1549301742-The_Theory_of_Difference_Schemes__Samarskii

268 Difference Schemes for Elliptic Equations

p

where e = p-^1 L BC,. Here eQ = 1 if a node x E w;, is regular along the

a=l

direction xa. Thus, Y is a solution of the problem

AY = -F(x),

• 0 -
  where F(x) = 2pf{ for x E wh and F(x) = 2pB f{ for x E w;,.
  0
  Comparison with problem (7), where F = 'P, that is, F = 0 for


• 0
  x E w;, and vl,,h = 0, shows that F(x) > I F(x) I = I 'P(x) I if we accept
  0
  f{ = 2 ~ II 'P lie. Here the conditions of the comparison theorem in Section
  2 are valid as long as F( x) > I F( x) I = 0 for x E w*, assuring the relation
  llvlle < llYlle·
  It is easily seen from the expression for Y that II Y lie < f{ R^2. So, for
  a solution of problem (7) the estimate

(9)

is valid in the next norm 11 'P II e a =max xEwh a I 'P(x) I·

Our next step is the estimation of the function w(x). First, we are

going to show that for problem (8)

( 10)

( 11)

1

D(x) > h2,

D(x) = 0 for

where h =max Q ha,

Assertion (11) is simple to follow.

After that, we look at equation (8) at a near-boundary node x E w7,

A(x) w(x) = B(x, ~) w(O + F(x),

( 12) ~EPatt'(x)

F(x) = <p*(x), wl -Yh = 0.

If one of the nodes ~ = ~ 0 , say ~ 0 x( +l,,), happens to be a boundary
node, then w(~ 0 ) = 0 and the neighborhood Patt'(x) contains no point ~o·