Smoothing Analysis of Distributive Red-Black Jacobi Relaxation for Solving 2 D Stokes Flow by Multigrid Method

Smoothing analysis process of distributive red-black Jacobi relaxation in multigrid method for solving 2D Stokes flow is mainly investigated on the nonstaggered grid by using local Fourier analysis (LFA). For multigrid relaxation, the nonstaggered discretizing scheme of Stokes flow is generally stabilized by adding an artificial pressure term. Therefore, an important problem is how to determine the zone of parameter in adding artificial pressure term in order to make stabilization of the algorithm for multigrid relaxation. To end that, a distributive red-black Jacobi relaxation technique for the 2D Stokes flow is established. According to the 2h-harmonics invariant subspaces in LFA, the Fourier representation of the distributive red-black Jacobi relaxation for discretizing Stokes flow is given by the form of square matrix, whose eigenvalues are meanwhile analytically computed. Based on optimal onestage relaxation, a mathematical relation of the parameter in artificial pressure term between the optimal relaxation parameter and related smoothing factor is well yielded. The analysis results show that the numerical schemes for solving 2D Stokes flow by multigridmethod on the distributive red-black Jacobi relaxation have a specified convergence parameter zone of the added artificial pressure term.

In multigrid methods, smoothing relaxations play an important role.Several multigrid relaxation methods were developed for solving PDEs, which are roughly classified into two categories, collective and decoupled relaxations [8].The collective relaxations are considered as a straightforward generalization of the scalar case [2].The early decoupled relaxation is on a distributive Gauss-Seidel relaxation [9].Gradually, it is generalized to an incomplete LU factorization relaxation [10].Recently, Stokes system with distributive Gauss-Seidel relaxation based on the least squares commutator has been researched [11].Much of the relaxations for Stokes system is seen in [12,13].
For multigrid methods, LFA is a very useful tool to design efficient algorithms and to predict convergence factors for solving PDEs with high order accuracy [1][2][3][4][5][6][7].Distributive relaxation for poroelasticity equations is optimized by LFA [14].Using LFA, textbook efficiency multigrid solver for compressible Navier-Stokes equations is designed [15].Allat-once multigrid approach for optimality systems with LFA is discussed in detail, and an analytical expression of the convergence factors is given by using symbolic computation [16][17][18].
The smoothing analysis of the distributive relaxations for solving 2D Stokes flow is investigated with LFA.As we know, the discretizing Stokes flow in computational domain is not stable by means of standard central differencing on nonstaggered grid.Thus, in order to overcome the stability problem, an artificial pressure term is generally added by the method in [1,2].The optimal one-stage relaxation parameter and related smoothing factor of the distributive relaxation with the red-black Jacobi point relaxation need to be developed.In deriving an explicit formulation of the smoothing factor for the multigrid method, the symbolic operation process is carried out by using the MATLAB and Mathematica software, especially, by the cylindrical algebraic decomposition (CAD) function in the Mathematica build-in command [19].

Elements of LFA in Multigrid.
In LFA, a current approximation and its corresponding error and residual are represented by a linear combination of certain exponential functions, for example, Fourier modes, which form a unitary basis in space on a bounded infinite grid functions [1][2][3][4][5][6][7].

Distributive Relaxation of System
where  ℎ is the unit operator with discrete stencil [1] ℎ .From ( 14), the discrete system ( 6) is transformed as where the discrete stencils of Δ 2 ℎ and −Δ 2ℎ are From ( 9)-( 11), the Fourier modes of the scalar discrete operators of ( 16) are where

Optimal One-Stage Relaxation.
For the discrete scalar operator of (15), standard coarsening and an ideal coarse grid correction operator [2] are applied as where _  2ℎ ℎ is the Fourier representation of the operator  2ℎ ℎ with subspace (13), which suppresses the low frequency error components and makes the high frequency components unchanged.Then, from [2], the smoothing factor for discrete operator ( 9) is defined by It implies that the asymptotic error reduction of the high frequency error components is given by n sweeps of the relaxation method, where _  ℎ (  ⇀  , ) is the Fourier representation of the relaxation operator  ℎ () on subspace (13) and  is the relaxation parameter.
From [2,14], a good smoothing factor is obtained by using one-stage parameter  in the relaxation operator  ℎ (); the optimal smoothing factor and related smoothing parameter are given by where  max and  min are the max and min eigenvalues of the product matrix ) with the relaxation parameter  = 1 for  ∈ Θ 2ℎ low .From [2,19], the smoothing factor of (6) with the distributive relaxation ( 14) is determined by the diagonal blocks of the transformed system (15), which is given by

Optimal Smoothing for Stokes Flow.
The red-black Jacobi point relaxation   ℎ is applied to (15) to discuss the optimal smoothing problems for Stokes flow.From [1,2,14], the operator   ℎ makes the 2ℎ-harmonics subspace (13) invariant; that is, where is the Fourier representation of   ℎ () with relaxation parameter  = 1 and is given as in which denotes the Fourier mode of the point Jacobi relaxation for the discrete operator (9) on subspace (13) and ) is the Fourier mode of the discrete operator with the stencil [ (0,0) ] ℎ in (9).For the sake of convenient discussion in the following, two variables are introduced as Thus, Theorem 1.For the Poisson operator −Δ ℎ , the optimal onestage relaxation parameter and related smoothing factor of the red-black Jacobi point relaxation are stated as Proof.For the red-black Jacobi point relaxation for the Poisson operator  ℎ = −Δ ℎ , substituting (12), (18), and (28) into ( 26) and ( 27), and from (5), the product of ( 21) and ( 25) is written as ) .