Abstract
This paper is devoted to the analysis of the cascadic multigrid algorithm for solving the linear system arising from the weak Galerkin finite element method. The proposed cascadic multigrid method is optimal for conjugate gradient iteration and quasi-optimal for Jacobi, Gauss-Seidel, and Richardson iterations. Numerical results are also provided to validate our theoretical analysis.
1. 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 [5–8].
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.
2. Model Problem and Its WG Finite Element Approximation
Consider the following second-order elliptic problem:where is a convex polygonal domain with boundary in , , . Furthermore, assume that is symmetric uniformly positive definite and uniformly bounded-above diffusion; namely, there exist positive constants and such that
Here and thereafter, for any subset , we use the standard notations for the Sobolev spaces and with . The inner-product, norm, and seminorm in are denoted by , and , respectively, and we skip the subscript when .
Since the domain is convex, the unique solution of problem (1) exists and satisfies the full regularity assumption [14]
Let be a polygonal domain with interior and boundary . Denote by the space of weak functions on ; that is,
For any , the weak gradient of is defined as ,where .
The discrete weak gradient, denoted by , is defined as follows:
Let be a shape-regular, quasi-uniform triangular mesh of the domain , with the mesh size . denotes a generic positive constant independent of the mesh size throughout this paper. Denote the weak function space on by ; that is,For any given integer , 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] thatFor the discrete weak gradient, we will drop the subscript in the notation for simplicity.
The weak Galerkin finite element method can be written as to find such thatwherefor any , .
For each element , denote by the projection from onto . is the set of all edges in . For each edge , let be the projection from onto . Define Denote by the projection onto the local discrete gradient space . Lemma in [3] shows that, on each ,For and defined above, the following lemmas provide some estimates.
Lemma 1 (see [3], Lemma ). For any , we have
Lemma 2 (see [3], Lemma ). For any and , we havewhere is the energy norm; that is, for any , .
Lemma 3 (see [3], Theorem , Theorem ). Assume the exact solution ; then we have
For any , , we define an inner-product byDefine the following discrete norms:It is clear that .
With the above estimates, the following lemma can be proved, which is needed in Section 3.
Lemma 4. Let be the exact solution of problem (1), and let be the weak Galerkin finite element solution of problem (11); then we have
Proof. Apparently, for any edge , we have is an element with as an edge. For any function , the following trace inequality is well knownUsing the trace inequalities (26) and (15), we haveThen it follows from (14) and (16) thatThus, we haveBy using the triangle inequality, we get from (19) thatAccording to the definition of and , we haveIt follows from (15) and (20) thatCombining the above three inequalities, we obtain the aimed result (24).
3. Cascadic Multigrid Algorithm
In this section, the error estimate and computational complexity of the cascadic multigrid method are analyzed.
Assume that is a triangular partition of with the mesh size , is the set of all edges in , and is the corresponding weak discrete space on mesh . Noting that is obtained by connecting the midpoints of three edges of all triangles in , we have , where is the mesh size of . For simplicity, define , , , , and . The weak Galerkin finite element approximation of problem (11) on level can be rewritten as to find such that
Define an intergrid transfer operator , for any (1)If and the element is obtained by refining , then (2)If and edge locates in the interior of , then (3)If and edge is part of edge , then
Then the cascadic multigrid method can be written as follows.
Step 1. Set .
Step 2. For , set , and do iterations
Step 3. Set .
The notation in Step 2 denotes the iterative operator on level . For the operator , we assume that there exists a linear operator such thatwhere represents the number of iteration steps on level and the parameter . As a matter of fact, the assumptions above hold for the Richardson, Jacobi, and Gauss-Seidel iterations with and for conjugate gradient iteration with . 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 , we have
Proof. For any element , let be the collection of edges located in . It follows from the definition of thatThus, we haveFor any , from the definition of weak gradient (5), we haveIf is a polynomial in , from trace inequality (26) and the standard inverse inequality, we haveSetting , with (44), we haveThis impliesNote that consisted of some elements in ; that is, there exist , , such that .
Application of (44) yieldsBy the definition of , (46) and (47), we haveSincewe obtainwhich completes the proof of this lemma.
Lemma 6. Assume that and are the weak Galerkin finite element solutions associated with and , respectively; then
Proof. Let and be the finite element solutions associated with and , respectively. By the triangle inequality, (39), (24), and the regularity (3), we havewhich completes the proof of this lemma.
Let be the standard finite element space associated with . Define the projection operator byIt is easy to check thatThe following lemma is needed in the convergence analysis.
Lemma 7. For the projection operator , we have
Proof. Since is a convex polygonal domain, for a given , we introduce an auxiliary problem, that is, to find such thatThe solution satisfies the following inequality:Let . By the definition of , we haveThen (58), the definition of weak gradient, and (14) give Sincewe get from (18) that For any , we have which implies Thus, combining (16) and (18), we haveBy using the regularity (60), we obtainThen it follows from (40) and (56) thatIt is easy to show thatCombining (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] based on the above lemmas.
Theorem 8. If we take the CG iteration as the smoother, and the number of iterations on the level is the minimum integer satisfyingwith some fixed and , then the cascadic multigrid method is optimal: that is, the error 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 satisfyingwhile the number of iterations on the level is the minimum integer satisfying with some fixed , then the cascadic multigrid method is quasi-optimal: that is, the error and the computational complexity
4. 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 ; that is, the weak Galerkin finite element space is and the weak gradient space is
For the model problem (1) in the domain , we give the following numerical examples.
Case 1. Let the exact solution and set . The numerical results are reported in Table 1.
Case 2. Let the exact solution and set . The numerical results are reported in Table 2.
Case 3. Let the exact solution and set The numerical results are reported in Table 3.
The right-hand function in the performed numerical examples is chosen to match the given and . We take the parameters of the cascadic multigrid method as and . 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 . We can observe that the proposed cascadic multigrid method is optimal for both convergence rate and computation complexity, which confirms our theoretical results.
5. Conclusion
This paper gives error and complexity estimates of the cascadic multigrid algorithm of the weak Galerkin finite element. The numerical solution is convergent with rate and the computing time is proportional to the number of unknowns, which verify the theoretical results.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by the National Science Foundation of China (Grant no. 11101311) and Applied Mathematics Chair Fund of China-German College (Grant no. 0900101021).