1549301742-The_Theory_of_Difference_Schemes__Samarskii

266 Difference Schemes for Elliptic Equations

where

Aa Y = Y"' •"a x· a '

p

A*y= L A~y,

a=l

Other ideas are connected with the operator A, which coincides with A* at

the near-boundary nodes and with A at the remaining inner nodes, leading

to an alternative form of writing

(2) Ay= -<p, YI -Yh =μ(x).

In conformity with Section 1, problem ( 1) can be recast as

(3) A(x) y(x) = B(x,~) y(~) + F(x), xEw, Yl,,h =μ(x),

~EPatt'(x)

where

A(x) > 0, B(x,~) > 0, D(x) = A(;i:) - B(x,~)>0.

~EPatt'(x)

We now represent a solution of problem (1) as a sum

y=f;+f;,

where f; and f; are,· respectively, solutions of the appropriate problems

(4)

(5)

Ay=O,

Af;=-<p,

xEwh, f;=μ for XE/hi

xEwh, f;=O for XE/h·

An estimate for a solution of problem ( 4) such as

(6) II fJ lie < IIμ lie.,