1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternative-triangular 111ethod 701

conditionality of the systen1 (72) of equations along with decreasing the

ratio h 1 / h.

It seems clear that in solving the system (72) the number of the it-

erations within the framework of the explicit scheme with optimal set of

Chebyshev's parameters or of the sirnple iteration sche1ne is proportional

to 1/ V1j or 1/17, thus causing an enormous growth as h 1 ---+ 0.

The generalized proble1n on eigenvalues of the sa111e operator A is of

the form

Ay+>..Dy=O,

where Dis a diagonal matrix such that Dy= d(x)y, d(x) > 0. In the case

of interest

and the problem of determining >.. an10unts to

Instead of (73) we rnust solve the quadratic equation

For the choice d 1 = l/t and d 2 = 1 we deduce from the foregoing that

17 = 1/(2 + t) and 17 ~ 0.5 when t ~ 1, that is, 17 remains finite as h 1 ---+ 0.

From such reasoning it seems clear that the scheme

(74) D ~--=Ayk+f i!J.:+1 - Y1.:

offers more advantages in comparison with the explicit scheme.
A sin1ilar situation exists during the course of ATM for .solving elliptic
equations on nonequidistant grids or in arbitrary c0111plex domains, giving
rise to obvious modifications of ATM with the intervention of the operator
D = D > 0 built into the structure of the operator B. Making a substanti-
ated choice of the operator D is stipulated by econon1y reasoning for every
iteration as well as by a rnini111al nm11ber of iterations. Let D = D > 0
be an arbitrary operator and the operator A = A* > 0 fr0111 the equation
Au= f be a sun1 of mutually adjoint operators A 1 and A 2 :

(75) A = A* > 0 , A = A 1 + A2 , A;' = A2 ,