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 themulticolor 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 tomulticolor 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.


Introduction
Multigrid methods [1][2][3][4][5][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][8][9][10][11][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 [11] provides an excellent background for experimenting with Fourier analysis.Recent advances in this context included LFA for triangular grids [13,14], hexagonal meshes [15], semistructured meshes [16], multigrid with overlapping smoothers [17], multigrid with a preconditioner as parameters [18], and full multigrid method [19].In [8], 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 [20].Smoothing analysis of the two-color relaxation on LFA was given in [21][22][23][24], and the four-color relaxation with tetrahedral grids was presented in [16,25].In [26], a parallel multigrid method for solving Navier-Stokes equation was investigated and a multigrid Poisson equation solver was employed in [27].A parallel successive overrelaxation (SOR) algorithm for solving the Poisson problem was discussed in [28], and multicolor SOR methods were studied in [29].

Mathematical Problems in Engineering
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.

LFA in Multigrid
2.1.General Definition.A rigorous base of the local mode analysis in multigrid was elaborated [12].Herein, we are following [11] as a starting point of our framework.
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  ⇀ ℎ = (ℎ 1 , ℎ 2 , . . ., ℎ  ) is the mesh size,  denotes the dimension of  ⇀  , the discrete operator is given by and  ⇀  ∈ R with  ⇀  ∈  is the stencil coefficients [3][4][5] of  ℎ for (2),  ⊂ Z  containing (0, 0, . . ., 0), and  ⇀  ⋅  ⇀ ℎ = ( 1 ℎ 1 ,  2 ℎ 2 , . . .,   ℎ  ).From [11,20], the Fourier eigenfunctions of the constant-coefficient infinite grid operator  ℎ in (1) are given by where and  ℎ (  ⇀  ,  ⇀  ) 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  ℎ are expressed by with called Fourier symbol of  ℎ .Further, a Fourier subspace with the bounded infinite grid function  ℎ ∈ ( ℎ ), that is ( ℎ ) ⊆  ℎ , is defined as in which Θ low = (−/2, /2]  is referred to the low frequency and Θ high = Θ \ Θ low is referred to the high frequency.As a standard multigrid coarsening [11], a case of  ⇀  = 2  ⇀ ℎ is considered, and infinite coarse grid   is stated as  [11], (4) are no longer the eigenfunctions of relaxation operator  ℎ .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 2ℎ-harmonics of ( 4) is defined as that is,  ℎ :  2ℎ →  2ℎ , the matrix Ŝℎ (  ⇀  ) is called Fourier representation of  ℎ .Furthermore, an idea coarse-grid correction operator   ℎ is introduced [11] to drop out the lowfrequency modes and to keep the high-frequency modes.So, it is clear that   ℎ 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 [11] is stated as Consequently, a smoothing factor [11] on the Fourier modes for the multigrid relaxation,  ℎ () and   ℎ , is yielded as where  is the relaxation parameter,

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   ℎ and   ℎ , referring to as the red and black points, respectively.Two process steps [11] are required to construct a complete two-color relaxation   ℎ ().In the first step (  ℎ ()), 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 (  ℎ ()), 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.

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][21][22][23][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 where  = 0, 1.According to (16), the subspace of the 2ℎharmonics ( 9) is redefined as with + (, )) mod 2,  = 0, 1.Thus, the constitutions among the various Fourier modes defined by ( 16) and ( 17) are presented as follows.
Subsequently, the smoothing analysis process of the twocolor relaxation on the subspace of the 2ℎ-harmonics ( 17) is conducted.By (15) and ( 16) and without loss of generality, let  0 ℎ and  1 ℎ correspond to   ℎ and   ℎ , respectively; thus (15) is rewritten as Theorem 3. The iteration operator  01 ℎ () for the two-color relaxation leaves the subspace of the 2ℎ-harmonics (17) to be invariant.

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 2ℎ-harmonics with four Fourier modes.By following (9), for 2D system, another subspace of the 2ℎ-harmonics is given as with For the sake of convenient analysis, taking Meanwhile, the grid  ℎ is divided into four subsets [11] as where and  ⇀  = ( 1 ,  2 ) ∈ Λ = {00, 11, 10, 01}.The red and black grid points corresponding with G ℎ are thus obtained as The proof of Propositions 5 and 4 is similar to Propositions 2 and 1.
Subsequently, a smoothing analysis process of the twocolor relaxation on the subspace of the 2ℎ-harmonics (35) is obtained.Theorem 6.The iteration operator (15) for the two-color relaxation leaves the subspace of the 2h-harmonics (35) to be invariant 0.
Proof.Similar to the proof of Theorem 3, from process of the two-color relaxation and (15), operators   ℎ () and   ℎ () of the grid (37) are (43) 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, ) .
From Theorems 3 and 6, two ways to carry out smoothing analysis of the two-color relaxation are obtained.

Poisson Equation and Optimal Smoothing Parameter. 2D
Poisson equation to be considered is stated as For using uniform grids of mesh size ℎ to solve this equation, a central discretization stencil is introduced as From ( 3)-( 6), the Fourier symbol of (50) is From [1], the damped Jacobi relaxation  JAC ℎ is defined as where  ℎ = [1] ℎ is the identity operator,  is the smoothing parameter, and  ℎ = (1/ℎ 2 ) [4] ℎ is the diagonal part of the discrete operator  ℎ .Thus, the Fourier symbol of ( 52) is given as For the operators  ℎ () and   ℎ in ( 14) with a relaxation parameter  and according to the optimal one-stage relaxation [11], smoothing parameter and a related smoothing factor are given by where  (17).According to (32), (33), and (53), for point Jacobi relaxation,    in ( 17) is expressed as

Extending Two-Color to Multicolor Relaxation
Herein, the proposed smoothing analysis process of twocolor 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 2 3 × 2 3 diagonal block matrix.
For a -color relaxation ( > 2), the infinite grid  ℎ is subdivided into  types of the grid points  0 ℎ ,  1 ℎ , . . .,  −1 ℎ for presenting  different colors [11,20].Thus a complete analyzing step of the -color relaxation consists of  substeps: at the th step ( = 0, 1, . . .,  − 1), the unknowns located at only  ⇀  ∈   ℎ are changed by using updated data at the previous step.For example, for the -color relaxation of a 2D system, the infinite grid  ℎ is stated as with where  ∈ Λ  := {0, 1, . . .,  − 1}.In the subdivisions of the infinite grids  ℎ , there are ∀,  ∈ Λ  ,  ̸ = , and   ℎ ∩  ℎ = .For the standard coarsening [11,20], the subspace of the 2ℎ-harmonics is defined as In order to obtain a Fourier representation of the color point relaxation, let   ℎ () be the above complete color point relaxation operator and let   ℎ () be the th subrelaxation ( ∈ Λ  ); thus, the -color point relaxation is expressed as , ))(mod 2).
max and  min are the maximum and minimum .The proof of this process is analogous to the two-color case.In fact, as we know, the subspace of the 2ℎharmonics   2ℎ with  Fourier modes remains to be invariant for -color point relaxation operator   ℎ (); that is,   ℎ () :   2ℎ →   2ℎ .So, the Fourier representation of the -color point relaxation (80) is given as Ŝ ℎ () = ∏ −1 =0 Ŝ ℎ (), where Ŝ ℎ () is a Fourier representation of   ℎ () in   2ℎ .