Economical factorized schernes 573In turn, the supplen1entary boundary conditions become( 30)The second sche1ne:( 31)with the supplementary boundary conditions for fj:(32)Then problems (29)-(30) and (31)-(32) can be solved by the alter-
nating direction method in just the same way as was clone in Section 1
for problem (46)-(47) using Gaussian elimination along the rows as well as
along the columns of the grid w h.
Thus, three different examples of interest bring out the indisputable
merit of the alternating direction method and unveil its potential. From
what has been said above it follows that the factorized schen1es find s01ne
range of applications solely in rectangles and parallelepipeds and no more.
The only exception is the case B = B1B2, Bo:= E + rRo:, where Ri and
R 2 are "triangle" operators, by means of which it is possible to generate a
lower-order approximation only under the condition T = 0( h^2 ).
- Three-layer factorized schemes. Being concerned with economical three-
layer sche1nes, we confine ourselves here to
A first step towards the solution of this difference problem is to solve it
with respect to yj +l, leading toBecause of this forn1, the operator B + 2 TR on the upper layer is yet to be
factorized for economical reasons.