1549301742-The_Theory_of_Difference_Schemes__Samarskii

292 Difference Schemes for Elliptic Equations

assuming that v( x) has a continuous second derivative on the segment

[x-h,x+h];

(5) Av= vxx = v^11 (x) + ~~ v(^4 l(~*), ~· = x + e• h, I e• I < 1,

v(x) has a continuous fourth derivative on the segment [x - h, x + h]. By

relating x 1 to be fixed we might have

h; 84 u

A2 u = L2 u(x 1 , x 2 ) + 12 ox 4 (x1, ~2),

2

Other ideas are connected with the expression

h2 [J4u

A 1 A 2u(x 1 ,x 2 )=A1L2u(x 1 ,x 2 )+ 1 ; A1 oxi(x1,~2)·

Applying formula (5) with v = L2 u and x = x 1 to the first summand yields

h2 [J4

A 1 L 2 u(x 1 , x 2 ) = L1 L2 u(x 1 , x 2 ) + 1 ; oxf (~;, x2)

I e; I < 1.

The same procedure works for the second summand with respect to formula
( 4):

\tVhat should be done is to bring together the results obtained:

Substituting into (3) the expression for A 1 A2 tl in place of L 1 L 2 u

and involving the equation Lu= -f(x), we finally get

(6)

h2+h2 h2 h2

Au = Lu -^1 12 2 A 1 A2 u - 1 ; L1 f - 1 ~ L2 f + O(J h J^4 )