702 Methods for Solving Grid Equationsby means of which the factorized operator B is selected by the rule(76)where w > 0 is the iteration parameter. The operator B so defined 1s
self-adjoint and positive. Once supplemented with the conditions
(77) A>bD or (Ax,x)>b(Dx,x), b>O,
(78) ~ > 0,
which are valid for all x E H, Theorem 2 and formulas (27)-(30) continue
to hold with the operator A standing in place of the operator R, so there
is no need to rewrite them once again in a common setting.
Of special interest is the well-founded choice of the operator D under
the agreement that the appropriate matrix D is diagonal:(79) Dy= d(x) y, d(x) > 0.
Because of this form, tl1P 111e111ber cl(:r) is so chosen as to max11111ze the
ratio TJ = b / ~ in the modified alternative-triangular method (MATM) with
reasonable efficiency.
Adopting those ideas, some progress has been achieved by means of
IVIATM in tackling the Dirichlet problem in an arbitrary complex d0111ain
G with the boundary r for an elliptic equation with variable coefficients:(80)u(x)=μ(x), x Er,
C\'=l,2.
The usual assmnptions are made here saying that the boundary r is smooth
enough and the intersection of the domain G and a .straight line passing
through any point x E G and in parallel to the axes Ox°', C\' = 1, 2, consists
of a unique interval. The latter should not confine generality.
The design of a difference scheme for problem (80) is mostly based on
a nonequidistant grid wh (generally speaking, nonequidistant everywhere,
not only near the boundary r). When drawing up the family of straight