Comparison of Geometric and Algebraic Multigrid Methods in Edge-Based Finite-Element Analysis

1672IEEE TRANSACTIONS ON MAGNETICS,VOL.41,NO.5,MAY2005 Comparison of Geometric and Algebraic Multigrid Methods in Edge-Based Finite-Element Analysis

K.Watanabe,H.Igarashi,and T.Honma,Member,IEEE

Division of Systems Science and Informatics,Graduate School of Information Science and Technology,Hokkaido University,

Sapporo060-0814,Japan

This paper discusses the comparison between the geometric multigrid(GMG)method and the algebraic multigrid(AMG)method in edge-based?nite-element(FE)analysis.The GMG method requires the hierarchical meshes.On the other hand,the AMG method requires the only a single mesh information.The system matrices of the coarse grids are generated using algebraic operation in AMG. The numerical results show that both multigrid methods are faster than the conventional solvers in large-scale analysis.Although multi-grid methods require the setup procedures,the calculation time of these procedures is comparatively short and increase linearly with the number of unknowns.

Index Terms—Algebraic multigrid(AMG),edge element,?nite-element method(FEM),geometric multigrid(GMG).

I.I NTRODUCTION

T HE MULTIGRID method has been applied to electromag-

netic?eld problems so far,to show that it can signi?-

cantly reduce computational time in comparison with conven-

tional linear solvers such as incomplete Cholesky conjugate gra-

dient(ICCG).The geometric multigrid(GMG)method requires

the hierarchical meshes.The nested GMG method[1],in which

the?ner meshes are automatically obtained by dividing each

coarse element into several?ner elements,is used in our study.

This lightens the load of mesh generators.On the other hand,the

algebraic multigrid(AMG)method requires the only a single

mesh information.The system matrices on the coarse meshes

are generated using algebraic operations in AMG.We compare

the computation time of the two methods in a?nite-element

(FE)analysis to show the effectiveness for large-scale analysis.

In this paper,we evaluate these setup process of multigrid in

large-scale FE analysis.

II.F ORMULATION

A.Magnetostatic Problem

Let us consider magnetostatic?eld governed

by

(1)

(2)

where is the magnetic

reluctivity,is the vector potential,

and

is the current density.The current vector

potential

(3)

is introduced for satisfaction of(2).Equation(1)now leads

to

(4)

Digital Object Identi?er10.1109/TMAG.2005.846092

FE discretization of(4)results in the system of linear

equations

(5)

where is a positively semide?nite matrix which is the

discrete counterpart of the operator in the left side of

(4),

and denote column vectors corresponding

to

and,

respectively.

B.GMG Method

It is known that the linear solvers such as Gauss–Seidel and

CG methods tend to eliminate the high-frequency components

of the residue in(5)more rapidly than the low-frequency com-

ponents.The multigrid method is based on this property,that

is,the high-frequency residual components are eliminated on a

?ne mesh by small numbers of iterations of the linear solver

(smoother).The remained residual components are then pro-

jected onto a more coarse mesh,in which they now have high

frequency that can again be eliminated by small numbers of the

iterations.The multigrid method solves(5)successively per-

forming these processes.This procedure is usually called the

coarse grid correction.Although there are many variations in

multigrid method,all these variations are based on the coarse

grid correction.The procedure of the two-grid V-cycle method

that is the simplest multigrid method is described in the fol-

lowing.

Step1(Smoothing):The smoothing operation is applied to

the system

equation

(6)

for the?ne mesh to obtain approximate

solution,

where

denotes the system matrix de?ned on the?ne mesh.In this

step,the high-frequency components in the solution error are

eliminated.

Step2:The residual

vector corresponding to the ap-

proximate

solution,is

calculated

(7)

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