic multigrid algorithm of the weak Galerkin finite
based on the above lemmas.

Table 3 :
Results of Case 3:   = 30,  = 3.The numerical solution is convergent with rate (ℎ) and the computing time is proportional to the number of unknowns, which verify the theoretical results.