1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternating direction n1ethod 551

On the other hand, a solution of problem (20)-(21) applies equally well to

problem (9)-(14). Indeed, by specifying y by formula (18) we deduce from

(18) that

and insert then the resulting expression in (20). By n1inor changes we are

led to equation (9), which in c01nbination with ( 18) gives ( 10). This pro-

vides enough reason to establish the equivalence between problen1 (9)-(14)

and proble1n (20)-(21) with compliance of the boundary values jJ assigned

by formulas (13)-(14) Careful analysis of scheme (9)-(14) is accompanied

by more a detailed exploration of sche1ne (20)-(21) in "integer steps".

The general theory of two-layer schemes is quite applicable in such a

setting. By regarding the boundary conditions to be homogeneous we turn

to the problem

y(x, 0) = u 0 (x), vi 'Yh = o,

with further reference to the space H of all grid functions given on the set

wh and vanishing on the boundary ih under the inner product structure

Ni-1 N2-l

(y,u)= L y(x)u(x)h 1 h 2 = LL y(i 1 h 1 ,i 2 h 2 )u(i 1 h 1 ,i)i 2 )h 1 h 2 •

The associated norm is taken, as usual, to be II y II = ~· We refer
to the operator A= -A= -(A 1 + A 2 ), which, by definition, is self-adjoint
and positive in the space H. The norm on the energetic space HA is defined
either by

Ni N2-l N,-1 N2

llYll~ =LL (Yx,(i1h1,i2h2))

2

h1h2+ LL (Yx 2 (i1h1,i2h2))

2

h1h2

or by

(23)

When treating y = y(t) as an abstract function of the argmnent t E w 7
with the values in the space H, scheme (22) admits an alternative form

(24) B Yt + Ay = VJ(t), 0 < t = nr < t 0 , y(O) = u 0 ,