TWO--LEVEL ANALYSIS OF AUTOMATIC MULTIGRID FOR
	     NON-NORMAL AND INDEFINITE PROBLEMS


		       Yair Shapira

		Computer Science Department
	Technion -- Israel Institute of Technology
		   Haifa 32000, Israel


			SUMMARY

A two-level analysis method for several automatic multigrid methods
(implemented with the red-black smoother) is presented.
Unlike standard two-level analysis methods, based on Fourier analysis,
it is based on spectral analysis; hence, it is also applicable to
non-normal problems. For indefinite equations, it provides
a prediction of the optimal mesh size for the coarsest grid used.
Furthermore, it motivates the definition of an improved version
of Black-Box multigrid. Numerical examples confirming the analysis
are presented.


A two-level analysis method for Black-Box Multigrid and AutoMUG
(implemented with the red-black smoother) is presented. Unlike 
standard two-level analysis methods, based on Fourier analysis,
it is based on spectral analysis; hence it is applicable to 
non-normal as well as normal linear systems. For indefinite
Helmholtz equations, it provides a prediction for the optimal 
mesh size for the coarsest grid used. Numerical experiments 
applying AutoMUG (accelerated by a Lanczos-type method)
to nearly singular highly indefinite Helmholtz equations
show the validity of this prediction.