1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternating direction inethod 563

Here the operators A 1 and A 2 are supposed to be self-adjoint, positiYe

and commuting:

A~ = Ai > 0, A; = A2 > 0, Ai A2 = A2 A 1 ,

implying that (A 1 A2)* = A1 A2 > 0.

The simple observations that 11Aa11 < 4/h; and x°' II Aa II < 1/3 may

be of he! p in verifying the stab iii ty condition in the space HA

1

: B 2': 0. 5 TA'.

Furthermore, with the aid of the inequality Aa < II Aa 11 E we obtain

1neaning B > 0.5 TA'+ E/3 and confirming the stability of the scheme in

view in the space HA,.

In particular, Theorem 7 in Chapter 6, Section 2 states that scheme

( 54) satisfies the a priori esti1nate

(55)

In trying to evaluate the accuracy of scheme (53) we set up the problem
for the error zn+l = yn+l - un+l:

z(O)=O,

whose solution obeys estin1ate (55)

(56) llP' llA <'. ~ (~ T II IJ' II') •fi

In passing fr0111 (55) to ( 56) we have taken into account that z^0 = y^0 -1t^0 =
U and that the operator A' 1nust satisfy the inequalities
2
A'> - A
3

or