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Abstract: In this work it is proposed an efficient and robust algorithm for the solution of a physical anisotropy problem. This algorithm is based on the choice of the most efficient restriction operator and on an incomplete LU decomposition suited for each anisotropy direction. Local Fourier analysis is carried out in order to assist the project of an efficient multigrid method. The model considered is pure diffusion with anisotropy aligned to the coordinate axis x. Finite difference method with uniform grid and second-order numerical scheme is used for the discretization of equations. Problems are solved with geometric multigrid method, correction scheme, V-cycle and standard coarsening ratio. The asymptotic convergence factor is calculated for different multigrid components, such as restriction operators, prolongation operators and solvers. Based on the optimum components obtained by LFA, experiments are carried out for the analysis of complexity and computational cost for the algorithm proposed. The main conclusion is that the methodology proposed is efficient for the resolution of problems with strong anisotropy.

Keywords: Physical anisotropy; Multigrid components; Diffusion; Local Fourier analysis.

1. Introduction

Computational Fluid Dynamics (CFD) is a branch of Computational Science that studies numerical methods for the simulation of fluid flows problems. It is known that these methods usually require a high computational cost. In general, this happens because the problems that have to be solved require the resolution of algebraic equation systems whose coefficients matrices are large and sparse [1].

Linear systems are obtained with discretization of the mathematical model, which consists in approximating, through algebraic equations, each term of the mathematical model for each grid node or point. This process leads to an algebraic equation system of the form

where ${\displaystyle A\in {R^{NxN}}}$ is the coefficient matrix, ${\textstyle T\in {R^{N}}}$ is the variable of interest and ${\textstyle b\in {R^{N}}}$ is the independent vector.

In CFD, the methods traditionally employed in this process are: Finite Difference Method (FDM) [2], Finite Volume Method (FVM) [3] and Finite Element Method (FEM) [4].

The algebraic equation system given by Eq. (1) can be solved using direct or iterative methods. In this work, iterative methods were employed due to the aforementioned characteristics of linear systems in CFD (sparse and large coefficients matrices), for which iterative methods are recommended. The multigrid method is one of the most effective methods to accelerate the convergence of iterative methods for the resolution of linear or nonlinear systems, isotropic or anisotropic problems, among others [5-7].

Anisotropic problems are fairly common in Engineering and appear in many phenomena, such as in the case of a material with different heat conduction behaviors for different directions. In this case, the coefficients of the differential equations are distinct among themselves and generate, what is called, physical anisotropy. Anisotropies can also appear from discretization of grids with different spacing in each direction, for instance, boundary layer problems. This is called geometric anisotropy [5,6]. The efficiency of the multigrid method has not yet been fully achieved for problems with strong anisotropy, either physical or geometric [8].

Physical anisotropy problems have been investigated by Rabi and de Lemos [9], who discretized the two-dimensional advection-diffusion equation combining null and constant coefficients. Finite volume method was used in the discretization of the equations. Multigrid method was employed using correction scheme, Gauss-Seidel solver and V- and W-cycles. The authors presented a study of the different speed ranges (null and constant), number of grids, number of smoothing steps at each grid level and different solvers. They concluded that there was a significant reduction in the computational effort when the speed range values were increased.

Wienands and Joppich [10] presented an in-depth study on the Local Fourier Analysis (LFA) and its application on several problems, which included anisotropic problems. The authors calculated the convergence factor of the multigrid method for the anisotropic diffusion equation with different solvers and restriction and prolongation operators.

Johannsen [11] solved an anisotropic diffusion problem using finite volume method for discretizing the equations and 9-point incomplete LU (ILU) decomposition method for solving the linear equations systems. The author employed LFA to demonstrate the superior smoothing properties of ILU.

Oliveira et al. [8] assessed geometric anisotropy for different grids and aspect ratios. They also assessed some components of the multigrid method, such as: solvers, type of restriction, number of levels and number of inner iterations, among several coarsening algorithms. They concluded that Partial Weighting (PW) had a good performance.

Vinogradova and Krukier [12] solved a three-dimensional advection-diffusion problem with intermediate anisotropy using FDM for the discretization of the equations and ILU for the resolution of the linear equation systems. They concluded that the proposed methodology is efficient, but it has the disadvantage of having limitations in the coefficients of the mixed derivatives, which has no physical meaning.

Vassoler-Rutz et al. [11] analyzed the effect of physical anisotropy on multigrid method for two anisotropic diffusion problems. They used FAS scheme, V-cycle, and Modified Strongly Implicit (MSI) and Gauss-Seidel (GS) solvers. They concluded that, for strong anisotropies, the complexity order of the multigrid method is not suitable.

Several works on the implementation of the multigrid method found in the literature demonstrate that the choice of the multigrid components is crucial for the convergence or not of the method. Trottenberg et al. [6] state that these choices are difficult and thus small changes can considerably improve convergence. In this sense, LFA can assist these choices as it allows to predict the performance of the multigrid method, since it provides estimates of the converge rates based on the variation of the multigrid method components.

Pinto et. al. [14] solved the anisotropic diffusion problem using ILU in triangular grids. They used LFA to show the good smoothing properties of the solver and the asymptotic convergence of the multigrid.

Franco et. al. [13] performed LFA in transient problems and obtained the critical value of the parameter that represents the level of space-time anisotropy for the 1D and 2D Fourier equations.

In this work, an efficient and robust method for solving physical anisotropy problems using LFA is proposed. A two-dimensional diffusion mathematical model is considered, in which physical anisotropy appears in the coefficients and it will be denominated diffusion anisotropy. Equations were discretized using FDM with second-order central difference scheme.

The asymptotic convergence factor (${\textstyle \rho _{loc}}$) of the multigrid method was calculated through assessment of the ILU solver in different directions [7] and several restriction and prolongation operators. The results obtained by means of LFA were used to assess the influence of physical anisotropy on computational cost by means of CPU time and number of operations in each V-cycle and at the restriction step.

The remainder of this work is organized as follows: section 2 presents the mathematical and numerical models. Section 3 discusses considerations regarding the LFA used. Section 4 presents the results of the convergence analysis and complexity analysis. Section 5 presents the conclusion.

2. Mathematical and Numerical Models

For the problem presented below, the calculus domain used is given by ${\textstyle {0}\leq {x}\leq {1}}$, ${\textstyle {0}\leq {y}\leq {1}}$and the discretization of the equations is done using a uniform grid with a number of points given by ${\textstyle N=N_{x}.N_{y}}$ , where ${\textstyle N_{x}}$ and ${\textstyle N_{y}}$ are the number of points in the directions x and y, respectively, including the boundaries.

2.1. Mathematical Model and Discretization

The diffusion anisotropy problem will be assessed through the two-dimension diffusion equation given by Eq. (2) [5,6]

where T is the temperature, ${\textstyle T_{xx}}$ is the second derivative of T as a function of ${\textstyle x}$, ${\textstyle T_{yy}}$ is the second derivative as a function of ${\textstyle y}$ and ${\textstyle \varepsilon }$ .

The source term S and the analytical solution T are given by

Eq. (2) was discretized using FDM with CDS, resulting in

where ${\displaystyle T}$ is the system unknown.

Fig. 1(b) depicts the notation of the grid points in Fig. 1(a). The points P (central), W (west), E (east), N (north) and S (south) in Fig. 1(b) correspond to the points ${\textstyle (i,j),(i-1,j),(i+1,j),(i,j+1),(i,j-1)}$ in Fig. 1(a), respectively.

(a) (b)

The discretization of Eq. (2) results in Eq. (4), and for the inner points, considering ${\textstyle h_{x}^{}={\frac {1}{N_{x}^{}-1}}}$ and

${\textstyle h_{y}^{}={\frac {1}{N_{y}^{}-1}}}$ ,

For the boundaries north, south, east and west,${\textstyle {{a}_{P}}=1}$, ${\displaystyle {{a}_{W}}={{a}_{E}}={{a}_{N}}={{a}_{S}}=0}$.

2.2. Multigrid method and computational details

The multigrid method accelerates the convergence rate of iterative methods. It consists in employing a group of grids with different refinement levels. At each grid refinement level, the more oscillatory errors are smoothed, and only low frequency errors remain. After passing to another grid, the remaining low frequency errors become more oscillatory. The efficiency of this process, called smoothing, depends on the choice of a suitable solver.

For employing multigrid, besides a solver with good smoothing properties, grid transfer operators are required (restriction and prolongation).

In [8] and [16], restriction through partial weighting in the directions ${\displaystyle x}$ and ${\displaystyle y}$, henceforth denoted by ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ and ${\textstyle {\mbox{PW}}_{\mbox{y}}}$, respectively, applied in problems involving geometric anisotropy was proposed. These operators, are given in stencil notation as

In this work, such restriction was combined with 7-point incomplete LU decomposition (henceforth denoted by ILU). According to [17], this decomposition has a better convergence factor than 5-point incomplete LU decomposition for anisotropic problems.

In Eq. (2), discretized using FDM, the stencil for 5-point Laplacian operator is given by

The ILU decomposition of this same operator will be given by

note that in this case, ${\textstyle b=f=0}$.

Thus, the ILU decomposition is represented by:

where ${\displaystyle {{{\overset {\hat {\ }}{\mathop {L}}}\,}_{h}}}$ is a lower triangular matrix, ${\displaystyle {{{\overset {\hat {\ }}{\mathop {U}}}\,}_{h}}}$ is an upper triangular matrix and ${\displaystyle {{R}_{h}}}$ is the residual matrix.

The iterative process for solving Eq. (4) can be:

Depending on the ordination of the grid points, different directions can be obtained for the ILU decomposition. In lexicographical order, ${\textstyle {\mbox{ILU}}_{\mbox{EN}}}$ [17], is given by

Another example of ordination for ILU, ${\textstyle {\mbox{ILU}}_{\mbox{NE}}}$ [17], is given by

The linear equation system given by Eq. (1) was solved using the geometric multigrid method [5,6] with correction scheme (CS), V-cycle and zero initial estimate.

The coarsening ratio is given by ${\displaystyle r=2}$ (standard coarsening) [17]. The grid transfer operators employed were: Injection Restriction (INJ), Half Weighting (HW), Full Weighting (FW) (see [6]), Partial Weighting in ${\textstyle x}$ (${\textstyle {\mbox{PW}}_{\mbox{x}}}$ ), Partial Weighting in ${\textstyle y}$ (${\textstyle {\mbox{PW}}_{\mbox{y}}}$) and prolongation by bilinear interpolation and 7-point interpolation. The equation systems obtained by means of discretization were resolved using 7-point ILU solvers in different directions (${\textstyle {\mbox{ILU}}_{\mbox{EN}}}$, ${\textstyle {\mbox{ILU}}_{\mbox{NE}}}$, among others).

The stop criterion used to interrupt the iterative process is based on the dimensionless residual norm. The residual of the algebraic equation system is defined by

where ${\displaystyle T^{m}}$ is the solution of the unknown in the iteration ${\displaystyle m}$.

Considering ${\displaystyle {{L}^{m}}={{\left\|r_{}^{m}\right\|}_{1}}}$ and ${\displaystyle {{L}^{0}}={{\left\|r_{}^{0}\right\|}_{1}}}$ , if ${\displaystyle {\frac {{L}^{m}}{{L}^{0}}}\leq tol}$ the iterative process is interrupted if ${\displaystyle tol={{10}^{-10}}}$ .

Double precision arithmetic was used for the simulations. The numerical codes were implemented in the Fortran 2003 language, using the Intel 9.1 Visual Fortran application. All numerical results were obtained in a computer with Intel Core i7 2.66 GHz processor, 16 GB RAM and Windows 8 operating system, 64-bit version.

3. Local Fourier Analysis

LFA allows to predict the performance of the multigrid method as it supplies estimates of the convergence rate of its components. Therefore, LFA becomes a powerful tool in quantitative analysis and for the project of efficient multigrid methods.

In order to perform the LFA, general discrete linear operators with constant coefficients are considered, which are defined in an infinite grid ${\displaystyle G_{h}}$ , where the influence of the boundaries can be dismissed [6].

Consider the grid functions of the form of

where ${\displaystyle x}$ varies in the infinite grid ${\displaystyle G_{h}}$ and ${\displaystyle \theta }$ is a parameter that characterize the frequency of the function concerning to grid ${\displaystyle G_{h}}$ .

For ${\displaystyle -\pi \leq \theta <\pi }$ , every function of the grid ${\displaystyle {{\phi }_{h}}\left(\theta ,x\right)}$ are eigenfunctions of a discrete operator that can be written by a stencil.

Thus,

where

For the smoothing and the analysis of two grids, it necessary to distinguish components of high and low frequency of ${\displaystyle G_{h}}$ in relation to ${\displaystyle G_{2h}}$ .

It is known that [6] only if

are visible in ${\displaystyle G_{h}}$ .

For each ${\textstyle {\bar {\theta }}\in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)\times \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)}$, three other frequency components ${\displaystyle {{\phi }_{h}}\left(\theta ,x\right)}$ with ${\textstyle \theta \in [-\pi ,\pi )\times [-\pi ,\pi )}$ coincide in ${\displaystyle G_{2h}}$ with ${\displaystyle {{\phi }_{h}}\left({\bar {\theta }},x\right)}$ and are not visible in ${\displaystyle G_{2h}}$ . Therefore, the low- and high-frequency components are defined as follows:

${\displaystyle \varphi }$ is a low-frequency component${\displaystyle \Leftrightarrow \theta \in {{T}^{low}}=\left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)\times \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)}$

${\displaystyle \varphi }$ is a high-frequency component${\displaystyle \theta \in {{T}^{high}}=\left[-\pi ,\pi \right)\times \left[-\pi ,\pi \right)\backslash \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)\times \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)}$. See Fig. 2.

Figure 2: Low- (inner white area) and high-frequency (hatched area) areas.

Considering the frequencies

The operator ${\displaystyle K_{h}^{2h}}$ is represented by a 4x4 matrix ${\displaystyle {\hat {K}}_{h}^{2h}}$ , as follows

where ${\displaystyle {\hat {I}}_{h}^{}}$ is represented by a 4x4 identity matrix, ${\displaystyle {{\hat {L}}_{h}}(\theta )}$ is the 4x4 matrix, ${\displaystyle {\hat {I}}_{h}^{2h}(\theta )}$ is a 1x4 matrix, ${\displaystyle {\hat {I}}_{2h}^{h}(\theta )}$ is a 4x1 matrix and ${\displaystyle {{\left({{\hat {L}}_{2h}}(2\theta )\right)}^{-1}}}$ is a 1x1 matrix.

A representation for ${\textstyle M_{h}^{2h}}$ can be obtained by a matrix ${\displaystyle {\hat {M}}_{h}^{2h}(\theta )}$ of the form

where ${\displaystyle {\hat {K}}_{h}^{2h}(\theta )}$ is given by Eq. (19) and ${\displaystyle {{\hat {S}}_{h}}(\theta )}$ is a 4x4 matrix and represents ${\displaystyle {{S}_{h}}(\theta )}$.

Let it be

The asymptotic convergence factor ${\displaystyle \rho \left(M_{h}^{2h}\right)}$ can be calculated by

where ${\displaystyle \rho \left({\hat {M}}_{h}^{2h}(\theta )\right)}$ is the spectral radius of the 4x4 matrix ${\displaystyle {\hat {M}}_{h}^{2h}(\theta )}$.

In this work, LFA was used to determine the asymptotic convergence factor of the multigrid method (${\displaystyle \rho \left(M_{h}^{2h}\right)={{\rho }_{loc}}}$) combining ILU solvers in several directions (such as ${\textstyle {\mbox{ILU}}_{\mbox{EN}}}$ and ${\textstyle {\mbox{ILU}}_{\mbox{NE}}}$), FW, HW, INJ, ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ and ${\textstyle {\mbox{PW}}_{\mbox{y}}}$ restriction operators and bilinear and 7-point prolongation operators.

For the FW restriction operator, ${\displaystyle {\overset {\tilde {\ }}{\mathop {I_{h}^{2h}}}}\,\left({{\theta }^{\alpha }}\right)={\frac {1}{4}}(1+\cos {{\bar {\theta }}_{1}})(1+\cos {{\bar {\theta }}_{2}})}$, for the ${\textstyle {\mbox{PW}}_{\mbox{x}}}$, ${\displaystyle {\overset {\tilde {\ }}{\mathop {I_{h}^{2h}}}}\,\left({{\theta }^{\alpha }}\right)={\frac {1}{2}}(1+\cos {{\bar {\theta }}_{1}})}$ and for the ${\textstyle {\mbox{PW}}_{\mbox{y}}}$, ${\displaystyle {\overset {\tilde {\ }}{\mathop {I_{h}^{2h}}}}\,\left({{\theta }^{\alpha }}\right)={\frac {1}{2}}(1+\cos {{\bar {\theta }}_{2}})}$. For the other restriction operators, see [4].

For the bilinear prolongation operator, ${\displaystyle {\overset {\tilde {\ }}{\mathop {I_{2h}^{h}}}}\,\left({{\theta }^{\alpha }}\right)=(1+\cos {{\bar {\theta }}_{1}})(1+\cos {{\bar {\theta }}_{2}})}$ and for the 7-point prolongation operator, ${\displaystyle {\overset {\tilde {\ }}{\mathop {I_{2h}^{h}}}}\,\left({{\theta }^{\alpha }}\right)=\left(1+\cos {{\bar {\theta }}_{1}}+\cos {{\bar {\theta }}_{2}}+\cos({{\bar {\theta }}_{1}}-{{\bar {\theta }}_{2}})\right)}$ [10].

In order to perform the LFA using the ${\textstyle {\mbox{ILU}}}$ solver, the eigen-functions of ${\displaystyle {{L}_{h}}}$, ${\displaystyle {{{\overset {\hat {\ }}{\mathop {L}}}\,}_{h}}}$, ${\displaystyle {{{\overset {\hat {\ }}{\mathop {U}}}\,}_{h}}}$ and ${\displaystyle {{R}_{h}}}$ are given by

The smoothing operator ${\displaystyle {{S}_{h}}}$, according to [6], is given by

with

For ${\textstyle {\mbox{ILU}}_{\mbox{EN}}}$, ${\displaystyle \lambda _{h}^{R}(\theta )\,={{p}_{1}}{{e}^{i(2{{\theta }_{1}}-{{\theta }_{2}})}}+{{p}_{3}}+{{p}_{2}}{{e}^{i(-2{{\theta }_{1}}+{{\theta }_{2}})}}}$ and for ${\textstyle {\mbox{ILU}}_{\mbox{NE}}}$, ${\displaystyle \lambda _{h}^{R}(\theta )\,={{p}_{1}}{{e}^{i(-{{\theta }_{1}}+2{{\theta }_{2}})}}+{{p}_{3}}+{{p}_{2}}{{e}^{i({{\theta }_{1}}-2{{\theta }_{2}})}}}$.

4. Numerical Results

In this work, the anisotropic diffusion equation was solved using 7-point ILU solver in different directions. Several restriction operators and two prolongation operators were employed. It is proposed an algorithm that presents the lowest asymptotic convergence factor values and the lowest computational cost for the multigrid method.

Eq. (2) was assessed for${\displaystyle \varepsilon ={{10}^{\kappa }}}$ and ${\displaystyle \varepsilon ={{10}^{-\kappa }}}$, with ${\displaystyle \kappa \in \mathrm {K} =\{0,\,\,1,\,\,2,\,\,3,\,\,4,\,\,5,\,\,6,\,\,7\}}$. When ${\displaystyle \varepsilon ={{10}^{\kappa }}}$ or ${\displaystyle \varepsilon ={{10}^{-\kappa }}}$ in this work, there is symmetric anisotropy. For instance, ${\displaystyle \varepsilon ={{10}^{2}}}$ is an anisotropy symmetric to ${\displaystyle \varepsilon ={{10}^{-2}}}$.

Section 4.1 presents the convergence analysis by means of LFA. Only the optimum components obtained via LFA will be used in the complexity analysis in section 4.2.

4.1. Convergence Analysis

Fig. 3 depicts ${\displaystyle {{\rho }_{loc}}}$ , given by Eq. (22), with ${\textstyle {\mbox{ILU}}}$ in the EN, NE, ES, SE directions, FW restriction, bilinear prolongation, number of inner iterations ${\displaystyle v=2}$,${\displaystyle \varepsilon ={{10}^{\kappa }}}$and ${\displaystyle \varepsilon ={{10}^{-\kappa }}}$, with ${\displaystyle \kappa \in K}$.

Figure 3: ${\displaystyle {{\rho }_{loc}}}$ versus anisotropy coefficients ( ) with ${\textstyle {\mbox{ILU}}}$ in different directions.

It is noticed that for , ${\textstyle {\mbox{ILU}}_{\mbox{EN}}}$ has a good performance and for

${\textstyle {\mbox{ILU}}_{\mbox{NE}}}$

has a good performance as well, that is, . By using ${\textstyle {\mbox{ILU}}}$ solvers in the ES and SE directions, the multigrid did not present a good performance for any of the anisotropy coefficients studied.

For the analyses presented below, tests were carried out using only the solvers that had the best performances in the previous analysis.

Fig. 4 presents ${\displaystyle {{\rho }_{loc}}}$ using as solvers, ${\textstyle {\mbox{ILU}}_{\mbox{EN}}}$ for and ${\textstyle {\mbox{ILU}}_{\mbox{NE}}}$ for  ;  ; FW restriction; 7-point and bilinear prolongation; and , with .

Figure 4: ${\displaystyle {{\rho }_{loc}}}$ versus anisotropy coefficients ( ) with different interpolation operators.

It is observed that the restriction|prolongation combinations FW|bilinear and FW|7-points had a good performance ( ) and presented very similar convergence factors.

Bilinear prolongation operator was used in the following analysis as it is easy to program and demands fewer memory resources. The asymptotic convergence factors were compared with different restriction operators.

Fig. 5 presents ${\displaystyle {{\rho }_{loc}}}$ using as solvers, ${\textstyle {\mbox{ILU}}_{\mbox{EN}}}$ for and ${\textstyle {\mbox{ILU}}_{\mbox{NE}}}$ for  ;  ; FW, HW, INJ, ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ and ${\textstyle {\mbox{PW}}_{\mbox{y}}}$ restriction; bilinear prolongation; and , with .

Figure 5: ${\displaystyle {{\rho }_{loc}}}$ versus anisotropy coefficients ( ) with different restriction operators.

Fig. 5 shows that for and , with (symmetric anisotropies), presents very similar values. It is noted that for anisotropic problems ( ), the lowest values for ${\displaystyle {{\rho }_{loc}}}$ are achieved with FW and ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ restriction, which present very similar values.

Based on the results shown, it is proposed Algorithm 1, which combines ILU solver in different directions with FW and ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ restrictions. The algorithm will be presented now. The abbreviation REST, used in the algorithm, represents any of the restrictions (FW or ${\textstyle {\mbox{PW}}_{\mbox{x}}}$) previously defined.

Algorithm1:

___________________________________________________________________________

if then

Apply ${\textstyle {\mbox{ILU}}_{\mbox{NE}}}$ smoothing with REST restriction

else if then

Apply ${\textstyle {\mbox{ILU}}_{\mbox{NE}}}$ smoothing with FW restriction

else

Apply ${\textstyle {\mbox{ILU}}_{\mbox{EN}}}$ REST restriction

end if

___________________________________________________________________________

Next, it is presented the asymptotic convergence factor ${\displaystyle {{\rho }_{loc}}}$, calculated via LFA, and the empiric asymptotic convergence factor , for different anisotropy coefficients.

Fig. 6 shows Algorithm 1 with REST ${\textstyle ={\mbox{PW}}_{\mbox{x}}}$ , and bilinear prolongation.

Figure 6: ${\displaystyle {{\rho }_{loc}}}$ and versus anisotropy coefficients ( ).

It is observed that for every anisotropy coefficients assessed. Moreover, ${\displaystyle {{\rho }_{loc}}}$ calculated via LFA is in accordance with calculated experimentally.

Fig. 7 depicts the numerical asymptotic convergence factor ${\displaystyle {{\rho }_{loc}}}$ lculated via LFA and the experimental asymptotic convergence factor , for different anisotropy coefficients for different grids. Algorithm 1 with REST = ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ , and bilinear prolongation was used. As the grid becomes more refined, for every anisotropy coefficient analyzed, what demonstrates the robustness of the methodology assessed.

Figure 7: ${\displaystyle {{\rho }_{loc}}}$and versus anisotropy coefficients ( ) for different number of grid points.

Next section presents the complexity analysis of multigrid. For the analysis, multigrid was built with the components that had the best convergence factors, according to LFA. In addition, a comparison between partial weighting and full weighting will be presented.

4.2. Complexity Analysis

In order to assess the effect of the number of unknowns on CPU time, optimum components obtained via LFA were used. Figures 8 (a) and 8 (b) show, for FW and ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ restrictions respectively, that for the anisotropic problem ( ), the ${\textstyle t_{CPU}}$ is not lower. As the problem becomes more anisotropic ( or ), the decreases for every N value assessed. For every anisotropy coefficient assessed, the with FW restriction is very similar to the with ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ restriction, that is, ${\textstyle t_{CPU}\left(FW\right)\approx t_{CPU}\left({\mbox{PW}}_{\mbox{x}}\right)}$ .

It is also observed for symmetric anisotropies ( and ) with , that the values of obtained are extremely similar.

In order to assess the performance of the multigrid with different anisotropy coefficients, a curve adjustment of the form [18] was made, where p represents the complexity order of the solver, N is the number of grid points and c is a constant that depends on the method. The closer the value of p is to one, the better the performance of method used. Ideally, multigrid presents p=1, what means that the CPU time grows linearly with the increase of N. Results are shown in Table 1 for both restrictions assessed (FW and ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ ).

One can observe in Tab. 1 that, for every anisotropy employed, the multigrid method has a good performance, since p ≈ 1 in every case. These results prove the efficiency and robustness of Algorithm 1, proposed in this work.

	1.07747	1.07733
	1.05023	1.05894
1	1.05940	1.04380
	1.07255	1.06199
	1.07627	1.07775

Table 1. Complexity order (p) for different anisotropy coefficients ( ).

The values obtained for the convergence factor, presented in section 4.1, and for the complexity order concerning to the ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ and FW restriction operators (Tab. 1) are quite similar and thus, insufficient to decide which one results in a more efficient algorithm.

To complement the analysis, the number of arithmetical operations performed in each V-cycle and at the restriction step for each one of the operators was assessed.

These arithmetical operations concern to floating point operations (flops) performed during the iterative process and are not affect by the hardware used. Each addition, multiplication and division operation correspond to 1 flop.

Tests were carried out for ( is the number of points of the finest grid, considering a problem whose maximum number of levels is ) and for some values of . Tab. 2 presents the ratio between number of flops of a V-cycle and number of points of the finest grid . Tab. 3 shows the ratio between number of flops performed in each restriction and the number of points of the finest grid .

		${\textstyle {\mbox{flops}}({\mbox{V-cycle}}\mid {\mbox{FW}})/N_{10}}$
	1510.221554	1543.296057
	1798.299366	1837.988770
	2368.277388	2126.896785
	1860.997604	1900.687008
	1564.958717	1598.033220

	${\textstyle {\mbox{flops}}({\mbox{PW}}_{\mbox{x}})/N_{10}}$	${\textstyle {\mbox{flops}}({\mbox{FW}})/N_{10}}$
	11.57607615	44.65057942
	13.89129130	53.58069530
	18.37833483	62.02688007
	13.89129137	53.58069530
	11.57607615	44.65057942

According to Tab. 2, the multigrid cycle that requires the lowest number of flops is the cycle with ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ operator, except for the isotropic case ( ). Considering only the restriction step, results presented in Tab. 3 show a great advantage of the ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ operator over the FW operator regarding the number of flops performed. For every case, the number of flops for ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ is roughly 75% lower than the number of flops for FW.

The remaining anisotropy coefficients assessed showed similar performances to those presented in Fig. 5 and Tables 2 and 3, what confirms the efficiency and robustness of the algorithm proposed, in addition to the low computational cost with the use of the ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ operator.

5. Conclusions

- Among the ILU directions assessed (EN, NE, ES, SE) for standard multigrid (FW restriction operator and bilinear prolongation), it is concluded that ${\textstyle {\rho }_{loc}\approx 0.02}$ when using ${\textstyle {\mbox{ILU}}_{\mbox{EN}}}$ for and ${\textstyle {\mbox{ILU}}_{\mbox{NE}}}$ for .

- With FW restriction, ${\textstyle {\mbox{ILU}}_{\mbox{EN}}}$ for and

${\textstyle {\mbox{ILU}}_{\mbox{NE}}}$

, it can be concluded that the 7-point and bilinear prolongation operators presented a similar performance, and for every value of assessed, was obtained.

- With bilinear interpolation, ${\textstyle {\mbox{ILU}}_{\mbox{EN}}}$ for and

${\textstyle {\mbox{ILU}}_{\mbox{NE}}}$

, it can be stated that the lowest values of ${\displaystyle {{\rho }_{loc}}}$ are obtained with FW and ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ restriction operators, and that the values of ${\displaystyle {{\rho }_{loc}}}$ with these operators are very similar.

- Using Algorithm 1, for every anisotropy coefficient assessed and as the grid becomes more refined.

- The ${\textstyle t_{CPU}\left({\mbox{FW}}\right)\approx t_{CPU}\left({\mbox{PW}}_{\mbox{x}}\right)}$ for every value of assessed.

- The complexity order p of the multigrid method with Algorithm 1 is close to one for every anisotropy assessed. For , for example, p=1.07747 with this algorithm.

- The computational cost of multigrid depends on the number of flops of the restriction in a V-cycle. Using Algorithm 1 with ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ restriction, the computational cost is 75% lower than with FW restriction.

- The Algorithm 1 with ${\textstyle {\mbox{PW}}_{\mbox{x}}}$ restriction proposed in this work is efficient, robust and has low computational cost.

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