Computational Methods for Engineering ScienceView this Special Issue
Research Article | Open Access
A Smoothing Process of Multicolor Relaxation for Solving Partial Differential Equation by Multigrid Method
This paper is concerned with a novel methodology of smoothing analysis process of multicolor point relaxation by multigrid method for solving elliptically partial differential equations (PDEs). The objective was firstly focused on the two-color relaxation technique on the local Fourier analysis (LFA) and then generalized to the multicolor problem. As a key starting point of the problems under consideration, the mathematical constitutions among Fourier modes with various frequencies were constructed as a base to expand two-color to multicolor smoothing analyses. Two different invariant subspaces based on the 2h-harmonics for the two-color relaxation with two and four Fourier modes were constructed and successfully used in smoothing analysis process of Poisson’s equation for the two-color point Jacobi relaxation. Finally, the two-color smoothing analysis was generalized to the multicolor smoothing analysis problems by multigrid method based on the invariant subspaces constructed.
Multigrid methods [1–6] are generally considered as one of the fastest numerical methods for solving complex partial differential equations (PDEs), for example, Navier-Stokes equation in computational fluid dynamics (CFD). As we know, the speed of the multigrid computational convergence depends closely on the numerical properties of the underlying problem of PDEs, for example, equating type and discretizing stencil. Meanwhile, a variety of algorithms for the components in multigrid are of great importance, for example, the processing methods based on smoothing, restriction, prolongation processes, and so on. So, an appropriate choice for the available components has a great impact on the overall performance for specific problems.
Local Fourier analysis (LFA) [5, 7–12] is a very useful tool to predict asymptotic convergence factors of the multigrid methods for PDEs with high order accuracy. Therefore it is widely used to design efficient multigrid algorithms. In LFA an infinite regular grid needs to be considered and boundary conditions need to be ignored. On an infinite grid, the discrete solutions and the corresponding errors are represented by linear combinations of certain complex exponential functions. Thus, Fourier modes are often used to form a unitary basis of the subspace of the grid functions with bounded norms [5, 7, 12]. The LFA monograph by Wienands and Joppich  provides an excellent background for experimenting with Fourier analysis. Recent advances in this context included LFA for triangular grids [13, 14], hexagonal meshes , semistructured meshes , multigrid with overlapping smoothers , multigrid with a preconditioner as parameters , and full multigrid method . In , an LFA for multigrid methods on the finite element discretization of a 2D curl-curl equation with a quadrilateral grid was introduced.
A general definition on the multicolor relaxation was provided in . Smoothing analysis of the two-color relaxation on LFA was given in [21–24], and the four-color relaxation with tetrahedral grids was presented in [16, 25]. In , a parallel multigrid method for solving Navier-Stokes equation was investigated and a multigrid Poisson equation solver was employed in . A parallel successive overrelaxation (SOR) algorithm for solving the Poisson problem was discussed in , and multicolor SOR methods were studied in .
In the present paper, a novel smoothing analysis process of multicolor relaxation on LFA is provided with details. An important coupled relation among Fourier modes with various frequencies is constructed and expanded to the multicolor smoothing analysis. The roles of the Fourier modes with the high and low frequencies in the proposed method are well characterized. Thus, by the two invariant subspaces based on the 2h-harmonics the two-color smoothing analysis process is well generalized to the multicolor problems.
2. LFA in Multigrid
2.1. General Definition
A generally linear scalar constant-coefficient system without boundary conditions is described with a discrete problem with infinite grid; that is, in which an infinite grid is stated as where is the mesh size, denotes the dimension of , the discrete operator is given by and with is the stencil coefficients [3–5] of for (2), containing , and . From [11, 20], the Fourier eigenfunctions of the constant-coefficient infinite grid operator in (1) are given by where , denotes the Fourier frequency, , and is called Fourier mode [3, 5, 20], which is orthogonal with respect to the scaled Euclidean inner product [3, 5, 10]. On grid (2), the corresponding eigenvalues of are expressed by with called Fourier symbol of . Further, a Fourier subspace with the bounded infinite grid function , that is , is defined as in which is referred to the low frequency and is referred to the high frequency. As a standard multigrid coarsening , a case of is considered, and infinite coarse grid is stated as
2.2. Smoothing Analysis of Multigrid Relaxation
For multigrid relaxation of discrete operator on the infinite grid (2), if (4) are the eigenfunctions of , then is the Fourier symbol of . For pattern relaxation , (4) are no longer the eigenfunctions of relaxation operator . However, it leaves certain low-dimensional subspaces of (4) invariant yielding a block-diagonal matrix of smoothing operator consisting of small blocks. As presented in [10, 11], the -harmonics of (4) is defined as where and . If relaxation operator satisfies that is, , the matrix is called Fourier representation of . Furthermore, an idea coarse-grid correction operator is introduced  to drop out the low-frequency modes and to keep the high-frequency modes. So, it is clear that is a projection operator onto the subspace of the high-frequency modes By the same way, a subspace of the low-frequency modes is defined as Thus, a general coarsening strategy  is stated as Consequently, a smoothing factor  on the Fourier modes for the multigrid relaxation, and , is yielded as where is the relaxation parameter, denotes the sum of pre- and postsmoothing steps, and are the Fourier representations of and , respectively, and denotes the spectral radius of the matrix .
3. Smoothing Analysis of Two-Color Relaxation
To develop two different processes of LFA for the two-color relaxation, grid (2) is divided into two disjoint subsets and , referring to as the red and black points, respectively. Two process steps  are required to construct a complete two-color relaxation . In the first step (), the unknowns located at the red points are only smoothed, whereas the unknowns at the black points remain to be unchanged. Then, in the second step (), the unknowns at the black points are changed by using the new values calculated with the red points in the first step. So, a complete red-black point process is obtained by iteration From the process mentioned above, it is noted that the Fourier modes (4) are no longer eigenfunctions of (15) on grid (2) because the relaxation operator is used.
3.1. Invariant Subspaces for Two-Color Relaxation
A new smoothing analysis process of the two-color relaxation is proposed with details. The proposed process is different with [11, 20–24]. A novel constitution among the Fourier modes with various frequencies is developed as a base of the smoothing analysis process. The analysis process is proved to be valuable.
The grid is divided into two disjoint subsets and ; that is, with where . According to (16), the subspace of the 2-harmonics (9) is redefined as with , where , . Thus, the constitutions among the various Fourier modes defined by (16) and (17) are presented as follows.
Proposition 1. For , , and , if , then the following formulation holds:
Proposition 2. For , , and , if , then the following formulation holds:
Subsequently, the smoothing analysis process of the two-color relaxation on the subspace of the -harmonics (17) is conducted. By (15) and (16) and without loss of generality, let and correspond to and , respectively; thus (15) is rewritten as
Theorem 3. The iteration operator for the two-color relaxation leaves the subspace of the 2-harmonics (17) to be invariant.
Proof. From the process of the two-color relaxation, the operator of grid (16) is where is Fourier symbol of on grid (16) with . From (10) and (25), now it is proved that the subspace of the 2-harmonics (17) is invariant for the iteration operator (24). Because of (17), we need to find out two complex numbers and with , and make them subjected to From (25), the right hand side of (26) is written as By Propositions 1 and 2, the right hand side of (27) is expressed as Taking , (28) is written as From Propositions 1 and 2, the left hand side of (26) is written as Hence, from (26), (29), and (30), a set of two linear equations on and is given as where , . Therefore, from (31), it is concluded that there exist two complex numbers and that are subjected to (26). From (10), (17), and (26), solving linear equation (31), the Fourier representations of the iteration operators and are obtained as where and , . Furthermore, from (32), the Fourier representations of the two-color relaxation are From (10), Theorem 3 holds.
3.2. Invariant Subspaces on Four Fourier Modes for Two-Color Relaxation
We need to develop a Fourier representation of the two-color relaxation in the subspace of the 2-harmonics with four Fourier modes. By following (9), for 2D system, another subspace of the 2-harmonics is given as with , .
For the sake of convenient analysis, taking , for example, , then is defined as Meanwhile, the grid is divided into four subsets  as where and . The red and black grid points corresponding with are thus obtained as Therefore, a constitutive relationship among the various Fourier 2-harmonics is constructed.
Proposition 4. For , , and , if , the following equation is yielded as
Subsequently, a smoothing analysis process of the two-color relaxation on the subspace of the -harmonics (35) is obtained.
Proof. Similar to the proof of Theorem 3, from process of the two-color relaxation and (15), operators and of the grid (37) are
where and are Fourier symbols of and with on the corresponding grids (37), respectively, and . From (15), in order to prove with , we need to find out four complex numbers , , , and subjected to
Meanwhile, we also need to find other four complex numbers , , , and and make them subjected to
Firstly, we prove (42) as follows.
From (36) and (40), as well as Propositions 5 and 4, the right and left hand sides of (42) are written as, respectively, where , . Hence, by using (42) and (44), linear equations with respect to the complex numbers , , , and are obtained as In the same way, equations with respect to the complex numbers , , , and are obtained as where , . From (10), (35), (42), and (43), solving (45) and (46), the Fourier representations of the iteration operators and are obtained as Furthermore, from (47), the Fourier representation of the iteration operators is where , , . Theorem 6 holds.
4. Two-Color Jacobi Relaxation for 2D Poisson Equation
4.1. Poisson Equation and Optimal Smoothing Parameter
2D Poisson equation to be considered is stated as For using uniform grids of mesh size to solve this equation, a central discretization stencil is introduced as From (3)–(6), the Fourier symbol of (50) is From , the damped Jacobi relaxation is defined as where is the identity operator, is the smoothing parameter, and is the diagonal part of the discrete operator . Thus, the Fourier symbol of (52) is given as
For the operators and in (14) with a relaxation parameter and according to the optimal one-stage relaxation , smoothing parameter and a related smoothing factor are given by where and are the maximum and minimum eigenvalues of the matrix for and is the Fourier representation of with .
4.2. Two-Color Relaxation on (17)
According to (32), (33), and (53), for point Jacobi relaxation, in (17) is expressed as which denotes that both red and black points are swept by the Jacobi point relaxation method, where , , and is the smoothing parameter. Further, when , (55) is rewritten as where , . For simplification, let By substituting (56) and (57) into (33), (56) is given as Further, by using (13) and (17), the Fourier representation of is given as . From (59), the product of and (59) is Therefore, a unique nonzero eigenvalue of the matrix is yielded as Because of , thus, from (57), we know . So, by using (54), the optimal smoothing parameters for the two-color relaxation are given as
4.3. Two-Color Jacobi Relaxation on (35)
By using (48) and (53), for point Jacobi relaxation, and for (35) are expressed as which denotes that both red and black points are swept by the Jacobi point relaxation method, where . Further, substituting (57) into (64), when , (64) is written as Substituting (65) into (48), the Fourier representation of with is expressed as where From (13) and (35), the Fourier representation of operator is given as where , . Therefore, the product of (66) and (68) is obtained as in which the diagonal blocks are expressed as The eigenvalues of the matrix (69) are obtained as When , the maximum and minimum eigenvalues of the matrix are as follows: Therefore, by using (54), the values of the optimal smoothing parameters for the Poisson equation are obtained as
5. Extending Two-Color to Multicolor Relaxation
Herein, the proposed smoothing analysis process of two-color relaxation is generalized to a 3D system. The Fourier representation of the smoothing operator for two-color relaxation is still a 2-order square matrix in (17). The result in (35) for a 3D case is changed to a diagonal block matrix.
For a -color relaxation (), the infinite grid is subdivided into types of the grid points for presenting different colors [11, 20]. Thus a complete analyzing step of the -color relaxation consists of substeps: at the th step (), the unknowns located at only are changed by using updated data at the previous step. For example, for the -color relaxation of a 2D system, the infinite grid is stated as with where . In the subdivisions of the infinite grids , there are , , , and .
In order to obtain a Fourier representation of the -color point relaxation, let be the above complete -color point relaxation operator and let be the th subrelaxation (); thus, the -color point relaxation is expressed as with where denotes the Fourier symbol of , , . The proof of this process is analogous to the two-color case. In fact, as we know, the subspace of the 2-harmonics with Fourier modes remains to be invariant for -color point relaxation operator ; that is, . So, the Fourier representation of the -color point relaxation (80) is given as , where is a Fourier representation of in .
Proposition 7. For , , , if , the following equation holds: The proof is similar to Proposition 1.
Theorem 8. The iteration operator (80) for -color point relaxation makes the subspace of the 2-harmonics (79) invariant.
The proof is similar to Theorem 3. In fact, in order to prove , one only needs to do , , which is equivalent to finding out complex numbers , , subject to
Being similar to the proof of Theorems 3 and 6 and according to (81), (83) and Proposition 7, linear equations on are obtained as Letting , with , the equations are simplified as where , , denotes transposition of matrix or vector, and is the Vander monde matrix; namely, Because of , , the determinant of the matrix is nonzero. Therefore, from (83)–(86), for , , the Fourier representation of the th substep relaxation in is obtained as in which is a square matrix which is obtained by substituting for the th row of the matrix , and . Therefore, from (80) and (87), the Fourier representation of the -color point relaxation in the subspace of the 2-harmonics is stated as Theorem 8 holds.
A novel smoothing analysis process of the two-color point relaxation for a 2D system is presented. The results are generalized to the -color point relaxation and extended to a 3D system. The applications to the 2D and 3D Poisson equations show that the computational domain over multigrids needs to be divided into the multisubsets to correspond with the different frequency modes in partial differential equations and to use the corresponding discretizing stencils. Meanwhile, the definition of the subspace based on the 2-harmonics has to be agreeable to the subdomains of the multigrids. It is an important fact that establishes a mathematical constitution among the various Fourier modes with the different 2-harmonics and constructs a usable Fourier representation of the -color point relaxation in subspace of the 2-harmonics.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors were supported by the National Natural Science Foundation of China (NSFC) (Grant no. 51279071) and the Doctoral Foundation of Ministry of Education of China (Grant no. 20135314130002).
- W. L. Briggs, V. E. Henson, and S. McCormick, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, 2nd edition, 2000.
- W. Hackbusch, Multigrid Methods and Applications, Springer, Berlin, Germany, 1985.
- U. Trottenberg, C. W. Oosterlee, and A. schuller, Multigrid, Academic Press, New York, NY, USA, 2001.
- P. Wesseling, An Introduction to Multigrid Methods, John Wiley, Chichester , UK, 1992.
- K. Stüben and U. Trottenberg, “Multigrid methods: fundamental algorithms, model problem analysis and applications,” in Multigrid Methods, W. Hackbusch and U. Trottenberg, Eds., vol. 960 of Lecture Notes in Mathematics, pp. 1–176, Springer, Berlin, Germany, 1982.
- A. Brandt and O. E. Livne, 1984 Guide to Multigrid Development in Multigrid Methods, Society for Industrial and Applied Mathematics, 2011.
- A. Brandt, “Multi-level adaptive solutions to boundary-value problems,” Mathematics of Computation, vol. 31, no. 138, pp. 333–390, 1977.
- T. Boonen, J. van lent, and S. Vandewalle, “Local Fourier analysis of multigrid for the curl-curl equation,” SIAM Journal on Scientific Computing, vol. 30, no. 4, pp. 1730–1755, 2008.