Abstract
This paper proposes a two-level additive Schwarz preconditioning algorithm for the weak Galerkin approximation of the second-order elliptic equation. In the algorithm, a 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 two-phase subsurface flow problems [3]. Chen et al. introduced a posteriori error estimates for the WG method for the second-order 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 , where denotes the mesh size. Recently, Li and Xie [5] proposed a multigrid algorithm for the WG method using 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 second-order 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 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 and is of , where denotes the coarse mesh size and 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.
2. The Weak Galerkin Method for the Second-Order Elliptic Equation
For the sake of simplicity, we consider the model problem as follows:where is bounded polygonal domain and . Assume that the matrix is symmetric positive definite (SPD); namely, there exist positive constants , such thatThe variational formulations of (1) is to find such thatwhere It follows from Lax-Milgram’s theorem that problem (3) has a unique solution.
Let be a quasiuniform triangulation of the domain with the mesh size . On each , the space denotes the collections of weak functions, each of which is consisted of an interior part and a boundary part; that is, For any weak function , the weak gradient of , denoted by , satisfies where is the unit outer normal of . It is easy to see that is the classical gradient if it actions on a function .
For each element , we denote by and the interior and the boundary of . The notation stands for the set of edges of . We also use and to denote the edges of in and on . Then the discrete WG space is defined as where or and denotes the set of polynomials on with degree no more than . To define a discrete weak gradient, we use to denote the RT element space if and the BDM element space if , where is the homogeneous polynomial with the degree .
For each , the discrete weak gradient is defined aswhere and denote the values of in the interior and on the boundary of . Then the discrete problem for (3) is to seek such thatwhere
For any and in , we use to indicate a special inner product defined as where and denote the inner product on and . Accordingly, a norm and seminorm are introduced for any by where and .
Let be an operator It can be proved that is symmetric and positive definite and the condition number is of (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 by overlapping subdomains such that each point in belongs to no more than subdomains. We assume that the boundary of each subdomain does not cut through any elements in the triangulation , and there exist nonnegative functions satisfying the following properties: () in ; () ; () there exists a positive constant , such that , where is a constant independent of , , and
On each subdomain , the notation stands for the triangulation inherited from . The corresponding weak Galerkin subspace on is defined as We introduce an operator byIt is easy to see that is symmetric and positive definite. Since is a subset of , we use to denote a natural injection from to .
To define a coarse subproblem, we define a coarse triangulation with mesh size such that each element in is a subdivision of the one in . The notation stands for conforming finite element space associated with . We also introduce an operator satisfyingNote that the coarse space is a subspace of if the degree of the piecewise polynomials in is greater than , and we can choose a natural injection as the intergird transfer operator from to . For the piecewise constant space case, the intergrid transfer operator is defined as where and satisfyfor any and . We also need the transpose of the intergrid transfer operators from to the subspace for , defined as
Then our two-level additive Schwarz preconditioner is stated as
3.2. Analysis
Our analysis is based on the standard Schwarz framework (see, e.g., [7, 9]).
Lemma 1 ([9, Theorem ]). The eigenvalues of are positive, and one has the following characterizations of the maximum and minimum eigenvalues:where the sum is taken over
Lemma 2. For the intergrid transfer operator , it holds for any that
Proof. We only need consider the case that the functions in 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 Similarly, for the lower-order term, we have which completes the proof.
Lemma 3. There exists an operator from to such that for any in it holds that
Proof. Let be the conforming finite element space defined on . Then, for any in , we can construct a satisfying ([6, Lemma ])Define , where is the projection operator from to . Using the standard properties of (see, e.g., [7]) and the inequality (24), we have The proof is completed.
Lemma 4. Given any , there exists a decomposition where and , such that
Proof. Let , , and , where and are the piecewise projection to the interior polynomial space and the edge polynomial space , respectively. It is easy to check that It follows from Lemma 3 that On the other hand, we have for that By the inverse inequality, the stability of projection, the Poincaré inequality, and the scaling argument, we deduce that which, together with the triangle inequality and the assumption of , yieldsFor the last term, the triangle inequality givesWe estimate , , and , respectively, as follows.
From the trace theorem, the scaling argument, and the stability and approximation of , we have Denote . The trace theorem, the stability of , and the Poincaré inequality imply that Simiarily, we obtain Combining the above five inequalities, we find which leads toUsing (27) and (38), we achieve This ends the proof.
Lemma 5. Let and , . For any , it is true that
Proof. The inequality can be obtained directly by using the triangle inequality and the assumption on the finite cover of subdomains.
As an immediate consequence, we have the following theorem.
Theorem 6. There exists a positive constant , independent of , , , and , such that
Remark 7. According the theorem, the two-level additive preconditioner is optimal if is bounded above by a constant. In particular, when , we have
4. Numerical Experiments
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:
We choose to satisfy the exact solution on . Let be a uniform triangulation with the mesh size , and let be the refinement of . All the tests are stopped when the relative error is less than .
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 are fixed. If the overlap becomes small, that is, the rate 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 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 and () 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 (). The iteration numbers, with different mesh sizes 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.